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+1
+Do Performance Aspirations Matter for Guiding Software
+Configuration Tuning?
+TAO CHEN, Loughborough University, United Kingdom
+MIQING LI, University of Birmingham, United Kingdom
+Configurable software systems can be tuned for better performance. Leveraging on some Pareto optimizers,
+recent work has shifted from tuning for a single, time-related performance objective to two intrinsically
+different objectives that assess distinct performance aspects of the system, each with varying aspirations to be
+satisfied, e.g., “the latency is less than 10s” while “the memory usage is no more than 1GB”. Before
+we design better optimizers, a crucial engineering decision to make therein is how to handle the performance
+requirements with clear aspirations in the tuning process. For this, the community takes two alternative
+optimization models: either quantifying and incorporating the aspirations into the search objectives that
+guide the tuning, or not considering the aspirations during the search but purely using them in the later
+decision-making process only. However, despite being a crucial decision that determines how an optimizer
+can be designed and tailored, there is a rather limited understanding of which optimization model should be
+chosen under what particular circumstance, and why.
+In this paper, we seek to close this gap. Firstly, we do that through a review of over 426 papers in the
+literature and 14 real-world requirements datasets, from which we summarize four performance requirement
+patterns that quantify the aspirations in the configuration tuning. Drawing on these, we then conduct a
+comprehensive empirical study that covers 15 combinations of the state-of-the-art performance requirement
+patterns, four types of aspiration space, three Pareto optimizers, and eight real-world systems/environments,
+leading to 1,296 cases of investigation. Our findings reveal that (1) the realism of aspirations is the key factor
+that determines whether they should be used to guide the tuning; (2) the given patterns and the position
+of the realistic aspirations in the objective landscape are less important for the choice, but they do matter
+to the extents of improvement; (3) the available tuning budget can also influence the choice for unrealistic
+aspirations but it is insignificant under realistic ones. To promote open science practice, we make our code
+and dataset publicly available at: https://github.com/ideas-labo/aspiration-study.
+CCS Concepts: • Software and its engineering → Search-based software engineering; Empirical software
+validation; Software performance.
+Additional Key Words and Phrases: Search-based software engineering, software configuration tuning, perfor-
+mance requirement, performance aspiration, multi-objective optimization..
+ACM Reference Format:
+Tao Chen and Miqing Li. 2023. Do Performance Aspirations Matter for Guiding Software Configuration Tuning?.
+ACM Trans. Softw. Eng. Methodol. 1, 1, Article 1 (January 2023), 41 pages. https://doi.org/10.1145/3571853
+1
+INTRODUCTION
+Many software systems are highly configurable, such that there is a daunting number of config-
+uration options (e.g., the max_spout in Apache Storm), which the software engineers can tune
+Authors’ addresses: Tao Chen, t.t.chen@lboro.ac.uk, Loughborough University, Loughborough, United Kingdom; Miqing Li,
+University of Birmingham, Birmingham, United Kingdom, m.li.8@bham.ac.uk.
+Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee
+provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the
+full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored.
+Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires
+prior specific permission and/or a fee. Request permissions from permissions@acm.org.
+© 2023 Copyright held by the owner/author(s). Publication rights licensed to ACM.
+1049-331X/2023/1-ART1 $15.00
+https://doi.org/10.1145/3571853
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+arXiv:2301.03290v1  [cs.SE]  9 Jan 2023
+
+1:2
+Chen and Li.
+to meet the requirements of some performance objectives, e.g., improving latency, throughput,
+and resource consumption [12, 15, 36, 52, 78]. Configuration tuning for software systems plays an
+integral role in Software Engineering as a recent interview reveals that industrial practitioners
+have recognized it as a key to the success of software products [67]. Indeed, it has been reported
+that globally 59% of the software performance issues—wherein the performance requirements were
+severely violated—are related to ill-suited configuration rather than code [39], leading to serious
+consequences. For example, in 2017-2018, configuration-related performance issues cost at least
+400,000 USD per hour for more than 50% of the software companies worldwide1.
+Finding good configurations (i.e., the possible combinational settings of the configuration options)
+is challenging, because:
+• The default configuration is often far from ideal. Jamshidi and Casale [44] show that the
+defaults for Apache Storm can lead to 480 times worse performance than some others.
+• The configuration space can be large and the measurement is often expensive [62], rendering
+greedy search unrealistic.
+• While traditionally software configuration tuning has been focusing on a single performance
+objective [6, 11, 37, 58, 64, 66, 71, 77, 79], recent work raises the necessity of simultaneously
+tuning for multiple performance objectives. Our review (see Section 3) found that considering
+two performance objectives is the most common case [16, 32, 41, 62]. For example, naturally,
+improving the image quality while reducing the energy consumption are both critical for video
+encoders like x264; higher accuracy with shorter training time are two inherent performance
+objectives for deep learning models, e.g., the deep neural network supported by frameworks
+such as Keras. This further complicates the tuning process as the performance objectives
+may be conflicting and the extents to such a conflict are often unknown a priori [16, 62].
+To automatically tune software configuration for better performance, different approaches have
+been proposed, such as rule-based [31, 35], learning-based [3, 45], and search-based [8, 9, 16, 50,
+62, 70]. Among these, search-based approach, primarily relying on the Pareto optimizers widely
+used in the Search-Based Software Engineering (SBSE) paradigm [40], has been a promising way
+to handle all the aforementioned challenges in software configuration tuning, especially in the
+presence of more than one performance objective [8, 9, 16, 70]. In a nutshell, a Pareto optimizer
+most commonly maintains a population (or at least an archive) of configurations, which can be
+repeatedly reproduced and evaluated by directly profiling the software, aiming to find the Pareto
+optimal ones. The output is a set of configurations that are nondominated to each other, which
+approximates the Pareto front of the software system.
+1.1
+The Problem and Significance
+An important factor in software configuration tuning is the possible requirements with clear
+aspirations for the performance objectives [8, 9, 27, 65], for which we distinguish two important
+notions in this work:
+• Aspiration: The information that allows us to quantify the extent to which the performance
+is considered satisfactory (or unsatisfactory).
+• Performance requirement: The context under which the preference of the performance is
+defined.
+For example, according to the research in the Requirement Engineering community [4, 75], it
+is not uncommon to have performance requirements from the requirement documents, such as
+“the latency shall be less than 𝑥” while “the memory usage shall be no more than 𝑦”,
+1https://www.evolven.com/blog/downtime-outages-and-failures-understanding-their-true-costs.html
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+Do Performance Aspirations Matter for Guiding Software Configuration Tuning?
+1:3
+where “less than 𝑥” and “no more than 𝑦” are the clear aspirations therein. It is worth noting
+that not all performance requirements would contain aspiration, e.g., “the latency shall be
+low” is a requirement with no aspiration since nothing can be quantified with respect to the level of
+satisfaction. Indeed, given a scenario with clear aspirations in the performance requirements, it has
+been well-acknowledged that the information provided serves as useful metrics for the software
+engineers to conduct a posterior cherry-picking after the tuning completes, extracting the satisficing
+configuration(s) from the set produced by a Pareto optimizer [55]. The natural motivations behind
+this are:
+• Given a fixed tuning budget, finding the optimal performance is not always feasible or even
+desirable to the stakeholders.
+• The clear aspiration levels allow an implicit trade-off/preferences between the conflicting
+performance objectives according to the stakeholders.
+Regardless of the Pareto optimizer used, in the tuning process, existing work takes one of two
+intrinsically different optimization models to handle aspirations when tuning for two performance
+objectives, namely: Pareto search with aspirations (denoted as PS-w) [8, 9, 33, 59] and Pareto
+search without aspirations (denoted as PS-w/o) [16, 49, 62, 70]. In PS-w, the performance re-
+quirements with aspirations are quantified in certain forms (we will elaborate on this in Section 3),
+which then serve as new search objectives in the tuning. The motivation is simple: since the
+aspirations provide information on the degree of satisficing, one can exploit this advantage to guide
+the tuning process. PS-w/o, in contrast, is more classic and simply ignores the aspirations in the
+tuning. The assumption here is that, since the search in whatever a Pareto optimizer is essentially
+an optimization process that seeks to find the Pareto optimal configurations, the tuning always
+aims to achieve the best possible performance, which preserves the tendency towards satisficing
+whatever aspirations2. For example, finding the Pareto optimal configuration latency=10s and
+memory usage=1GB will certainly meet the requirement and aspiration of “latency shall be less
+than 20s” while “memory usage shall be no more than 2GB”. This matches with Odhnoff’s
+argument that “optimizing” and “satisficing” are merely stylistically different but fundamentally
+the same [63].
+Despite either of the two optimization models being respectively used by their corresponding
+research groups, the choice was mostly ad-hoc and there is often an implied belief that “they do not
+differ much hence can be used arbitrarily.” As such, there remains a rather limited understanding of
+which optimization model should be chosen under what particular circumstance, and why. This has
+been well-echoed by some researchers. Ghanbari et al. [34] have stated that it is important to consider
+the choice, as the shape of the function that guides the tuning, especially after passing the aspirations,
+may impact the behavior of the optimizer; but they did not discuss what implication that would be.
+Yet another example from a recent work by Fekry et al. [28] recommends that studying whether
+to leverage aspirations for guiding the optimizers and measuring its effectiveness is an important
+future challenge for software configuration tuning. Indeed, understanding in this regard is non-
+trivial as it will help practitioners to make more informed-decision, especially when given
+the expensive measurements of configurable software systems, it is unrealistic to always empirically
+compare the two models in a case-by-case manner. Furthermore, the insights can hint at future
+research directions for software configuration tuning: if the PS-w/o is more promising, then
+we can largely simplify the research to the design of an effective optimizer without considering the
+given requirements since the human inputs (i.e., the requirements/aspirations) are less important
+in the overall tuning process. On the other hand, if PS-w is overall more effective, then the problem
+2This assumes the most common case that the best possible performance is at least equally preferred than some other
+values.
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+1:4
+Chen and Li.
+can become more complicated but also provide more opportunities, e.g., future research can largely
+focus on how to better quantify those performance requirements and aspirations, together with
+how to better embed them into more specialized optimizers.
+To understand this, we have also tuned into the literature on general multi-objective optimization,
+with a particular focus on preference-driven multi-objective optimization [5, 51, 74, 80]. However, we
+did not find answers that are directly relevant to our case, due to two reasons: (1) the representation
+of the preferences (e.g., weights and ranks) in preference-driven multi-objective optimization is
+different from the requirement patterns we summarized from the work for software configuration
+tuning; (2) they mainly develop algorithms/optimizers that are tailored to a specific preference
+representation while software configuration tuning often relies on a vanilla optimizer [8, 9].
+Our work is, therefore, motivated by the desirability of the community to understand the
+following:
+Should we incorporate requirements and aspirations to guide the software configuration
+tuning process? If so, in what context and why?
+1.2
+Research Questions
+In this paper, we seek to fill the above gap via an empirical study that systematically compares PS-w
+and PS-w/o for tuning software configuration under two performance objectives. Suppose that
+there are some realistic aspirations (i.e., all the aspirations are achievable by tuning the configuration
+of the software system), the first research question (RQ) we wish to answer is:
+RQ1: Given performance requirements with realistic aspirations, of PS-w and PS-w/o, which
+can find a better set of configurations?
+RQ1 seeks to provide a global picture of the comparison between the two optimization models.
+However, the diverse possible requirement scenarios imply that the specific aspirations can be
+radically different in the objective landscape. For example, one may have higher expectations
+on latency while lower needs on throughput, or vice versus. Therefore, what we would like to
+understand in more detail is:
+RQ2: How do different realistic aspirations influence the result?
+RQ1 and RQ2 investigate under the normal context where the given aspirations are reasonable
+and achievable. However, since the actual aspirations are negotiated by the software engineers
+and stakeholders a priori, they could turn out to be unrealistic and may require attention beyond
+configurations, i.e., no configuration in the search landscape can reach the required aspiration levels
+for all performance objectives simultaneously, despite that may be possible for a single objective.
+This brings our next RQ, in which we ask:
+RQ3: What if the given aspirations are unrealistic?
+While we are interested in cases where the tuning budget is reasonably sufficient to achieve a
+good convergence, it is possible that, in real-world scenarios, there is a limited resource for tuning
+software configuration due to, e.g., pressure for quick release or task prioritization. Therefore, our
+last question aims to explore:
+RQ4: Is the given tuning resource (tuning budget) important to the choice between PS-w and
+PS-w/o?
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+Do Performance Aspirations Matter for Guiding Software Configuration Tuning?
+1:5
+1.3
+Contributions
+To address these RQs , we conducted an extensive empirical study on 15 combinations of patterns
+to quantify aspirations, four types of aspiration space in the objective landscape, three Pareto
+optimizers, and eight real-world systems/environments with diverse performance objectives, leading
+to 1,296 cases of investigations. Briefly, the first contribution in this paper is a set of performance
+requirement patterns (for individual performance objectives) summarized from 426 papers in the
+literature from the Software Engineering community and 14 widely-used real-world requirements
+datasets. These patterns are:
+(1) No aspiration is given but assuming that the optimal possible performance is preferred, e.g.,
+“the lower latency is preferred”, meaning that one prefers the best possible latency.
+(2) The performance in the aspiration space is equally good or otherwise there is a certain degree
+of tolerance, e.g., “the minimum latency shall ideally be 500ms”, implying that anything better
+than 500ms is equally good while a performance worse than that is acceptable but not ideal.
+(3) The performance in the aspiration space is equally good while anything outside the space
+is unacceptable, e.g., “the latency shall be 500ms”. This suggests that a latency better than
+500ms is equally good and no tolerance is allowed for performance worse than that.
+(4) Preferring the optimal performance while anything outside the aspiration space is unaccept-
+able, e.g., “the latency shall be at most 500ms”, reflecting that no tolerance is allowed for
+worse than 500ms while the lower the latency, the better.
+Our second contribution is the pragmatic findings that answer the aforementioned RQs over the
+1,296 cases as follows:
+• To RQ1: PS-w performs considerably better or similar to PS-w/o on 84% of the cases, out of
+which over 60% show statistically significant improvement.
+• To RQ2: The improvement of PS-w over PS-w/o is often largely biased to a certain position
+of the aspiration space in the objective landscape, e.g., centered or left-shifted.
+• To RQ3: PS-w/o is no worse than PS-w for 70% cases, wherein the difference is considerable
+with statistical significance for more than 85%.
+• To RQ4: Under realistic aspirations, PS-w obtains consistently better outcomes than PS-w/o
+throughout the trajectory and with a speedup up to 10×. When the aspirations are unrealistic,
+in contrast, the two optimization models are competitive in the early stage of tuning but
+soon PS-w/o would lead to better results with considerably high speedup.
+Hence, we conjecture that the performance aspirations do matter for guiding bi-objective
+software configuration tuning in general. Yet, depending on the context, it can either be
+helpful or harmful. We provide, as part of the third contribution, some in-depth analysis and
+discussions on the reasons behind the above observations. More importantly, these findings allow us
+to derive our fourth contribution: the key lessons learned on the choice between PS-w and PS-w/o
+for bi-objective software configuration tuning, which are:
+• Lesson 1: The choice on whether to exploit aspirations for guiding the tuning is primarily
+dependent on their realism.
+• Lesson 2: It is unlikely that the combinations of patterns can change the decision on whether
+to incorporate aspiration in the tuning, but it can influence the benefit/detriment of aspiration-
+guided tuning.
+• Lesson 3: The positions of realistic aspiration space in the objective space can largely affect
+the benefits brought by considering aspirations within tuning, but it is less likely to influence
+the choice.
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+1:6
+Chen and Li.
+• Lesson 4: The given tuning budget has a marginal impact on the choice when the aspirations
+are realistic. However, it can be an important factor to consider under unrealistic aspirations.
+Drawing on those lessons, our fifth contribution outlines three future opportunities for this field
+of research, namely:
+• Landscape Analysis for Configurable Software Systems.
+• Requirement-Robust Optimizer for Configuration Tuning.
+• Study on the Relative Impact between Requirement Patterns to the Tuning.
+To promote open science practice, all the code, dataset, and necessary supplementary documents
+for this work can be publicly accessed at: https://github.com/ideas-labo/aspiration-study.
+The rest of this paper is organized as follows: Section 2 formalizes the problem and presents the
+motivating example. Section 3 discusses the patterns that quantify performance requirements with
+aspirations and how they were identified. Section 4 elaborates the design of our empirical study.
+Section 5 presents and analyzes the experiment results. Thereafter, Section 6 discusses the lessons
+learned and future opportunities, followed by threats to validity in Section 7. Finally, Sections 8
+and 9 review the related work and conclude the paper, respectively.
+2
+THEORY
+In this section, we present the theoretical knowledge for understanding the purpose of this work.
+2.1
+Formal Definition
+2.1.1
+Background and Problem Formalization. In the DevOps era, software configuration tuning
+involves two fundamental roles that interact frequently [67] — the stakeholders (whose benefit is
+directly affected by the software performance) negotiate their performance requirements with the
+software engineers, who then act as the operators to tune the configurations for satisfying these
+requirements. Beyond a single performance concern, recently there has been an increasing demand
+for considering multiple performance objectives [19, 41]. Among those, our literature review from
+Section 3 shows that 90% of the recent work has considered two performance objectives [16, 32, 62],
+such as the latency versus throughput for Storm; image quality versus energy usage for x264. This
+makes software configuration tuning with requirements in mind even more complex.
+Without loss of generality, we assume that a configurable software comes with a set of configura-
+tion options, whereby the 𝑖th option is denoted as 𝑐𝑖, which can be a binary, integer, or enumerate
+variable. A particular configuration is denoted as 𝒄. The search space, 𝒞, is the Cartesian product of
+the possible values for all the 𝑐𝑖. Formally, given a scenario of requirements with clear aspirations
+for two performance objectives, the goal of PS-w for software configuration tuning is to find the
+configuration(s) that achieve:
+𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑝𝑥 (𝑓1(𝒄)), 𝑝𝑦(𝑓2(𝒄)), 𝒄 ∈ 𝒞
+(1)
+whereby 𝑓 is the raw measurement of the performance value achieved by 𝒄; 𝑝 is the corresponding
+requirement pattern, which quantifies the degree of satisficing given 𝑓 (𝒄) (see Section 3). In this
+work, we consider cases where at least one 𝑝 contains a clear aspiration level3.
+In contrast, the goal of PS-w/o is to:
+𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓1(𝒄), 𝑓2(𝒄), 𝒄 ∈ 𝒞
+(2)
+3We use 𝑝𝑥 and 𝑝𝑦 to distinguish two performance requirement patterns.
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+Do Performance Aspirations Matter for Guiding Software Configuration Tuning?
+1:7
+Fig. 1. A performance requirement snippet from the requirement document of a real-world project in the
+PURE dataset [29].
+�5.5 �5 �4.5 �4 �3.5
+0
+1
+2
+3
+log10 Throughput�1
+log10 Latency
+Aspiration level
+Aspiration level
+Aspiration 
+space
+Fig. 2. The aspiration space (highlighted by color) and aspiration levels within the bi-objective space (latency
+and throughput) for Storm under the Rolling Sort benchmark.
+As can be seen, PS-w explicitly leverages information about the given requirements with clear
+aspirations to guide the search and tuning while PS-w/o assumes the basic Pareto optimality4.
+2.1.2
+Aspiration Space. Following the normal software engineering practice of requirement ne-
+gotiation, it is likely that a single performance requirement can come with a clear aspiration, in
+which case we define aspiration space as the portion of performance points that are not inferior
+to the given aspiration level. A real-world example has been shown in Figure 1. Here, “the system
+shall support at least 1,000 concurrent users” contains a clear aspiration level of 1,000
+users, meaning that the aspiration space covers throughput between 1,000 (inclusive) and the true
+optimum (which is case-dependent). Beyond such a one-dimensional case, it is easy to know that
+the aspiration space can be generalized to a two-dimensional case when the aspiration levels of
+two performance objectives are involved. For example, Figure 2 shows the aspiration space for the
+requirements “the system shall perform with 39800 users at a time” while “the latency
+shall be no worse than 160 seconds” for Storm (with log-transformed values (log10) and all
+performance objectives are to be minimized as we consider the reciprocal of Throughput). This
+forms the foundation of our analysis in what follows.
+2.1.3
+Pareto search with and without Aspirations for Tuning Software. To illustrate the difference
+between PS-w and PS-w/o, a pseudo-code using NSGA-II as the underlying optimizer has been
+4We assume that all performance objectives are to be minimized; maximizing ones can be easily converted.
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+4.3
+Performance
+4.3.1
+The system should support at least 1000 concurrent users.
+This statement provides a general sense of reliability when the system is under load. It is
+important that a substantial number of actors be able to access the system at the same time,
+since a courseware system is important to the courses that employ it. The times when the
+system will be under the most stress are likely during midterm and finals weeks. Therefore, it
+must be able to handle at least 1,000 concurrent users.
+Priority: 11:8
+Chen and Li.
+Algorithm 1: Unified code for PS-w and PS-w/o with NSGA-II.
+Input: Configuration space V; the system F; a matrix of fitness quantified by the requirements Γ
+Output: A set of nondominated configurations S′
+1 Randomly initialize a population of 𝑛 configurations P
+2 /* measuring on the actual configurable system
+*/
+3 measure(P, F)
+4 /* for PS-w, the fitness that guides the search is computed according to Equation (1)
+*/
+5 if PS-w then Γ ←getFitnessBasedonReqirements(P)
+6 while The search budget is not exhausted do
+7
+P′ = ∅
+8
+while P′ < 𝑛 do
+9
+/* for PS-w, selecting parents with respect to their compliance to the requirements */
+10
+if PS-w then {𝑠𝑥,𝑠𝑦 } ←mating(P, Γ)
+11
+else {𝑠𝑥,𝑠𝑦 } ←mating(P)
+12
+{𝑜𝑥,𝑜𝑦 } ←doCrossoverAndMutation(V,𝑠𝑥,𝑠𝑦)
+13
+measure(𝑜𝑥,𝑜𝑦, F)
+14
+if PS-w then Γ ←getFitnessBasedonReqirements(𝑜𝑥,𝑜𝑦)
+15
+P′ ← P′ �{𝑜𝑥,𝑜𝑦 }
+16
+/* for PS-w, the configurations are preserved according to the fitness computed with
+respect to the requirements
+*/
+17
+if PS-w then U ←nondominatedSorting(P � P′, Γ)
+18
+else U ←nondominatedSorting(P � P′)
+19
+P ←top 𝑛 configurations from U
+20 if PS-w then return S′ ←nondominatedConfigurations(P, Γ)
+21 else return S′ ←nondominatedConfigurations(P)
+shown in Algorithm 1. As can be seen, PS-w and PS-w/o mainly differ in the fact that the former is
+guided by the information extracted from the given requirements and aspirations (denoted as Γ)
+while the latter runs without, i.e., it uses the raw values of the measured performance objectives.
+This means that all the fitness of configurations evaluated in the PS-w makes use of the Γ while
+that of the configuration in PS-w/o does not. For example, under the raw performance, a latency of
+500ms is certainly more preferred than the case of 700ms. However, under the requirement and
+aspiration that any latency less than 900ms is equally preferred, they are actually equivalent therein
+and hence PS-w reflects precisely that.
+As a result, the above generates two differences between PS-w and PS-w/o. Firstly, the process of
+deciding on which two configurations to be selected as parents for generating new configurations
+is guided differently (i.e., lines 10–11). Secondly, the environmental selection that determines what
+configurations to be preserved in the next iteration is also guided by different fitness (i.e., lines
+17–18).
+As we will show, even with such a simple deviation the leading results can be radically different
+depending on the circumstances.
+2.2
+Motivating Scenario
+Taking x264 — a configurable video encoder — as a concrete example, a possible requirement
+scenario could involve performance requirements (denoted as 𝒫1) “the PNSR5 shall be
+at least 40dB” and “the energy usage shall be at most 80 watts”. Here, there is a
+clear aspiration level 40dB and 80 watts for the performance attribute PNSR and energy usage,
+5PNSR stands for Peak signal-to-noise ratio, which measures the reconstruction quality for images; the larger the PNSR, the
+better.
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+Do Performance Aspirations Matter for Guiding Software Configuration Tuning?
+1:9
+�80
+�60
+�40
+�20
+20
+40
+60
+80
+100
+120
+140
+�PNSR (dB)
+Energy (watt)
+Preferred configurations
+(b) PS-w with P2
+�80
+�60
+�40
+�20
+20
+40
+60
+80
+100
+120
+140
+�PNSR (dB)
+Energy (watt)
+Preferred configurations
+(a) PS-w with P1
+�80
+�60
+�40
+�20
+20
+40
+60
+80
+100
+120
+140
+�PNSR (dB)
+Energy (watt)
+Preferred configurations
+(c) PS-w/o
+Aspiration space
+Aspiration space
+Aspiration space 
+(assuming P1 or P2)
+Fig. 3. The preferred configurations for x264 by PS-w/o and PS-w given different requirement scenarios.
+respectively. Indeed, depending on the requirement scenario, the preference for performance
+deviating from the aspiration level could vary even with a clear aspiration level (as we will discuss
+in Section 3). For instance, the above example may imply that one would not accept any performance
+worse than 40dB or 80 watts but prefers any configurations with better PNSR and energy usage. This
+means that, suppose there are three configurations 𝑨 = {65𝑑𝐵, 30𝑤𝑎𝑡𝑡𝑠}, 𝑩 = {80𝑑𝐵, 25𝑤𝑎𝑡𝑡𝑠}, and
+𝑪 = {35𝑑𝐵, 10𝑤𝑎𝑡𝑡𝑠}, the 𝑪, although it has the best energy usage, would be ruled out as it fails to
+meet aspiration for PNSR; 𝑩 would certainly be more ideal under such a requirement scenario since
+it has better results on both performance objectives than 𝑨. In a different requirement scenario, the
+requirements (denoted as 𝒫2) may become “the PNSR shall be no worse than 40dB” while
+“the energy usage shall be no worse than 80 watts”, which implies that one would not
+accept any performance worse than 40dB or 80 watts, but equally prefer anything that goes beyond
+40dB and 80 watts. Here, 𝑪 is ruled out again but 𝑨 and 𝑩 would become equally preferred as their
+PNSR and energy usages are better than 40dB and 80 watts, respectively. Of course, the given
+40dB and/or 80 watts may well be unrealistic aspirations, i.e., none of the configurations would
+reach them (or at least no one can be found under the possible tuning budget).
+To make the meaning of the above clear for PS-w and PS-w/o, Figure 3 illustrates what configura-
+tions are preferred when using PS-w and PS-w/o in the tuning under 𝒫1 or 𝒫2. Here, the quality of
+the configurations produced would need to be evaluated with respect to the requirements and PS-w
+prefers precisely what is needed therein. PS-w/o, in contrast, naturally prefers all configurations
+on the Pareto front. Intuitively, we note that PS-w/o would also prefer some configurations that
+are preferred by its PS-w counterpart. For example, when comparing Figure 3a and 3c, all the
+points preferred by PS-w are also preferred by PS-w/o (but not vice versus), hence they should
+converge to the same satisfiability under 𝒫1. In Figure 3b and 3c, although PS-w/o prefers different
+points to that of PS-w in the aspiration space, they should be able to reach the same degree of
+satisfaction with respect to the requirements because all configurations within the aspiration space
+are deemed equivalent when being evaluated by 𝒫2. Indeed, if both PS-w and PS-w/o can find all
+their preferred points in the space, then the engineers can simply cherry-pick the fully satisfied
+ones according to the given performance requirements from the final set of configurations returned.
+Yet, the unanswered question would be: is the above assumption true and hence there would be no
+difference regarding whether PS-w or PS-w/o is chosen?
+The rest of this paper provides an empirical understanding of the above confusion.
+3
+HOW REQUIREMENTS ARE HANDLED
+Here we describe the process of mining, classifying, and analyzing the real-world performance
+requirements with aspirations. We use Cohen’s Kappa coefficient (𝜅) [60] to mitigate bias between
+authors — the classification is often regarded as unbiased and sustainable when 𝜅 > 0.7 . In a
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+(a) 
+PS-w with Pi(c) 
+PS-w/o(b)
+PS-w with P21:10
+Chen and Li.
+Filtering
+Inclusion 
+criteria
+Exclusion 
+criteria
+Filtering
+Selection
+Identifying 
+requirements
+Analyzing 
+Implication
+Extracting 
+patterns
+Mappings
+14
+9
+5
+393
+107
+29
+386
+426
+Fig. 4. Overview of dataset analysis and literature review.
+nutshell, Cohen’s Kappa coefficient is generally thought to be a more robust measure than a simple
+percent agreement calculation between the raters, as it takes into account the possibility of the
+agreement occurring by chance. In this work, we use the coefficient in two aspects:
+• Measure the agreement on which implication category a requirement belongs to (we have
+𝜅 = 0.85).
+• Measure the agreement on which patterns that a paper assumes (we have 𝜅 = 0.76).
+3.1
+Real-world Requirements with Aspirations
+To understand what are the common real-world performance requirements with aspirations and
+their implications in the industry, in Jan 2021, we mined the publicly available requirement dataset
+from Zenodo (under the Empirical Software Engineering label), GitHub, and the Google Dataset
+Search, using a keyword “requirement dataset”, as shown in Figure 4. The results led to 386 items,
+including duplication and many irrelevant ones which can be easily identified from their titles. As
+such, we filtered the candidates down to 14, within which we followed the criteria below to extract
+the most relevant ones for this study:
+• The dataset has clearly documented requirement statements for the software systems to be
+built.
+• The dataset contains labeled requirements for performance objectives or there is readily
+available code to do so.
+• To ensure external validity, the dataset contains performance requirements for systems from
+different domains.
+The process has resulted in nine shortlisted datasets, based on which we attempted to identify
+the statements of performance requirements according to the following rules:
+• The performance requirement should contain a quantifiable aspiration level, such as “the
+system shall perform with 1500 users at a time”. In contrast, “the system shall
+be fast” is too vague to be quantified.
+• To ensure fairness when comparing with the PS-w/o, we eliminate the performance require-
+ments that do not prefer one extreme of the objective, such as “the display shall be
+refreshed every 60 seconds”. This is because such requirements prefer the performance
+to reach a clear aspiration (e.g., 60 seconds) instead of a maximum/minimum of the perfor-
+mance objective. Therefore, in such a case, PS-w should always be preferred, since there is
+no point to use PS-w/o which naturally maximizes/minimizes the objectives while does not
+take aspiration into account6.
+6Note that, indeed, in some cases, the preference of this kind of requirement can be derived by inferring from the context.
+Using the same example, if the display could not be refreshed because some long-running analyses could not be terminated
+within 60 seconds, then the preference would be to guarantee the ability to refresh every 60-sec or less. However, in our
+cases, most of those requirements come from the PROMISE dataset, which has no extra information other than some
+sentences describing the requirement. This makes it difficult for us to correctly infer the preferences implied. Hence, in the
+above example, we stick with the literal meaning that one would prefer and only prefer a refresh rate of 60 seconds; no
+more and no less.
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+GDo Performance Aspirations Matter for Guiding Software Configuration Tuning?
+1:11
+Table 1. Performance requirements with aspirations.
+Dataset
+# Requirements
+Link
+Do et al. [23]
+52
+https://github.com/aqd14/ICSR-2019
+PROMISE [61]
+48
+https://zenodo.org/record/268542
+PURE [29]
+28
+https://zenodo.org/record/1414117
+Shaukat et al. [69]
+13
+https://zenodo.org/record/1209601
+Dalpiaz et al. [21]
+10
+https://zenodo.org/record/3309669
+Table 2. Identified papers with aspiration quantification.
+Venue
+# Papers
+Venue
+# Papers
+Venue
+# Papers
+TSE (journal)
+3
+JSS (journal)
+6
+TAAS (journal)
+3
+ASE (journal)
+2
+ESE (journal)
+1
+ICPE (conference)
+1
+ICSE (conference)
+1
+FSE (conference)
+3
+ASE (conference)
+1
+SEAMS (symposium)
+6
+ICSA (conference)
+1
+MODELS (conference)
+1
+The above has led us to rule out four datasets that contain no appropriate requirements. Table 1
+shows details of the final five datasets used in our study (removing duplication).
+3.2
+Literature Search of Patterns
+As from Figure 4, we also conducted a literature search according to the best practice of a systematic
+literature review in software engineering [48], containing search protocol, inclusion, and exclusion
+criteria. Our goal is to understand a single question: how are the implications of real-world perfor-
+mance requirements with aspirations, which are generic to the software systems as identified from
+Section 3.1, have been specifically quantified in current bi-objective software configuration tuning
+work? Note that we do not intend to be comprehensive, but rather to gather representatives.
+In Feb 2021, we conducted a full-text search over Google Scholar for papers published since
+2010 from the software engineering community (we exclude the system-related papers for better
+representation in the community), using a focused search string below:
+“requirement” AND (“multi objective” OR “multi goal” OR “multi criteria”) AND (“per-
+formance” OR “non-functional”) AND (“configurable software” OR “adaptive software”)
+AND (“tuning” OR “optimization”)
+This gives us 426 papers. We then filtered patents, inaccessible papers, and any non-English
+documents, leading to 393 papers. Next, we further extracted the papers by using the following
+inclusion criteria on the title and abstract:
+• The paper is relevant to tuning the configuration of the software system.
+• The paper seeks to improve or evaluate the performance objectives of the software system.
+• The paper considers performance requirements.
+• The paper is peer-reviewed and is not a survey or tutorial.
+A paper was ruled out if it does not meet all the above criteria, which resulted in 107 papers.
+Then, we removed papers based on the following exclusion criteria by reviewing the content:
+• The considered performance requirements do not have a clear aspiration level.
+• The paper tackle only a single performance objective.
+• The paper does not have quantitatively experimental results with clear instructions on how
+the results were obtained.
+A paper was ruled out if it met any of the above criteria. Finally, we obtained 29 papers, as shown
+in Table 2.
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+1:12
+Chen and Li.
+I1
+93
+I2
+18
+I3
+40
+(a) # requirements per implication
+p1
+17
+p2
+11
+p3
+7
+(b) # papers per pattern
+Pattern Implication
+𝒑1
+I1, I3
+𝒑2
+I1, I2, I3
+𝒑3
+I2, I3
+(c) Mappings
+Fig. 5. Distribution of implications, patterns and their mappings (six papers consider more than one pattern).
+3.3
+Results Analysis
+3.3.1
+Number of Performance Objectives. From the review, we found that 26 out of 29 (90%) of
+the papers considered two performance objectives in their tuning process. The remaining three
+papers take into account three or more. This is a clear sign that two performance objectives remain
+a state-of-the-art setting for tuning software configuration, which is consistent with the finding
+from the recent survey for a related field [19]. Therefore, in this work, we focus on bi-objective
+software configuration tuning.
+3.3.2
+Implications. We analyzed all 151 performance requirements with aspirations from Sec-
+tion 3.1, and found three possible implications on the aspiration space for a given performance
+objective:
+• I1: Anything in the aspiration space is equally preferred. This gives a clear upper
+aspiration bound without other information, e.g., “the server will support a maximum
+of 1,000 simultaneous users”; or there is a lower aspiration bound but clear information
+has been given for the cases when the performance reaches the aspiration space, e.g., “results
+shall be returned in under 15 seconds”.
+• I2: Anything not in the aspiration space is equally non-preferred. For example, “the
+system shall allow for a minimum of 6 users at the same time”, in which case
+there is only information for a clear lower aspiration bound.
+• I3: No information is available with respect to the aspiration space. This often refers
+to the requirements where there is a clear aspiration level, but no indication about whether
+it is an upper or lower aspiration bound while any other information is unavailable. For
+example “the system shall cater to 10 simultaneous users”.
+The distribution of the implications can be found in Figure 5a and we achieve a Kappa coefficient
+𝜅 = 0.85 for this.
+3.3.3
+Patterns. Next, with the above implications in mind, we seek to understand how they are
+quantified within the 29 papers identified. This led to three state-of-the-art patterns on the functions
+to quantify requirements with aspiration level (assuming the lower bound is optimum). Suppose
+that 𝛼 and 𝛽 denote the lower and upper bound of the performance objective, respectively; 𝑑 is
+the aspiration level, the patterns, and their quantification have been shown in Figure 6 and are
+explained below:
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+Do Performance Aspirations Matter for Guiding Software Configuration Tuning?
+1:13
+↵
+d
+�
+0
+0.5
+1
+Raw measurement
+(c) p2(x) =
+(
+0
+x > d
+1
+x  d
+↵
+d
+�
+0
+0.5
+1
+Raw measurement
+(b) p1(x) =
+( ��x
+��d
+x > d
+1
+x  d
+↵
+d
+�
+0
+0.5
+1
+Raw measurement
+(d) p3(x) =
+(
+0
+x > d
+d�x
+d�↵
+x  d
+↵
+�
+0
+0.5
+1
+Raw measurement
+Satisficing value
+(a) p0(x) = ��x
+��↵
+Fig. 6. Requirement patterns with (and without) aspiration from the literature. 𝛼 and 𝛽 denote the lower and
+upper bound of the performance objective, respectively. 𝑑 is the aspiration level and the aspiration space has
+been shaded.
+• 𝒑1: The performance in the aspiration space is equally good or otherwise there is a certain
+degree of tolerance (Figure 6b). The function can be formulated as:
+𝒑1(𝑥) =
+� 𝛽−𝑥
+𝛽−𝑑
+𝑥 > 𝑑
+1
+𝑥 ≤ 𝑑
+(3)
+• 𝒑2: The performance in the aspiration space is equally good while anything outside the space
+is unacceptable (Figure 6c), such that:
+𝒑2(𝑥) =
+�
+0
+𝑥 > 𝑑
+1
+𝑥 ≤ 𝑑
+(4)
+• 𝒑3: Preferring the optimal performance while anything outside the aspiration space is unac-
+ceptable (Figure 6d), which is defined as:
+𝒑3(𝑥) =
+�
+0
+𝑥 > 𝑑
+𝑑−𝑥
+𝑑−𝛼
+𝑥 ≤ 𝑑
+(5)
+Similarly, we can also formalize the requirement with no clear aspiration level involved (e.g.,
+“the latency shall be small”), denoted as 𝒑0, which is illustrated in Figure 6a and can be
+formulated as follow:
+𝒑0(𝑥) = 𝛽 − 𝑥
+𝛽 − 𝛼
+(6)
+The distribution of the patterns has been shown in Figure 5b where we have 𝜅 = 0.76, which is
+sustainable [60].
+Through normalization in those patterns, the raw measurement of a performance objective is
+transformed into the satisficing degree with respect to a given aspiration space (if any), ranging
+between 0 and 1 where the latter means fully satisfied. As such, the transformation depends on the
+assumption of satisficing over measurements included or excluded by the aspiration space, which
+distinguishes the patterns. At this point, we can immediately see the mappings between the patterns
+and the extracted implications from the real-world dataset. Such mappings have been illustrated in
+Figure 5c, from which we see that each pattern, except 𝒑0, can fit with at least two implications
+from the real-world requirements. For example, 𝒑1 can fit with I1 and I3, because the former prefers
+anything within the aspiration space and specifies nothing on the other extreme, while the latter
+has no information at all and thus one needs to rely on an assumption when quantifying I3 to
+guide the search, meaning that it has the possibility to fit with all the three patterns.
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+1:14
+Chen and Li.
+�80
+�60
+�40
+�20
+20
+40
+60
+80
+100
+120
+140
+�PNSR (dB) with p3
+Energy (watt) with p1
+Original Space
+Aspiration space
+0
+0.2
+0.4
+0.2
+0.4
+0.8
+PNSR p3
+Energy p1
+Transformed Space
+For both performance objectives, the smaller the better
+For both performance objectives, the larger the better
+Best configuration under the 
+requirements and aspirations
+Best configuration under the 
+requirements and aspirations
+Transformed Space
+Original Space
+Fig. 7. Example of using the patterns for evaluating the goodness of configurations and guiding PS-w in the
+transformed space.
+From the above, it is confirmed that there exist patterns from current work which can reflect the
+implication of real-world performance requirements and their aspirations. We, therefore, will seek
+to examine all of them in our empirical study.
+3.4
+Respecting Requirements and Aspiration in Software Configuration Tuning
+While the above requirement patterns are the key to evaluating the “better” or “worse” in the set of
+configurations produced by any Pareto optimizer and optimization model, they directly influence
+the behavior of PS-w (they correspond to the 𝑝𝑛 in Equation (1)) but not that of the PS-w/o.
+Most importantly, those patterns allow us to precisely quantify what is the best configuration(s)
+amongst the configurations produced by those two optimization models given a set of requirements
+and aspirations. Figure 7 shows an example of evaluating the configurations (and guiding PS-w)
+in a transformed space when taking the requirements and aspirations into account, i.e., energy
+usage with 𝒑1 and aspiration of 80 watts while PNSR with 𝒑3 and aspiration of 40dB. Here, we
+certainly prefer the points within the aspiration space in contrast to those outside. However, for
+those points within the aspiration space, we only prefer those with better PNSR while the energy
+usage is deemed as equivalent (due to the implication of 𝒑1 and 𝒑3).
+The above is difficult to assess and quantify in the original space (Figure 7 left), since naturally
+the points that are non-dominated by each other (in the sense of the original objective values)
+are considered as equivalent when the requirements and aspiration are not involved. Therefore,
+the actual most preferred point (arrow highlighted) is not considered the best. In contrast, the
+evaluation becomes immediately obvious on what is the best point in the transformed space, where
+the energy and PNSR are converted by the equations for 𝒑1 and 𝒑3, respectively. Now, clearly, the
+most preferred point is the only non-dominated point therein (Figure 7 right).
+4
+EMPIRICAL STUDY DESIGN
+As shown in Figure 8, our methodology consists of the following steps:
+Step 1: Assume that a requirement scenario has been negotiated by the software engineers
+and stakeholders, we quantify the requirements such that they are ready for the Pareto
+optimizers, i.e., in the forms of a combination of patterns from Section 3 and their
+aspiration space. In particular, to form a requirement scenario, the given combination of
+the patterns is denoted as a two-dimensional vector 𝒫, such that there is at least one
+that comes with a clear aspiration level, e.g., 𝒫 = {𝒑0, 𝒑3}. In this work, we examine
+all possible combinations of the patterns (including 𝒑0). Under each combination, we
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+Do Performance Aspirations Matter for Guiding Software Configuration Tuning?
+1:15
+3 Pareto optimizers
+8 software systems
+and environments
+evaluate quality of 
+configuration set
+Pareto search 
+with 
+aspiration
+Pareto search 
+without 
+aspiration
+�5
+�4
+�3
+0
+2
+4
+log10 Throughput�1
+log10 Latency
+u
+r
+l
+c
+15 combinations of patterns 
+and 3 (or 4) aspiration space
+ 
+Next combination of patterns/aspirations
+Scenario 
+Identification
+Configuration 
+Tuning
+Quality 
+Evaluation
+1
+2
+2
+4
+5
+3
+↵
+d
+�
+0
+0.5
+1
+Raw measurement
+p3 )
+p(x) =
+(
+0
+x > d
+x�↵
+d�↵
+x  d
+(d)
+↵
+d
+�
+0
+0.5
+1
+Raw measurement
+Satisficing value
+p2 )
+p(x) =
+(
+0
+x > d
+1
+x  d
+(c)
+↵
+d
+�
+0
+0.5
+1
+p1 )
+p(x) =
+(
+x�d
+��d
+x > d
+1
+x  d
+(b)
+↵
+�
+0
+0.5
+1
+Satisficing value
+p0 ) p(x) = x�↵
+��↵
+(a)
+100 repeats
+with
+with
+with
+Fig. 8. Overview of the empirical study.
+also consider different aspiration spaces for our RQs; this will be further elaborated in
+Section 4.2.
+Step 2: Run both PS-w/o and PS-w on different software systems. Particularly, when formulating
+the performance objectives, PS-w/o is steered by the raw measurements only7 while
+PS-w is designed to be guided by the given vector of patterns 𝒫 as the new objectives. To
+ensure fairness, both optimization models are examined under the same optimizer and
+we consider three representative optimizers in this work, i.e., NSGA-II [22], IBEA [85],
+and MOEA/D [82].
+Step 3: Measure the system as the search proceeds until the tuning budget has been exhausted;
+repeat 100 times.
+Step 4: Evaluate the set of configurations thereafter using 𝒫 as part of the Quality Evaluation
+phase.
+Step 5: Go back to Step 1 if there are more combinations of patterns and aspirations to examine.
+It is worth noting that, although the patterns from Section 3 are for single performance objective,
+they can be arbitrarily combined for the bi-objective software configuration tuning in the Scenario
+Identification phase of Step 1 [8, 9, 33, 59], as illustrated in Table 3.
+In the Configuration Tuning phase (Step 2 and 3), the patterns require normalization using
+the lower and/or upper bound (except for 𝒑0 and 𝒑2). However, since these are often unknown,
+we adopt a dynamic method wherein the raw measurements are normalized using the maximal
+and minimal values found so far as the tuning proceeds, which is common in SBSE for software
+configuration tuning [7, 68]. We record the raw measurements of each configuration throughout
+the tuning to efficiently utilize the tuning budget (Section 4.3.2).
+To mitigate stochastic bias, we repeat each experiment 100 runs. The study is conducted on a
+cluster of machines each with Intel i5 six cores CPU at 2.9GHz and 8GB memory, running numerous
+experiments in parallel over the course of five months (24 × 7).
+4.1
+Subject Software Systems
+We conduct our study on a set of real-world highly configurable software systems and environments
+that have been widely studied in existing work [17, 44, 45, 62]. These are selected according to the
+criteria below:
+(1) To ensure that the search landscape is not too trivial to be explored, the system should contain
+a mix of binary and enumerative configuration options.
+7This is effectively identical to using 𝒑0 for all performance objectives.
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+PPP1:16
+Chen and Li.
+Table 3. The considered requirement scenarios (in terms of the combination of the patterns identified from
+Section 3) and their example interpretations. The interpretations are based on the assumption that the
+performance objectives are {𝑙𝑎𝑡𝑒𝑛𝑐𝑦,𝑡ℎ𝑟𝑜𝑢𝑔ℎ𝑝𝑢𝑡} with possible aspiration levels 𝑑1 and 𝑑2, respectively.
+Possible 𝒫
+Example Interpretation
+{𝒑0, 𝒑1}
+Prefer better latency and throughput better than 𝑑2, but any configurations better than 𝑑2 are equally
+preferred; willing to accept throughput worse than 𝑑2.
+{𝒑1, 𝒑0}
+Prefer better throughput and latency better than 𝑑1, but any configurations better than 𝑑1 are equally
+preferred; willing to accept latency worse than 𝑑1.
+{𝒑0, 𝒑2}
+Prefer better latency and throughput better than 𝑑2, but any configurations better than 𝑑2 are equally
+preferred; do not accept throughput worse than 𝑑2.
+{𝒑2, 𝒑0}
+Prefer better throughput and latency better than 𝑑1, but any configurations better than 𝑑1 are equally
+preferred; do not accept latency worse than 𝑑1.
+{𝒑0, 𝒑3}
+Prefer better latency and throughput; do not accept throughput worse than 𝑑2.
+{𝒑3, 𝒑0}
+Prefer better latency and throughput; do not accept latency worse than 𝑑1.
+{𝒑1, 𝒑1}
+Prefer latency better than 𝑑1 and throughput better than 𝑑2, but any configurations better than 𝑑1 and
+𝑑2 are equally preferred; willing to accept latency and throughput worse than 𝑑1 and 𝑑2, respectively.
+{𝒑2, 𝒑2}
+Prefer latency better than 𝑑1 and throughput better than 𝑑2, but any configurations better than 𝑑1 and
+𝑑2 are equally preferred; do not accept latency and throughput worse than 𝑑1 and 𝑑2, respectively.
+{𝒑3, 𝒑3}
+Prefer better latency and throughput; do not accept latency and throughput worse than 𝑑1 and 𝑑2,
+respectively.
+{𝒑1, 𝒑2}
+Prefer latency better than 𝑑1 and throughput better than 𝑑2, but any configurations better than 𝑑1 and
+𝑑2 are equally preferred; willing to accept latency worse than 𝑑1 but do not accept throughput worse
+than 𝑑2.
+{𝒑2, 𝒑1}
+Prefer latency better than 𝑑1 and throughput better than 𝑑2, but any configurations better than 𝑑1 and
+𝑑2 are equally preferred; willing to accept throughput worse than 𝑑2 but do not accept latency worse
+than 𝑑1.
+{𝒑1, 𝒑3}
+Prefer better throughput and latency better than 𝑑1, but any configurations better than 𝑑1 are equally
+preferred; willing to accept latency worse than 𝑑1 but do not accept throughput worse than 𝑑2.
+{𝒑3, 𝒑1}
+Prefer better latency and throughput better than 𝑑2, but any configurations better than 𝑑2 are equally
+preferred; willing to accept throughput worse than 𝑑2 but do not accept latency worse than 𝑑1.
+{𝒑2, 𝒑3}
+Prefer better throughput and latency better than 𝑑1, but any configurations better than 𝑑1 are equally
+preferred; do not accept latency worse than 𝑑1 nor throughput worse than 𝑑2.
+{𝒑3, 𝒑2}
+Prefer better latency and throughput better than 𝑑2, but any configurations better than 𝑑2 are equally
+preferred; do not accept latency worse than 𝑑1 nor throughput worse than 𝑑2.
+(2) A full exploration of the search space is infeasible, i.e., it cannot be done within 24 hours.
+(3) There are clear instructions on how to set up the benchmark under which the system will be
+measured.
+(4) If the same system of an environment has been used with a different set of configuration
+options, choose those with relatively higher complexity, i.e., larger search space and more
+configuration options. For example, Storm can be tuned under different workload bench-
+marks, and we choose WordCount and RollingSort as the two that satisfy the above
+criteria.
+We firstly eliminated LLVM from [62], as it violates Criterion (1). Similarly, sort-256 and noc-
+CM-log is also ruled out due to their rather small search space which can be exhaustively explored
+in 24 hours, i.e., Criterion (2). We cannot consider the system SaC as there is no clear instruction
+on under what benchmark it can be profiled, which violates Criterion (3). We also noticed that
+Storm and Keras (with DNN or LSTM) have been much more commonly used than others, but
+with different configuration options and environments. Therefore, according to Criterion (4), we
+use the settings that lead to a much larger search space and more options. As shown in Table 4, the
+selected software systems come from diverse domains, e.g., video encoding, stream processing, and
+deep/machine learning, while having different performance objectives, scales, and search spaces.
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+Do Performance Aspirations Matter for Guiding Software Configuration Tuning?
+1:17
+Table 4. Configurable software systems studied. We run all software systems under their standard benchmarks.
+Storm and Keras (with DNN) use two benchmarks and three dataset, respectively.
+Software
+Domain
+Performance Objectives
+# Options
+Search Space
+Used By
+Trimesh
+Mesh solver
+Latency and # Iteration
+13
+239,260
+[17, 62]
+x264
+Video encoding
+PSNR and Energy Usage
+17
+53,662
+[17, 62]
+Storm/WC
+Stream processing
+Latency and Throughput
+6
+2,880
+[17, 44, 45, 62]
+Storm/RS
+Stream processing
+Latency and Throughput
+6
+3,839
+[17, 44, 45, 62]
+Keras/Adiac
+Deep learning
+AUC and Inference Time
+13
+3.99×1013
+[45]
+Keras/DSR
+Deep learning
+AUC and Inference Time
+13
+3.32×1013
+[17, 45]
+Keras/SA
+Deep learning
+AUC and Inference Time
+13
+2.66×1013
+[45]
+XGBoost
+Machine learning
+Accuracy and Training Time
+13
+2.88×1010
+[45]
+Their measurements are also expensive8, e.g., XGBoost needs 2,807 hours to explore less than 1%
+of its search space.
+We keep the same performance objectives, configuration options, and their ranges as studied in
+the prior work that made use of them, e.g., [17, 44, 45, 62], since those have been shown to be the
+key ones for the software systems under the related environment. As a result, although the software
+systems are the same, the actual search spaces are different, such as Storm/WC and Storm/RS. In
+particular, following what has been used in previous work, the environment/workload we consider
+are:
+• Trimesh: we use the Shapenet dataset that contains 51,300 unique 3D models. In this work,
+we randomly sample 100 models as the standard benchmark.
+• x264: for this, the benchmark used is a standard video of 1GB size, which was chosen
+randomly.
+• Storm/WC: we use the WordCount as the benchmark. This is a typical simple streaming
+example where Storm is used to keep track of the words and their counts streaming in.
+WordCount generates a CPU-intensive workload.
+• Storm/RS: similar to Storm/WC, here we use the RollingSort as the benchmark. Unlike
+WordCount, RollingSort generates a memory intensive workload.
+• Keras/Adiac: we use the Deep Neural Network (DNN) from the Keras software and run it
+on the Adiac dataset. Generally, the dataset contains a task of automatic identification of
+diatoms (unicellular algae) among 31 classes with a training and testing size of 390 and 391,
+respectively.
+• Keras/DSR: we use the DNN from the Keras software and run it on the DiatomSizeReduc-
+tion dataset. The dataset concerns the prediction of four types of diatoms with a training
+and testing size of 16 and 306, respectively.
+• Keras/SA: we use the DNN from the Keras software and run it on the ShapesAll dataset.
+Generally, the dataset aims to test contour/image and skeleton-based descriptors; there are
+60 classes with a training and testing size of 600 each.
+• XGBoost: we use the Covertype dataset that contains 54 forest cover type from cartographic
+variables only. The size of the dataset is 581,012 and we follow a 70%-30% training and testing
+split.
+Indeed, the analyzed dataset and literature in Section 3 may not specifically target the software
+systems considered in this work. However, the extracted implication and patterns are rather generic
+such that they can be applied to different cases. Further, some widely studied performance objectives
+(from both the dataset and literature) are overwhelmingly applicable. For example, latency- and
+8Each measurement consists of 5 repeated samples and the median value is used.
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+1:18
+Chen and Li.
+Table 5. Aspiration levels and spaces for the configurable software systems studied (used for all combinations
+of patterns). 𝑙, 𝑟, 𝑐, and 𝑢 denote left-shifted, right-shifted, centered, and unrealistic aspirations, respectively.
+Software
+Performance Objectives
+𝑙
+𝑟
+𝑐
+𝑢
+Trimesh
+{Latency (s), # Iterations}
+{81, 4}
+{461, 15}
+{135, 7}
+{37, 501}
+x264
+{PSNR (dB), Energy Usage (W)}
+{50, 3680}
+{37, 462}
+{46, 1260}
+{100, 34}
+Storm/WC
+{Throughput (msgs/m), Latency (ms)}
+{16473, 15677}
+{994, 5}
+{8982, 101}
+{34740, 3}
+Storm/RS
+{Throughput (msgs/m), Latency (ms)} {1.3 × 105, 7819}
+{3006, 5}
+{3.7 × 104, 126} {2.3 × 105, 1.9}
+Keras/Adiac
+{AUC, Inference Time (ms)}
+{0.030, 44}
+{0.017, 0.05}
+{0.028, 3}
+{0.292, 0.03}
+Keras/DSR
+{AUC, Inference Time (ms)}
+{0.307, 123}
+{0.107, 0.12}
+{0.300, 25}
+{0.581, 0.031}
+Keras/SA
+{AUC, Inference Time (ms)}
+{0.167, 21}
+{0.157, 0.07}
+{0.160, 6}
+{0.325, 0.04}
+XGBoost
+{Accuracy (%), Training Time (s)}
+{80, 42}
+{54, 3}
+{72, 8}
+{92, 1}
+throughput-related requirements (with different aspiration levels) are prevalent for a wide range of
+software [62].
+4.2
+Aspiration Space
+To improve external validity, we consider aspiration levels that draw two types of aspiration space
+under two performance objectives: realistic and unrealistic ones. To that end, for each software
+system, we run all the Pareto optimizers for three hours each to obtain a landscape that contains an
+approximated Pareto front. We do so by ensuring that the obtained front is reasonably converged,
+i.e., increasing the budget only marginally changes the results. We then set the aspiration space
+based on such a front as summarized in Table 5.
+4.2.1
+Realistic Aspiration Space. For software configuration tuning with two performance objec-
+tives, we say an aspiration space is realistic if there is at least one configuration that can reach the
+aspiration levels of both performance objectives.
+Using Storm/RS as an example in Figure 9, for the realistic ones under each combination of
+patterns, we set three aspiration space based on their positions in the objective space: left-shifted (𝑙),
+right-shifted (𝑟) and centered (𝑐). In particular, 𝑙 is defined as using the value of the 20𝑡ℎ percentile
+for throughput and the value of the 80𝑡ℎ percentile for latency as their corresponding aspiration
+levels; similarly,𝑟 uses the value of the 20𝑡ℎ percentile for latency and the value of the 80𝑡ℎ percentile
+for throughput; finally, 𝑐 uses the values of the 50𝑡ℎ percentile for both performance objectives.
+Clearly, despite covering diverse regions in the overall space of performance objectives, all those
+spaces contain at least one point (configuration). Note that the aspiration space is applicable to any
+combination of patterns with and without 𝒑0 (in Figure 9a and Figure 9b respectively), as long as
+there is a clear aspiration level for at least one performance objective.
+4.2.2
+Unrealistic Aspiration Space. Since the aspiration level/space is negotiated beforehand, it
+may be unrealistic. In this work, we refer to an unrealistic aspiration space as the situation wherein
+the aspiration levels of two performance objectives can be at most reached one at a time, but not
+both simultaneously. For example, in the case of two performance objectives from Figure 9a, 𝑢 is
+an unrealistic aspiration space such that the level is achievable for either of the two objectives
+individually (as indicated by the dashed lines), but not for both, as there is no point (configuration)
+residing in the space. As a result, it is not applicable when only one performance objective contains
+clear aspiration, e.g., in Figure 9b. To define such a space, we set the value of 5𝑡ℎ percentile of both
+performance objectives as the corresponding aspiration levels, which we have found as sufficient
+to create an unrealistic aspiration space for each system.
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+Do Performance Aspirations Matter for Guiding Software Configuration Tuning?
+1:19
+�5
+�4
+�3
+0
+2
+4
+log10 Throughput�1 (p1)
+log10 Latency (p0)
+r
+l
+c
+(b)
+�5
+�4
+�3
+0
+2
+4
+log10 Throughput�1 (p1)
+log10 Latency (p1)
+u
+r
+l
+c
+(a)
+Fig. 9. Distant aspiration space (shaded by different colors) under different combinations of patterns for
+Storm/RS.
+Table 6. Population size and measurement tuning budget.
+Software
+Population Size
+# Measurements
+Software
+Population Size
+# Measurements
+Trimesh
+10
+500
+x264
+50
+1,500
+Storm/WC
+50
+500
+Storm/RS
+30
+700
+Keras/Adiac
+50
+700
+Keras/DSR
+60
+500
+Keras/SA
+60
+500
+XGBoost
+30
+300
+4.3
+Tuning Settings
+4.3.1
+Pareto Optimizer. We consider three Pareto optimizers, i.e., NSGA-II [22], IBEA [85], and
+MOEA/D [82], because:
+• They have been widely used for software configuration tuning in prior work [8, 9, 16, 33, 59,
+70].
+• They are the representatives of three fundamentally different frameworks for Pareto search [26].
+• They can, but do not have to, rely on a surrogate model(s) [16], which greatly reduces the
+noise in our empirical study.
+All the above optimizers are adopted for both PS-w and PS-w/o based on the implementations in
+jMetal [25].
+4.3.2
+Tuning Budget. In this work, we set a budget of one hour for each run as commonly used for
+expensive SBSE problems [53]. However, directly relying on the time as a termination criterion can
+suffer severe interference during the tuning as numerous experiments need to be run in parallel.
+To prevent this, for each software system, we did the following to convert the one-hour tuning
+budget into the number of unique measurements:
+(1) incrementally (100 each step) measuring distinct configurations on a dedicated machine using
+random sampling until the one-hour time budget is exhausted.
+(2) repeating the above 5 times and collect the number of measurements.
+(3) the median of the 5 repeats serves as the key termination criterion of the tuning thereafter
+(in Table 6).
+Note that in each run of the tuning, we cached the measurement of every distinct configuration
+for direct reuse. Hence, only the distinct configurations would consume the budget.
+4.3.3
+Parameters. For the three optimizers in all cases, we apply the binary tournament, boundary
+mutation, and uniformed crossover, as used in prior work [16, 20]. The mutation and crossover
+rates are set to 0.1 and 0.9, respectively, which also follows the most common setting for software
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+1:20
+Chen and Li.
+configuration tuning [16, 20]. Other specific settings for IBEA and MOEA/D are kept as default
+values, which have been shown to be effective [16].
+For each system, we pragmatically set the population size via:
+(1) examining different sizes in pilot runs under the budget in Table 6, i.e., {10, 20, ..., 100}, over
+all optimizers, combinations of patterns and aspiration (on both PS-w and PS-w/o).
+(2) recording the average change rate of population over the last 10% generations using 𝑔 =
+1
+𝑘 × �𝑘
+𝑖=0
+𝑐𝑖
+𝑠 , where 𝑘 is the number of the last 10% generations; 𝑐𝑖 denotes the number of
+different configurations in the ith generation compared with those in the i-1th generation; 𝑠
+is the population size.
+(3) the largest population size where 𝑔 ≤ 0.1 across all conditions (or 10, if no size satisfies the
+above constraints) will be used.
+The results are also shown in Table 6. In this way, we seek to reach a balance between convergence
+(smaller population change) and diversity (larger population size) under the given tuning budget.
+That is, increasing the budget will unlikely change the result. This has been practiced in [17, 32].
+4.4
+Analysis and Comparison
+4.4.1
+Metric. To make a comparison and determine which optimization model is better in this
+work, we need to measure the “best” with two conditions in mind:
+• Condition 1: The metric needs to be able to comprehensively compare the different sets of
+configurations as produced by the Pareto optimizers, covering diverse quality aspects, such
+as convergence and diversity.
+• Condition 2: The metric should be able to reflect the given requirement scenarios, i.e., taking
+the given patterns identified from Section 3 into account when conducting the comparisons
+and evaluations.
+To that end, we use Hypervolume (HV) [84, 86] as the basic metric to assess the quality of the
+configuration set produced in each run. In a nutshell, HV measures the volume between all points
+of a configuration set and a reference point (usually a nadir point); the larger the volume, the better
+convergence, and diversity that the set achieves. HV is chosen in this work because:
+• HV is a comprehensive metric that covers all quality aspects of a configuration set, i.e.,
+convergence, uniformity, spread, and cardinality [55, 57], which meets Condition 1.
+• HV also does not require a reference Pareto front and is Pareto compliant9, which fits our
+case as the true Pareto front is unknown.
+• By following the guidelines proposed by [54, 55], we landed on HV as the appropriate metric
+for our SBSE problem.
+Since we are interested in a requirement scenario that has a specific combination of patterns and
+aspiration space (𝒫) in the objective space, the original HV, which always favors the configurations
+that are close to the entire Pareto front, is no longer suitable. Therefore, we need to transfer these
+preferences into the HV following the guidance by Li et al. [55] and leveraging the patterns and
+quantification from Section 3 (for satisfying Condition 2). Using the same example from Section 2.2,
+as shown in Figure 10, the requirement scenario is that: the stakeholders prefer better PNSR and
+energy usage better than 80 watts, but any configurations better than 80 watts are equally preferred;
+willing to accept energy usage worse than 80 watts but do not accept PNSR worse than 40dB. This
+means for any points in the aspiration space, the ones with better PNSR would be preferred more.
+9Generally speaking, a quality indicator being Pareto compliant means that its evaluation result does not conflict with
+the Pareto dominance relation between two solution sets. More strictly, if a solution set 𝐴 is better [87] than 𝐵 (i.e., for
+any solution in 𝐵, there exists one solution in 𝐴 that covers (dominates or is equivalent to) it, and there exists at least one
+solution in 𝐴 that is not covered by any solution in 𝐵), then 𝐴 is always evaluated better than 𝐵 by the indicator.
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+Do Performance Aspirations Matter for Guiding Software Configuration Tuning?
+1:21
+�80
+�60
+�40
+�20
+20
+40
+60
+80
+100
+120
+140
+160
+PNSR p3
+Energy p1
+Original Space with HV
+�80
+�60
+�40
+�20
+20
+40
+60
+80
+100
+120
+140
+�PNSR (dB) with p3
+Energy (watt) with p1
+Original Space
+Aspiration space
+Reference
+Hypervolume area
+Original Space with HV
+Original Space
+Best point A
+Best point A
+(a) Evaluating with HV in the original space
+�0.1
+0
+0.2
+0.4
+�0.1
+0.2
+0.4
+0.8
+PNSR p3
+Energy p1
+Transformed Space with HV
+0
+0.2
+0.4
+0.2
+0.4
+0.8
+PNSR p3
+Energy p1
+Transformed Space
+Reference
+Hypervolume area
+Transformed Space with HV
+Transformed Space
+Best point A
+Best point A
+(b) Evaluating with HV in the transformed space
+Fig. 10. Evaluation of HV with and without requirements/aspirations.
+Therefore, point 𝑨 is the best based on the requirements and should contribute the most to the
+chosen metric. However, directly applying HV would make some configurations, which are less
+preferred to the requirements, contribute significantly to the HV value (Figure 10a). This would
+misleadingly evaluate some sets that have many non-preferred points to have a very good HV value.
+In contrast, when transferring the information of patterns before using HV (i.e., in the transformed
+space), the above requirements and aspirations can be better complied with, as 𝑨 is certainly the
+one that contributes the most and other non-preferred points tend to have no or little contributions.
+(Figure 10b)
+To that end, we extend the HV in this work. Suppose that there are 𝑚 performance objectives
+(we have 𝑚 = 2 in this work) and 𝒜 is a produced configuration set wherein the vector of a
+configuration’s raw measurements is 𝒙𝒊 = {𝑥1,𝑥2, ...𝑥𝑚}, we calculate HV based on the converted
+satisficing value of 𝒙𝒊 according to the given 𝒫. We call it aspiration-aware HV (dubbed A-HV),
+which is formulated as:
+𝐴-𝐻𝑉 (𝒜) = 𝜆(
+�
+𝒙𝒊 ∈𝒜
+{𝒗|𝒫(𝒙𝒊) ≺ 𝒗 ≺ 𝒓})
+(7)
+where 𝜆 is the Lebesgue measure that quantifies the volume [86] as used in the original HV; 𝒓 is
+the reference nadir point, which is often taken as the 1.1 times of the range of the nondominated
+set [55], hence in our case, this would be {−0.1, −0.1} as 𝒫(𝒙𝒊) converts the outputs to [0, 1]. Like
+HV, a higher A-HV value is better. To ensure fair comparison with A-HV, we use the minimum
+and/or maximum values (of each performance objective) from all experiments for the posterior
+normalization in the patterns.
+To enable more intuitive exposition, we report on the % gain of the A-HV for considering
+requirements and aspirations in the tuning, i.e., PS-w, over that for PS-w/o on each run, which is
+defined as:
+% Gain = 𝑥𝑖 − 𝑦𝑖
+𝑦𝑖
+× 100
+(8)
+whereby 𝑥𝑖 and 𝑦𝑖 are the A-HV value at the 𝑖th run for PS-w and PS-w/o, respectively, in their
+sorted lists. Clearly, a positive % gain indicates that the aspirations are helpful (PS-w is better) while
+a negative value implies they are harmful (PS-w/o is better); zero gain means identical result.
+4.4.2
+Statistical Validation. We use the standard methods to interpret the significance of the results
+over 100 runs in each case [1, 47]:
+• Wilcoxon test: We apply the Wilcoxon test [76] with 𝑎 = 0.05 [1] to investigate the statistical
+significance of the A-HV comparisons over all 100 runs, as it is a non-parametric statistical
+test that makes little assumption about the data distribution and has been recommended in
+software engineering research for pair-wise comparisons [1].
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+1:22
+Chen and Li.
+90
+17
+20
+48
+Large ˆA12 (p < 0.05)
+Medium ˆA12 (p < 0.05)
+Small ˆA12 (p < 0.05)
+Trivial ˆA12 or p � 0.05
+(c) Distribution on the 175 cases that PS-w/o win
+454
+66
+52
+85
+Large ˆA12 (p < 0.05)
+Medium ˆA12 (p < 0.05)
+Small ˆA12 (p < 0.05)
+Trivial ˆA12 or p � 0.05
+(b) Distribution on the 657 cases that PS-w win
+657
+175
+248
+PS-w win
+PS-w/o win
+Tie
+(a) Distribution on all 1,080 cases
+Fig. 11. Summary of the wins by PS-w and PS-w/o together with their detailed statistics validation results.
+• ˆA12 effect size: To ensure that a 𝑝 < 0.05 is not caused by a trivial amount of the samples, we
+apply ˆ𝐴12 [72] to measure the effect size. In this work, ˆ𝐴12 > 0.5 denotes PS-w wins wherein
+it has better A-HV for more than 50% of the runs. ˆ𝐴12 ≥ 0.6 or ˆ𝐴12 ≤ 0.4 indicate a non-trivial
+effect size. Since there are 100 runs (instead of the commonly used 30), we use a stricter
+interpretation by which 0.6 ≤ ˆ𝐴12 < 0.7 (0.3 < ˆ𝐴12 ≤ 0.4), 0.7 ≤ ˆ𝐴12 < 0.8 (0.2 < ˆ𝐴12 ≤ 0.3),
+and ˆ𝐴12 ≥ 0.8 ( ˆ𝐴12 ≤ 0.2) indicate small, medium, and large effect, respectively.
+5
+RESULTS AND FINDINGS
+In this section, we present the results of our empirical study and address the research questions
+posed in Section 1.
+5.1
+RQ1: Which is Better under Requirements with Realistic Aspirations?
+5.1.1
+Method. To answer RQ1, we compare PS-w and PS-w/o across 15 combinations of patterns,
+three realistic aspiration spaces (𝑙, 𝑟, and 𝑐), three optimizers and eight subject systems, leading to
+15 × 3 × 3 × 8 = 1, 080 cases. Since we are interested in a pair-wise comparison of the A-HV under
+each case, the Wilcoxon test and ˆ𝐴12 are used to verify the statistical significance over 100 runs.
+5.1.2
+Results. As an overview, Figure 11 shows a summary of the ˆ𝐴12 outcomes across the cases.
+Clearly, we see that PS-w performs overwhelmingly better than its PS-w/o counterpart. In particular,
+PS-w wins for 61% (657/1080) of the cases and loses for 16% (175/1080), while there is a 23% (248/1080)
+tie. In other words, PS-w is better or similar for 84% (905/1080) of the cases in contrast to the 39%
+(423/1080) when using PS-w/o. Statistically, PS-w wins 572 cases with ˆ𝐴12 ≥ 0.6 and 𝑝 < 0.05, while
+there are only 127 significant cases when it loses.
+To provide a more comprehensive view on the different systems and requirement scenarios, in
+Table 7, we see that PS-w performs considerably better in general, as it achieves reasonably well
+positive gains on the majority of the cases (up to 145% improvement on A-HV in average) with
+generally more statistically significance wins. It is worth noting that we observed particularly
+high gains on PS-w under Storm (Table 7c and Table 7d). This is attributed to the highly diverse
+performance between configurations for the system, as what has been reported in prior work [17,
+44, 62].
+It is exciting to see that the superiority of PS-w is consistent across the given requirement
+patterns — a clear sign to confirm that the requirements can offer important guidance to steer the
+tuning. However, as expected, when the scenario requires 𝒑1 or 𝒑0 only, PS-w and PS-w/o perform
+mostly identical (or very similar). As for the very few cases where PS-w is inferior to PS-w/o, the
+results can be the cause of some accidentally encountered local optima issues, which we will discuss
+in greater detail in what follow.
+Therefore, we say:
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+a.
+Distribution on all 1.080 casesC
+ Distribution on the 175 cases that PS-w/o winDo Performance Aspirations Matter for Guiding Software Configuration Tuning?
+1:23
+Table 7. Comparing PS-w and PS-w/o under realistic requirements and aspirations over 100 runs.
+sc
+and
+sc
+denote the average (Avg) and standard error (SE) of the positive and negative % gain, respectively.
+sc
+means zero gain overall. The column “PS-w” and “PS-w/o” show the number of cases that the corresponding
+optimization model wins. 9 (6) means one wins on 9 cases within which 6 shows statistical significance, i.e.,
+ˆ𝐴12 ≥ 0.6 (or ˆ𝐴12 ≤ 0.4) and 𝑝 < 0.05 (each combination of requirement patterns has 9 cases in total, as there
+are 3 aspiration space and 3 optimizers). The blue cells denote PS-w wins more while red cells mean it loses
+more.
+PS-w PS-w/o Tie
+Avg (SE) of A-HV Gain
+{𝒑0,𝒑1}
+8 (7)
+1 (1)
+0
+1.1% (0.3%)
+y
+i
+{𝒑1,𝒑0}
+7 (7)
+2 (2)
+0
+-4.6% (1.4%)
+y
+i
+{𝒑0,𝒑2}
+5 (5)
+4 (4)
+0
+36.4% (10.4%)
+y
+i
+{𝒑2,𝒑0}
+7 (7)
+2 (2)
+0
+-3.5% (2.4%)
+y
+i
+{𝒑0,𝒑3}
+6 (5)
+3 (2)
+0
+29.0% (7.2%)
+y
+i
+{𝒑3,𝒑0}
+8 (8)
+1 (1)
+0
+2.9% (0.6%)
+y
+i
+{𝒑1,𝒑1}
+3 (0)
+2 (1)
+4
+-4.6% (1.4%)
+y
+i
+{𝒑2,𝒑2}
+1 (0)
+5 (1)
+3
+-7.6% (5.1%)
+y
+i
+{𝒑3,𝒑3}
+6 (5)
+3 (3)
+0
+17.8% (7.0%)
+y
+i
+{𝒑1,𝒑2}
+3 (0)
+3 (1)
+3
+-3.5% (5.2%)
+y
+i
+{𝒑2,𝒑1}
+2 (0)
+4 (1)
+3
+-6.2% (1.6%)
+y
+i
+{𝒑1,𝒑3}
+6 (6)
+3 (3)
+0
+6.0% (5.2%)
+y
+i
+{𝒑3,𝒑1}
+9 (9)
+0 (0)
+0
+3.1% (1.3%)
+y
+i
+{𝒑2,𝒑3}
+6 (6)
+3 (3)
+0
+-1.4% (5.9%)
+y
+i
+{𝒑3,𝒑2}
+4 (4)
+5 (3)
+0
+28.1% (8.2%)
+y
+i
+PS-w PS-w/o Tie
+Avg (SE) of A-HV Gain
+{𝒑0,𝒑1}
+9 (9)
+0 (0)
+0
+0.2% (0.0%)
+y
+i
+{𝒑1,𝒑0}
+6 (6)
+2 (2)
+1
+0.1% (0.0%)
+y
+i
+{𝒑0,𝒑2}
+8 (6)
+1 (1)
+0
+4.9% (4.2%)
+y
+i
+{𝒑2,𝒑0}
+6 (5)
+3 (3)
+0
+0.1% (0.0%)
+y
+i
+{𝒑0,𝒑3}
+3 (2)
+6 (4)
+0
+7.2% (2.1%)
+y
+i
+{𝒑3,𝒑0}
+6 (6)
+3 (3)
+0
+2.1% (0.3%)
+y
+i
+{𝒑1,𝒑1}
+1 (0)
+0 (0)
+8
+0.1% (0.0%)
+y
+i
+{𝒑2,𝒑2}
+1 (0)
+0 (0)
+8
+5.0% (4.5%)
+y
+i
+{𝒑3,𝒑3}
+3 (3)
+6 (6)
+0
+3.6% (1.1%)
+y
+i
+{𝒑1,𝒑2}
+1 (0)
+0 (0)
+8
+5.0% (4.5%)
+y
+i
+{𝒑2,𝒑1}
+1 (0)
+0 (0)
+8
+0.1% (0.0%)
+y
+i
+{𝒑1,𝒑3}
+5 (5)
+4 (3)
+0
+10.0% (2.8%)
+y
+i
+{𝒑3,𝒑1}
+9 (9)
+0 (0)
+0
+2.8% (0.3%)
+y
+i
+{𝒑2,𝒑3}
+6 (6)
+3 (3)
+0
+9.3% (2.8%)
+y
+i
+{𝒑3,𝒑2}
+8 (6)
+1 (1)
+0
+6.3% (3.5%)
+y
+i
+PS-w PS-w/o Tie
+Avg (SE) of A-HV Gain
+{𝒑0,𝒑1}
+8 (8)
+1 (1)
+0
+0.1% (0.0%)
+y
+i
+{𝒑1,𝒑0}
+8 (8)
+1 (1)
+0
+0.1% (0.0%)
+y
+i
+{𝒑0,𝒑2}
+8 (8)
+0 (0)
+1
+82.6% (17.2%)
+y
+i
+{𝒑2,𝒑0}
+6 (6)
+3 (3)
+0
+-0.1% (0.0%)
+y
+i
+{𝒑0,𝒑3}
+7 (6)
+2 (1)
+0
+90.4% (17.4%)
+y
+i
+{𝒑3,𝒑0}
+9 (9)
+0 (0)
+0
+67.4% (13.4%)
+y
+i
+{𝒑1,𝒑1}
+1 (1)
+0 (0)
+8
+0.1% (0.0%)
+y
+i
+{𝒑2,𝒑2}
+1 (1)
+0 (0)
+8
+83.0% (17.4%)
+y
+i
+{𝒑3,𝒑3}
+9 (8)
+0 (0)
+0
+142.1% (18.8%)
+y
+i
+{𝒑1,𝒑2}
+1 (1)
+0 (0)
+8
+83.0% (17.4%)
+y
+i
+{𝒑2,𝒑1}
+1 (1)
+0 (0)
+8
+0.1% (0.0%)
+y
+i
+{𝒑1,𝒑3}
+7 (7)
+2 (2)
+0
+91.5% (17.6%)
+y
+i
+{𝒑3,𝒑1}
+9 (9)
+0 (0)
+0
+67.9% (13.4%)
+y
+i
+{𝒑2,𝒑3}
+6 (6)
+3 (3)
+0
+91.5% (17.6%)
+y
+i
+{𝒑3,𝒑2}
+8 (8)
+1 (0)
+0
+142.9% (19.6%)
+y
+i
+(a). Trimesh
+(b). x264
+(c). Storm/WC
+PS-w PS-w/o Tie
+Avg (SE) of A-HV Gain
+{𝒑0,𝒑1}
+8 (8)
+1 (1)
+0
+0.1% (0.0%)
+y
+i
+{𝒑1,𝒑0}
+6 (5)
+3 (2)
+0
+-0.1% (0.0%)
+y
+i
+{𝒑0,𝒑2}
+8 (7)
+1 (1)
+0
+1.0% (2.0%)
+y
+i
+{𝒑2,𝒑0}
+8 (7)
+1 (0)
+0
+114.8% (19.9%)
+y
+i
+{𝒑0,𝒑3}
+5 (5)
+4 (4)
+0
+1.5% (1.9%)
+y
+i
+{𝒑3,𝒑0}
+9 (9)
+0 (0)
+0
+145.6% (18.5%)
+y
+i
+{𝒑1,𝒑1}
+2 (1)
+0 (0)
+7
+0.1% (0.0%)
+y
+i
+{𝒑2,𝒑2}
+2 (1)
+0 (0)
+7
+117.0% (20.2%)
+y
+i
+{𝒑3,𝒑3}
+8 (8)
+1 (1)
+0
+95.5% (14.3%)
+y
+i
+{𝒑1,𝒑2}
+2 (1)
+0 (0)
+7
+1.0% (2.0%)
+y
+i
+{𝒑2,𝒑1}
+2 (1)
+0 (0)
+7
+116.0% (20.1%)
+y
+i
+{𝒑1,𝒑3}
+5 (4)
+4 (3)
+0
+1.5% (1.9%)
+y
+i
+{𝒑3,𝒑1}
+9 (9)
+0 (0)
+0
+151.4% (19.4%)
+y
+i
+{𝒑2,𝒑3}
+8 (7)
+1 (0)
+0
+112.5% (19.3%)
+y
+i
+{𝒑3,𝒑2}
+8 (7)
+1 (1)
+0
+139.5% (19.2%)
+y
+i
+PS-w PS-w/o Tie
+Avg (SE) of A-HV Gain
+{𝒑0,𝒑1}
+6 (5)
+3 (2)
+0
+-0.1% (0.0%)
+y
+i
+{𝒑1,𝒑0}
+9 (9)
+0 (0)
+0
+0.1% (0.0%)
+y
+i
+{𝒑0,𝒑2}
+7 (7)
+2 (1)
+0
+5.0% (4.5%)
+y
+i
+{𝒑2,𝒑0}
+9 (9)
+0 (0)
+0
+0.1% (0.0%)
+y
+i
+{𝒑0,𝒑3}
+6 (4)
+3 (2)
+0
+8.3% (4.5%)
+y
+i
+{𝒑3,𝒑0}
+2 (1)
+7 (6)
+0
+-1.2% (0.6%)
+y
+i
+{𝒑1,𝒑1}
+1 (0)
+0 (0)
+8
+0.1% (0.0%)
+y
+i
+{𝒑2,𝒑2}
+1 (0)
+0 (0)
+8
+5.0% (4.5%)
+y
+i
+{𝒑3,𝒑3}
+6 (5)
+2 (1)
+1
+10.6% (5.0%)
+y
+i
+{𝒑1,𝒑2}
+1 (0)
+0 (0)
+8
+5.0% (4.5%)
+y
+i
+{𝒑2,𝒑1}
+1 (0)
+0 (0)
+8
+0.1% (0.0%)
+y
+i
+{𝒑1,𝒑3}
+9 (9)
+0 (0)
+0
+8.4% (4.5%)
+y
+i
+{𝒑3,𝒑1}
+6 (4)
+3 (2)
+0
+-1.4% (0.6%)
+y
+i
+{𝒑2,𝒑3}
+9 (9)
+0 (0)
+0
+9.6% (4.5%)
+y
+i
+{𝒑3,𝒑2}
+6 (5)
+3 (1)
+0
+5.5% (5.0%)
+y
+i
+PS-w PS-w/o Tie
+Avg (SE) of A-HV Gain
+{𝒑0,𝒑1}
+6 (3)
+3 (2)
+0
+0.2% (0.1%)
+y
+i
+{𝒑1,𝒑0}
+8 (8)
+1 (0)
+0
+0.1% (0.0%)
+y
+i
+{𝒑0,𝒑2}
+6 (5)
+2 (1)
+1
+1.1% (1.9%)
+y
+i
+{𝒑2,𝒑0}
+8 (7)
+1 (0)
+0
+9.1% (6.0%)
+y
+i
+{𝒑0,𝒑3}
+6 (4)
+3 (2)
+0
+1.3% (1.9%)
+y
+i
+{𝒑3,𝒑0}
+6 (5)
+3 (1)
+0
+22.6% (5.3%)
+y
+i
+{𝒑1,𝒑1}
+2 (1)
+0 (0)
+7
+0.1% (0.0%)
+y
+i
+{𝒑2,𝒑2}
+2 (1)
+0 (0)
+7
+18.1% (7.3%)
+y
+i
+{𝒑3,𝒑3}
+8 (7)
+1 (1)
+0
+23.9% (3.6%)
+y
+i
+{𝒑1,𝒑2}
+2 (1)
+0 (0)
+7
+1.0% (2.0%)
+y
+i
+{𝒑2,𝒑1}
+2 (1)
+0 (0)
+7
+9.1% (6.0%)
+y
+i
+{𝒑1,𝒑3}
+9 (8)
+0 (0)
+0
+1.5% (2.0%)
+y
+i
+{𝒑3,𝒑1}
+8 (8)
+1 (1)
+0
+49.1% (6.4%)
+y
+i
+{𝒑2,𝒑3}
+8 (8)
+1 (1)
+0
+22.6% (7.3%)
+y
+i
+{𝒑3,𝒑2}
+9 (8)
+0 (0)
+0
+44.6% (6.9%)
+y
+i
+(d). Storm/RS
+(e). Keras/Adiac
+(f). Keras/DSR
+PS-w PS-w/o Tie
+Avg (SE) of A-HV Gain
+{𝒑0,𝒑1}
+5 (5)
+3 (2)
+1
+0.1% (0.0%)
+y
+i
+{𝒑1,𝒑0}
+9 (9)
+0 (0)
+0
+0.1% (0.0%)
+y
+i
+{𝒑0,𝒑2}
+4 (3)
+5 (4)
+0
+0.1% (0.0%)
+y
+i
+{𝒑2,𝒑0}
+9 (9)
+0 (0)
+0
+0.1% (0.0%)
+y
+i
+{𝒑0,𝒑3}
+6 (5)
+3 (3)
+0
+0.5% (0.1%)
+y
+i
+{𝒑3,𝒑0}
+6 (3)
+3 (0)
+0
+-0.5% (0.3%)
+y
+i
+{𝒑1,𝒑1}
+0 (0)
+0 (0)
+9
+0.0% (0.0%)
+y
+i
+{𝒑2,𝒑2}
+0 (0)
+0 (0)
+9
+0.0% (0.0%)
+y
+i
+{𝒑3,𝒑3}
+6 (6)
+2 (1)
+1
+0.4% (0.1%)
+y
+i
+{𝒑1,𝒑2}
+0 (0)
+0 (0)
+9
+0.0% (0.0%)
+y
+i
+{𝒑2,𝒑1}
+0 (0)
+0 (0)
+9
+0.0% (0.0%)
+y
+i
+{𝒑1,𝒑3}
+9 (9)
+0 (0)
+0
+0.7% (0.1%)
+y
+i
+{𝒑3,𝒑1}
+4 (3)
+3 (1)
+2
+-0.3% (0.3%)
+y
+i
+{𝒑2,𝒑3}
+9 (9)
+0 (0)
+0
+0.7% (0.1%)
+y
+i
+{𝒑3,𝒑2}
+4 (3)
+5 (4)
+0
+-0.2% (0.3%)
+y
+i
+PS-w PS-w/o Tie
+Avg (SE) of A-HV Gain
+{𝒑0,𝒑1}
+8 (5)
+0 (0)
+1
+0.1% (0.0%)
+y
+i
+{𝒑1,𝒑0}
+9 (9)
+0 (0)
+0
+0.1% (0.0%)
+y
+i
+{𝒑0,𝒑2}
+8 (3)
+0 (0)
+1
+0.1% (0.0%)
+y
+i
+{𝒑2,𝒑0}
+9 (9)
+0 (0)
+0
+0.1% (0.0%)
+y
+i
+{𝒑0,𝒑3}
+4 (4)
+5 (5)
+0
+-0.1% (0.0%)
+y
+i
+{𝒑3,𝒑0}
+6 (5)
+3 (1)
+0
+1.7% (0.2%)
+y
+i
+{𝒑1,𝒑1}
+0 (0)
+0 (0)
+9
+0.0% (0.0%)
+y
+i
+{𝒑2,𝒑2}
+0 (0)
+0 (0)
+9
+0.0% (0.0%)
+y
+i
+{𝒑3,𝒑3}
+5 (3)
+4 (4)
+0
+0.3% (0.1%)
+y
+i
+{𝒑1,𝒑2}
+0 (0)
+0 (0)
+9
+0.0% (0.0%)
+y
+i
+{𝒑2,𝒑1}
+0 (0)
+0 (0)
+9
+0.0% (0.0%)
+y
+i
+{𝒑1,𝒑3}
+9 (9)
+0 (0)
+0
+0.1% (0.0%)
+y
+i
+{𝒑3,𝒑1}
+9 (5)
+0 (0)
+0
+1.9% (0.2%)
+y
+i
+{𝒑2,𝒑3}
+9 (9)
+0 (0)
+0
+0.1% (0.0%)
+y
+i
+{𝒑3,𝒑2}
+8 (4)
+0 (0)
+1
+1.9% (0.2%)
+y
+i
+(g). Keras/SA
+(h). XGBoost
+RQ1: Given realistic aspiration space, PS-w is 84% of the time similar or better than PS-w/o with
+considerable improvements, suggesting that the requirements and aspirations are beneficial for
+guiding the tuning in such a situation. Yet, the benefits can vary depending on some particular
+combinations of the patterns, i.e., it tends to be blurred when only the 𝒑1 and/or 𝒑0 is given.
+5.1.3
+Discussion. To understand what causes the results under realistic aspiration space, in Fig-
+ure 12 we show a common example from x264, where all PSNR values better than its aspiration are
+equally preferred and no worse results are acceptable (𝒑2), while the energy usage is desired to be
+as low as possible, even if its aspiration has already been exceeded (𝒑3). Figure 12a is a superior case
+of PS-w, in which we see that the aspirations drive the tuning to focus more on the local regions
+within the objective space, hence the points of PS-w is much less spread than those of PS-w/o (as
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+1:24
+Chen and Li.
+�1.75
+�1.67
+�1.6
+2.4
+2.6
+2.8
+3
+3.2
+3.4
+3.6
+log10 PSNR�1 (p2)
+log10 Energy (p3)
+PS-w
+PS-w/o
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+PSNR (p2)
+Energy (p3)
+PS-w
+PS-w/o
+Reference
+A-HV area
+Original Space
+(a) x264, PS-w wins
+�1.75
+�1.67
+�1.6
+2.4
+2.6
+2.8
+3
+3.2
+3.4
+3.6
+log10 PSNR�1 (p2)
+log10 Energy (p3)
+PS-w
+PS-w/o
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+PSNR (p2)
+Energy (p3)
+PS-w
+PS-w/o
+Reference
+A-HV area
+Original Space
+(b) x264, PS-w/o wins
+Fig. 12. Example runs of the final configuration sets (with NSGA-II) under realistic aspiration space indicated
+by the shaded areas.
+see in the Original Space). Such a “focused pressure” is mostly sufficient to help find some more
+preferred regions by the scenario under a fixed tuning budget, hence the PS-w has better A-HV
+than PS-w/o (larger volume, as seen in the A-HV area).
+However, PS-w is not always beneficial. As reported by Chen and Li [17], Nair et al. [62], and the
+others [38, 44], configurable software systems are known to exhibit a high degree of sparsity, i.e.,
+the close configurations can also have radically different performance, thus only a small amount of
+them may achieve certain performance range, causing rather sparse objective points (e.g., Figure 9).
+For example, switching the wait_strategy in Storm can have dramatic impacts on the performance,
+despite that it is merely a single change on an option. This is because the wait_strategy conserves
+CPU usage depending on whether the wait is a fixed interval or is progressively determined based
+on the length of the queue at runtime, therefore it has a large impact on latency and throughput.
+However, in the tuning, it is represented as a single configuration option with a value chosen
+from {0, 1, 2, 3} where each value represents a distinct wait strategy. The presence of high sparsity
+exacerbates the problem of local optima traps — some undesired regions that are difficult to escape
+from by an optimizer. Occasionally, searching focally under high sparsity does cause PS-w to
+overemphasize the less desired local optima, which harms the results. This is why there are some
+cases where the PS-w show no advantage, as illustrated in Figure 12b where the points of PS-w are
+too densely populated compared with those of the PS-w/o (as see in the Original Space), causing
+the volume covered by PS-w is smaller than that of PS-w/o and smaller A-HV (as see in the A-HV
+area).
+It is interesting to observe that under certain combinations of patterns, i.e., with 𝒑0 and/or 𝒑1 only,
+both optimization models perform similarly. This makes sense, as in those cases the requirements
+would create similar discriminative power between configurations to that of PS-w/o (which is
+essentially guided by {𝒑0, 𝒑0}), generating configurations that are equally preferred under the
+given needs.
+5.2
+RQ2: How do Different Aspirations Influence the Comparisons?
+5.2.1
+Method. To understand RQ2, we follow the procedure used for RQ1, but with particular
+focus on the results with respect to the three aspiration spaces used (i.e., 𝑙, 𝑐, and 𝑟).
+5.2.2
+Results. Figure 13 plots the sensitivity of A-HV to the different aspiration spaces. While the
+overall conclusion is consistent with that for RQ1 over different patterns and systems, we see that
+there is often a strong bias on the gains for a certain position of the aspiration spaces. For example,
+on x264 and Keras/Adiac, the improvement of PS-w is particularly high for aspiration space located
+at the centered area of the objective space. In contrast, the gain is particularly high on left-shifted
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+Do Performance Aspirations Matter for Guiding Software Configuration Tuning?
+1:25
+0
+5 10
+Keras/Adiac
+10 20 30
+Keras/RSD
+0.1
+0.3
+Keras/SA
+0 80 180
+Storm/RS
+0 50 120
+Storm/WC
+l
+c
+r 0 5 10 15
+Trimesh
+0
+5
+10
+x264
+0.1 0.4
+0.8
+XGBoost
+Fig. 13. Sensitivity of the % gain on PS-w over PS-w/o to different positions of the realistic aspiration space.
+Each point is the average and standard error over all combinations of patterns and optimizers. 𝑙, 𝑐, and 𝑟
+denote left-shifted, centered, and right-shifted position in the performance landscape, respectively.
+5
+10
+15
+2
+4
+6
+800
+900
+# Counters
+# Splitters
+Latency (ms)
+r
+c
+Fig. 14. A projected landscape of the performance objective Latency with respect to configuration options
+Splitters and Counters for Storm/WC. 𝑐 and 𝑟 denote centered and right-shifted aspiration space, respec-
+tively. Note that the aspirations spaces are bounded because the throughput objective is also considered; it is
+however not showed here for simpler exposition.
+aspiration space under Storm/RS and centered space for Storm/WC, which is possible depending
+on the landscape of a system (as we will discuss next). Indeed, some aspiration spaces can easily
+cause the PS-w to be trapped at the local optima, making its improvements over PS-w/o blurred.
+For example, on Storm/WC with right-shifted aspiration space, this effect is largely detrimental
+and hence severely influence the benefits of PS-w.
+In summary, we have:
+RQ2: The improvement of PS-w over PS-w/o is often largely biased to certain position of
+the aspiration space in the performance landscape, e.g., centered or left-shifted. Yet, PS-w still
+performs more advantageously in general.
+5.2.3
+Discussion. As discussed for RQ1, the main reason that PS-w can perform better than PS-w/o
+is due to the “focused search pressure”. However, this may not be always helpful if the tuning
+encounters complex local optima that are difficult for the optimizer to escape from. The high
+sensitivity of the gains to the positions of aspiration space suggests that the local optima can be
+distributed unevenly across the landscape. If the aspiration space covers many local optima regions,
+then certainly the gains of PS-w would be marginal.
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+% Gain1:26
+Chen and Li.
+83
+28
+18
+10
+Large ˆA12 (p < 0.05)
+Medium ˆA12 (p < 0.05)
+Small ˆA12 (p < 0.05)
+Trivial ˆA12 or p � 0.05
+(c) Distribution on the 139 cases that PS-w/o win
+25
+10
+13
+17
+Large ˆA12 (p < 0.05)
+Medium ˆA12 (p < 0.05)
+Small ˆA12 (p < 0.05)
+Trivial ˆA12 or p � 0.05
+(b) Distribution on the 65 cases that PS-w win
+65
+139
+12
+PS-w win
+PS-w/o win
+Tie
+(a) Distribution on all 216 cases
+Fig. 15. Summary of the wins by PS-w and PS-w/o together with their detailed statistics validation results.
+Table 8. Comparing PS-w and PS-w/o under unrealistic requirements and aspirations over 100 runs. Formats
+are the same as Table 7.
+PS-w PS-w/o Tie
+Avg (SE) of A-HV Gain
+{𝒑1,𝒑1}
+2 (2)
+1 (1)
+0
+-1.1% (0.7%)
+y
+i
+{𝒑2,𝒑2}
+0 (0)
+3 (3)
+0
+-1.3% (22.8%)
+y
+i
+{𝒑3,𝒑3}
+0 (0)
+3 (2)
+0
+-30.7% (5.3%)
+y
+i
+{𝒑1,𝒑2}
+1 (1)
+2 (1)
+0
+-18.1% (8.0%)
+y
+i
+{𝒑2,𝒑1}
+1 (1)
+2 (2)
+0
+15.7% (17.2%)
+y
+i
+{𝒑1,𝒑3}
+1 (1)
+2 (1)
+0
+-24.6% (4.5%)
+y
+i
+{𝒑3,𝒑1}
+2 (2)
+1 (1)
+0
+0.1% (0.1%)
+y
+i
+{𝒑2,𝒑3}
+0 (0)
+3 (2)
+0
+12.3% (26.1%)
+y
+i
+{𝒑3,𝒑2}
+0 (0)
+3 (2)
+0
+-36.0% (5.5%)
+y
+i
+PS-w PS-w/o Tie
+Avg (SE) of A-HV Gain
+{𝒑1,𝒑1}
+1 (1)
+2 (2)
+0
+-3.8% (0.4%)
+y
+i
+{𝒑2,𝒑2}
+0 (0)
+0 (0)
+3
+0.0% (0.0%)
+y
+i
+{𝒑3,𝒑3}
+0 (0)
+0 (0)
+3
+0.0% (0.0%)
+y
+i
+{𝒑1,𝒑2}
+1 (1)
+2 (2)
+0
+-2.7% (0.3%)
+y
+i
+{𝒑2,𝒑1}
+1 (1)
+2 (2)
+0
+-0.8% (0.1%)
+y
+i
+{𝒑1,𝒑3}
+1 (1)
+2 (2)
+0
+-2.7% (0.3%)
+y
+i
+{𝒑3,𝒑1}
+1 (1)
+2 (2)
+0
+-0.7% (0.1%)
+y
+i
+{𝒑2,𝒑3}
+0 (0)
+0 (0)
+3
+0.0% (0.0%)
+y
+i
+{𝒑3,𝒑2}
+0 (0)
+0 (0)
+3
+0.0% (0.0%)
+y
+i
+PS-w PS-w/o Tie
+Avg (SE) of A-HV Gain
+{𝒑1,𝒑1}
+2 (1)
+1 (1)
+0
+-0.0% (0.0%)
+y
+i
+{𝒑2,𝒑2}
+2 (2)
+1 (1)
+0
+70.0% (29.8%)
+y
+i
+{𝒑3,𝒑3}
+1 (1)
+2 (1)
+0
+-95.5% (224.5%)
+y
+i
+{𝒑1,𝒑2}
+2 (0)
+1 (1)
+0
+53.0% (30.4%)
+y
+i
+{𝒑2,𝒑1}
+1 (1)
+2 (2)
+0
+-90.4% (218.8%)
+y
+i
+{𝒑1,𝒑3}
+2 (2)
+1 (1)
+0
+119.1% (37.0%)
+y
+i
+{𝒑3,𝒑1}
+1 (1)
+2 (2)
+0
+-52.7% (136.9%)
+y
+i
+{𝒑2,𝒑3}
+2 (1)
+1 (1)
+0
+74.2% (30.0%)
+y
+i
+{𝒑3,𝒑2}
+2 (0)
+1 (1)
+0
+21.2% (21.7%)
+y
+i
+(a). Trimesh
+(b). x264
+(c). Storm/WC
+PS-w PS-w/o Tie
+Avg (SE) of A-HV Gain
+{𝒑1,𝒑1}
+2 (2)
+1 (1)
+0
+0.0% (0.0%)
+y
+i
+{𝒑2,𝒑2}
+1 (1)
+2 (2)
+0
+-243.4% (541.0%)
+y
+i
+{𝒑3,𝒑3}
+1 (1)
+2 (2)
+0
+-147.7% (336.0%)
+y
+i
+{𝒑1,𝒑2}
+1 (1)
+2 (2)
+0
+6.2% (23.3%)
+y
+i
+{𝒑2,𝒑1}
+2 (2)
+1 (1)
+0
+195.3% (44.2%)
+y
+i
+{𝒑1,𝒑3}
+1 (1)
+2 (2)
+0
+-10.3% (19.1%)
+y
+i
+{𝒑3,𝒑1}
+2 (1)
+1 (1)
+0
+81.9% (28.0%)
+y
+i
+{𝒑2,𝒑3}
+1 (1)
+2 (2)
+0
+-214.2% (478.1%)
+y
+i
+{𝒑3,𝒑2}
+1 (1)
+2 (2)
+0
+-139.8% (320.2%)
+y
+i
+PS-w PS-w/o Tie
+Avg (SE) of A-HV Gain
+{𝒑1,𝒑1}
+1 (1)
+2 (2)
+0
+-0.0% (0.0%)
+y
+i
+{𝒑2,𝒑2}
+0 (0)
+3 (3)
+0
+-56.6% (5.0%)
+y
+i
+{𝒑3,𝒑3}
+0 (0)
+3 (3)
+0
+-42.4% (5.1%)
+y
+i
+{𝒑1,𝒑2}
+1 (1)
+2 (2)
+0
+1.4% (20.4%)
+y
+i
+{𝒑2,𝒑1}
+2 (2)
+1 (1)
+0
+55.8% (28.3%)
+y
+i
+{𝒑1,𝒑3}
+0 (0)
+3 (2)
+0
+-9.7% (10.8%)
+y
+i
+{𝒑3,𝒑1}
+1 (0)
+2 (2)
+0
+-26.4% (4.6%)
+y
+i
+{𝒑2,𝒑3}
+0 (0)
+3 (3)
+0
+-45.4% (5.1%)
+y
+i
+{𝒑3,𝒑2}
+0 (0)
+3 (3)
+0
+-57.9% (4.8%)
+y
+i
+PS-w PS-w/o Tie
+Avg (SE) of A-HV Gain
+{𝒑1,𝒑1}
+2 (1)
+1 (1)
+0
+0.1% (0.1%)
+y
+i
+{𝒑2,𝒑2}
+1 (1)
+2 (2)
+0
+-47.3% (122.7%)
+y
+i
+{𝒑3,𝒑3}
+1 (1)
+2 (2)
+0
+-33.6% (89.6%)
+y
+i
+{𝒑1,𝒑2}
+0 (0)
+3 (3)
+0
+-1.7% (26.9%)
+y
+i
+{𝒑2,𝒑1}
+2 (1)
+1 (0)
+0
+53.3% (24.2%)
+y
+i
+{𝒑1,𝒑3}
+1 (1)
+2 (2)
+0
+-2.4% (18.5%)
+y
+i
+{𝒑3,𝒑1}
+2 (1)
+1 (0)
+0
+69.8% (25.8%)
+y
+i
+{𝒑2,𝒑3}
+2 (1)
+1 (1)
+0
+5.9% (11.0%)
+y
+i
+{𝒑3,𝒑2}
+1 (1)
+2 (2)
+0
+-33.3% (93.2%)
+y
+i
+(d). Storm/RS
+(e). Keras/Adiac
+(f). Keras/DSR
+PS-w PS-w/o Tie
+Avg (SE) of A-HV Gain
+{𝒑1,𝒑1}
+3 (2)
+0 (0)
+0
+-1.0% (0.0%)
+y
+i
+{𝒑2,𝒑2}
+0 (0)
+3 (3)
+0
+-64.6% (4.6%)
+y
+i
+{𝒑3,𝒑3}
+0 (0)
+3 (3)
+0
+-39.1% (4.7%)
+y
+i
+{𝒑1,𝒑2}
+0 (0)
+3 (3)
+0
+6.7% (17.0%)
+y
+i
+{𝒑2,𝒑1}
+0 (0)
+3 (3)
+0
+-46.2% (5.2%)
+y
+i
+{𝒑1,𝒑3}
+0 (0)
+3 (3)
+0
+4.9% (15.3%)
+y
+i
+{𝒑3,𝒑1}
+1 (0)
+2 (2)
+0
+-26.6% (4.3%)
+y
+i
+{𝒑2,𝒑3}
+0 (0)
+3 (3)
+0
+-67.7% (4.3%)
+y
+i
+{𝒑3,𝒑2}
+0 (0)
+3 (3)
+0
+-38.3% (5.1%)
+y
+i
+PS-w PS-w/o Tie
+Avg (SE) of A-HV Gain
+{𝒑1,𝒑1}
+2 (1)
+1 (1)
+0
+-1.0% (0.0%)
+y
+i
+{𝒑2,𝒑2}
+0 (0)
+3 (3)
+0
+-58.3% (4.4%)
+y
+i
+{𝒑3,𝒑3}
+0 (0)
+3 (3)
+0
+-68.4% (3.8%)
+y
+i
+{𝒑1,𝒑2}
+0 (0)
+3 (3)
+0
+-51.0% (5.1%)
+y
+i
+{𝒑2,𝒑1}
+1 (0)
+2 (1)
+0
+106.3% (36.4%)
+y
+i
+{𝒑1,𝒑3}
+0 (0)
+3 (3)
+0
+-60.3% (3.9%)
+y
+i
+{𝒑3,𝒑1}
+2 (0)
+1 (1)
+0
+-0.0% (0.0%)
+y
+i
+{𝒑2,𝒑3}
+0 (0)
+3 (3)
+0
+-63.2% (3.7%)
+y
+i
+{𝒑3,𝒑2}
+0 (0)
+3 (3)
+0
+-64.8% (4.7%)
+y
+i
+(g). Keras/SA
+(h). XGBoost
+For example, in Figure 14, clearly the aspiration space 𝑐 (which covers the requirements for
+latency and throughput) would be bounded on some regions in the landscape with a much more
+smooth surface for the latency. However, for 𝑟, the region becomes highly rugged and steep, which
+involves some very difficult local optima. Unfortunately, we did not see consistent patterns of such
+a sensitivity across the configurable software systems, which makes sense as the performance
+landscape of those systems can be very different too.
+5.3
+RQ3: What if the Aspirations are Unrealistic?
+5.3.1
+Method. To investigate RQ3, we omit the scenarios with 𝒑0 as they cannot create an unreal-
+istic aspiration space. This has left us with nine combinations of patterns, which, together with
+three Pareto optimizers and eight subjects, provide 9 × 3 × 8 = 216 cases. All other settings are
+identical to those for RQ1.
+5.3.2
+Results. As the summary from Figure 15, we see that PS-w/o is generally better across all
+the cases, as it wins on 64% (139/216) while loses on 30% (65/216). There is also a 6% (12/216) tie.
+This means that PS-w/o is better or similar on 70% (151/216) of the cases against the 36% (77/216)
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+b
+Distribution on the 65 cases that PS-w winC
+Distribution on the 139 cases that PS-w/o winDistribution on all 216 casesDo Performance Aspirations Matter for Guiding Software Configuration Tuning?
+1:27
+�5.6
+�5.2
+�4.8
+�4.4
+0
+0.5
+1
+1.5
+2
+2.5
+3
+log10 Throughput�1 (p3)
+log10 Latency (p3)
+PS-w
+PS-w/o
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+Throughput (p3)
+Latency (p3)
+PS-w
+PS-w/o
+Reference
+A-HV area
+Original Space
+(a) Storm/RS, PS-w/o wins
+�5.6
+�5.2
+�4.8
+�4.4
+0
+0.5
+1
+1.5
+2
+2.5
+3
+log10 Throughput�1 (p3)
+log10 Latency (p2)
+PS-w
+PS-w/o
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+Throughput (p3)
+Latency (p2)
+PS-w
+PS-w/o
+Reference
+A-HV area
+Original Space
+(b) Storm/RS, PS-w wins
+Fig. 16. Example runs of the final configuration sets (with NSGA-II) under unrealistic aspiration space
+indicated by the shaded areas.
+for PS-w. Among these, PS-w/o wins 129 cases with ˆ𝐴12 ≤ 0.4 and 𝑝 < 0.05 comparing with 41 of
+such cases when it loses.
+Similar results can be confirmed in Table 8 when inspecting specific system and requirement
+scenarios. Albeit there is a limited number of cases where PS-w is still advantageous, it is more
+common to show no improvement at all or even cause fairly negative gains, which could be up to
+an average of −243%. It also has overall much less statistically significant wins across the cases.
+Particularly, we found that under {𝒑1, 𝒑1} on all systems, the two optimization models perform
+similarly but PS-w tends to obtain more wins. This is because such a combination pattern is the only
+case where the unrealism of aspiration does not lead to too many incomparable configurations.
+Overall, we conclude that:
+RQ3: When the aspiration space is unrealistic, PS-w/o is safer as it is similar or reasonably
+better than PS-w for 70% of the time, meaning that the requirements and aspirations are more
+harmful for guiding the tuning in this case. Yet, the only exception applied to {𝒑1, 𝒑1}.
+5.3.3
+Discussion. Given unrealistic aspiration space, the most common cases are similar to the
+Storm/RS example in Figure 16, where the PS-w is commonly inferior to PS-w/o when the diversity
+tends to be high (Figure 16a), but sometimes superior to PS-w/o under limited diversity (Figure 16b).
+This is because in most of the cases, after being transformed using the requirements with unrealistic
+aspirations, PS-w tends to find too many incomparable configurations from the beginning (as in
+the cases other than {𝒑1, 𝒑1}, most configurations are fully unsatisfied on at least one performance
+objective), implying that the guidance provided by an unrealistic aspiration space is dramatically
+weakened. Such an incomparability, although may prompt slightly better diversity to escape from
+the local regions, can often severely harm the tendency towards more preferred configurations
+that reach/exceed the aspirations, leading to worse A-HV (the smaller volume) than PS-w/o in
+Figure 16a. This is because no selection pressure (i.e., discriminative power) can be generated in
+such a case. It is also the reason why PS-w is not deteriorated by the unrealistic aspirations under
+{𝒑1, 𝒑1}, which can still ensure that the configurations are comparable. Yet sometimes (Figure 16b),
+such a high incomparability does help PS-w to find a good configuration by chance (e.g., better
+than the aspiration of latency), which is more desired than those of PS-w/o, leading to better HV
+(the larger volume). Hence the PS-w remains better for certain cases, despite that the tuning would
+be easily trapped at that configuration due to the high sparsity.
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+1:28
+Chen and Li.
+100
+200
+300
+400
+500
+600
+700
+0.85
+0.9
+0.95
+# Measurements
+A-HV
+PS-w
+PS-w/o
+(e) Keras/Adiac
+100
+200
+300
+400
+500
+0.75
+0.8
+0.85
+# Measurements
+A-HV
+PS-w
+PS-w/o
+(f) Keras/DSR
+100
+200
+300
+400
+500
+0.9
+0.92
+0.94
+0.96
+0.98
+# Measurements
+A-HV
+PS-w
+PS-w/o
+(g) Keras/SA
+100
+200
+300
+400
+500
+600
+0.85
+0.9
+0.95
+# Measurements
+A-HV
+PS-w
+PS-w/o
+(d) Storm/RS
+100
+200
+300
+400
+500
+0.9
+0.95
+# Measurements
+A-HV
+PS-w
+PS-w/o
+(c) Storm/WC
+100
+200
+300
+400
+500
+0.6
+0.7
+0.8
+# Measurements
+A-HV
+PS-w
+PS-w/o
+(a) Trimesh
+200
+400
+600
+800
+1,000 1,200 1,400
+0.68
+0.7
+0.72
+0.74
+# Measurements
+A-HV
+PS-w
+PS-w/o
+(b) x264
+50
+100
+150
+200
+250
+300
+0.96
+0.98
+# Measurements
+A-HV
+PS-w
+PS-w/o
+(h) XGBoost
+1.08X speedup
+2.08X speedup
+10X speedup
+10X speedup
+1.19X speedup
+2.17X speedup
+1.05X speedup
+2.31X speedup
+Fig. 17. Speedup on PS-w over PS-w/o under realistic aspirations (each point is the average and standard
+error over all combinations of patterns, aspiration space, optimziers and their runs).
+100
+200
+300
+400
+500
+600
+700
+0.2
+0.3
+# Measurements
+A-HV
+PS-w
+PS-w/o
+(e) Keras/Adiac
+100
+200
+300
+400
+500
+0.2
+0.3
+# Measurements
+A-HV
+PS-w
+PS-w/o
+(f) Keras/DSR
+100
+200
+300
+400
+500
+0.2
+0.3
+# Measurements
+A-HV
+PS-w
+PS-w/o
+(g) Keras/SA
+100
+200
+300
+400
+500
+600
+0.2
+0.4
+0.6
+# Measurements
+A-HV
+PS-w
+PS-w/o
+(d) Storm/RS
+100
+200
+300
+400
+500
+0.2
+0.3
+0.4
+# Measurements
+A-HV
+PS-w
+PS-w/o
+(c) Storm/WC
+100
+200
+300
+400
+500
+0.1
+0.15
+0.2
+0.25
+# Measurements
+A-HV
+PS-w
+PS-w/o
+(a) Trimesh
+200
+400
+600
+800
+1,000 1,200 1,400
+0.1
+0.12
+0.14
+0.16
+0.18
+# Measurements
+A-HV
+PS-w
+PS-w/o
+(b) x264
+50
+100
+150
+200
+250
+300
+0.2
+0.3
+0.4
+0.5
+# Measurements
+A-HV
+PS-w
+PS-w/o
+(h) XGBoost
+3.13X speedup
+10X speedup
+1.5X speedup
+1.49X speedup
+1.18X speedup
+1.36X speedup
+1.61X speedup
+1.65X speedup
+Fig. 18. Speedup on PS-w/o over PS-w under unrealistic aspirations (each point is the average and standard
+error over all combinations of patterns, optimziers and their runs).
+5.4
+RQ4: Does the Given Tuning Resource Important?
+5.4.1
+Method. To understand the resource efficiency of both optimization models in RQ4, for each
+system, we use the following procedure:
+(1) Plot the trajectories of A-HV along with the number of measurements for both PS-w and
+PS-w/o, where each point is the average of all requirement patterns, aspiration spaces, and
+optimizers.
+(2) Identify a baseline, 𝑏, taken as the smallest number of measurements that the baseline model
+consumes to achieve its best A-HV (say 𝑇).
+(3) For the other model, find the smallest number of measurements, denoted as 𝑚, at which the
+average A-HV is equivalent to or better than 𝑇.
+(4) Calculate the speedup over the baseline model, i.e., 𝑠 = 𝑏
+𝑚, according to the metric used by
+Gao et al. [30].
+Since we found that the generally better optimization model differs depending on the realism of
+the aspiration space, we use PS-w/o and PS-w as the baseline for realistic and unrealistic aspiration
+situations, respectively.
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+Do Performance Aspirations Matter for Guiding Software Configuration Tuning?
+1:29
+5.4.2
+Results. From the results under realistic aspirations as shown in Figure 17, we see that PS-w
+outperforms PS-w/o throughout the trajectories over different configurable systems, which further
+strengthen our findings for RQ1. The improvement in resource efficiency has been remarkable:
+there is a speedup between 1.05× and 10×. In contrast, when the given aspirations are unrealistic
+(Figure 18), PS-w/o is much more resource-efficient, as it enables a speedup from 1.18× to 10×. This
+again complies with the findings for RQ3. However, under unrealistic aspirations, the advantages
+of PS-w/o may not be obvious at the early stage of the tuning; on some systems (e.g., Figure 18c
+and Figure 18f), it is even inferior to the PS-w until around 250 configurations have been measured.
+In summary, we found that:
+RQ4: Under realistic aspirations, PS-w often obtains consistently better A-HV than PS-w/o
+throughout the trajectory and with a speedup up to 10×. When the aspirations are unrealistic,
+in contrast, the two optimization models are competitive in the early stage of tuning but soon
+PS-w/o would lead to better results with considerably high speedup.
+5.4.3
+Discussion. Under realistic aspirations, the reasons that PS-w has better A-HV throughout the
+trajectory with remarkably high speedup are two folds: firstly, as what we have already discussed
+for RQ1, the guidance provided by the requirements and aspirations are often helpful to enable the
+tuning to be more focus-driven, hence better utilizing the resources to explore the more promising
+area. Secondly, PS-w/o would waste the valuable tuning budget to explore those configurations that
+it favors, but would never be preferred under the given requirements, since it is naturally interested
+in the whole Pareto front. Therefore, the above difference enables PS-w to be a particularly attractive
+model for some systems, such as Storm, where the performance of diverse configurations can be
+radically different.
+The situation is completely different when the given aspirations are unrealistic and it is mainly
+due to the high incomparability in PS-w as mentioned for RQ3 — many of the configurations are
+incomparable when transformed using the requirements with unrealistic aspirations. It has been
+shown that this situation can cause severe issues for any Pareto optimizer [56], as the resources
+would have been spent mainly on exploration. However, such an incomparability can occasionally
+be helpful to explore some preferred configurations by chance, especially at the early stage of the
+tuning where the PS-w/o has yet explored enough space to pursue the Pareto front. As such, we
+see that at the beginning PS-w performs similarly to PS-w/o and, sometimes, even better.
+6
+LESSONS LEARNED
+In this section, we discuss how our findings can be useful for the practitioners in the field in light
+of the lessons learned and future opportunities discovered.
+Lesson 1: The choice on whether to exploit aspirations for guiding the tuning is primarily dependent
+on their realism.
+It is interesting to find that we cannot draw the conclusion to choose between PS-w and PS-w/o
+arbitrarily for software configuration tuning with two performance objectives, as opposed to
+what has been overwhelmingly assumed in existing work. Instead, from RQ1, RQ3, and RQ4, we
+discovered that the realism of the given aspirations is crucial to the choice: PS-w is more beneficial
+for realistic aspirations while PS-w/o is safer when the aspirations are unrealistic (given that the
+tuning budget is also sufficient). This raises the importance of understanding whether the given
+requirements and aspirations can be realistic, or the assumption therein, prior to choosing the right
+optimization model for tuning software configuration with two performance objectives.
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+1:30
+Chen and Li.
+Lesson 2: Little combinations of patterns can change the decision on whether to incorporate
+aspiration in the tuning, but it can influence the benefit/detriment of aspiration-guided tuning.
+Although from RQ1, we noticed that the benefits of PS-w is blurred when the given combination
+of patterns contain 𝒑1 and/or 𝒑0 only, this does not change the decision as PS-w remains outperform
+its PS-w/o counterpart. The only definitive case is when the aspirations are unrealistic, PS-w should
+be chosen under a patterns of {𝒑1, 𝒑1}. Therefore, we envisage that the sensitivity of given patterns
+to the choice between PS-w and PS-w/o is marginal and we have discovered other more important
+factors. However, we do see that the extent of improvement/degradation from PS-w can be sensitive
+to the given patterns.
+Lesson 3: The positions of realistic aspiration space in the objective space can largely affect the
+benefits brought by considering aspirations within tuning, but it is less likely to influence the choice.
+An unexpected discovery from RQ2 is that, when given realistic aspirations, the position of
+the aspiration space can largely influence the benefits of PS-w. While this is unlikely to affect the
+choice between PS-w and PS-w/o, it does raise the need to systematically analyze the correlation
+between the aspiration space and the configuration landscape of the system, particularly on the
+likelihood of covering some difficult local optima and their implication.
+Lesson 4: The given tuning budget has marginal impact to the choice when the aspirations are
+realistic. However, it can be an important factor to consider under unrealistic aspirations.
+According to RQ4 we have also revealed that, given realistic aspirations, the choice between
+PS-w and PS-w/o is marginally sensitive to the tuning budget, but it can be influenced by the
+budget when the aspirations are unrealistic. This adds an extra layer of consideration for unrealistic
+aspirations. In this case, what we observed, in general, is that for a small tuning budget, the benefit
+of PS-w/o is much less justified, hence using either of the two optimization models may not lead
+to significantly different results. However, given sufficient budget, PS-w/o is likely to dominate its
+PS-w counterparts. Unfortunately, with the current evidence, it remains very difficult to precisely
+quantify how “small” or “large” the tuning budget is required to make such a distinction.
+The above lessons not only reveal the important factors for the practitioners to consider when
+choosing PS-w and PS-w/o for bi-objective software configuration tuning but also hint at a few
+future research opportunities in this regard. These are:
+• Landscape analysis for configurable software systems: We have found that the realism
+of aspiration space, its position in the objective landscape, and tuning budget can be the key
+factors to consider when choosing between PS-w and PS-w/o. All of those are relevant to
+the landscape analysis of the configurable system itself. Indeed, by systematically analyzing
+any collected data, we are able to obtain more knowledge about the above factors, and hence
+make more informed decisions on whether to incorporate requirements into the tuning.
+• Requirement-robust optimizer for configuration tuning: The realism of the aspiration
+is certainly the key factor in the choice between PS-w and PS-w/o. However, it may not be
+always possible to obtain such knowledge in advance, leaving uncertainty to the decision. In
+this regard, it would be desirable to combine the strength of PS-w and PS-w/o to design an
+optimizer that is robust to such an uncertainty in the requirements. Again, the landscape
+analysis from the previous opportunity can provide insights into the designs.
+• Rigorous analysis of requirement patterns and their relationships to the tuning:
+Although we see little implication of the requirement patterns to the choice between PS-w
+and PS-w/o, it is important to better understand why they work more diversely on some
+of the patterns and how exactly they can affect the performance of PS-w. In fact, on the
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+Do Performance Aspirations Matter for Guiding Software Configuration Tuning?
+1:31
+theoretical side, the quantification from Section 3 provides the foundation of theoretical
+reasoning for switching between patterns, which is important in the topic of requirement
+relaxation. For example, this can be achieved in two aspects:
+– With the quantification of the patterns, one can formally show the relations between them.
+For example, since all points in 𝒑1 have a higher satisficing value than those of 𝒑3, we can
+say that 𝒑1 is a “relaxed” form of 𝒑3.
+– Similarly, we can quantify the relationships between a pattern with two different aspiration
+levels.
+With the above understanding, we allow the software engineers to achieve more explainability
+in terms of the given requirements during the tuning. For example, once the tuning completes,
+one would know how to relax or tighten the requirements, such that the most preferred
+configuration can be found under the requirements. This can be a unified process that
+combines both requirement negotiations and the tuning itself.
+• Interactive configuration tuning: On the empirical side, our findings provide a few insights
+on what to do under different circumstances during interactive tuning. For example, if the
+software engineers find that the tuning never (or rarely) produces configurations that satisfy
+the requirements/aspirations under PS-w, then one can immediately switch to PS-w/o instead
+before concerns about the suitability of the underlying optimizer. Similarly, one can influence
+the results produced by PS-w (or PS-w/o) by changing the position of the aspiration space.
+7
+THREATS TO VALIDITY
+As with other empirical studies in software engineering, our work may contain threats to construct
+validity in the following aspects:
+• Metric: Pareto search produces a set of configurations, and thus the comparisons need to
+work on a set rather than a single configuration. We used HV, which is a comprehensive
+metric for evaluating solution sets, following the methodology proposed by Li et al. [55].
+Since there can be different given sets of requirements with aspirations, the configuration
+sets ought to be compared under such a scenario. To that end, we extend the HV to explicitly
+consider the patterns of requirements, as discussed in Section 4.4.1.
+• Statistics: The stochastic nature of the Pareto optimizers can raise threats to the stability of
+results. To mitigate such, we repeat the experiments 100 runs and use Wilcoxon test along
+with ˆ𝐴12 to verify all pairwise comparisons. All the above methods have been recommended
+and widely used for Software Engineering research [1].
+Two factors may form threats to internal validity in our study:
+• Tuning budget: Given the size of our study, we set a one-hour budget for each case, which
+is a common setting for expensive problems in SBSE [53]. To mitigate the interference of
+our experiments, this is then converted into the number of unique measurements following
+systematic steps (Section 4.3.2). We have also analyzed the trajectories of A-HV, in Section 5.4,
+showing what would happen if a smaller budget is used. Admittedly, investigating a larger
+tuning budget may affect some of the results, but confirming this would need even more
+computational resources and time (due to the expensive tuning), which we will plan as part
+of future work.
+• Optimizer setting: In this work, we follow what has been shown to be effective for a SBSE
+problem in the literature, as our aim is to compare the most common practices. The only part
+we could not have found for sure is the population size, which is highly problem-dependent.
+To tackle this, we have followed carefully designed criteria (Section 4.3.3) that strike a balance
+between reasonable convergence and the time required under the tuning budget. However,
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+1:32
+Chen and Li.
+we do agree that exploring alternative parameter settings can be a thread that requires further
+exploration, which we leave as part of future work.
+Threats to external validity can come from various sources, including:
+• Software systems: In this work, we select the eight most representative systems/environ-
+ments from existing work on software configuration tuning based on carefully codified rules
+(Section 4.1). Those subject systems come from diverse domains and with different scales,
+performance objectives, and search spaces. A worth noting point is that the requirements
+extracted include those for more complex systems, such as Cyber-Physical systems, while
+the subjects we examined are mainly software systems. This does not severely invalid our
+conclusion because the extracted implication and patterns are rather generic such that they
+can be applied to different cases while there exist some performance attributes that are of
+relevance to a wide range of systems, e.g., latency- and throughput-related requirements (with
+different aspiration levels) [62]. Nonetheless, we agree that this list of the studied systems is
+not exhaustive and we may miss some particular situations that can only become clear for
+more complex systems. Experimenting with more systems that are of diverse types may prove
+fruitful. A relevant point is that we did not examine our results on highly complex software
+systems that cut across the software and hardware layers. In those cases, the interaction
+between cross-layered configuration options can be more complex, leading to some different
+configuration landscapes [42]. Therefore, examining those highly complex systems may
+provide new insights and further consolidate our findings.
+It is worth noting that it can be particularly attractive to relate the results with respect to
+the different types of software systems. However, unfortunately, we have not yet observed
+consistent patterns in the results according to the domain of systems, hence unable to draw a
+general conclusion thereupon. This can be attributed to two reasons:
+– The workload and benchmark under which each of the systems runs are rather different,
+creating a distinct configuration landscape.
+– Because of the above, the appropriate aspirations (levels) used are also different even for
+systems that are of the same domain.
+Again, using even more software systems may help us to achieve such, which we certainly
+plan to do for future work. However, this does not invalidate the conclusions drawn regarding
+the comparison between PS-w and PS-w/o.
+• Configuration options: The discretization level of each configuration option can have a
+non-trivial impact on the tuning. In this work, we use exactly the same configuration options
+and their values as used in previous work [17, 44, 45, 62]. However, it is necessary to note that
+changing the discretization level may disclose new insights, which we will seek to investigate
+as part of future work.
+• Requirement patterns and aspiration space: To emulate real-world requirement sce-
+narios, as shown in Section 3, we capture the implications and how they are quantified
+by surveying the relevant datasets and papers. This has enabled us to concentrate on four
+patterns that cover a wide range of situations, leading to 15 combinations of the patterns. As
+for the aspiration space, we cover both realistic and unrealistic aspirations, and for the former,
+we set three types of aspiration space including two skewed spaces and a more balanced one.
+Yet, admittedly, unintentionally ignored cases are always possible.
+• Optimizers: In this study, three common Pareto optimizers based on evolutionary search
+are used, each of which is a distinct representative of its own kind. Admittedly, there are
+other popular multi-objective optimization approaches used in SBSE, such as exact methods
+and Bayesian optimization methods.
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+Do Performance Aspirations Matter for Guiding Software Configuration Tuning?
+1:33
+When the given multi-objective optimization problems are of special characteristics (e.g.,
+linearity in both objective functions and constraints and the scale is small or moderate),
+then exact methods (e.g., integer linear programming) can be very good choices, where
+solutions of the Pareto front can be iteratively found by specifying different weights or
+desirable/tolerant values. Such optimization problems have been commonly seen in the next
+release problem, and well-established exact methods, such as 𝜖-constraint and augmented
+Tchebycheff methods, have shown promising results [24, 73].
+Another popular kind of optimizers, particularly used in software configuration tuning, is
+Bayesian optimization. Recently, there are multi-objective Bayesian optimizers (e.g., [43])
+which search for the whole Pareto front of the problem. Compared to evolutionary algorithms,
+Bayesian optimizers are usually more sample efficient [44, 46]. However, one issue with
+such approaches is that they may not be as straightforward as evolutionary algorithms to
+incorporate the stakeholders’ performance aspirations (e.g., for Bayesian optimization this
+may need a careful design of the acquisition function). Different incorporation ideas may
+lead to different results, thus likely affecting the reliability of the conclusions drawn from
+the direct comparison between Pareto search with and without performance aspirations.
+In addition, it is necessary to point out that the conclusions drawn from multi-objective
+evolutionary algorithms may not apply to other optimization approaches. Optimizers, which
+are guided “heavily” by the aspirations, may find them quickly if they are realistic, but may
+end up with undesirable solutions if unrealistic. Optimizers, which can strike a good balance
+between exploitation and exploration (under limited budgets) like Bayesian optimization,
+may bring different results, though it depends on the incorporation of the aspirations in
+the optimization process. Consequently, it is desirable to investigate different optimization
+approaches to study the generalizability of our findings, particularly those with fruitful
+theoretical results (e.g., convergence rate regarding the response surface’s smoothness as
+well as the regret bounds in Bayesian optimization), which may help support our empirical
+conclusions. This will be an important part of our future work.
+• Number of objectives: Our study covers the case of two performance objectives for software
+configuration tuning, which, as we have found from our review in Section 3, tends to be the
+most common situation when multiple objectives are considered. The results may not be
+generalizable to higher dimension cases of the objectives though. Extending the study to
+more objectives can be part of future work, but there would also be exponentially increasing
+factors to consider, e.g., the number of pattern combinations. Our results from this work
+serve as the very first step to raising the importance of studying whether to use aspiration to
+guide software configuration tuning with more than one performance objective.
+Overall, the above settings have provided us with more than 1,000 cases to generalize our findings
+in this study.
+8
+RELATED WORK
+Here, we discuss the related work in light of the purpose of our empirical study.
+8.1
+PS-w/o for Software Configuration Tuning
+Search-based approaches for software configuration tuning have been commonly studied under a
+wide range of optimizers, such as random search [64], hill climbing [77], genetic algorithm [68],
+and ant colony optimization [14]. In the presence of more than one objective, Pareto search has
+been shown to be highly effective. Among others, Chen et al. [16] and Singh et al. [70] leverage
+different multi-objective evolutionary algorithms to search the Pareto optimal configurations. Nair
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+1:34
+Chen and Li.
+et al. [62] also aim for the same, but their approach applies Bayesian optimization wherein the
+two performance objectives are handled similarly to MOEA/D. More recently, Zhu et al. [30] also
+propose an extended Bayesian optimization approach to reach a given performance aspiration
+while considering both latency and resource consumption; however, such information has not been
+used to explicitly guide the search.
+The above work has one thing in common: they have ignored the aspirations in the search
+process. The assumption therein is that the concept of optimization can obtain whatever best
+configuration that satisfies any given patterns of requirements.
+8.2
+PS-w for Software Configuration Tuning
+In contrast, Calinescu et al. [8, 32] explicitly quantifies aspirations as part of the objectives (as 𝒑1) to
+guide the Pareto search. Martens et al. [59] also bear similar idea, but their pattern matches with 𝒑3.
+Ghanbari et al. [34] has also been relying aspiration to guide the tuning, and they assume a smoother
+curve over the requirement patterns. However, there has been no study that justifies the importance
+of aspirations in guiding the Pareto search for bi-objective software configuration tuning. Indeed,
+a recent discussion paper from Fekry et al. [28] commented that studying the aspirations for
+guiding the optimizers and measuring its effectiveness is an important future challenge for software
+configuration tuning.
+These are typical examples of the PS-w optimization model, such that the performance require-
+ments are precisely quantified as part of the search and tuning process. It is also worth noting that
+the actual requirements patterns used can vary depending on the assumption, and none of the
+existing work for PS-w has considered all the patterns we summarized in our study.
+8.3
+Tuning with or without Surrogate
+From another perspective, existing search-based approaches, regardless the number of performance
+objectives considered, can be classified as model-based (e.g., EvoChecker [32], FLASH [62], and
+BOCA [10]) and measurement-based (e.g., FEMOSAA [16], eQual [68], and Plato [66]), by which
+the former relies on surrogate models [13] to guide the search while the latter do so via direct
+measurements from the software. The key difference between those two are the landscape upon
+which the search is conducted: the measurement-based approaches do so directly on the configura-
+tion landscape while the model-based approaches searches in a surrogate landscape, which is an
+approximation of the true configuration landscape.
+This work focuses on tuning by directly measuring the systems without using the surrogate,
+because for the following reasons:
+• We seek to avoid the noises caused by the surrogate models as they would inevitably introduce
+errors, which, as demonstrated by Zhu et al. [83], can severely affect the search and tuning
+behavior.
+• In fact, we treat these two categories as complementary rather than alternative. For example,
+a measurement-based approach that works well can be also applied in Bayesian optimization,
+which is model-based, to search for the acquisition. This means that our findings on whether
+aspirations matter are also applicable therein.
+8.4
+General Multi-objective Optimization
+Conceptually, our work can be relevant to the theme of preference-driven multi-objective optimiza-
+tion. In this regard, as surveyed by Wang et al. [74], the preferences on the objective values (i.e.,
+the performance requirements in this work) can form the following categories:
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+Do Performance Aspirations Matter for Guiding Software Configuration Tuning?
+1:35
+• Weights: In this case, a weight vector representing the relative importance of the objectives
+is given. Most commonly, this would convert the multi-objective problem into a single one via
+some form of aggregation, e.g., weighted sum. Indeed, both the weights and the aspirations
+in this work are some forms of preferences. However, they are very different because the
+weights need to be specified in-between the performance objectives, representing an explicit
+trade-off [68]. The aspiration level, in contrast, serves as the expectation for a single objective,
+which is often easier to specify, and no explicit trade-off is required. In fact, using the weight
+can be thought as a special case of the kind of preference we consider in this work: in the
+case of weight, the best configuration is typically a particular point on the Pareto front. In
+contrast, under the requirements and aspirations considered, there are often more than one
+best point, including a proportion of the points on the Pareto front. Interestingly, given a set
+of weights, it has been shown that the Pareto search (which runs without the weights) can
+generally find better configurations than the search guided by the weights [18].
+• Objective relation: This refers to the case where a full or partial rank of the objectives has
+been provided. For example, one may prefer to satisfy the requirement of throughput first
+before considering latency. The combinations of patterns and aspirations in this work, in
+contrast, have no direct ranking between the objectives.
+• Area of objective space: One may provide a rough notion of a particularly preferred area
+in the objective space, e.g., knee points or extreme points. Compared with the combinations
+of patterns from Section 3, this preference is vague — the knee or extreme points are relative
+among the solutions found, while the patterns are still guided by a clearly defined aspiration
+space.
+• Reference points [51, 80]: Here, a vector representing the expectations of the objectives are
+given. In this regard, the concept is indeed similar to the aspiration levels we discussed in this
+work. However, a major difference is that reference point-based multi-objective optimization
+is always under the assumption that Pareto optimality needs to be considered first, i.e., the
+Pareto optimal solutions close to the reference point (along certain direction specified by the
+decision-maker) are preferred [2]. Moreover, additional parameters are required to specify the
+spread of the preferred solutions [51]. In contrast, the combinations of patterns we discovered
+for software configuration tuning can be rather different from the above, as the solutions in
+(outside) the aspiration space may be equally preferred (unpreferred) while do not favor the
+solutions close to the aspiration vector. Further, no other parameters are needed besides the
+aspiration levels.
+Therefore, all the above forms of preferences differ from the requirement aspirations and patterns
+used for software configuration tuning, as we summarized in Section 3. This is important as the
+summarized patterns are derived from empirical findings for the characteristics of the problems
+— they may not be generalizable to other problems but are significant to software configuration
+tuning.
+An empirical study on the importance of considering preferences in the search also exists from
+the general optimization community [51, 81]. However, they differ from our work in two aspects:
+• They focus on reference points, which, as we discussed, are rather different from the patterns
+considered in this work.
+• The studied optimizers are specialized algorithms that take specific forms of preferences
+into account. In contrast, our study considers classic optimizers which are equipped with
+the requirement patterns to guide the search, as what has been commonly used for software
+configuration tuning.
+ACM Trans. Softw. Eng. Methodol., Vol. 1, No. 1, Article 1. Publication date: January 2023.
+
+1:36
+Chen and Li.
+In summary, this work is, to the best of our knowledge, the first empirical study to understand
+whether, when, and why aspirations matter for guiding bi-objective software configuration tuning,
+according to the characteristics of requirement patterns and practice summarized for the problem.
+9
+CONCLUSION
+In this paper, we conduct a comprehensive empirical study, which offers an in-depth understanding
+of whether performance aspirations matter to bi-objective tuning for configurable software systems.
+Our study covers 15 combinations of patterns that quantify the aspirations, four types of aspiration
+space, three search algorithms, and eight software systems/environments, leading to 1,296 cases of
+investigation. The results challenge the belief of “it does not matter whether to use aspiration in the
+tuning” and reveal that:
+• the aspirations are generally helpful in guiding the tuning when they are realistic.
+• but, they can often be harmful to the tuning when these aspirations are unrealistic.
+• the different patterns and position of the aspiration space do not change the above conclusion,
+but can affect the extent of benefits/detriments generated.
+• when the aspirations are realistic, the tuning budget has marginal implication on their benefits
+during tuning. In contrast, it is an important factor when the aspirations are unrealistic.
+Our findings in this work provide useful insights for the practitioners in this particular field
+of research, particularly on the factors/information needed for answering the question of “when
+aspirations should be considered during the tuning?”.
+We outline some exciting future research opportunities:
+• To better understand the likelihood of realism and position of the given aspirations, we shall
+analyze the landscape of the configurable software systems even based on some limited
+samples.
+• To cater for the uncertainty of given requirements and aspirations (e.g., possible to be unreal-
+istic), we should investigate requirement-robust optimizers for tuning software configuration.
+• To explain the impact of requirements patterns for guiding the tuning, we ought to provide
+the foundation to rigorously analyze their relationships and how they may be able to switch
+to one another for achieving relaxation.
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf,len=2776
+page_content='1  Do Performance Aspirations Matter for Guiding Software  Configuration Tuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  TAO CHEN, Loughborough University, United Kingdom  MIQING LI, University of Birmingham, United Kingdom  Configurable software systems can be tuned for better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Leveraging on some Pareto optimizers,  recent work has shifted from tuning for a single, time-related performance objective to two intrinsically  different objectives that assess distinct performance aspects of the system, each with varying aspirations to be  satisfied, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', “the latency is less than 10s” while “the memory usage is no more than 1GB”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Before  we design better optimizers, a crucial engineering decision to make therein is how to handle the performance  requirements with clear aspirations in the tuning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For this, the community takes two alternative  optimization models: either quantifying and incorporating the aspirations into the search objectives that  guide the tuning, or not considering the aspirations during the search but purely using them in the later  decision-making process only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, despite being a crucial decision that determines how an optimizer  can be designed and tailored, there is a rather limited understanding of which optimization model should be  chosen under what particular circumstance, and why.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  In this paper, we seek to close this gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Firstly, we do that through a review of over 426 papers in the  literature and 14 real-world requirements datasets, from which we summarize four performance requirement  patterns that quantify the aspirations in the configuration tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Drawing on these, we then conduct a  comprehensive empirical study that covers 15 combinations of the state-of-the-art performance requirement  patterns, four types of aspiration space, three Pareto optimizers, and eight real-world systems/environments,  leading to 1,296 cases of investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Our findings reveal that (1) the realism of aspirations is the key factor  that determines whether they should be used to guide the tuning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' (2) the given patterns and the position  of the realistic aspirations in the objective landscape are less important for the choice, but they do matter  to the extents of improvement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' (3) the available tuning budget can also influence the choice for unrealistic  aspirations but it is insignificant under realistic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' To promote open science practice, we make our code  and dataset publicly available at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='com/ideas-labo/aspiration-study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  CCS Concepts: • Software and its engineering → Search-based software engineering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Empirical software  validation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Software performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Additional Key Words and Phrases: Search-based software engineering, software configuration tuning, perfor-  mance requirement, performance aspiration, multi-objective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='.  ACM Reference Format:  Tao Chen and Miqing Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='.  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' 1, 1, Article 1 (January 2023), 41 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1145/3571853  1  INTRODUCTION  Many software systems are highly configurable, such that there is a daunting number of config-  uration options (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=', the max_spout in Apache Storm), which the software engineers can tune  Authors’ addresses: Tao Chen, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='uk, Loughborough University, Loughborough, United Kingdom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Miqing Li,  University of Birmingham, Birmingham, United Kingdom, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='8@bham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Copyrights for components of this work owned by others than the author(s) must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='00  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1145/3571853  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='03290v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='SE]  9 Jan 2023  1:2  Chen and Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  to meet the requirements of some performance objectives, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', improving latency, throughput,  and resource consumption [12, 15, 36, 52, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Configuration tuning for software systems plays an  integral role in Software Engineering as a recent interview reveals that industrial practitioners  have recognized it as a key to the success of software products [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Indeed, it has been reported  that globally 59% of the software performance issues—wherein the performance requirements were  severely violated—are related to ill-suited configuration rather than code [39], leading to serious  consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For example, in 2017-2018, configuration-related performance issues cost at least  400,000 USD per hour for more than 50% of the software companies worldwide1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Finding good configurations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', the possible combinational settings of the configuration options)  is challenging, because:  The default configuration is often far from ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Jamshidi and Casale [44] show that the  defaults for Apache Storm can lead to 480 times worse performance than some others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  The configuration space can be large and the measurement is often expensive [62], rendering  greedy search unrealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  While traditionally software configuration tuning has been focusing on a single performance  objective [6, 11, 37, 58, 64, 66, 71, 77, 79], recent work raises the necessity of simultaneously  tuning for multiple performance objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Our review (see Section 3) found that considering  two performance objectives is the most common case [16, 32, 41, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For example, naturally,  improving the image quality while reducing the energy consumption are both critical for video  encoders like x264;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' higher accuracy with shorter training time are two inherent performance  objectives for deep learning models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', the deep neural network supported by frameworks  such as Keras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This further complicates the tuning process as the performance objectives  may be conflicting and the extents to such a conflict are often unknown a priori [16, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  To automatically tune software configuration for better performance, different approaches have  been proposed, such as rule-based [31, 35], learning-based [3, 45], and search-based [8, 9, 16, 50,  62, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Among these, search-based approach, primarily relying on the Pareto optimizers widely  used in the Search-Based Software Engineering (SBSE) paradigm [40], has been a promising way  to handle all the aforementioned challenges in software configuration tuning, especially in the  presence of more than one performance objective [8, 9, 16, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In a nutshell, a Pareto optimizer  most commonly maintains a population (or at least an archive) of configurations, which can be  repeatedly reproduced and evaluated by directly profiling the software, aiming to find the Pareto  optimal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The output is a set of configurations that are nondominated to each other, which  approximates the Pareto front of the software system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1  The Problem and Significance  An important factor in software configuration tuning is the possible requirements with clear  aspirations for the performance objectives [8, 9, 27, 65], for which we distinguish two important  notions in this work:  Aspiration: The information that allows us to quantify the extent to which the performance  is considered satisfactory (or unsatisfactory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Performance requirement: The context under which the preference of the performance is  defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  For example, according to the research in the Requirement Engineering community [4, 75], it  is not uncommon to have performance requirements from the requirement documents, such as  “the latency shall be less than 𝑥” while “the memory usage shall be no more than 𝑦”,  1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='evolven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='com/blog/downtime-outages-and-failures-understanding-their-true-costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='html  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Do Performance Aspirations Matter for Guiding Software Configuration Tuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:3  where “less than 𝑥” and “no more than 𝑦” are the clear aspirations therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' It is worth noting  that not all performance requirements would contain aspiration, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', “the latency shall be  low” is a requirement with no aspiration since nothing can be quantified with respect to the level of  satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Indeed, given a scenario with clear aspirations in the performance requirements, it has  been well-acknowledged that the information provided serves as useful metrics for the software  engineers to conduct a posterior cherry-picking after the tuning completes, extracting the satisficing  configuration(s) from the set produced by a Pareto optimizer [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The natural motivations behind  this are:  Given a fixed tuning budget, finding the optimal performance is not always feasible or even  desirable to the stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  The clear aspiration levels allow an implicit trade-off/preferences between the conflicting  performance objectives according to the stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Regardless of the Pareto optimizer used, in the tuning process, existing work takes one of two  intrinsically different optimization models to handle aspirations when tuning for two performance  objectives, namely: Pareto search with aspirations (denoted as PS-w) [8, 9, 33, 59] and Pareto  search without aspirations (denoted as PS-w/o) [16, 49, 62, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In PS-w, the performance re-  quirements with aspirations are quantified in certain forms (we will elaborate on this in Section 3),  which then serve as new search objectives in the tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The motivation is simple: since the  aspirations provide information on the degree of satisficing, one can exploit this advantage to guide  the tuning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' PS-w/o, in contrast, is more classic and simply ignores the aspirations in the  tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The assumption here is that, since the search in whatever a Pareto optimizer is essentially  an optimization process that seeks to find the Pareto optimal configurations, the tuning always  aims to achieve the best possible performance, which preserves the tendency towards satisficing  whatever aspirations2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For example, finding the Pareto optimal configuration latency=10s and  memory usage=1GB will certainly meet the requirement and aspiration of “latency shall be less  than 20s” while “memory usage shall be no more than 2GB”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This matches with Odhnoff’s  argument that “optimizing” and “satisficing” are merely stylistically different but fundamentally  the same [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Despite either of the two optimization models being respectively used by their corresponding  research groups, the choice was mostly ad-hoc and there is often an implied belief that “they do not  differ much hence can be used arbitrarily.” As such, there remains a rather limited understanding of  which optimization model should be chosen under what particular circumstance, and why.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This has  been well-echoed by some researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Ghanbari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' [34] have stated that it is important to consider  the choice, as the shape of the function that guides the tuning, especially after passing the aspirations,  may impact the behavior of the optimizer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' but they did not discuss what implication that would be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Yet another example from a recent work by Fekry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' [28] recommends that studying whether  to leverage aspirations for guiding the optimizers and measuring its effectiveness is an important  future challenge for software configuration tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Indeed, understanding in this regard is non-  trivial as it will help practitioners to make more informed-decision, especially when given  the expensive measurements of configurable software systems, it is unrealistic to always empirically  compare the two models in a case-by-case manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Furthermore, the insights can hint at future  research directions for software configuration tuning: if the PS-w/o is more promising, then  we can largely simplify the research to the design of an effective optimizer without considering the  given requirements since the human inputs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', the requirements/aspirations) are less important  in the overall tuning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' On the other hand, if PS-w is overall more effective, then the problem  2This assumes the most common case that the best possible performance is at least equally preferred than some other  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:4  Chen and Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  can become more complicated but also provide more opportunities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', future research can largely  focus on how to better quantify those performance requirements and aspirations, together with  how to better embed them into more specialized optimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  To understand this, we have also tuned into the literature on general multi-objective optimization,  with a particular focus on preference-driven multi-objective optimization [5, 51, 74, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, we  did not find answers that are directly relevant to our case, due to two reasons: (1) the representation  of the preferences (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', weights and ranks) in preference-driven multi-objective optimization is  different from the requirement patterns we summarized from the work for software configuration  tuning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' (2) they mainly develop algorithms/optimizers that are tailored to a specific preference  representation while software configuration tuning often relies on a vanilla optimizer [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Our work is, therefore, motivated by the desirability of the community to understand the  following:  Should we incorporate requirements and aspirations to guide the software configuration  tuning process?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' If so, in what context and why?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  Research Questions  In this paper, we seek to fill the above gap via an empirical study that systematically compares PS-w  and PS-w/o for tuning software configuration under two performance objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Suppose that  there are some realistic aspirations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', all the aspirations are achievable by tuning the configuration  of the software system), the first research question (RQ) we wish to answer is:  RQ1: Given performance requirements with realistic aspirations, of PS-w and PS-w/o, which  can find a better set of configurations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  RQ1 seeks to provide a global picture of the comparison between the two optimization models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  However, the diverse possible requirement scenarios imply that the specific aspirations can be  radically different in the objective landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For example, one may have higher expectations  on latency while lower needs on throughput, or vice versus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Therefore, what we would like to  understand in more detail is:  RQ2: How do different realistic aspirations influence the result?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  RQ1 and RQ2 investigate under the normal context where the given aspirations are reasonable  and achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, since the actual aspirations are negotiated by the software engineers  and stakeholders a priori, they could turn out to be unrealistic and may require attention beyond  configurations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', no configuration in the search landscape can reach the required aspiration levels  for all performance objectives simultaneously, despite that may be possible for a single objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  This brings our next RQ, in which we ask:  RQ3: What if the given aspirations are unrealistic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  While we are interested in cases where the tuning budget is reasonably sufficient to achieve a  good convergence, it is possible that, in real-world scenarios, there is a limited resource for tuning  software configuration due to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', pressure for quick release or task prioritization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Therefore, our  last question aims to explore:  RQ4: Is the given tuning resource (tuning budget) important to the choice between PS-w and  PS-w/o?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Do Performance Aspirations Matter for Guiding Software Configuration Tuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3  Contributions  To address these RQs , we conducted an extensive empirical study on 15 combinations of patterns  to quantify aspirations, four types of aspiration space in the objective landscape, three Pareto  optimizers, and eight real-world systems/environments with diverse performance objectives, leading  to 1,296 cases of investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Briefly, the first contribution in this paper is a set of performance  requirement patterns (for individual performance objectives) summarized from 426 papers in the  literature from the Software Engineering community and 14 widely-used real-world requirements  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' These patterns are:  (1) No aspiration is given but assuming that the optimal possible performance is preferred, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=',  “the lower latency is preferred”, meaning that one prefers the best possible latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  (2) The performance in the aspiration space is equally good or otherwise there is a certain degree  of tolerance, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', “the minimum latency shall ideally be 500ms”, implying that anything better  than 500ms is equally good while a performance worse than that is acceptable but not ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  (3) The performance in the aspiration space is equally good while anything outside the space  is unacceptable, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', “the latency shall be 500ms”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This suggests that a latency better than  500ms is equally good and no tolerance is allowed for performance worse than that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  (4) Preferring the optimal performance while anything outside the aspiration space is unaccept-  able, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', “the latency shall be at most 500ms”, reflecting that no tolerance is allowed for  worse than 500ms while the lower the latency, the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Our second contribution is the pragmatic findings that answer the aforementioned RQs over the  1,296 cases as follows:  To RQ1: PS-w performs considerably better or similar to PS-w/o on 84% of the cases, out of  which over 60% show statistically significant improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  To RQ2: The improvement of PS-w over PS-w/o is often largely biased to a certain position  of the aspiration space in the objective landscape, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', centered or left-shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  To RQ3: PS-w/o is no worse than PS-w for 70% cases, wherein the difference is considerable  with statistical significance for more than 85%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  To RQ4: Under realistic aspirations, PS-w obtains consistently better outcomes than PS-w/o  throughout the trajectory and with a speedup up to 10×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' When the aspirations are unrealistic,  in contrast, the two optimization models are competitive in the early stage of tuning but  soon PS-w/o would lead to better results with considerably high speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Hence, we conjecture that the performance aspirations do matter for guiding bi-objective  software configuration tuning in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Yet, depending on the context, it can either be  helpful or harmful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' We provide, as part of the third contribution, some in-depth analysis and  discussions on the reasons behind the above observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' More importantly, these findings allow us  to derive our fourth contribution: the key lessons learned on the choice between PS-w and PS-w/o  for bi-objective software configuration tuning, which are:  Lesson 1: The choice on whether to exploit aspirations for guiding the tuning is primarily  dependent on their realism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Lesson 2: It is unlikely that the combinations of patterns can change the decision on whether  to incorporate aspiration in the tuning, but it can influence the benefit/detriment of aspiration-  guided tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Lesson 3: The positions of realistic aspiration space in the objective space can largely affect  the benefits brought by considering aspirations within tuning, but it is less likely to influence  the choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:6  Chen and Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Lesson 4: The given tuning budget has a marginal impact on the choice when the aspirations  are realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, it can be an important factor to consider under unrealistic aspirations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Drawing on those lessons, our fifth contribution outlines three future opportunities for this field  of research, namely:  Landscape Analysis for Configurable Software Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Requirement-Robust Optimizer for Configuration Tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Study on the Relative Impact between Requirement Patterns to the Tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  To promote open science practice, all the code, dataset, and necessary supplementary documents  for this work can be publicly accessed at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='com/ideas-labo/aspiration-study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  The rest of this paper is organized as follows: Section 2 formalizes the problem and presents the  motivating example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Section 3 discusses the patterns that quantify performance requirements with  aspirations and how they were identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Section 4 elaborates the design of our empirical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Section 5 presents and analyzes the experiment results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Thereafter, Section 6 discusses the lessons  learned and future opportunities, followed by threats to validity in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Finally, Sections 8  and 9 review the related work and conclude the paper, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  2  THEORY  In this section, we present the theoretical knowledge for understanding the purpose of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1  Formal Definition  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1  Background and Problem Formalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In the DevOps era, software configuration tuning  involves two fundamental roles that interact frequently [67] — the stakeholders (whose benefit is  directly affected by the software performance) negotiate their performance requirements with the  software engineers, who then act as the operators to tune the configurations for satisfying these  requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Beyond a single performance concern, recently there has been an increasing demand  for considering multiple performance objectives [19, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Among those, our literature review from  Section 3 shows that 90% of the recent work has considered two performance objectives [16, 32, 62],  such as the latency versus throughput for Storm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' image quality versus energy usage for x264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This  makes software configuration tuning with requirements in mind even more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Without loss of generality, we assume that a configurable software comes with a set of configura-  tion options, whereby the 𝑖th option is denoted as 𝑐𝑖, which can be a binary, integer, or enumerate  variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' A particular configuration is denoted as 𝒄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The search space, 𝒞, is the Cartesian product of  the possible values for all the 𝑐𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Formally, given a scenario of requirements with clear aspirations  for two performance objectives, the goal of PS-w for software configuration tuning is to find the  configuration(s) that achieve:  𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑝𝑥 (𝑓1(𝒄)), 𝑝𝑦(𝑓2(𝒄)), 𝒄 ∈ 𝒞  (1)  whereby 𝑓 is the raw measurement of the performance value achieved by 𝒄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 𝑝 is the corresponding  requirement pattern, which quantifies the degree of satisficing given 𝑓 (𝒄) (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In this  work, we consider cases where at least one 𝑝 contains a clear aspiration level3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  In contrast, the goal of PS-w/o is to:  𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓1(𝒄), 𝑓2(𝒄), 𝒄 ∈ 𝒞  (2)  3We use 𝑝𝑥 and 𝑝𝑦 to distinguish two performance requirement patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Do Performance Aspirations Matter for Guiding Software Configuration Tuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' A performance requirement snippet from the requirement document of a real-world project in the  PURE dataset [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  �5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='5 �5 �4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='5 �4 �3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='5  0  1  2  3  log10 Throughput�1  log10 Latency  Aspiration level  Aspiration level  Aspiration  space  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The aspiration space (highlighted by color) and aspiration levels within the bi-objective space (latency  and throughput) for Storm under the Rolling Sort benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  As can be seen, PS-w explicitly leverages information about the given requirements with clear  aspirations to guide the search and tuning while PS-w/o assumes the basic Pareto optimality4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  Aspiration Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Following the normal software engineering practice of requirement ne-  gotiation, it is likely that a single performance requirement can come with a clear aspiration, in  which case we define aspiration space as the portion of performance points that are not inferior  to the given aspiration level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' A real-world example has been shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Here, “the system  shall support at least 1,000 concurrent users” contains a clear aspiration level of 1,000  users, meaning that the aspiration space covers throughput between 1,000 (inclusive) and the true  optimum (which is case-dependent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Beyond such a one-dimensional case, it is easy to know that  the aspiration space can be generalized to a two-dimensional case when the aspiration levels of  two performance objectives are involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For example, Figure 2 shows the aspiration space for the  requirements “the system shall perform with 39800 users at a time” while “the latency  shall be no worse than 160 seconds” for Storm (with log-transformed values (log10) and all  performance objectives are to be minimized as we consider the reciprocal of Throughput).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This  forms the foundation of our analysis in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3  Pareto search with and without Aspirations for Tuning Software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' To illustrate the difference  between PS-w and PS-w/o, a pseudo-code using NSGA-II as the underlying optimizer has been  4We assume that all performance objectives are to be minimized;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' maximizing ones can be easily converted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3  Performance  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1  The system should support at least 1000 concurrent users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  This statement provides a general sense of reliability when the system is under load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' It is  important that a substantial number of actors be able to access the system at the same time,  since a courseware system is important to the courses that employ it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The times when the  system will be under the most stress are likely during midterm and finals weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Therefore, it  must be able to handle at least 1,000 concurrent users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Priority: 11:8  Chen and Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Algorithm 1: Unified code for PS-w and PS-w/o with NSGA-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Input: Configuration space V;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' the system F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' a matrix of fitness quantified by the requirements Γ  Output: A set of nondominated configurations S′  1 Randomly initialize a population of 𝑛 configurations P  2 /* measuring on the actual configurable system  /  3 measure(P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' F)  4 /* for PS-w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' the fitness that guides the search is computed according to Equation (1)  /  5 if PS-w then Γ ←getFitnessBasedonReqirements(P)  6 while The search budget is not exhausted do  7  P′ = ∅  8  while P′ < 𝑛 do  9  /* for PS-w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' selecting parents with respect to their compliance to the requirements */  10  if PS-w then {𝑠𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='𝑠𝑦 } ←mating(P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Γ)  11  else {𝑠𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='𝑠𝑦 } ←mating(P)  12  {𝑜𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='𝑜𝑦 } ←doCrossoverAndMutation(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='𝑠𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='𝑠𝑦)  13  measure(𝑜𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='𝑜𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' F)  14  if PS-w then Γ ←getFitnessBasedonReqirements(𝑜𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='𝑜𝑦)  15  P′ ← P′ �{𝑜𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='𝑜𝑦 }  16  /* for PS-w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' the configurations are preserved according to the fitness computed with  respect to the requirements  /  17  if PS-w then U ←nondominatedSorting(P � P′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Γ)  18  else U ←nondominatedSorting(P � P′)  19  P ←top 𝑛 configurations from U  20 if PS-w then return S′ ←nondominatedConfigurations(P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Γ)  21 else return S′ ←nondominatedConfigurations(P)  shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' As can be seen, PS-w and PS-w/o mainly differ in the fact that the former is  guided by the information extracted from the given requirements and aspirations (denoted as Γ)  while the latter runs without, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', it uses the raw values of the measured performance objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  This means that all the fitness of configurations evaluated in the PS-w makes use of the Γ while  that of the configuration in PS-w/o does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For example, under the raw performance, a latency of  500ms is certainly more preferred than the case of 700ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, under the requirement and  aspiration that any latency less than 900ms is equally preferred, they are actually equivalent therein  and hence PS-w reflects precisely that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  As a result, the above generates two differences between PS-w and PS-w/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Firstly, the process of  deciding on which two configurations to be selected as parents for generating new configurations  is guided differently (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', lines 10–11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Secondly, the environmental selection that determines what  configurations to be preserved in the next iteration is also guided by different fitness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', lines  17–18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  As we will show, even with such a simple deviation the leading results can be radically different  depending on the circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  Motivating Scenario  Taking x264 — a configurable video encoder — as a concrete example, a possible requirement  scenario could involve performance requirements (denoted as 𝒫1) “the PNSR5 shall be  at least 40dB” and “the energy usage shall be at most 80 watts”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Here, there is a  clear aspiration level 40dB and 80 watts for the performance attribute PNSR and energy usage,  5PNSR stands for Peak signal-to-noise ratio, which measures the reconstruction quality for images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' the larger the PNSR, the  better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Do Performance Aspirations Matter for Guiding Software Configuration Tuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:9  �80  �60  �40  �20  20  40  60  80  100  120  140  �PNSR (dB)  Energy (watt)  Preferred configurations  (b) PS-w with P2  �80  �60  �40  �20  20  40  60  80  100  120  140  �PNSR (dB)  Energy (watt)  Preferred configurations  (a) PS-w with P1  �80  �60  �40  �20  20  40  60  80  100  120  140  �PNSR (dB)  Energy (watt)  Preferred configurations  (c) PS-w/o  Aspiration space  Aspiration space  Aspiration space  (assuming P1 or P2)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The preferred configurations for x264 by PS-w/o and PS-w given different requirement scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Indeed, depending on the requirement scenario, the preference for performance  deviating from the aspiration level could vary even with a clear aspiration level (as we will discuss  in Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For instance, the above example may imply that one would not accept any performance  worse than 40dB or 80 watts but prefers any configurations with better PNSR and energy usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This  means that, suppose there are three configurations 𝑨 = {65𝑑𝐵, 30𝑤𝑎𝑡𝑡𝑠}, 𝑩 = {80𝑑𝐵, 25𝑤𝑎𝑡𝑡𝑠}, and  𝑪 = {35𝑑𝐵, 10𝑤𝑎𝑡𝑡𝑠}, the 𝑪, although it has the best energy usage, would be ruled out as it fails to  meet aspiration for PNSR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 𝑩 would certainly be more ideal under such a requirement scenario since  it has better results on both performance objectives than 𝑨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In a different requirement scenario, the  requirements (denoted as 𝒫2) may become “the PNSR shall be no worse than 40dB” while  “the energy usage shall be no worse than 80 watts”, which implies that one would not  accept any performance worse than 40dB or 80 watts, but equally prefer anything that goes beyond  40dB and 80 watts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Here, 𝑪 is ruled out again but 𝑨 and 𝑩 would become equally preferred as their  PNSR and energy usages are better than 40dB and 80 watts, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Of course, the given  40dB and/or 80 watts may well be unrealistic aspirations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', none of the configurations would  reach them (or at least no one can be found under the possible tuning budget).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  To make the meaning of the above clear for PS-w and PS-w/o, Figure 3 illustrates what configura-  tions are preferred when using PS-w and PS-w/o in the tuning under 𝒫1 or 𝒫2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Here, the quality of  the configurations produced would need to be evaluated with respect to the requirements and PS-w  prefers precisely what is needed therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' PS-w/o, in contrast, naturally prefers all configurations  on the Pareto front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Intuitively, we note that PS-w/o would also prefer some configurations that  are preferred by its PS-w counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For example, when comparing Figure 3a and 3c, all the  points preferred by PS-w are also preferred by PS-w/o (but not vice versus), hence they should  converge to the same satisfiability under 𝒫1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In Figure 3b and 3c, although PS-w/o prefers different  points to that of PS-w in the aspiration space, they should be able to reach the same degree of  satisfaction with respect to the requirements because all configurations within the aspiration space  are deemed equivalent when being evaluated by 𝒫2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Indeed, if both PS-w and PS-w/o can find all  their preferred points in the space, then the engineers can simply cherry-pick the fully satisfied  ones according to the given performance requirements from the final set of configurations returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Yet, the unanswered question would be: is the above assumption true and hence there would be no  difference regarding whether PS-w or PS-w/o is chosen?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  The rest of this paper provides an empirical understanding of the above confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  3  HOW REQUIREMENTS ARE HANDLED  Here we describe the process of mining, classifying, and analyzing the real-world performance  requirements with aspirations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' We use Cohen’s Kappa coefficient (𝜅) [60] to mitigate bias between  authors — the classification is often regarded as unbiased and sustainable when 𝜅 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In a  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  (a)  PS-w with Pi(c)  PS-w/o(b)  PS-w with P21:10  Chen and Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Filtering  Inclusion  criteria  Exclusion  criteria  Filtering  Selection  Identifying  requirements  Analyzing  Implication  Extracting  patterns  Mappings  14  9  5  393  107  29  386  426  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Overview of dataset analysis and literature review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  nutshell, Cohen’s Kappa coefficient is generally thought to be a more robust measure than a simple  percent agreement calculation between the raters, as it takes into account the possibility of the  agreement occurring by chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In this work, we use the coefficient in two aspects:  Measure the agreement on which implication category a requirement belongs to (we have  𝜅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='85).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Measure the agreement on which patterns that a paper assumes (we have 𝜅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='76).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1  Real-world Requirements with Aspirations  To understand what are the common real-world performance requirements with aspirations and  their implications in the industry, in Jan 2021, we mined the publicly available requirement dataset  from Zenodo (under the Empirical Software Engineering label), GitHub, and the Google Dataset  Search, using a keyword “requirement dataset”, as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The results led to 386 items,  including duplication and many irrelevant ones which can be easily identified from their titles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' As  such, we filtered the candidates down to 14, within which we followed the criteria below to extract  the most relevant ones for this study:  The dataset has clearly documented requirement statements for the software systems to be  built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  The dataset contains labeled requirements for performance objectives or there is readily  available code to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  To ensure external validity, the dataset contains performance requirements for systems from  different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  The process has resulted in nine shortlisted datasets, based on which we attempted to identify  the statements of performance requirements according to the following rules:  The performance requirement should contain a quantifiable aspiration level, such as “the  system shall perform with 1500 users at a time”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In contrast, “the system shall  be fast” is too vague to be quantified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  To ensure fairness when comparing with the PS-w/o, we eliminate the performance require-  ments that do not prefer one extreme of the objective, such as “the display shall be  refreshed every 60 seconds”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This is because such requirements prefer the performance  to reach a clear aspiration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', 60 seconds) instead of a maximum/minimum of the perfor-  mance objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Therefore, in such a case, PS-w should always be preferred, since there is  no point to use PS-w/o which naturally maximizes/minimizes the objectives while does not  take aspiration into account6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  6Note that, indeed, in some cases, the preference of this kind of requirement can be derived by inferring from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Using the same example, if the display could not be refreshed because some long-running analyses could not be terminated  within 60 seconds, then the preference would be to guarantee the ability to refresh every 60-sec or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, in our  cases, most of those requirements come from the PROMISE dataset, which has no extra information other than some  sentences describing the requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This makes it difficult for us to correctly infer the preferences implied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Hence, in the  above example, we stick with the literal meaning that one would prefer and only prefer a refresh rate of 60 seconds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' no  more and no less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  GDo Performance Aspirations Matter for Guiding Software Configuration Tuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:11  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Performance requirements with aspirations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Dataset  # Requirements  Link  Do et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' [23]  52  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='com/aqd14/ICSR-2019  PROMISE [61]  48  https://zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='org/record/268542  PURE [29]  28  https://zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='org/record/1414117  Shaukat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' [69]  13  https://zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='org/record/1209601  Dalpiaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' [21]  10  https://zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='org/record/3309669  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Identified papers with aspiration quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Venue  # Papers  Venue  # Papers  Venue  # Papers  TSE (journal)  3  JSS (journal)  6  TAAS (journal)  3  ASE (journal)  2  ESE (journal)  1  ICPE (conference)  1  ICSE (conference)  1  FSE (conference)  3  ASE (conference)  1  SEAMS (symposium)  6  ICSA (conference)  1  MODELS (conference)  1  The above has led us to rule out four datasets that contain no appropriate requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Table 1  shows details of the final five datasets used in our study (removing duplication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  Literature Search of Patterns  As from Figure 4, we also conducted a literature search according to the best practice of a systematic  literature review in software engineering [48], containing search protocol, inclusion, and exclusion  criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Our goal is to understand a single question: how are the implications of real-world perfor-  mance requirements with aspirations, which are generic to the software systems as identified from  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1, have been specifically quantified in current bi-objective software configuration tuning  work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Note that we do not intend to be comprehensive, but rather to gather representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  In Feb 2021, we conducted a full-text search over Google Scholar for papers published since  2010 from the software engineering community (we exclude the system-related papers for better  representation in the community), using a focused search string below:  “requirement” AND (“multi objective” OR “multi goal” OR “multi criteria”) AND (“per-  formance” OR “non-functional”) AND (“configurable software” OR “adaptive software”)  AND (“tuning” OR “optimization”)  This gives us 426 papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' We then filtered patents, inaccessible papers, and any non-English  documents, leading to 393 papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Next, we further extracted the papers by using the following  inclusion criteria on the title and abstract:  The paper is relevant to tuning the configuration of the software system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  The paper seeks to improve or evaluate the performance objectives of the software system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  The paper considers performance requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  The paper is peer-reviewed and is not a survey or tutorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  A paper was ruled out if it does not meet all the above criteria, which resulted in 107 papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Then, we removed papers based on the following exclusion criteria by reviewing the content:  The considered performance requirements do not have a clear aspiration level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  The paper tackle only a single performance objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  The paper does not have quantitatively experimental results with clear instructions on how  the results were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  A paper was ruled out if it met any of the above criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Finally, we obtained 29 papers, as shown  in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:12  Chen and Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  I1  93  I2  18  I3  40  (a) # requirements per implication  p1  17  p2  11  p3  7  (b) # papers per pattern  Pattern Implication  𝒑1  I1, I3  𝒑2  I1, I2, I3  𝒑3  I2, I3  (c) Mappings  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Distribution of implications, patterns and their mappings (six papers consider more than one pattern).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3  Results Analysis  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1  Number of Performance Objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' From the review, we found that 26 out of 29 (90%) of  the papers considered two performance objectives in their tuning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The remaining three  papers take into account three or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This is a clear sign that two performance objectives remain  a state-of-the-art setting for tuning software configuration, which is consistent with the finding  from the recent survey for a related field [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Therefore, in this work, we focus on bi-objective  software configuration tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  Implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' We analyzed all 151 performance requirements with aspirations from Sec-  tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1, and found three possible implications on the aspiration space for a given performance  objective:  I1: Anything in the aspiration space is equally preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This gives a clear upper  aspiration bound without other information, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', “the server will support a maximum  of 1,000 simultaneous users”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' or there is a lower aspiration bound but clear information  has been given for the cases when the performance reaches the aspiration space, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', “results  shall be returned in under 15 seconds”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  I2: Anything not in the aspiration space is equally non-preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For example, “the  system shall allow for a minimum of 6 users at the same time”, in which case  there is only information for a clear lower aspiration bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  I3: No information is available with respect to the aspiration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This often refers  to the requirements where there is a clear aspiration level, but no indication about whether  it is an upper or lower aspiration bound while any other information is unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For  example “the system shall cater to 10 simultaneous users”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  The distribution of the implications can be found in Figure 5a and we achieve a Kappa coefficient  𝜅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='85 for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3  Patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Next, with the above implications in mind, we seek to understand how they are  quantified within the 29 papers identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This led to three state-of-the-art patterns on the functions  to quantify requirements with aspiration level (assuming the lower bound is optimum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Suppose  that 𝛼 and 𝛽 denote the lower and upper bound of the performance objective, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 𝑑 is  the aspiration level, the patterns, and their quantification have been shown in Figure 6 and are  explained below:  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Do Performance Aspirations Matter for Guiding Software Configuration Tuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:13  ↵  d  �  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='5  1  Raw measurement  (c) p2(x) =  (  0  x > d  1  x \uf8ff d  ↵  d  �  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='5  1  Raw measurement  (b) p1(x) =  ( ��x  ��d  x > d  1  x \uf8ff d  ↵  d  �  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='5  1  Raw measurement  (d) p3(x) =  (  0  x > d  d�x  d�↵  x \uf8ff d  ↵  �  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='5  1  Raw measurement  Satisficing value  (a) p0(x) = ��x  ��↵  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Requirement patterns with (and without) aspiration from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 𝛼 and 𝛽 denote the lower and  upper bound of the performance objective, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 𝑑 is the aspiration level and the aspiration space has  been shaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  𝒑1: The performance in the aspiration space is equally good or otherwise there is a certain  degree of tolerance (Figure 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The function can be formulated as:  𝒑1(𝑥) =  � 𝛽−𝑥  𝛽−𝑑  𝑥 > 𝑑  1  𝑥 ≤ 𝑑  (3)  𝒑2: The performance in the aspiration space is equally good while anything outside the space  is unacceptable (Figure 6c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' such that:  𝒑2(𝑥) =  �  0  𝑥 > 𝑑  1  𝑥 ≤ 𝑑  (4)  𝒑3: Preferring the optimal performance while anything outside the aspiration space is unac-  ceptable (Figure 6d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' which is defined as:  𝒑3(𝑥) =  �  0  𝑥 > 𝑑  𝑑−𝑥  𝑑−𝛼  𝑥 ≤ 𝑑  (5)  Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' we can also formalize the requirement with no clear aspiration level involved (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=',  “the latency shall be small”), denoted as 𝒑0, which is illustrated in Figure 6a and can be  formulated as follow:  𝒑0(𝑥) = 𝛽 − 𝑥  𝛽 − 𝛼  (6)  The distribution of the patterns has been shown in Figure 5b where we have 𝜅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='76, which is  sustainable [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Through normalization in those patterns, the raw measurement of a performance objective is  transformed into the satisficing degree with respect to a given aspiration space (if any), ranging  between 0 and 1 where the latter means fully satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' As such, the transformation depends on the  assumption of satisficing over measurements included or excluded by the aspiration space, which  distinguishes the patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' At this point, we can immediately see the mappings between the patterns  and the extracted implications from the real-world dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Such mappings have been illustrated in  Figure 5c, from which we see that each pattern, except 𝒑0, can fit with at least two implications  from the real-world requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For example, 𝒑1 can fit with I1 and I3, because the former prefers  anything within the aspiration space and specifies nothing on the other extreme, while the latter  has no information at all and thus one needs to rely on an assumption when quantifying I3 to  guide the search, meaning that it has the possibility to fit with all the three patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:14  Chen and Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  �80  �60  �40  �20  20  40  60  80  100  120  140  �PNSR (dB) with p3  Energy (watt) with p1  Original Space  Aspiration space  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='8  PNSR p3  Energy p1  Transformed Space  For both performance objectives, the smaller the better  For both performance objectives, the larger the better  Best configuration under the  requirements and aspirations  Best configuration under the  requirements and aspirations  Transformed Space  Original Space  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Example of using the patterns for evaluating the goodness of configurations and guiding PS-w in the  transformed space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  From the above, it is confirmed that there exist patterns from current work which can reflect the  implication of real-world performance requirements and their aspirations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' We, therefore, will seek  to examine all of them in our empirical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  Respecting Requirements and Aspiration in Software Configuration Tuning  While the above requirement patterns are the key to evaluating the “better” or “worse” in the set of  configurations produced by any Pareto optimizer and optimization model, they directly influence  the behavior of PS-w (they correspond to the 𝑝𝑛 in Equation (1)) but not that of the PS-w/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Most importantly, those patterns allow us to precisely quantify what is the best configuration(s)  amongst the configurations produced by those two optimization models given a set of requirements  and aspirations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Figure 7 shows an example of evaluating the configurations (and guiding PS-w)  in a transformed space when taking the requirements and aspirations into account, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', energy  usage with 𝒑1 and aspiration of 80 watts while PNSR with 𝒑3 and aspiration of 40dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Here, we  certainly prefer the points within the aspiration space in contrast to those outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, for  those points within the aspiration space, we only prefer those with better PNSR while the energy  usage is deemed as equivalent (due to the implication of 𝒑1 and 𝒑3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  The above is difficult to assess and quantify in the original space (Figure 7 left), since naturally  the points that are non-dominated by each other (in the sense of the original objective values)  are considered as equivalent when the requirements and aspiration are not involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Therefore,  the actual most preferred point (arrow highlighted) is not considered the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In contrast, the  evaluation becomes immediately obvious on what is the best point in the transformed space, where  the energy and PNSR are converted by the equations for 𝒑1 and 𝒑3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Now, clearly, the  most preferred point is the only non-dominated point therein (Figure 7 right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  4  EMPIRICAL STUDY DESIGN  As shown in Figure 8, our methodology consists of the following steps:  Step 1: Assume that a requirement scenario has been negotiated by the software engineers  and stakeholders, we quantify the requirements such that they are ready for the Pareto  optimizers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', in the forms of a combination of patterns from Section 3 and their  aspiration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In particular, to form a requirement scenario, the given combination of  the patterns is denoted as a two-dimensional vector 𝒫, such that there is at least one  that comes with a clear aspiration level, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', 𝒫 = {𝒑0, 𝒑3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In this work, we examine  all possible combinations of the patterns (including 𝒑0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Under each combination, we  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Do Performance Aspirations Matter for Guiding Software Configuration Tuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:15  3 Pareto optimizers  8 software systems  and environments  evaluate quality of  configuration set  Pareto search  with  aspiration  Pareto search  without  aspiration  �5  �4  �3  0  2  4  log10 Throughput�1  log10 Latency  u  r  l  c  15 combinations of patterns  and 3 (or 4) aspiration space  Next combination of patterns/aspirations  Scenario  Identification  Configuration  Tuning  Quality  Evaluation  1  2  2  4  5  3  ↵  d  �  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='5  1  Raw measurement  p3 )  p(x) =  (  0  x > d  x�↵  d�↵  x \uf8ff d  (d)  ↵  d  �  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='5  1  Raw measurement  Satisficing value  p2 )  p(x) =  (  0  x > d  1  x \uf8ff d  (c)  ↵  d  �  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='5  1  p1 )  p(x) =  (  x�d  ��d  x > d  1  x \uf8ff d  (b)  ↵  �  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='5  1  Satisficing value  p0 ) p(x) = x�↵  ��↵  (a)  100 repeats  with  with  with  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Overview of the empirical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  also consider different aspiration spaces for our RQs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' this will be further elaborated in  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Step 2: Run both PS-w/o and PS-w on different software systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Particularly, when formulating  the performance objectives, PS-w/o is steered by the raw measurements only7 while  PS-w is designed to be guided by the given vector of patterns 𝒫 as the new objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' To  ensure fairness, both optimization models are examined under the same optimizer and  we consider three representative optimizers in this work, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', NSGA-II [22], IBEA [85],  and MOEA/D [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Step 3: Measure the system as the search proceeds until the tuning budget has been exhausted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  repeat 100 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Step 4: Evaluate the set of configurations thereafter using 𝒫 as part of the Quality Evaluation  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Step 5: Go back to Step 1 if there are more combinations of patterns and aspirations to examine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  It is worth noting that, although the patterns from Section 3 are for single performance objective,  they can be arbitrarily combined for the bi-objective software configuration tuning in the Scenario  Identification phase of Step 1 [8, 9, 33, 59], as illustrated in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  In the Configuration Tuning phase (Step 2 and 3), the patterns require normalization using  the lower and/or upper bound (except for 𝒑0 and 𝒑2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, since these are often unknown,  we adopt a dynamic method wherein the raw measurements are normalized using the maximal  and minimal values found so far as the tuning proceeds, which is common in SBSE for software  configuration tuning [7, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' We record the raw measurements of each configuration throughout  the tuning to efficiently utilize the tuning budget (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  To mitigate stochastic bias, we repeat each experiment 100 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The study is conducted on a  cluster of machines each with Intel i5 six cores CPU at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='9GHz and 8GB memory, running numerous  experiments in parallel over the course of five months (24 × 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1  Subject Software Systems  We conduct our study on a set of real-world highly configurable software systems and environments  that have been widely studied in existing work [17, 44, 45, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' These are selected according to the  criteria below:  (1) To ensure that the search landscape is not too trivial to be explored, the system should contain  a mix of binary and enumerative configuration options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  7This is effectively identical to using 𝒑0 for all performance objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  PPP1:16  Chen and Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The considered requirement scenarios (in terms of the combination of the patterns identified from  Section 3) and their example interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The interpretations are based on the assumption that the  performance objectives are {𝑙𝑎𝑡𝑒𝑛𝑐𝑦,𝑡ℎ𝑟𝑜𝑢𝑔ℎ𝑝𝑢𝑡} with possible aspiration levels 𝑑1 and 𝑑2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Possible 𝒫  Example Interpretation  {𝒑0, 𝒑1}  Prefer better latency and throughput better than 𝑑2, but any configurations better than 𝑑2 are equally  preferred;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' willing to accept throughput worse than 𝑑2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  {𝒑1, 𝒑0}  Prefer better throughput and latency better than 𝑑1, but any configurations better than 𝑑1 are equally  preferred;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' willing to accept latency worse than 𝑑1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  {𝒑0, 𝒑2}  Prefer better latency and throughput better than 𝑑2, but any configurations better than 𝑑2 are equally  preferred;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' do not accept throughput worse than 𝑑2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  {𝒑2, 𝒑0}  Prefer better throughput and latency better than 𝑑1, but any configurations better than 𝑑1 are equally  preferred;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' do not accept latency worse than 𝑑1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  {𝒑0, 𝒑3}  Prefer better latency and throughput;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' do not accept throughput worse than 𝑑2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  {𝒑3, 𝒑0}  Prefer better latency and throughput;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' do not accept latency worse than 𝑑1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  {𝒑1, 𝒑1}  Prefer latency better than 𝑑1 and throughput better than 𝑑2, but any configurations better than 𝑑1 and  𝑑2 are equally preferred;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' willing to accept latency and throughput worse than 𝑑1 and 𝑑2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  {𝒑2, 𝒑2}  Prefer latency better than 𝑑1 and throughput better than 𝑑2, but any configurations better than 𝑑1 and  𝑑2 are equally preferred;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' do not accept latency and throughput worse than 𝑑1 and 𝑑2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  {𝒑3, 𝒑3}  Prefer better latency and throughput;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' do not accept latency and throughput worse than 𝑑1 and 𝑑2,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  {𝒑1, 𝒑2}  Prefer latency better than 𝑑1 and throughput better than 𝑑2, but any configurations better than 𝑑1 and  𝑑2 are equally preferred;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' willing to accept latency worse than 𝑑1 but do not accept throughput worse  than 𝑑2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  {𝒑2, 𝒑1}  Prefer latency better than 𝑑1 and throughput better than 𝑑2, but any configurations better than 𝑑1 and  𝑑2 are equally preferred;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' willing to accept throughput worse than 𝑑2 but do not accept latency worse  than 𝑑1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  {𝒑1, 𝒑3}  Prefer better throughput and latency better than 𝑑1, but any configurations better than 𝑑1 are equally  preferred;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' willing to accept latency worse than 𝑑1 but do not accept throughput worse than 𝑑2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  {𝒑3, 𝒑1}  Prefer better latency and throughput better than 𝑑2, but any configurations better than 𝑑2 are equally  preferred;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' willing to accept throughput worse than 𝑑2 but do not accept latency worse than 𝑑1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  {𝒑2, 𝒑3}  Prefer better throughput and latency better than 𝑑1, but any configurations better than 𝑑1 are equally  preferred;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' do not accept latency worse than 𝑑1 nor throughput worse than 𝑑2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  {𝒑3, 𝒑2}  Prefer better latency and throughput better than 𝑑2, but any configurations better than 𝑑2 are equally  preferred;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' do not accept latency worse than 𝑑1 nor throughput worse than 𝑑2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  (2) A full exploration of the search space is infeasible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', it cannot be done within 24 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  (3) There are clear instructions on how to set up the benchmark under which the system will be  measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  (4) If the same system of an environment has been used with a different set of configuration  options, choose those with relatively higher complexity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', larger search space and more  configuration options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For example, Storm can be tuned under different workload bench-  marks, and we choose WordCount and RollingSort as the two that satisfy the above  criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  We firstly eliminated LLVM from [62], as it violates Criterion (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Similarly, sort-256 and noc-  CM-log is also ruled out due to their rather small search space which can be exhaustively explored  in 24 hours, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Criterion (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' We cannot consider the system SaC as there is no clear instruction  on under what benchmark it can be profiled, which violates Criterion (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' We also noticed that  Storm and Keras (with DNN or LSTM) have been much more commonly used than others, but  with different configuration options and environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Therefore, according to Criterion (4), we  use the settings that lead to a much larger search space and more options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' As shown in Table 4, the  selected software systems come from diverse domains, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', video encoding, stream processing, and  deep/machine learning, while having different performance objectives, scales, and search spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Do Performance Aspirations Matter for Guiding Software Configuration Tuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:17  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Configurable software systems studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' We run all software systems under their standard benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Storm and Keras (with DNN) use two benchmarks and three dataset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Software  Domain  Performance Objectives  # Options  Search Space  Used By  Trimesh  Mesh solver  Latency and # Iteration  13  239,260  [17, 62]  x264  Video encoding  PSNR and Energy Usage  17  53,662  [17, 62]  Storm/WC  Stream processing  Latency and Throughput  6  2,880  [17, 44, 45, 62]  Storm/RS  Stream processing  Latency and Throughput  6  3,839  [17, 44, 45, 62]  Keras/Adiac  Deep learning  AUC and Inference Time  13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='99×1013  [45]  Keras/DSR  Deep learning  AUC and Inference Time  13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='32×1013  [17, 45]  Keras/SA  Deep learning  AUC and Inference Time  13  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='66×1013  [45]  XGBoost  Machine learning  Accuracy and Training Time  13  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='88×1010  [45]  Their measurements are also expensive8, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', XGBoost needs 2,807 hours to explore less than 1%  of its search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  We keep the same performance objectives, configuration options, and their ranges as studied in  the prior work that made use of them, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', [17, 44, 45, 62], since those have been shown to be the  key ones for the software systems under the related environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' As a result, although the software  systems are the same, the actual search spaces are different, such as Storm/WC and Storm/RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In  particular, following what has been used in previous work, the environment/workload we consider  are:  Trimesh: we use the Shapenet dataset that contains 51,300 unique 3D models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In this work,  we randomly sample 100 models as the standard benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  x264: for this, the benchmark used is a standard video of 1GB size, which was chosen  randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Storm/WC: we use the WordCount as the benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This is a typical simple streaming  example where Storm is used to keep track of the words and their counts streaming in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  WordCount generates a CPU-intensive workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Storm/RS: similar to Storm/WC, here we use the RollingSort as the benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Unlike  WordCount, RollingSort generates a memory intensive workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Keras/Adiac: we use the Deep Neural Network (DNN) from the Keras software and run it  on the Adiac dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Generally, the dataset contains a task of automatic identification of  diatoms (unicellular algae) among 31 classes with a training and testing size of 390 and 391,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Keras/DSR: we use the DNN from the Keras software and run it on the DiatomSizeReduc-  tion dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The dataset concerns the prediction of four types of diatoms with a training  and testing size of 16 and 306, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Keras/SA: we use the DNN from the Keras software and run it on the ShapesAll dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Generally, the dataset aims to test contour/image and skeleton-based descriptors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' there are  60 classes with a training and testing size of 600 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  XGBoost: we use the Covertype dataset that contains 54 forest cover type from cartographic  variables only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The size of the dataset is 581,012 and we follow a 70%-30% training and testing  split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Indeed, the analyzed dataset and literature in Section 3 may not specifically target the software  systems considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, the extracted implication and patterns are rather generic  such that they can be applied to different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Further, some widely studied performance objectives  (from both the dataset and literature) are overwhelmingly applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For example, latency- and  8Each measurement consists of 5 repeated samples and the median value is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:18  Chen and Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Aspiration levels and spaces for the configurable software systems studied (used for all combinations  of patterns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 𝑙, 𝑟, 𝑐, and 𝑢 denote left-shifted, right-shifted, centered, and unrealistic aspirations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Software  Performance Objectives  𝑙  𝑟  𝑐  𝑢  Trimesh  {Latency (s), # Iterations}  {81, 4}  {461, 15}  {135, 7}  {37, 501}  x264  {PSNR (dB), Energy Usage (W)}  {50, 3680}  {37, 462}  {46, 1260}  {100, 34}  Storm/WC  {Throughput (msgs/m), Latency (ms)}  {16473, 15677}  {994, 5}  {8982, 101}  {34740, 3}  Storm/RS  {Throughput (msgs/m), Latency (ms)} {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3 × 105, 7819}  {3006, 5}  {3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='7 × 104, 126} {2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3 × 105, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='9}  Keras/Adiac  {AUC, Inference Time (ms)}  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='030, 44}  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='017, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05}  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='028, 3}  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='292, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='03}  Keras/DSR  {AUC, Inference Time (ms)}  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='307, 123}  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='107, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='12}  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='300, 25}  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='581, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='031}  Keras/SA  {AUC, Inference Time (ms)}  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='167, 21}  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='157, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='07}  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='160, 6}  {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='325, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='04}  XGBoost  {Accuracy (%), Training Time (s)}  {80, 42}  {54, 3}  {72, 8}  {92, 1}  throughput-related requirements (with different aspiration levels) are prevalent for a wide range of  software [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  Aspiration Space  To improve external validity, we consider aspiration levels that draw two types of aspiration space  under two performance objectives: realistic and unrealistic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' To that end, for each software  system, we run all the Pareto optimizers for three hours each to obtain a landscape that contains an  approximated Pareto front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' We do so by ensuring that the obtained front is reasonably converged,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', increasing the budget only marginally changes the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' We then set the aspiration space  based on such a front as summarized in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1  Realistic Aspiration Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For software configuration tuning with two performance objec-  tives, we say an aspiration space is realistic if there is at least one configuration that can reach the  aspiration levels of both performance objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Using Storm/RS as an example in Figure 9, for the realistic ones under each combination of  patterns, we set three aspiration space based on their positions in the objective space: left-shifted (𝑙),  right-shifted (𝑟) and centered (𝑐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In particular, 𝑙 is defined as using the value of the 20𝑡ℎ percentile  for throughput and the value of the 80𝑡ℎ percentile for latency as their corresponding aspiration  levels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' similarly,𝑟 uses the value of the 20𝑡ℎ percentile for latency and the value of the 80𝑡ℎ percentile  for throughput;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' finally, 𝑐 uses the values of the 50𝑡ℎ percentile for both performance objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Clearly, despite covering diverse regions in the overall space of performance objectives, all those  spaces contain at least one point (configuration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Note that the aspiration space is applicable to any  combination of patterns with and without 𝒑0 (in Figure 9a and Figure 9b respectively), as long as  there is a clear aspiration level for at least one performance objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  Unrealistic Aspiration Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Since the aspiration level/space is negotiated beforehand, it  may be unrealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In this work, we refer to an unrealistic aspiration space as the situation wherein  the aspiration levels of two performance objectives can be at most reached one at a time, but not  both simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For example, in the case of two performance objectives from Figure 9a, 𝑢 is  an unrealistic aspiration space such that the level is achievable for either of the two objectives  individually (as indicated by the dashed lines), but not for both, as there is no point (configuration)  residing in the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' As a result, it is not applicable when only one performance objective contains  clear aspiration, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', in Figure 9b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' To define such a space, we set the value of 5𝑡ℎ percentile of both  performance objectives as the corresponding aspiration levels, which we have found as sufficient  to create an unrealistic aspiration space for each system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Do Performance Aspirations Matter for Guiding Software Configuration Tuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:19  �5  �4  �3  0  2  4  log10 Throughput�1 (p1)  log10 Latency (p0)  r  l  c  (b)  �5  �4  �3  0  2  4  log10 Throughput�1 (p1)  log10 Latency (p1)  u  r  l  c  (a)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Distant aspiration space (shaded by different colors) under different combinations of patterns for  Storm/RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Population size and measurement tuning budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Software  Population Size  # Measurements  Software  Population Size  # Measurements  Trimesh  10  500  x264  50  1,500  Storm/WC  50  500  Storm/RS  30  700  Keras/Adiac  50  700  Keras/DSR  60  500  Keras/SA  60  500  XGBoost  30  300  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3  Tuning Settings  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1  Pareto Optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' We consider three Pareto optimizers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', NSGA-II [22], IBEA [85], and  MOEA/D [82], because:  They have been widely used for software configuration tuning in prior work [8, 9, 16, 33, 59,  70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  They are the representatives of three fundamentally different frameworks for Pareto search [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  They can, but do not have to, rely on a surrogate model(s) [16], which greatly reduces the  noise in our empirical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  All the above optimizers are adopted for both PS-w and PS-w/o based on the implementations in  jMetal [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  Tuning Budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In this work, we set a budget of one hour for each run as commonly used for  expensive SBSE problems [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, directly relying on the time as a termination criterion can  suffer severe interference during the tuning as numerous experiments need to be run in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  To prevent this, for each software system, we did the following to convert the one-hour tuning  budget into the number of unique measurements:  (1) incrementally (100 each step) measuring distinct configurations on a dedicated machine using  random sampling until the one-hour time budget is exhausted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  (2) repeating the above 5 times and collect the number of measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  (3) the median of the 5 repeats serves as the key termination criterion of the tuning thereafter  (in Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Note that in each run of the tuning, we cached the measurement of every distinct configuration  for direct reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Hence, only the distinct configurations would consume the budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3  Parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For the three optimizers in all cases, we apply the binary tournament, boundary  mutation, and uniformed crossover, as used in prior work [16, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The mutation and crossover  rates are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='9, respectively, which also follows the most common setting for software  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:20  Chen and Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  configuration tuning [16, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Other specific settings for IBEA and MOEA/D are kept as default  values, which have been shown to be effective [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  For each system, we pragmatically set the population size via:  (1) examining different sizes in pilot runs under the budget in Table 6, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', {10, 20, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', 100}, over  all optimizers, combinations of patterns and aspiration (on both PS-w and PS-w/o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  (2) recording the average change rate of population over the last 10% generations using 𝑔 =  1  𝑘 × �𝑘  𝑖=0  𝑐𝑖  𝑠 , where 𝑘 is the number of the last 10% generations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 𝑐𝑖 denotes the number of  different configurations in the ith generation compared with those in the i-1th generation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 𝑠  is the population size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  (3) the largest population size where 𝑔 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1 across all conditions (or 10, if no size satisfies the  above constraints) will be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  The results are also shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In this way, we seek to reach a balance between convergence  (smaller population change) and diversity (larger population size) under the given tuning budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  That is, increasing the budget will unlikely change the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This has been practiced in [17, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  Analysis and Comparison  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1  Metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' To make a comparison and determine which optimization model is better in this  work, we need to measure the “best” with two conditions in mind:  Condition 1: The metric needs to be able to comprehensively compare the different sets of  configurations as produced by the Pareto optimizers, covering diverse quality aspects, such  as convergence and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Condition 2: The metric should be able to reflect the given requirement scenarios, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', taking  the given patterns identified from Section 3 into account when conducting the comparisons  and evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  To that end, we use Hypervolume (HV) [84, 86] as the basic metric to assess the quality of the  configuration set produced in each run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In a nutshell, HV measures the volume between all points  of a configuration set and a reference point (usually a nadir point);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' the larger the volume, the better  convergence, and diversity that the set achieves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' HV is chosen in this work because:  HV is a comprehensive metric that covers all quality aspects of a configuration set, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=',  convergence, uniformity, spread, and cardinality [55, 57], which meets Condition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  HV also does not require a reference Pareto front and is Pareto compliant9, which fits our  case as the true Pareto front is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  By following the guidelines proposed by [54, 55], we landed on HV as the appropriate metric  for our SBSE problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Since we are interested in a requirement scenario that has a specific combination of patterns and  aspiration space (𝒫) in the objective space, the original HV, which always favors the configurations  that are close to the entire Pareto front, is no longer suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Therefore, we need to transfer these  preferences into the HV following the guidance by Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' [55] and leveraging the patterns and  quantification from Section 3 (for satisfying Condition 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Using the same example from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2,  as shown in Figure 10, the requirement scenario is that: the stakeholders prefer better PNSR and  energy usage better than 80 watts, but any configurations better than 80 watts are equally preferred;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  willing to accept energy usage worse than 80 watts but do not accept PNSR worse than 40dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This  means for any points in the aspiration space, the ones with better PNSR would be preferred more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  9Generally speaking, a quality indicator being Pareto compliant means that its evaluation result does not conflict with  the Pareto dominance relation between two solution sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' More strictly, if a solution set 𝐴 is better [87] than 𝐵 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', for  any solution in 𝐵, there exists one solution in 𝐴 that covers (dominates or is equivalent to) it, and there exists at least one  solution in 𝐴 that is not covered by any solution in 𝐵), then 𝐴 is always evaluated better than 𝐵 by the indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Do Performance Aspirations Matter for Guiding Software Configuration Tuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:21  �80  �60  �40  �20  20  40  60  80  100  120  140  160  PNSR p3  Energy p1  Original Space with HV  �80  �60  �40  �20  20  40  60  80  100  120  140  �PNSR (dB) with p3  Energy (watt) with p1  Original Space  Aspiration space  Reference  Hypervolume area  Original Space with HV  Original Space  Best point A  Best point A  (a) Evaluating with HV in the original space  �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='8  PNSR p3  Energy p1  Transformed Space with HV  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='8  PNSR p3  Energy p1  Transformed Space  Reference  Hypervolume area  Transformed Space with HV  Transformed Space  Best point A  Best point A  (b) Evaluating with HV in the transformed space  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Evaluation of HV with and without requirements/aspirations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Therefore, point 𝑨 is the best based on the requirements and should contribute the most to the  chosen metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, directly applying HV would make some configurations, which are less  preferred to the requirements, contribute significantly to the HV value (Figure 10a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This would  misleadingly evaluate some sets that have many non-preferred points to have a very good HV value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  In contrast, when transferring the information of patterns before using HV (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', in the transformed  space), the above requirements and aspirations can be better complied with, as 𝑨 is certainly the  one that contributes the most and other non-preferred points tend to have no or little contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  (Figure 10b)  To that end, we extend the HV in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Suppose that there are 𝑚 performance objectives  (we have 𝑚 = 2 in this work) and 𝒜 is a produced configuration set wherein the vector of a  configuration’s raw measurements is 𝒙𝒊 = {𝑥1,𝑥2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='𝑥𝑚}, we calculate HV based on the converted  satisficing value of 𝒙𝒊 according to the given 𝒫.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' We call it aspiration-aware HV (dubbed A-HV),  which is formulated as:  𝐴-𝐻𝑉 (𝒜) = 𝜆(  �  𝒙𝒊 ∈𝒜  {𝒗|𝒫(𝒙𝒊) ≺ 𝒗 ≺ 𝒓})  (7)  where 𝜆 is the Lebesgue measure that quantifies the volume [86] as used in the original HV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 𝒓 is  the reference nadir point, which is often taken as the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1 times of the range of the nondominated  set [55], hence in our case, this would be {−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1} as 𝒫(𝒙𝒊) converts the outputs to [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Like  HV, a higher A-HV value is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' To ensure fair comparison with A-HV, we use the minimum  and/or maximum values (of each performance objective) from all experiments for the posterior  normalization in the patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  To enable more intuitive exposition, we report on the % gain of the A-HV for considering  requirements and aspirations in the tuning, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', PS-w, over that for PS-w/o on each run, which is  defined as:  % Gain = 𝑥𝑖 − 𝑦𝑖  𝑦𝑖  × 100  (8)  whereby 𝑥𝑖 and 𝑦𝑖 are the A-HV value at the 𝑖th run for PS-w and PS-w/o, respectively, in their  sorted lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Clearly, a positive % gain indicates that the aspirations are helpful (PS-w is better) while  a negative value implies they are harmful (PS-w/o is better);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' zero gain means identical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  Statistical Validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' We use the standard methods to interpret the significance of the results  over 100 runs in each case [1, 47]:  Wilcoxon test: We apply the Wilcoxon test [76] with 𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05 [1] to investigate the statistical  significance of the A-HV comparisons over all 100 runs, as it is a non-parametric statistical  test that makes little assumption about the data distribution and has been recommended in  software engineering research for pair-wise comparisons [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:22  Chen and Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  90  17  20  48  Large ˆA12 (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05)  Medium ˆA12 (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05)  Small ˆA12 (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05)  Trivial ˆA12 or p � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05  (c) Distribution on the 175 cases that PS-w/o win  454  66  52  85  Large ˆA12 (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05)  Medium ˆA12 (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05)  Small ˆA12 (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05)  Trivial ˆA12 or p � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05  (b) Distribution on the 657 cases that PS-w win  657  175  248  PS-w win  PS-w/o win  Tie  (a) Distribution on all 1,080 cases  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Summary of the wins by PS-w and PS-w/o together with their detailed statistics validation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ˆA12 effect size: To ensure that a 𝑝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05 is not caused by a trivial amount of the samples, we  apply ˆ𝐴12 [72] to measure the effect size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In this work, ˆ𝐴12 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='5 denotes PS-w wins wherein  it has better A-HV for more than 50% of the runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' ˆ𝐴12 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6 or ˆ𝐴12 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4 indicate a non-trivial  effect size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Since there are 100 runs (instead of the commonly used 30), we use a stricter  interpretation by which 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6 ≤ ˆ𝐴12 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='7 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3 < ˆ𝐴12 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='7 ≤ ˆ𝐴12 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2 < ˆ𝐴12 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3),  and ˆ𝐴12 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='8 ( ˆ𝐴12 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2) indicate small, medium, and large effect, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  5  RESULTS AND FINDINGS  In this section, we present the results of our empirical study and address the research questions  posed in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1  RQ1: Which is Better under Requirements with Realistic Aspirations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1  Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' To answer RQ1, we compare PS-w and PS-w/o across 15 combinations of patterns,  three realistic aspiration spaces (𝑙, 𝑟, and 𝑐), three optimizers and eight subject systems, leading to  15 × 3 × 3 × 8 = 1, 080 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Since we are interested in a pair-wise comparison of the A-HV under  each case, the Wilcoxon test and ˆ𝐴12 are used to verify the statistical significance over 100 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' As an overview, Figure 11 shows a summary of the ˆ𝐴12 outcomes across the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Clearly, we see that PS-w performs overwhelmingly better than its PS-w/o counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In particular,  PS-w wins for 61% (657/1080) of the cases and loses for 16% (175/1080), while there is a 23% (248/1080)  tie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In other words, PS-w is better or similar for 84% (905/1080) of the cases in contrast to the 39%  (423/1080) when using PS-w/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Statistically, PS-w wins 572 cases with ˆ𝐴12 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6 and 𝑝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05, while  there are only 127 significant cases when it loses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  To provide a more comprehensive view on the different systems and requirement scenarios, in  Table 7, we see that PS-w performs considerably better in general, as it achieves reasonably well  positive gains on the majority of the cases (up to 145% improvement on A-HV in average) with  generally more statistically significance wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' It is worth noting that we observed particularly  high gains on PS-w under Storm (Table 7c and Table 7d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This is attributed to the highly diverse  performance between configurations for the system, as what has been reported in prior work [17,  44, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  It is exciting to see that the superiority of PS-w is consistent across the given requirement  patterns — a clear sign to confirm that the requirements can offer important guidance to steer the  tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, as expected, when the scenario requires 𝒑1 or 𝒑0 only, PS-w and PS-w/o perform  mostly identical (or very similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' As for the very few cases where PS-w is inferior to PS-w/o, the  results can be the cause of some accidentally encountered local optima issues, which we will discuss  in greater detail in what follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Therefore, we say:  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Distribution on all 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='080 casesC  Distribution on the 175 cases that PS-w/o winDo Performance Aspirations Matter for Guiding Software Configuration Tuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:23  Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Comparing PS-w and PS-w/o under realistic requirements and aspirations over 100 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  sc  and  sc  denote the average (Avg) and standard error (SE) of the positive and negative % gain, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  sc  means zero gain overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The column “PS-w” and “PS-w/o” show the number of cases that the corresponding  optimization model wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 9 (6) means one wins on 9 cases within which 6 shows statistical significance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=',  ˆ𝐴12 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6 (or ˆ𝐴12 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4) and 𝑝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05 (each combination of requirement patterns has 9 cases in total, as there  are 3 aspiration space and 3 optimizers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The blue cells denote PS-w wins more while red cells mean it loses  more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  PS-w PS-w/o Tie  Avg (SE) of A-HV Gain  {𝒑0,𝒑1}  8 (7)  1 (1)  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3%)  y  i  {𝒑1,𝒑0}  7 (7)  2 (2)  0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6% (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4%)  y  i  {𝒑0,𝒑2}  5 (5)  4 (4)  0  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4% (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4%)  y  i  {𝒑2,𝒑0}  7 (7)  2 (2)  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='4%)  y  i  {𝒑0,𝒑3}  6 (5)  3 (2)  0  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='2%)  y  i  {𝒑3,𝒑0}  8 (8)  1 (1)  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='6%)  y  i  {𝒑1,𝒑1}  3 (0)  2 (1)  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='4%)  y  i  {𝒑2,𝒑2}  1 (0)  5 (1)  3  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='1%)  y  i  {𝒑3,𝒑3}  6 (5)  3 (3)  0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='3%)  y  i  {𝒑2,𝒑3}  6 (6)  3 (3)  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='9%)  y  i  {𝒑3,𝒑2}  4 (4)  5 (3)  0  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='0%)  y  i  {𝒑3,𝒑2}  8 (4)  0 (0)  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='9% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2%)  y  i  (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Keras/SA  (h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' XGBoost  RQ1: Given realistic aspiration space, PS-w is 84% of the time similar or better than PS-w/o with  considerable improvements, suggesting that the requirements and aspirations are beneficial for  guiding the tuning in such a situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Yet, the benefits can vary depending on some particular  combinations of the patterns, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', it tends to be blurred when only the 𝒑1 and/or 𝒑0 is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3  Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' To understand what causes the results under realistic aspiration space, in Fig-  ure 12 we show a common example from x264, where all PSNR values better than its aspiration are  equally preferred and no worse results are acceptable (𝒑2), while the energy usage is desired to be  as low as possible, even if its aspiration has already been exceeded (𝒑3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Figure 12a is a superior case  of PS-w, in which we see that the aspirations drive the tuning to focus more on the local regions  within the objective space, hence the points of PS-w is much less spread than those of PS-w/o (as  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:24  Chen and Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='75  �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='67  �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='8  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6  log10 PSNR�1 (p2)  log10 Energy (p3)  PS-w  PS-w/o  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='8  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='8  1  PSNR (p2)  Energy (p3)  PS-w  PS-w/o  Reference  A-HV area  Original Space  (a) x264, PS-w wins  �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='75  �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='67  �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='8  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6  log10 PSNR�1 (p2)  log10 Energy (p3)  PS-w  PS-w/o  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='8  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='8  1  PSNR (p2)  Energy (p3)  PS-w  PS-w/o  Reference  A-HV area  Original Space  (b) x264, PS-w/o wins  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Example runs of the final configuration sets (with NSGA-II) under realistic aspiration space indicated  by the shaded areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  see in the Original Space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Such a “focused pressure” is mostly sufficient to help find some more  preferred regions by the scenario under a fixed tuning budget, hence the PS-w has better A-HV  than PS-w/o (larger volume, as seen in the A-HV area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  However, PS-w is not always beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' As reported by Chen and Li [17], Nair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' [62], and the  others [38, 44], configurable software systems are known to exhibit a high degree of sparsity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=',  the close configurations can also have radically different performance, thus only a small amount of  them may achieve certain performance range, causing rather sparse objective points (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  For example, switching the wait_strategy in Storm can have dramatic impacts on the performance,  despite that it is merely a single change on an option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This is because the wait_strategy conserves  CPU usage depending on whether the wait is a fixed interval or is progressively determined based  on the length of the queue at runtime, therefore it has a large impact on latency and throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  However, in the tuning, it is represented as a single configuration option with a value chosen  from {0, 1, 2, 3} where each value represents a distinct wait strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The presence of high sparsity  exacerbates the problem of local optima traps — some undesired regions that are difficult to escape  from by an optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Occasionally, searching focally under high sparsity does cause PS-w to  overemphasize the less desired local optima, which harms the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This is why there are some  cases where the PS-w show no advantage, as illustrated in Figure 12b where the points of PS-w are  too densely populated compared with those of the PS-w/o (as see in the Original Space), causing  the volume covered by PS-w is smaller than that of PS-w/o and smaller A-HV (as see in the A-HV  area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  It is interesting to observe that under certain combinations of patterns, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', with 𝒑0 and/or 𝒑1 only,  both optimization models perform similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This makes sense, as in those cases the requirements  would create similar discriminative power between configurations to that of PS-w/o (which is  essentially guided by {𝒑0, 𝒑0}), generating configurations that are equally preferred under the  given needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  RQ2: How do Different Aspirations Influence the Comparisons?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1  Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' To understand RQ2, we follow the procedure used for RQ1, but with particular  focus on the results with respect to the three aspiration spaces used (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', 𝑙, 𝑐, and 𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Figure 13 plots the sensitivity of A-HV to the different aspiration spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' While the  overall conclusion is consistent with that for RQ1 over different patterns and systems, we see that  there is often a strong bias on the gains for a certain position of the aspiration spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For example,  on x264 and Keras/Adiac, the improvement of PS-w is particularly high for aspiration space located  at the centered area of the objective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In contrast, the gain is particularly high on left-shifted  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Do Performance Aspirations Matter for Guiding Software Configuration Tuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:25  0  5 10  Keras/Adiac  10 20 30  Keras/RSD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3  Keras/SA  0 80 180  Storm/RS  0 50 120  Storm/WC  l  c  r 0 5 10 15  Trimesh  0  5  10  x264  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='8  XGBoost  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Sensitivity of the % gain on PS-w over PS-w/o to different positions of the realistic aspiration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Each point is the average and standard error over all combinations of patterns and optimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 𝑙, 𝑐, and 𝑟  denote left-shifted, centered, and right-shifted position in the performance landscape, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  5  10  15  2  4  6  800  900  # Counters  # Splitters  Latency (ms)  r  c  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' A projected landscape of the performance objective Latency with respect to configuration options  Splitters and Counters for Storm/WC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 𝑐 and 𝑟 denote centered and right-shifted aspiration space, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Note that the aspirations spaces are bounded because the throughput objective is also considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' it is  however not showed here for simpler exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  aspiration space under Storm/RS and centered space for Storm/WC, which is possible depending  on the landscape of a system (as we will discuss next).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Indeed, some aspiration spaces can easily  cause the PS-w to be trapped at the local optima, making its improvements over PS-w/o blurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  For example, on Storm/WC with right-shifted aspiration space, this effect is largely detrimental  and hence severely influence the benefits of PS-w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  In summary, we have:  RQ2: The improvement of PS-w over PS-w/o is often largely biased to certain position of  the aspiration space in the performance landscape, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', centered or left-shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Yet, PS-w still  performs more advantageously in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3  Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' As discussed for RQ1, the main reason that PS-w can perform better than PS-w/o  is due to the “focused search pressure”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, this may not be always helpful if the tuning  encounters complex local optima that are difficult for the optimizer to escape from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The high  sensitivity of the gains to the positions of aspiration space suggests that the local optima can be  distributed unevenly across the landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' If the aspiration space covers many local optima regions,  then certainly the gains of PS-w would be marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  % Gain1:26  Chen and Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  83  28  18  10  Large ˆA12 (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05)  Medium ˆA12 (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05)  Small ˆA12 (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05)  Trivial ˆA12 or p � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05  (c) Distribution on the 139 cases that PS-w/o win  25  10  13  17  Large ˆA12 (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05)  Medium ˆA12 (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05)  Small ˆA12 (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05)  Trivial ˆA12 or p � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05  (b) Distribution on the 65 cases that PS-w win  65  139  12  PS-w win  PS-w/o win  Tie  (a) Distribution on all 216 cases  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Summary of the wins by PS-w and PS-w/o together with their detailed statistics validation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Comparing PS-w and PS-w/o under unrealistic requirements and aspirations over 100 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Formats  are the same as Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  PS-w PS-w/o Tie  Avg (SE) of A-HV Gain  {𝒑1,𝒑1}  2 (2)  1 (1)  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='7%)  y  i  {𝒑2,𝒑2}  0 (0)  3 (3)  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3% (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='8%)  y  i  {𝒑3,𝒑3}  0 (0)  3 (2)  0  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='7% (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3%)  y  i  {𝒑1,𝒑2}  1 (1)  2 (1)  0  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='2%)  y  i  (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Storm/RS  (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Keras/Adiac  (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Keras/DSR  PS-w PS-w/o Tie  Avg (SE) of A-HV Gain  {𝒑1,𝒑1}  3 (2)  0 (0)  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='7%)  y  i  {𝒑3,𝒑2}  0 (0)  3 (3)  0  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='8% (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='7%)  y  i  (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Keras/SA  (h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' XGBoost  For example, in Figure 14, clearly the aspiration space 𝑐 (which covers the requirements for  latency and throughput) would be bounded on some regions in the landscape with a much more  smooth surface for the latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, for 𝑟, the region becomes highly rugged and steep, which  involves some very difficult local optima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Unfortunately, we did not see consistent patterns of such  a sensitivity across the configurable software systems, which makes sense as the performance  landscape of those systems can be very different too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3  RQ3: What if the Aspirations are Unrealistic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1  Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' To investigate RQ3, we omit the scenarios with 𝒑0 as they cannot create an unreal-  istic aspiration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This has left us with nine combinations of patterns, which, together with  three Pareto optimizers and eight subjects, provide 9 × 3 × 8 = 216 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' All other settings are  identical to those for RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' As the summary from Figure 15, we see that PS-w/o is generally better across all  the cases, as it wins on 64% (139/216) while loses on 30% (65/216).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' There is also a 6% (12/216) tie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  This means that PS-w/o is better or similar on 70% (151/216) of the cases against the 36% (77/216)  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  b  Distribution on the 65 cases that PS-w winC  Distribution on the 139 cases that PS-w/o winDistribution on all 216 casesDo Performance Aspirations Matter for Guiding Software Configuration Tuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:27  �5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6  �5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  �4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='8  �4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='5  3  log10 Throughput�1 (p3)  log10 Latency (p3)  PS-w  PS-w/o  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='8  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='8  1  Throughput (p3)  Latency (p3)  PS-w  PS-w/o  Reference  A-HV area  Original Space  (a) Storm/RS, PS-w/o wins  �5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6  �5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  �4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='8  �4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='5  3  log10 Throughput�1 (p3)  log10 Latency (p2)  PS-w  PS-w/o  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='8  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='8  1  Throughput (p3)  Latency (p2)  PS-w  PS-w/o  Reference  A-HV area  Original Space  (b) Storm/RS, PS-w wins  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Example runs of the final configuration sets (with NSGA-II) under unrealistic aspiration space  indicated by the shaded areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  for PS-w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Among these, PS-w/o wins 129 cases with ˆ𝐴12 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4 and 𝑝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05 comparing with 41 of  such cases when it loses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Similar results can be confirmed in Table 8 when inspecting specific system and requirement  scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Albeit there is a limited number of cases where PS-w is still advantageous, it is more  common to show no improvement at all or even cause fairly negative gains, which could be up to  an average of −243%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' It also has overall much less statistically significant wins across the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Particularly, we found that under {𝒑1, 𝒑1} on all systems, the two optimization models perform  similarly but PS-w tends to obtain more wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This is because such a combination pattern is the only  case where the unrealism of aspiration does not lead to too many incomparable configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Overall, we conclude that:  RQ3: When the aspiration space is unrealistic, PS-w/o is safer as it is similar or reasonably  better than PS-w for 70% of the time, meaning that the requirements and aspirations are more  harmful for guiding the tuning in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Yet, the only exception applied to {𝒑1, 𝒑1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3  Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Given unrealistic aspiration space, the most common cases are similar to the  Storm/RS example in Figure 16, where the PS-w is commonly inferior to PS-w/o when the diversity  tends to be high (Figure 16a), but sometimes superior to PS-w/o under limited diversity (Figure 16b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  This is because in most of the cases, after being transformed using the requirements with unrealistic  aspirations, PS-w tends to find too many incomparable configurations from the beginning (as in  the cases other than {𝒑1, 𝒑1}, most configurations are fully unsatisfied on at least one performance  objective), implying that the guidance provided by an unrealistic aspiration space is dramatically  weakened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Such an incomparability, although may prompt slightly better diversity to escape from  the local regions, can often severely harm the tendency towards more preferred configurations  that reach/exceed the aspirations, leading to worse A-HV (the smaller volume) than PS-w/o in  Figure 16a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This is because no selection pressure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', discriminative power) can be generated in  such a case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' It is also the reason why PS-w is not deteriorated by the unrealistic aspirations under  {𝒑1, 𝒑1}, which can still ensure that the configurations are comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Yet sometimes (Figure 16b),  such a high incomparability does help PS-w to find a good configuration by chance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', better  than the aspiration of latency), which is more desired than those of PS-w/o, leading to better HV  (the larger volume).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Hence the PS-w remains better for certain cases, despite that the tuning would  be easily trapped at that configuration due to the high sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:28  Chen and Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  100  200  300  400  500  600  700  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='95  # Measurements  A-HV  PS-w  PS-w/o  (e) Keras/Adiac  100  200  300  400  500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='85  # Measurements  A-HV  PS-w  PS-w/o  (f) Keras/DSR  100  200  300  400  500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='98  # Measurements  A-HV  PS-w  PS-w/o  (g) Keras/SA  100  200  300  400  500  600  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='95  # Measurements  A-HV  PS-w  PS-w/o  (d) Storm/RS  100  200  300  400  500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='95  # Measurements  A-HV  PS-w  PS-w/o  (c) Storm/WC  100  200  300  400  500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='8  # Measurements  A-HV  PS-w  PS-w/o  (a) Trimesh  200  400  600  800  1,000 1,200 1,400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='68  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='74  # Measurements  A-HV  PS-w  PS-w/o  (b) x264  50  100  150  200  250  300  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='98  # Measurements  A-HV  PS-w  PS-w/o  (h) XGBoost  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='08X speedup  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='08X speedup  10X speedup  10X speedup  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='19X speedup  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='17X speedup  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05X speedup  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='31X speedup  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Speedup on PS-w over PS-w/o under realistic aspirations (each point is the average and standard  error over all combinations of patterns, aspiration space, optimziers and their runs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  100  200  300  400  500  600  700  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3  # Measurements  A-HV  PS-w  PS-w/o  (e) Keras/Adiac  100  200  300  400  500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3  # Measurements  A-HV  PS-w  PS-w/o  (f) Keras/DSR  100  200  300  400  500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3  # Measurements  A-HV  PS-w  PS-w/o  (g) Keras/SA  100  200  300  400  500  600  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='6  # Measurements  A-HV  PS-w  PS-w/o  (d) Storm/RS  100  200  300  400  500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  # Measurements  A-HV  PS-w  PS-w/o  (c) Storm/WC  100  200  300  400  500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='25  # Measurements  A-HV  PS-w  PS-w/o  (a) Trimesh  200  400  600  800  1,000 1,200 1,400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='18  # Measurements  A-HV  PS-w  PS-w/o  (b) x264  50  100  150  200  250  300  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='5  # Measurements  A-HV  PS-w  PS-w/o  (h) XGBoost  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='13X speedup  10X speedup  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='5X speedup  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='49X speedup  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='18X speedup  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='36X speedup  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='61X speedup  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='65X speedup  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Speedup on PS-w/o over PS-w under unrealistic aspirations (each point is the average and standard  error over all combinations of patterns, optimziers and their runs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  RQ4: Does the Given Tuning Resource Important?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1  Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' To understand the resource efficiency of both optimization models in RQ4, for each  system, we use the following procedure:  (1) Plot the trajectories of A-HV along with the number of measurements for both PS-w and  PS-w/o, where each point is the average of all requirement patterns, aspiration spaces, and  optimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  (2) Identify a baseline, 𝑏, taken as the smallest number of measurements that the baseline model  consumes to achieve its best A-HV (say 𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  (3) For the other model, find the smallest number of measurements, denoted as 𝑚, at which the  average A-HV is equivalent to or better than 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  (4) Calculate the speedup over the baseline model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', 𝑠 = 𝑏  𝑚, according to the metric used by  Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Since we found that the generally better optimization model differs depending on the realism of  the aspiration space, we use PS-w/o and PS-w as the baseline for realistic and unrealistic aspiration  situations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Do Performance Aspirations Matter for Guiding Software Configuration Tuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:29  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' From the results under realistic aspirations as shown in Figure 17, we see that PS-w  outperforms PS-w/o throughout the trajectories over different configurable systems, which further  strengthen our findings for RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The improvement in resource efficiency has been remarkable:  there is a speedup between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05× and 10×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In contrast, when the given aspirations are unrealistic  (Figure 18), PS-w/o is much more resource-efficient, as it enables a speedup from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='18× to 10×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This  again complies with the findings for RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, under unrealistic aspirations, the advantages  of PS-w/o may not be obvious at the early stage of the tuning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' on some systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Figure 18c  and Figure 18f), it is even inferior to the PS-w until around 250 configurations have been measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  In summary, we found that:  RQ4: Under realistic aspirations, PS-w often obtains consistently better A-HV than PS-w/o  throughout the trajectory and with a speedup up to 10×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' When the aspirations are unrealistic,  in contrast, the two optimization models are competitive in the early stage of tuning but soon  PS-w/o would lead to better results with considerably high speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3  Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Under realistic aspirations, the reasons that PS-w has better A-HV throughout the  trajectory with remarkably high speedup are two folds: firstly, as what we have already discussed  for RQ1, the guidance provided by the requirements and aspirations are often helpful to enable the  tuning to be more focus-driven, hence better utilizing the resources to explore the more promising  area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Secondly, PS-w/o would waste the valuable tuning budget to explore those configurations that  it favors, but would never be preferred under the given requirements, since it is naturally interested  in the whole Pareto front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Therefore, the above difference enables PS-w to be a particularly attractive  model for some systems, such as Storm, where the performance of diverse configurations can be  radically different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  The situation is completely different when the given aspirations are unrealistic and it is mainly  due to the high incomparability in PS-w as mentioned for RQ3 — many of the configurations are  incomparable when transformed using the requirements with unrealistic aspirations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' It has been  shown that this situation can cause severe issues for any Pareto optimizer [56], as the resources  would have been spent mainly on exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, such an incomparability can occasionally  be helpful to explore some preferred configurations by chance, especially at the early stage of the  tuning where the PS-w/o has yet explored enough space to pursue the Pareto front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' As such, we  see that at the beginning PS-w performs similarly to PS-w/o and, sometimes, even better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  6  LESSONS LEARNED  In this section, we discuss how our findings can be useful for the practitioners in the field in light  of the lessons learned and future opportunities discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Lesson 1: The choice on whether to exploit aspirations for guiding the tuning is primarily dependent  on their realism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  It is interesting to find that we cannot draw the conclusion to choose between PS-w and PS-w/o  arbitrarily for software configuration tuning with two performance objectives, as opposed to  what has been overwhelmingly assumed in existing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Instead, from RQ1, RQ3, and RQ4, we  discovered that the realism of the given aspirations is crucial to the choice: PS-w is more beneficial  for realistic aspirations while PS-w/o is safer when the aspirations are unrealistic (given that the  tuning budget is also sufficient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This raises the importance of understanding whether the given  requirements and aspirations can be realistic, or the assumption therein, prior to choosing the right  optimization model for tuning software configuration with two performance objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:30  Chen and Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Lesson 2: Little combinations of patterns can change the decision on whether to incorporate  aspiration in the tuning, but it can influence the benefit/detriment of aspiration-guided tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Although from RQ1, we noticed that the benefits of PS-w is blurred when the given combination  of patterns contain 𝒑1 and/or 𝒑0 only, this does not change the decision as PS-w remains outperform  its PS-w/o counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The only definitive case is when the aspirations are unrealistic, PS-w should  be chosen under a patterns of {𝒑1, 𝒑1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Therefore, we envisage that the sensitivity of given patterns  to the choice between PS-w and PS-w/o is marginal and we have discovered other more important  factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, we do see that the extent of improvement/degradation from PS-w can be sensitive  to the given patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Lesson 3: The positions of realistic aspiration space in the objective space can largely affect the  benefits brought by considering aspirations within tuning, but it is less likely to influence the choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  An unexpected discovery from RQ2 is that, when given realistic aspirations, the position of  the aspiration space can largely influence the benefits of PS-w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' While this is unlikely to affect the  choice between PS-w and PS-w/o, it does raise the need to systematically analyze the correlation  between the aspiration space and the configuration landscape of the system, particularly on the  likelihood of covering some difficult local optima and their implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Lesson 4: The given tuning budget has marginal impact to the choice when the aspirations are  realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, it can be an important factor to consider under unrealistic aspirations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  According to RQ4 we have also revealed that, given realistic aspirations, the choice between  PS-w and PS-w/o is marginally sensitive to the tuning budget, but it can be influenced by the  budget when the aspirations are unrealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This adds an extra layer of consideration for unrealistic  aspirations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In this case, what we observed, in general, is that for a small tuning budget, the benefit  of PS-w/o is much less justified, hence using either of the two optimization models may not lead  to significantly different results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, given sufficient budget, PS-w/o is likely to dominate its  PS-w counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Unfortunately, with the current evidence, it remains very difficult to precisely  quantify how “small” or “large” the tuning budget is required to make such a distinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  The above lessons not only reveal the important factors for the practitioners to consider when  choosing PS-w and PS-w/o for bi-objective software configuration tuning but also hint at a few  future research opportunities in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' These are:  Landscape analysis for configurable software systems: We have found that the realism  of aspiration space, its position in the objective landscape, and tuning budget can be the key  factors to consider when choosing between PS-w and PS-w/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' All of those are relevant to  the landscape analysis of the configurable system itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Indeed, by systematically analyzing  any collected data, we are able to obtain more knowledge about the above factors, and hence  make more informed decisions on whether to incorporate requirements into the tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Requirement-robust optimizer for configuration tuning: The realism of the aspiration  is certainly the key factor in the choice between PS-w and PS-w/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, it may not be  always possible to obtain such knowledge in advance, leaving uncertainty to the decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In  this regard, it would be desirable to combine the strength of PS-w and PS-w/o to design an  optimizer that is robust to such an uncertainty in the requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Again, the landscape  analysis from the previous opportunity can provide insights into the designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Rigorous analysis of requirement patterns and their relationships to the tuning:  Although we see little implication of the requirement patterns to the choice between PS-w  and PS-w/o, it is important to better understand why they work more diversely on some  of the patterns and how exactly they can affect the performance of PS-w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In fact, on the  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Do Performance Aspirations Matter for Guiding Software Configuration Tuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:31  theoretical side, the quantification from Section 3 provides the foundation of theoretical  reasoning for switching between patterns, which is important in the topic of requirement  relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For example, this can be achieved in two aspects:  – With the quantification of the patterns, one can formally show the relations between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  For example, since all points in 𝒑1 have a higher satisficing value than those of 𝒑3, we can  say that 𝒑1 is a “relaxed” form of 𝒑3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  – Similarly, we can quantify the relationships between a pattern with two different aspiration  levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  With the above understanding, we allow the software engineers to achieve more explainability  in terms of the given requirements during the tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For example, once the tuning completes,  one would know how to relax or tighten the requirements, such that the most preferred  configuration can be found under the requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This can be a unified process that  combines both requirement negotiations and the tuning itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Interactive configuration tuning: On the empirical side, our findings provide a few insights  on what to do under different circumstances during interactive tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For example, if the  software engineers find that the tuning never (or rarely) produces configurations that satisfy  the requirements/aspirations under PS-w, then one can immediately switch to PS-w/o instead  before concerns about the suitability of the underlying optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Similarly, one can influence  the results produced by PS-w (or PS-w/o) by changing the position of the aspiration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  7  THREATS TO VALIDITY  As with other empirical studies in software engineering, our work may contain threats to construct  validity in the following aspects:  Metric: Pareto search produces a set of configurations, and thus the comparisons need to  work on a set rather than a single configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' We used HV, which is a comprehensive  metric for evaluating solution sets, following the methodology proposed by Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Since there can be different given sets of requirements with aspirations, the configuration  sets ought to be compared under such a scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' To that end, we extend the HV to explicitly  consider the patterns of requirements, as discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Statistics: The stochastic nature of the Pareto optimizers can raise threats to the stability of  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' To mitigate such, we repeat the experiments 100 runs and use Wilcoxon test along  with ˆ𝐴12 to verify all pairwise comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' All the above methods have been recommended  and widely used for Software Engineering research [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Two factors may form threats to internal validity in our study:  Tuning budget: Given the size of our study, we set a one-hour budget for each case, which  is a common setting for expensive problems in SBSE [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' To mitigate the interference of  our experiments, this is then converted into the number of unique measurements following  systematic steps (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' We have also analyzed the trajectories of A-HV, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4,  showing what would happen if a smaller budget is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Admittedly, investigating a larger  tuning budget may affect some of the results, but confirming this would need even more  computational resources and time (due to the expensive tuning), which we will plan as part  of future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Optimizer setting: In this work, we follow what has been shown to be effective for a SBSE  problem in the literature, as our aim is to compare the most common practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The only part  we could not have found for sure is the population size, which is highly problem-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  To tackle this, we have followed carefully designed criteria (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3) that strike a balance  between reasonable convergence and the time required under the tuning budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However,  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:32  Chen and Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  we do agree that exploring alternative parameter settings can be a thread that requires further  exploration, which we leave as part of future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Threats to external validity can come from various sources, including:  Software systems: In this work, we select the eight most representative systems/environ-  ments from existing work on software configuration tuning based on carefully codified rules  (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Those subject systems come from diverse domains and with different scales,  performance objectives, and search spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' A worth noting point is that the requirements  extracted include those for more complex systems, such as Cyber-Physical systems, while  the subjects we examined are mainly software systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This does not severely invalid our  conclusion because the extracted implication and patterns are rather generic such that they  can be applied to different cases while there exist some performance attributes that are of  relevance to a wide range of systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', latency- and throughput-related requirements (with  different aspiration levels) [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Nonetheless, we agree that this list of the studied systems is  not exhaustive and we may miss some particular situations that can only become clear for  more complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Experimenting with more systems that are of diverse types may prove  fruitful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' A relevant point is that we did not examine our results on highly complex software  systems that cut across the software and hardware layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In those cases, the interaction  between cross-layered configuration options can be more complex, leading to some different  configuration landscapes [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Therefore, examining those highly complex systems may  provide new insights and further consolidate our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  It is worth noting that it can be particularly attractive to relate the results with respect to  the different types of software systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, unfortunately, we have not yet observed  consistent patterns in the results according to the domain of systems, hence unable to draw a  general conclusion thereupon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This can be attributed to two reasons:  – The workload and benchmark under which each of the systems runs are rather different,  creating a distinct configuration landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  – Because of the above, the appropriate aspirations (levels) used are also different even for  systems that are of the same domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Again, using even more software systems may help us to achieve such, which we certainly  plan to do for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, this does not invalidate the conclusions drawn regarding  the comparison between PS-w and PS-w/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Configuration options: The discretization level of each configuration option can have a  non-trivial impact on the tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In this work, we use exactly the same configuration options  and their values as used in previous work [17, 44, 45, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, it is necessary to note that  changing the discretization level may disclose new insights, which we will seek to investigate  as part of future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Requirement patterns and aspiration space: To emulate real-world requirement sce-  narios, as shown in Section 3, we capture the implications and how they are quantified  by surveying the relevant datasets and papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This has enabled us to concentrate on four  patterns that cover a wide range of situations, leading to 15 combinations of the patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' As  for the aspiration space, we cover both realistic and unrealistic aspirations, and for the former,  we set three types of aspiration space including two skewed spaces and a more balanced one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Yet, admittedly, unintentionally ignored cases are always possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Optimizers: In this study, three common Pareto optimizers based on evolutionary search  are used, each of which is a distinct representative of its own kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Admittedly, there are  other popular multi-objective optimization approaches used in SBSE, such as exact methods  and Bayesian optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Do Performance Aspirations Matter for Guiding Software Configuration Tuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:33  When the given multi-objective optimization problems are of special characteristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=',  linearity in both objective functions and constraints and the scale is small or moderate),  then exact methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', integer linear programming) can be very good choices, where  solutions of the Pareto front can be iteratively found by specifying different weights or  desirable/tolerant values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Such optimization problems have been commonly seen in the next  release problem, and well-established exact methods, such as 𝜖-constraint and augmented  Tchebycheff methods, have shown promising results [24, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Another popular kind of optimizers, particularly used in software configuration tuning, is  Bayesian optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Recently, there are multi-objective Bayesian optimizers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', [43])  which search for the whole Pareto front of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Compared to evolutionary algorithms,  Bayesian optimizers are usually more sample efficient [44, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, one issue with  such approaches is that they may not be as straightforward as evolutionary algorithms to  incorporate the stakeholders’ performance aspirations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', for Bayesian optimization this  may need a careful design of the acquisition function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Different incorporation ideas may  lead to different results, thus likely affecting the reliability of the conclusions drawn from  the direct comparison between Pareto search with and without performance aspirations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  In addition, it is necessary to point out that the conclusions drawn from multi-objective  evolutionary algorithms may not apply to other optimization approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Optimizers, which  are guided “heavily” by the aspirations, may find them quickly if they are realistic, but may  end up with undesirable solutions if unrealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Optimizers, which can strike a good balance  between exploitation and exploration (under limited budgets) like Bayesian optimization,  may bring different results, though it depends on the incorporation of the aspirations in  the optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Consequently, it is desirable to investigate different optimization  approaches to study the generalizability of our findings, particularly those with fruitful  theoretical results (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', convergence rate regarding the response surface’s smoothness as  well as the regret bounds in Bayesian optimization), which may help support our empirical  conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This will be an important part of our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Number of objectives: Our study covers the case of two performance objectives for software  configuration tuning, which, as we have found from our review in Section 3, tends to be the  most common situation when multiple objectives are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The results may not be  generalizable to higher dimension cases of the objectives though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Extending the study to  more objectives can be part of future work, but there would also be exponentially increasing  factors to consider, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', the number of pattern combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Our results from this work  serve as the very first step to raising the importance of studying whether to use aspiration to  guide software configuration tuning with more than one performance objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Overall, the above settings have provided us with more than 1,000 cases to generalize our findings  in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  8  RELATED WORK  Here, we discuss the related work in light of the purpose of our empirical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1  PS-w/o for Software Configuration Tuning  Search-based approaches for software configuration tuning have been commonly studied under a  wide range of optimizers, such as random search [64], hill climbing [77], genetic algorithm [68],  and ant colony optimization [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In the presence of more than one objective, Pareto search has  been shown to be highly effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Among others, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' [16] and Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' [70] leverage  different multi-objective evolutionary algorithms to search the Pareto optimal configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Nair  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:34  Chen and Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' [62] also aim for the same, but their approach applies Bayesian optimization wherein the  two performance objectives are handled similarly to MOEA/D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' More recently, Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' [30] also  propose an extended Bayesian optimization approach to reach a given performance aspiration  while considering both latency and resource consumption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' however, such information has not been  used to explicitly guide the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  The above work has one thing in common: they have ignored the aspirations in the search  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The assumption therein is that the concept of optimization can obtain whatever best  configuration that satisfies any given patterns of requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2  PS-w for Software Configuration Tuning  In contrast, Calinescu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' [8, 32] explicitly quantifies aspirations as part of the objectives (as 𝒑1) to  guide the Pareto search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Martens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' [59] also bear similar idea, but their pattern matches with 𝒑3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Ghanbari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' [34] has also been relying aspiration to guide the tuning, and they assume a smoother  curve over the requirement patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, there has been no study that justifies the importance  of aspirations in guiding the Pareto search for bi-objective software configuration tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Indeed,  a recent discussion paper from Fekry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' [28] commented that studying the aspirations for  guiding the optimizers and measuring its effectiveness is an important future challenge for software  configuration tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  These are typical examples of the PS-w optimization model, such that the performance require-  ments are precisely quantified as part of the search and tuning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' It is also worth noting that  the actual requirements patterns used can vary depending on the assumption, and none of the  existing work for PS-w has considered all the patterns we summarized in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3  Tuning with or without Surrogate  From another perspective, existing search-based approaches, regardless the number of performance  objectives considered, can be classified as model-based (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', EvoChecker [32], FLASH [62], and  BOCA [10]) and measurement-based (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', FEMOSAA [16], eQual [68], and Plato [66]), by which  the former relies on surrogate models [13] to guide the search while the latter do so via direct  measurements from the software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The key difference between those two are the landscape upon  which the search is conducted: the measurement-based approaches do so directly on the configura-  tion landscape while the model-based approaches searches in a surrogate landscape, which is an  approximation of the true configuration landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  This work focuses on tuning by directly measuring the systems without using the surrogate,  because for the following reasons:  We seek to avoid the noises caused by the surrogate models as they would inevitably introduce  errors, which, as demonstrated by Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' [83], can severely affect the search and tuning  behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  In fact, we treat these two categories as complementary rather than alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For example,  a measurement-based approach that works well can be also applied in Bayesian optimization,  which is model-based, to search for the acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This means that our findings on whether  aspirations matter are also applicable therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='4  General Multi-objective Optimization  Conceptually, our work can be relevant to the theme of preference-driven multi-objective optimiza-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In this regard, as surveyed by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' [74], the preferences on the objective values (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=',  the performance requirements in this work) can form the following categories:  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Do Performance Aspirations Matter for Guiding Software Configuration Tuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:35  Weights: In this case, a weight vector representing the relative importance of the objectives  is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Most commonly, this would convert the multi-objective problem into a single one via  some form of aggregation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', weighted sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Indeed, both the weights and the aspirations  in this work are some forms of preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, they are very different because the  weights need to be specified in-between the performance objectives, representing an explicit  trade-off [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The aspiration level, in contrast, serves as the expectation for a single objective,  which is often easier to specify, and no explicit trade-off is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In fact, using the weight  can be thought as a special case of the kind of preference we consider in this work: in the  case of weight, the best configuration is typically a particular point on the Pareto front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In  contrast, under the requirements and aspirations considered, there are often more than one  best point, including a proportion of the points on the Pareto front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Interestingly, given a set  of weights, it has been shown that the Pareto search (which runs without the weights) can  generally find better configurations than the search guided by the weights [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Objective relation: This refers to the case where a full or partial rank of the objectives has  been provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' For example, one may prefer to satisfy the requirement of throughput first  before considering latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The combinations of patterns and aspirations in this work, in  contrast, have no direct ranking between the objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Area of objective space: One may provide a rough notion of a particularly preferred area  in the objective space, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', knee points or extreme points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Compared with the combinations  of patterns from Section 3, this preference is vague — the knee or extreme points are relative  among the solutions found, while the patterns are still guided by a clearly defined aspiration  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Reference points [51, 80]: Here, a vector representing the expectations of the objectives are  given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In this regard, the concept is indeed similar to the aspiration levels we discussed in this  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, a major difference is that reference point-based multi-objective optimization  is always under the assumption that Pareto optimality needs to be considered first, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', the  Pareto optimal solutions close to the reference point (along certain direction specified by the  decision-maker) are preferred [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Moreover, additional parameters are required to specify the  spread of the preferred solutions [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In contrast, the combinations of patterns we discovered  for software configuration tuning can be rather different from the above, as the solutions in  (outside) the aspiration space may be equally preferred (unpreferred) while do not favor the  solutions close to the aspiration vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Further, no other parameters are needed besides the  aspiration levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Therefore, all the above forms of preferences differ from the requirement aspirations and patterns  used for software configuration tuning, as we summarized in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' This is important as the  summarized patterns are derived from empirical findings for the characteristics of the problems  — they may not be generalizable to other problems but are significant to software configuration  tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  An empirical study on the importance of considering preferences in the search also exists from  the general optimization community [51, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' However, they differ from our work in two aspects:  They focus on reference points, which, as we discussed, are rather different from the patterns  considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  The studied optimizers are specialized algorithms that take specific forms of preferences  into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In contrast, our study considers classic optimizers which are equipped with  the requirement patterns to guide the search, as what has been commonly used for software  configuration tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:36  Chen and Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  In summary, this work is, to the best of our knowledge, the first empirical study to understand  whether, when, and why aspirations matter for guiding bi-objective software configuration tuning,  according to the characteristics of requirement patterns and practice summarized for the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  9  CONCLUSION  In this paper, we conduct a comprehensive empirical study, which offers an in-depth understanding  of whether performance aspirations matter to bi-objective tuning for configurable software systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Our study covers 15 combinations of patterns that quantify the aspirations, four types of aspiration  space, three search algorithms, and eight software systems/environments, leading to 1,296 cases of  investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The results challenge the belief of “it does not matter whether to use aspiration in the  tuning” and reveal that:  the aspirations are generally helpful in guiding the tuning when they are realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  but, they can often be harmful to the tuning when these aspirations are unrealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  the different patterns and position of the aspiration space do not change the above conclusion,  but can affect the extent of benefits/detriments generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  when the aspirations are realistic, the tuning budget has marginal implication on their benefits  during tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In contrast, it is an important factor when the aspirations are unrealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Our findings in this work provide useful insights for the practitioners in this particular field  of research, particularly on the factors/information needed for answering the question of “when  aspirations should be considered during the tuning?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  We outline some exciting future research opportunities:  To better understand the likelihood of realism and position of the given aspirations, we shall  analyze the landscape of the configurable software systems even based on some limited  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  To cater for the uncertainty of given requirements and aspirations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', possible to be unreal-  istic), we should investigate requirement-robust optimizers for tuning software configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  To explain the impact of requirements patterns for guiding the tuning, we ought to provide  the foundation to rigorously analyze their relationships and how they may be able to switch  to one another for achieving relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' In International Conference for High Performance  Computing, Networking, Storage and Analysis, SC’13, Denver, CO, USA - November 17 - 21, 2013, William Gropp and  Satoshi Matsuoka (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='2503278  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Do Performance Aspirations Matter for Guiding Software Configuration Tuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Bowers, Erik M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Fredericks, and Betty H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Cheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Automated Optimization of Weighted Non-functional  Objectives in Self-adaptive Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In Search-Based Software Engineering - 10th International Symposium, SSBSE 2018,  Montpellier, France, September 8-9, 2018, Proceedings (Lecture Notes in Computer Science, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 11036), Thelma Elita Colanzi  and Phil McMinn (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Springer, 182–197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  [8] Radu Calinescu, Milan Ceska, Simos Gerasimou, Marta Kwiatkowska, and Nicola Paoletti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Designing Ro-  bust Software Systems through Parametric Markov Chain Synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In IEEE International Conference on Software  Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  [9] Radu Calinescu, Milan Ceska Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Efficient synthesis  of robust models for stochastic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Efficient Compiler Autotuning via Bayesian Optimiza-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='  IEEE, 1198–1209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='00110  [11] Tao Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Lifelong Dynamic Optimization for Self-Adaptive Systems: Fact or Fiction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='. In IEEE International  Conference on Software Analysis, Evolution and Reengineering, SANER 2022, Honolulu, HI, USA, March 15-18, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' IEEE,  78–89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='00022  [12] Tao Chen and Rami Bahsoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Toward a Smarter Cloud: Self-Aware Autoscaling of Cloud Configurations and  Resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Computer 48, 9 (2015), 93–96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='278  [13] Tao Chen and Rami Bahsoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Self-Adaptive and Online QoS Modeling for Cloud-Based Software Services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' IEEE  Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Software Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 43, 5 (2017), 453–475.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='2608826  [14] Tao Chen and Rami Bahsoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Self-Adaptive Trade-off Decision Making for Autoscaling Cloud-Based Services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  IEEE Transactions on Services Computing 10, 4 (2017), 618–632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' A Survey and Taxonomy of Self-Aware and Self-Adaptive Cloud  Autoscaling Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 51, 3 (2018), 61:1–61:40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1145/3190507  [16] Tao Chen, Ke Li, Rami Bahsoon, and Xin Yao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' FEMOSAA: Feature Guided and Knee Driven Multi-Objective  Optimization for Self-Adaptive Software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' ACM Transactions on Software Engineering and Methodology 27, 2 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  [17] Tao Chen and Miqing Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' In ESEC/FSE ’21: 29th ACM Joint  European Software Engineering Conference and Symposium on the Foundations of Software Engineering, Athens, Greece,  August 23-28, 2021, Diomidis Spinellis, Georgios Gousios, Marsha Chechik, and Massimiliano Di Penta (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' ACM,  453–465.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' The Weights Can Be Harmful: Pareto Search versus Weighted Search in Multi-  Objective Search-Based Software Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' ACM Transactions on Software Engineering and Methodology (Jan 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='1145/3514233  [19] Tao Chen, Miqing Li, Ke Li, and Kalyanmoy Deb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Deb, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Pratap, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Agarwal, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Meyarivan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' A fast and elitist multiobjective genetic algorithm: NSGA-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  IEEE Transactions on Evolutionary Computation 6, 2 (2002), 182–197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  [23] Quoc Anh Do, Surendra Raju Chekuri, and Tanmay Bhowmik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='  Automated Support to Capture Creative  Requirements via Requirements Reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In Reuse in the Big Data Era - 18th International Conference on Software and  Systems Reuse, ICSR 2019, Cincinnati, OH, USA, June 26-28, 2019, Proceedings (Lecture Notes in Computer Science,  Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 11602), Xin Peng, Apostolos Ampatzoglou, and Tanmay Bhowmik (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Springer, 47–63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1007/  978-3-030-22888-0_4  [24] Miguel Ángel Domínguez-Ríos, Francisco Chicano, Enrique Alba, Isabel del Águila, and José del Sagrado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Efficient  anytime algorithms to solve the bi-objective Next Release Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Journal of Systems and Software 156 (2019), 217–231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  [25] Juan José Durillo and Antonio J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Nebro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' 42, 10 (2011), 760–771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='advengsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='  [27] Naeem Esfahani, Ehsan Kouroshfar, and Sam Malek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Taming uncertainty in self-adaptive software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In SIGSOFT-  /FSE’11 19th ACM SIGSOFT Symposium on the Foundations of Software Engineering (FSE-19) and ESEC’11: 13th European  Software Engineering Conference (ESEC-13), Szeged, Hungary, September 5-9, 2011, Tibor Gyimóthy and Andreas Zeller  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  1:38  Chen and Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' ACM, 234–244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Pasquier, Andrew Rice, and Andy Hopper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Towards Seamless  Configuration Tuning of Big Data Analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In 39th IEEE International Conference on Distributed Computing Systems,  ICDCS 2019, Dallas, TX, USA, July 7-10, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' IEEE, 1912–1919.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' PURE: A Dataset of Public Requirements  Documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In 25th IEEE International Requirements Engineering Conference, RE 2017, Lisbon, Portugal, September  4-8, 2017, Ana Moreira, João Araújo, Jane Hayes, and Barbara Paech (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' IEEE Computer Society, 502–505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' https:  //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' In 43rd IEEE/ACM International Conference on Software Engineering, ICSE 2021,  Madrid, Spain, 22-30 May 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' IEEE, 175–187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Computer 37, 10 (2004), 46–54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Search-Based Synthesis of Probabilistic Models for  Quality-of-Service Software Engineering (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In IEEE/ACM International Conference on Automated Software Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='  [34] Hamoun Ghanbari, Bradley Simmons, Marin Litoiu, and Gabriel Iszlai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' ATOM: Model-Driven Autoscaling for Microservices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In  39th IEEE International Conference on Distributed Computing Systems, ICDCS 2019, Dallas, TX, USA, July 7-10, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  IEEE, 1994–2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1109/ICDCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='00197  [36] Jingzhi Gong and Tao Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Does Configuration Encoding Matter in Learning Software Performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' An  Empirical Study on Encoding Schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In 19th IEEE/ACM International Conference on Mining Software Repositories,  MSR 2022, Pittsburgh, PA, USA, May 23-24, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' ACM, 482–494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Evaluating the role of optimization-specific search heuristics in effective  autotuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Technical report (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  [38] Huong Ha and Hongyu Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' DeepPerf: performance prediction for configurable software with deep sparse  neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In Proceedings of the 41st International Conference on Software Engineering, ICSE 2019, Montreal, QC,  Canada, May 25-31, 2019, Joanne M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Atlee, Tevfik Bultan, and Jon Whittle (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' IEEE / ACM, 1095–1106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  https:  //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='00113  [39] Xue Han and Tingting Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' An Empirical Study on Performance Bugs for Highly Configurable Software Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  In Proceedings of the 10th ACM/IEEE International Symposium on Empirical Software Engineering and Measurement,  ESEM 2016, Ciudad Real, Spain, September 8-9, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' ACM, 23:1–23:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Search-based software engineering: Trends, techniques  and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' ACM Computing Surveys (CSUR) 45, 1 (2012), 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' A Survey of Performance Optimization for Mobile  Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' IEEE Transactions on Software Engineering (2021), 1–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1109/TSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Unicorn:  reasoning about configurable system performance through the lens of causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In EuroSys ’22: Seventeenth European  Conference on Computer Systems, Rennes, France, April 5 - 8, 2022, Yérom-David Bromberg, Anne-Marie Kermarrec, and  Christos Kozyrakis (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' ACM, 199–217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Flexibo: Cost-aware multi-objective optimiza-  tion of deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' arXiv preprint arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' An Uncertainty-Aware Approach to Optimal Configuration of Stream  Processing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In 24th IEEE International Symposium on Modeling, Analysis and Simulation of Computer and  Telecommunication Systems, MASCOTS 2016, London, United Kingdom, September 19-21, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' IEEE Computer Society,  39–48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Learning to sample: exploiting  similarities across environments to learn performance models for configurable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In Proceedings of the 2018 ACM  Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software Engineering,  ESEC/SIGSOFT FSE 2018, Lake Buena Vista, FL, USA, November 04-09, 2018, Gary T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Leavens, Alessandro Garcia, and  Corina S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Pasareanu (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' ACM, 71–82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' In 2017 IEEE/ACM 12th International Symposium on  Software Engineering for Adaptive and Self-Managing Systems (SEAMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' IEEE, 31–41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Liao, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Min, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Does Preference Always Help?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' A Holistic Study on Preference-Based  Evolutionary Multiobjective Optimization Using Reference Points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' IEEE Transactions on Evolutionary Computation 24,  6 (2020), 1078–1096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='2987559  [52] Ke Li, Zilin Xiang, Tao Chen, and Kay Chen Tan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' BiLO-CPDP: Bi-Level Programming for Automated Model  Discovery in Cross-Project Defect Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In 35th IEEE/ACM International Conference on Automated Software  Engineering, ASE 2020, Melbourne, Australia, September 21-25, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' IEEE, 573–584.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Understanding the automated parameter  optimization on transfer learning for cross-project defect prediction: an empirical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In ICSE ’20: 42nd International  Conference on Software Engineering, Seoul, South Korea, 27 June - 19 July, 2020, Gregg Rothermel and Doo-Hwan Bae  (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' ACM, 566–577.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='3380360  [54] Miqing Li, Tao Chen, and Xin Yao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' A Critical Review of "A Practical Guide to Select Quality Indicators for  Assessing Pareto-Based Search Algorithms in Search-Based Software Engineering": Essay on Quality Indicator Selection  for SBSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In 2018 IEEE/ACM 40th International Conference on Software Engineering: New Ideas and Emerging Technologies  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' How to Evaluate Solutions in Pareto-based Search-Based Software Engineering?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  A Critical Review and Methodological Guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' IEEE Transactions on Software Engineering 48, 5 (2022), 1771–1799.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='3036108  [56] Miqing Li, Shengxiang Yang, Ke Li, and Xiaohui Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Evolutionary algorithms with segment-based search for  multiobjective optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' IEEE Transactions on Cybernetics 44, 8 (2014), 1295–1313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Quality Evaluation of Solution Sets in Multiobjective Optimisation: A Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Surveys 52, 2 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Fuller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' In The 23rd International Symposium on High-Performance Parallel and Dis-  tributed Computing, HPDC’14, Vancouver, BC, Canada - June 23 - 27, 2014, Beth Plale, Matei Ripeanu, Franck Cappello,  and Dongyan Xu (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' ACM, 165–176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='2600229  [59] Anne Martens, Heiko Koziolek, Steffen Becker, and Ralf H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Reussner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Automatically improve software architecture  models for performance, reliability, and cost using evolutionary algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In Proceedings of the first joint WOSP/SIPEW  International Conference on Performance Engineering, San Jose, California, USA, January 28-30, 2010, Alan Adamson,  Andre B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Bondi, Carlos Juiz, and Mark S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Squillante (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' ACM, 105–116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1145/1712605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='1712624  [60] Mary L McHugh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Interrater reliability: the kappa statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Biochemia medica 22, 3 (2012), 276–282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The promise  repository of empirical software engineering data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Finding faster configurations using FLASH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' On the techniques of optimizing and satisficing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' The Swedish Journal of Economics 67, 1 (1965),  24–39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Proceedings (Lecture Notes in Computer Science, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' ACM, 97–106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='  In ESEC/FSE ’20: 28th ACM Joint European Software Engineering Conference and Symposium on the Foundations of Software  Engineering, Virtual Event, USA, November 8-13, 2020, Prem Devanbu, Myra B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='  ACM, 1039–1051.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' ACM, 309–320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Cohen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' In Search-Based Software Engineering - 12th International Symposium, SSBSE 2020, Bari, Italy,  October 7-8, 2020, Proceedings (Lecture Notes in Computer Science, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 12420), Aldeida Aleti and Annibale Panichella  (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Springer, 137–152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content='  In Proceedings of the 2015 10th Joint Meeting on Foundations of Software Engineering, ESEC/FSE 2015, Bergamo, Italy,  August 30 - September 4, 2015, Elisabetta Di Nitto, Mark Harman, and Patrick Heymans (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' ACM, 307–319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' A recursive random search algorithm for large-scale network parameter  configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' In Proceedings of the International Conference on Measurements and Modeling of Computer Systems,  SIGMETRICS 2003, June 9-14, 2003, San Diego, CA, USA, Bill Cheng, Satish K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  Do Performance Aspirations Matter for Guiding Software Configuration Tuning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' In Parallel Problem Solving  from Nature - PPSN VIII, 8th International Conference, Birmingham, UK, September 18-22, 2004, Proceedings (Lecture Notes  in Computer Science, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+page_content=' Zitzler, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Thiele, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Laumanns, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Fonseca, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Da Fonseca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Performance assessment of multiobjective  optimizers: An analysis and review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' IEEE Transactions on Evolutionary Computation 7, 2 (2003), 117–132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' 1, Article 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
+page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQflwTE/content/2301.03290v1.pdf'}
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+Scalable Communication for Multi-Agent Reinforcement Learning
+via Transformer-Based Email Mechanism
+Xudong Guo , Daming Shi , Wenhui Fan
+Department of Automation, Tsinghua University
+{gxd20, shidm18}@mails.tsinghua.edu.cn, fanwenhui@tsinghua.edu.cn
+Abstract
+Communication can impressively improve co-
+operation in multi-agent reinforcement learning
+(MARL), especially for partially-observed tasks.
+However, existing works either broadcast the mes-
+sages leading to information redundancy, or learn
+targeted communication by modeling all the other
+agents as targets, which is not scalable when the
+number of agents varies. In this work, to tackle
+the scalability problem of MARL communication
+for partially-observed tasks, we propose a novel
+framework Transformer-based Email Mecha-
+nism (TEM). The agents adopt local communica-
+tion to send messages only to the ones that can
+be observed without modeling all the agents. In-
+spired by human cooperation with email forward-
+ing, we design message chains to forward informa-
+tion to cooperate with the agents outside the ob-
+servation range. We introduce Transformer to en-
+code and decode the message chain to choose the
+next receiver selectively. Empirically, TEM outper-
+forms the baselines on multiple cooperative MARL
+benchmarks. When the number of agents varies,
+TEM maintains superior performance without fur-
+ther training.
+1
+Introduction
+Multi-agent reinforcement learning (MARL) has achieved re-
+markable success in many complex challenges, especially in
+game playing [OpenAI et al., 2019; Vinyals et al., 2019].
+MARL shows great potential to solve cooperative multi-agent
+real-world tasks, such as autonomous vehicle teams [Shalev-
+Shwartz et al., 2016], robotics control [Kober et al., 2013]
+and intelligent traffic control [Wei et al., 2019]. However,
+some essential obstacles still exist for MARL to reach satis-
+factory performance. When training the MARL algorithms,
+the agents keep updating their policies and causing dynam-
+ics in the environment, which may hinder the model con-
+vergence. Worse still, in most cooperative multi-agent tasks,
+agents can only observe part of the environment. Partial ob-
+servability and non-stationarity make it harder to success-
+fully cooperate, even though some works employ centralized
+Figure 1:
+Message chain formed by email forwarding in the
+Transformer-based Email Mechanism (TEM). The agents (circles)
+are trying to surround and capture the target (square). The dotted
+circle is the observation range for the agent with the same color.
+The black lines are message chains. ⟨·⟩ demotes concatenating. The
+cross denotes not sending or forwarding after receiver selection. The
+agents a, b and c indirectly cooperate by sending (ma) and forward-
+ing (⟨ma, mb⟩) messages to capture the target T. As they are not in
+the same observation range, forwarding like emails is necessary.
+training and decentralized execution (CTDE) paradigm to im-
+port a critic to coordinate the whole team [Yu et al., 2021;
+Lowe et al., 2020; Son et al., 2019; Rashid et al., 2018].
+Inspired by the ways how humans and animals cooperate,
+communication is introduced to share information between
+agents.
+Some works broadcast the messages to all the
+other agents [Zhang et al., 2019; Sukhbaatar et al., 2016;
+Foerster et al., 2016], and other recent works try to learn
+targeted peer-to-peer communication to reduce the commu-
+nication bandwidth [Ding et al., 2020; Jiang and Lu, 2018;
+Yuan et al., 2022]. Attention mechanism from Transformer
+[Vaswani et al., 2017] is also employed to learn the commu-
+nication [Jiang and Lu, 2018]. However, the existing meth-
+ods rely on modeling every teammate in the environment by
+ID to decide whether to communicate, which will bring huge
+computational overhead when the number of agents is large.
+As the modeling network is trained by a specific amount of
+IDs, the learned communication mechanism is not scalable
+to reuse when the number of agents changes. In fact, the
+agent cannot know the state of an agent outside the observa-
+tion range, and cannot judge whether the information is useful
+for it, so it is unreasonable to directly share information with
+arXiv:2301.01919v1  [cs.MA]  5 Jan 2023
+
+d
+e
+区
+T
+ar
+Csuch an agent. For example, for applications to autonomous
+vehicles, only the vehicles nearby are worth communicating
+with to avoid collisions. Thus, global communication with
+all vehicles is unnecessary. Moreover, communication with
+other vehicles should adapt to different numbers of agents as
+the traffic situation varies a lot.
+In this work, to tackle this new problem - the scalability
+of MARL communication, we propose a scalable multi-agent
+communication mechanism via Transformer-based Email
+Mechanism (TEM) to tackle the abovementioned challenges
+as shown in Fig 1. We adopt local communication to send
+messages only to the agents in the observation range, with-
+out modeling all the agents. The agent will decide whom to
+communicate with by its own intention and by observing the
+agents in the range. Thus, no matter how the overall num-
+ber of the agents changes, the learned communication mech-
+anism is scalable. To better utilize the key information and
+indirectly cooperate with the agents outside the range, we de-
+sign a new communication mechanism like email forward-
+ing to form a message chain. The agents can send and for-
+ward the messages so that the chain connects agents from
+different ranges. For example, the agent a in Fig 1 would
+like to surround and capture the target T with other agents,
+thus the agent a may send a message to the agent b instead of
+d, though d is the nearest. Then the agent b can forward the
+message together with the information from itself to c, so that
+a, b and c can cooperate for the same goal though there is no
+direct communication between them. Similarly, in our daily
+life, cooperation in a big company or organization relies on
+such forwarding emails to share information, as it is always
+hard to directly find the exact contact in another department.
+To suit the unfixed length of the message chain and ensure
+the communication mechanism is scalable, we design a new
+message network and employ Transformer to encode and de-
+code the sequence of messages. Furthermore, augmented by
+the attention mechanism in the Transformer, the communi-
+cation is selective by modeling the correlation between the
+messages and the observation. The message network is inde-
+pendent and can be plugged into any CTDE method. What’s
+more, we design a loss to guide the agent to estimate the im-
+pact of the message on other agents. Note that we do not
+introduce the broadcast mechanism from email to keep the
+communication efficient.
+For evaluation, we test TEM on three partial-observation
+cooperative MARL benchmarks: the Starcraft multi-agent
+challenge (SMAC) [Samvelyan et al., 2019], Predator Prey
+(PP) [Kim et al., 2019] and Cooperative Navigation (CN)
+[Lowe et al., 2020], where TEM reaches better performance
+than the baselines. We also evaluate the scalability of TEM.
+Without extra training, TEM can suit both situations where
+the number of agents increases and decreases, and still out-
+performs the baselines.
+2
+Related Works
+Learning how to communicate is a popular research domain
+in multi-agent reinforcement learning. Researches in this do-
+main focus mainly on cooperative scenarios, where agents
+could communicate with each other explicitly. In the early
+works, RIAL and DIAL [Foerster et al., 2016] are designed
+to learn communication, where messages are delivered from
+one timestep to the next timestep in a broadcast way. Comm-
+Net [Sukhbaatar et al., 2016] proposes a hidden layer as com-
+munication and allows the agents to communicate repeatedly
+in each step. IC3Net [Singh et al., 2018] brings in the gat-
+ing mechanism to control communication based on Comm-
+Net. Both of BiC-Net [Peng et al., 2017] and ATOC [Jiang
+and Lu, 2018] implement the communication layer as bidi-
+rectional RNN, which inputs the observations of all agents
+and outputs the action or the integrated thought of each agent.
+However, these methods either broadcast the messages or rely
+on a centralized communication layer, which is high-cost and
+not stable. Communication should not only serve as an inte-
+gration of information, instead, the agents should share infor-
+mation selectively through peer-to-peer communication.
+To avoid broadcasting messages, recent works try to de-
+sign more intelligent communication mechanisms.
+CTDE
+paradigms are also imported to implement decentralized com-
+munication.
+Some works are based on QMIX [Rashid et
+al., 2018]: VBC [Zhang et al., 2019] proposes a request-
+reply mechanism and a variance-based regularizer to elimi-
+nate the noisy components in messages. NDQ [Wang et al.,
+2020] learns nearly decomposable value functions with com-
+munication. TMC [Zhang et al., 2020] maximizes the mu-
+tual information between the decentralized Q functions and
+the communication messages while minimizing the entropy
+of messages between agents. MAIC [Yuan et al., 2022] al-
+lows each agent to learn to generate incentive messages by
+modeling every teammate and bias other agents’ value func-
+tions directly. Some are based on another CTDE framework
+MADDPG [Lowe et al., 2020]: TarMAC [Das et al., 2019]
+proposes a targeted communication behavior via a signature-
+based soft attention mechanism. Besides the message, the
+sender broadcasts a key used by the receivers to gauge the
+message’s relevance. DAACMP [Mao et al., 2020] adds a
+double attention mechanism in the actor and critic network
+respectively to select and process the important messages.
+I2C [Ding et al., 2020] learns a prior net via causal inference
+for peer-to-peer communication. The influence of one agent
+on another is inferred via the joint action-value function and
+quantified to label the necessity of peer-to-peer communica-
+tion. Nevertheless, the methods above need to model every
+other agent in the environment to achieve individual commu-
+nication, which is not scalable and practical.
+To the best of our knowledge, none of the existing MARL
+communication methods considers the scalability of the com-
+munication mechanism and the forwarding protocol inspired
+by email.
+3
+Background
+3.1
+Policy Gradient (PG) Reinforcement Learning
+Policy Gradient (PG) reinforcement learning has the advan-
+tage of learning a policy network explicitly, in contrast to
+value-based reinforcement learning methods. PG methods
+optimize the policy parameter θ to maximize its objective
+function J(θ) = ES(Vπθ(s)). However, due to the variance
+of environments, it is hard to choose a subtle learning rate in
+
+reinforcement learning. To resolve this problem and ensure
+the safe optimization of policy learning, the Trust Region Pol-
+icy Optimization (TRPO) [Schulman et al., 2015] increases
+the constraint of the parameter difference between policy up-
+dates. Such constraint ensures the parameter changes in a
+small range, so that the collapse of value can be avoided and
+the policy can learn monotonically. The parameter update of
+TRPO is θk+1 = arg maxθL(θk, θ) s.t. ¯DKL(θ||θk) ≤ δ,
+where L(θk, θ) = E[ πθ(a|s)
+πθk (a|s)Aπθk (s, a)] is the approxima-
+tion of the original policy gradient object J(θ) within the
+constraint of KL divergence.
+Based on TRPO, a simplified version Proximal Policy Op-
+timization [Schulman et al., 2017] is carried out, maintaining
+the motivation to constrain the learning step while more ef-
+ficient and easy to be implemented. The object function of
+PPO can be written as:
+L(s, a, θk, θ) =min[ πθ(a|s)
+πθk(a|s)Aπθk (s, a),
+clip( πθ(a|s)
+πθk(a|s)), 1 − ϵ, 1 + ϵ)Aπθk (s, a)],
+(1)
+which forces the ratio of πθ(a|s)
+πθk (a|s) to locate in the interval (1−
+ϵ, 1 + ϵ), so that the new θ is not too far away from old θk.
+3.2
+MAPPO Algorithm
+Multi-agent PPO (MAPPO) introduces PPO into the multi-
+agent scenario [Yu et al., 2021]. MAPPO mainly considers
+decentralized partially observable Markov decision processes
+(DEC-POMDP). In an environment with n agents, s ∈ S de-
+notes the state of the environment. The agent i only has a
+local observation of environment oi = O(s) and chooses its
+action based on its observation and policy ai = πi(ai|oi).
+The joint action A = (a1, ..., an) denotes the set of actions of
+all agents. Then, the environment transits its state based on
+the transition probability P(s′|s, A). In MARL, all the agents
+will get rewards based on the transition of state and their ac-
+tions (or more likely joint action) ri = R(s, A). Each agent is
+supposed to get a higher accumulated reward �
+t rt
+i. There-
+fore, the agents optimize their policy to maximize the dis-
+count accumulated reward J(θ) = Eat,st[�
+t γtR(st, at)],
+where γ ∈ (0, 1] is the discount factor.
+MAPPO utilizes parameter sharing within homogeneous
+agents, i.e., homogeneous agents share the same set of net-
+work structure and parameters during training and testing.
+MAPPO is also a CTDE framework, namely, each PPO agent
+maintains an actor network πθ to learn the policy and a critic
+network Vφ(s) to learn the value function, where θ and φ are
+the parameters of policy network and value network, respec-
+tively. The value function requires the global state and only
+works during training procedures to reduce variance. In our
+work, we take MAPPO as our baseline and backbone, and
+add TEM as the communication mechanism into MAPPO.
+4
+Methods
+In this section, we introduce the detailed structure and de-
+sign of TEM. Before each action decision-making, the agents
+Figure 2: Workflow of TEM during one time step. A denotes the
+actor network, C denotes the critic network, E denotes the environ-
+ment. One training step has three phases: communication, action
+and learning. The execution only includes the first two phases and
+the critic will not work.
+communicate with each other following the designed proto-
+col, sharing the key information efficiently and selectively.
+We design a message module based on Transformer to en-
+code the messages received. At the same time, the module
+is able to decide whether to communicate and whom to com-
+municate with. The message module works together with the
+original action decision module from MAPPO, to form the
+actor network in the CTDE structure. The workflow of TEM
+is illustrated in Fig 2. We design an independent loss to en-
+courage the message module to maximize the messages’ im-
+pact on other agents. The whole model has the scalability to
+transfer from one scenario to another. As the message mod-
+ule is parallel to the action module, our model can be plugged
+into any CTDE structure.
+4.1
+Communication Protocol Design
+We design a communication protocol following the way how
+humans communicate by email. The information flow is like
+a forwarding chain: the chain starts with an agent with key in-
+formation to share, and the following agents merge their own
+information into the chain and then forward the new message
+to the next agent. The chain ends when the final agent finds
+the message useless for others, or there are no more potential
+communication objects.
+When designing the communication protocol, we mainly
+consider the following questions: (1) Whether to communi-
+cate? (2) Whom to communicate with? (3) What to commu-
+nicate? (4) How to utilize the messages?
+(1) Whether to communicate? As shown in Fig 2, in ev-
+ery step of execution and training, the first stage is commu-
+nication. When the communication stage is done, the actor
+networks for each agent will make the action decisions by
+the observations and messages. In the communication stage,
+each agent has the chance to decide whether to start a new
+chain and send a message. And the agents who receive mes-
+sages can decide whether to continue forwarding the mes-
+sages. Multiple message chains are allowed and the informa-
+tion from different chains is merged if there is a shared node
+for the chains.
+
+Communication !
+Action
+Critic & Actor
+Learning
+01'
+A
+a1
+m_a1 = 
+2
+m1
+02
+A
+a2
+m_a2
+m2
+E
++R
+V
+03
+A
+m_a3
+04
+A
+Execution
+TrainingFigure 3: Network structure of TEM. (a) Actor network of agent i, including an action network and a message network. Emb denotes the
+embedding network. (b) Encoder module. (c) Decoder module, where m_dec0
+i = o_fi.
+(2) Whom to communicate with?
+We think that for
+partial-observation (PO) problems, communication with all
+the agents is not reasonable and effective. The direct commu-
+nication with the agent outside the observation range may not
+bring helpful information as the sender does not even know
+the receiver’s state. Therefore, we do not model all the other
+agents to decide whether to communicate with them like in
+some previous works [Ding et al., 2020; Yuan et al., 2022].
+Instead, when the agent i chooses communication objects, we
+only consider the agents in the observation range Oi , and in
+our experiments, Oi includes the nearest several agents of the
+agent i. By training the message module, the agent can pre-
+dict the impact of the message on other agents, and is more
+likely to choose the one with the highest impact to communi-
+cate.
+We combine the two decisions (1) and (2) into one commu-
+nication action. The communication actions of agent i m_ai
+include not sending at all m_ai = 0, and sending to one agent
+j in the observation range m_ai = j, (j ∈ Oi). Namely, we
+have:
+P(m_ai = 0) +
+�
+j∈Oi
+(P(m_ai = j)) = 1.
+(2)
+This way, the agent can decide when and whom to communi-
+cate by one action, simplifying the modeling and learning.
+(3) What to communicate? To keep the information from
+the head nodes in the chain, and merge the information from
+different chains, every agent maintains a message buffer to
+store the messages. In practice, the message buffer is imple-
+mented as a queue, with a fixed storage length, but can flexi-
+bly push in and pop out elements as the communication goes
+(First Input First Output, FIFO). We use m_bi to denote all
+the messages inside the agent i ’s message buffer. When send-
+ing the new message, the agent i merges its own observation
+into the chain, then the message chain expands to ⟨m_bi, oi⟩.
+Here, the operation ⟨·⟩ demotes pushing into the queue. The
+buffer is clear when every step starts.
+(4) How to utilize the messages? Instead of some previ-
+ous works [Zhang et al., 2019; Yuan et al., 2022], we do not
+think the messages directly influence the value estimation of
+other agents is the natural way of communication. The infor-
+mation exchange should be separated from the information
+utilization. And the final effects of the messages should be
+determined by the receiver instead of the sender. Thus, in our
+model, messages are taken as a counterpart of the observa-
+tion, serving as part of the inputs of the actor network.
+4.2
+Network Design
+The schematics of the network design in our model are shown
+as Fig 3. Each agent has an actor network to observe the envi-
+ronment and communicate with other agents. The actor net-
+work of the agent i will output the action to interact with the
+environment ai, the action to communicate m_ai (whether
+to communicate and whom to communicate with), and the
+corresponding message to be sent mi. To better utilize the
+history information and get a smoother action sequence, an
+RNN is employed in the actor network. Thus the agent i also
+keeps a hidden state ht
+i, and updates it every time step.
+The actor network consists of two sub-networks, the ac-
+tion network and the message network. The action network
+mainly concentrates on the task itself and tries to get bet-
+ter rewards by outputting reasonable actions. The message
+network concentrates on the communication to share infor-
+mation with other agents instead. The two sub-networks ex-
+change the representation feature of the observations o_fi and
+that of the messages m_fi to merge the information.
+In the action network, o_fi is learned by a multi-layer
+perceptron (MLP), and then the action network concatenates
+o_fi and m_fi to input into the RNN together with the hidden
+state from the last time step ht−1
+i
+. Another MLP, in the end,
+processes the output of the RNN to generate the final action
+ai.
+On the other hand, the messages from other agents like
+mj · · · mk are stored in the message buffer, like the email
+inbox. The embedding layer (we implement it as a full con-
+nected (FC) layer by practice) converts the messages to fit
+
+Message Buffer
+FC
+FC
+FC
+Action
+Network
+Message
+Emb
+Self-Attention
+Network
+MLP
+m_enci
+Softmax
+o_fi
+Encoder
+× Ne
+m_fi
+MLP
+FC
+Attention
+cat
+m_fi
+Network
+o_f i
+Decoder
+Encoder Φ
+RNN
+m_deci
+MLP
+m_bi
+Decoder
+MLP
+i
+MLPthe input dimensions of the encoders. Ne sequential encoder
+modules and Nd sequential decoder modules are followed by
+the embedding layer. The output of encoder modules m_fi
+serves as the representation of all the messages in the buffer,
+with the key information emphasized by the attention mecha-
+nism. The decoder modules further combine the information
+from both of m_fi and o_fi to get the output m_deci. Fi-
+nally, one MLP produces the communication decision m_ai.
+For each encoder module, it takes in m_encne−1
+i
+from the
+embedding layer or the last encoder, then generates m_encne
+i
+as the input for the next layer. The transformer in the mod-
+ule can model the sequential information and is flexible to
+fit message chains with different lengths. Also, the attention
+mechanism will help the agent to pick out the key informa-
+tion from the chain. ne implies the position of the layer in
+the encoder sequence. To prevent gradient vanishing, the en-
+coder module employs the residual connections to link the
+self-attention mechanism and the MLP [Wen et al., 2022].
+The structure of the self-attention mechanism is the same as
+the attention network in the decoder while k, q and v are gen-
+erated from the same input m_encne−1
+i
+.
+In the decoder module, the first m_dec0
+i is the representa-
+tion of the observations o_fi. In the attention network, full
+connected layers generates key k and query q by m_decnd−1
+i
+and m_fi, respectively. Also, the third FC layer generates
+value v from m_decnd−1
+i
+. k and q are used for calculating
+the weights α of the value v as Equation 3.
+α = Softmax(exp(qkT
+√dk
+)).
+(3)
+In fact, the weight α learns the correlations between the
+m_decnd−1
+i
+and m_fi. By multiplying v and α, then we get
+the weighted representation of m_decnd−1
+i
+from the ending
+FC layer. With a similar structure of the residual connections
+and MLP, we get m_decnd
+i
+as the input for the next layer.
+4.3
+Loss Function Design
+The communication among the agents during a collaborative
+task aims to share the key information that one believes is
+useful for some specific other agents. So the learning of the
+message network is driven by the impact of the message to be
+sent.
+As the communication will not change either the action of
+other agents or the loss of the action network, an indepen-
+dent loss to model the influence of the messages on other
+agents’ actions is needed.
+We denote the communication
+loss as L(i)
+m (θ), where θ is the parameters of the actor net-
+work. The action of an agent j is sampled from the categorical
+distribution P(aj|oj, m_bj) learned by the action network.
+Then, when considering the new message from the agent i mi,
+we can estimate the distribution P(aj|oj, ⟨m_bj, mi⟩) as the
+consequence of the communication. Kullback-Leibler (KL)
+divergence is widely used to measure the discrepancy be-
+tween these two conditional probability distributions. Thus,
+the causal effect Γ(i)
+j
+of the message from agent i on agent j
+can be defined as:
+Γ(i)
+j
+= DKL (P(aj|oj, ⟨m_bj, mi⟩)||P(aj|oj, m_bj)) . (4)
+By considering all the possible agents to send the message to
+in the observation range, we can get the expectation of the
+causal effect of the message EΓ(i)(θ) by Equation 5:
+EΓ(i)(θ) =
+�
+j∈Oi
+�
+Pθ(m_ai = j|oi, m_bi)Γ(i)
+j
+�
+,
+(5)
+where Oi denotes the observation range of the agent i. The
+communication decision of agent i is sampled form the cat-
+egorical distribution Pθ(m_ai|oi, m_bi) learned by the mes-
+sage network. Pθ denotes that the gradient of this item should
+be propagated when training.
+The expectation EΓ(i)(θ) represents the overall effect the
+message mi can bring to the whole system, which we should
+maximize in the loss function.
+However, communication
+should also be sparse and efficient.
+If we do not control
+the communication times by the external guidance, the agents
+will tend to send as many messages as possible to get higher
+EΓ(i)(θ).
+Therefore, we also designed another item for
+communication loss to reduce the communication overhead.
+When the agent i chooses not to send the message to any
+agents in the observation range for most of the times, the
+probability Pθ(m_ai = 0|oi, m_bi) should be relatively high.
+So we need to maximize this probability at the same time.
+So far, we can get the final communication loss L(i)
+m (θ) by
+the following equation and maximize it when training.
+L(i)
+m (θ) = EΓ(i)(θ) + δPθ(m_ai = 0|oi, m_bi),
+(6)
+where δ is the weight of the communication reduction.
+The loss of the action network L(i)
+a (θ) is defined followed
+by Equation 1 in MAPPO as:
+L(i)
+a (θ) =min(r(i)
+θ A(i)
+πθold , clip(r(i)
+θ , 1 − ϵ, 1 + ϵ)A(i)
+πθold ),
+(7)
+where r(i)
+θ
+=
+πθ(a(i)|o(i))
+πθold(a(i)|o(i)), A(i)
+πθold is the advantage function.
+What’s more, to encourage more exploration when train-
+ing, we adopt an entropy loss L(i)
+e (θ) as [Yu et al., 2021]:
+L(i)
+e (θ) = S(πθ(oi)).
+(8)
+We can get the overall loss function for the actor network
+when training:
+L(θ) =
+n
+�
+i=1
+�
+L(i)
+a (θ) + λmL(i)
+m (θ) + λeL(i)
+e (θ)
+�
+,
+(9)
+where n is the number of the agents, and λm, λe are the coef-
+ficients to weight the corresponding losses.
+The critic network is trained to minimize the loss function
+L(φ) =
+n
+�
+i=1
+(max[(Vφ(s(i)) − R)2,
+(clip(Vφ(s(i)), Vφold(s(i)) − ϵ, Vφold(s(i)) + ϵ) − R)2]),
+(10)
+where R is the discounted accumulated reword.
+
+Figure 4: Test win rate for the SMAC map 5m vs. 6m, the shaded
+regions represent the 95% confidence intervals. FC: Full Communi-
+cation, RC: Randomly-stop Communication.
+5
+Experiments
+We evaluate the performance of TEM on three widely-used
+partially-observed multi-agent cooperative tasks: the Star-
+craft multi-agent challenge (SMAC), Predator Prey (PP) and
+Cooperative Navigation (CN). We compare the training pro-
+cess of TEM with the baselines and analyze the performance.
+We test the scalability of TEM to scenarios with different
+numbers of agents and targets when zero-shot transferring.
+5.1
+StarCraft II Micromanagement Benchmark
+In the SMAC task, N units controlled by the algorithm try
+to kill all the M enemies. There are usually more enemies
+than agents, or the enemies are more powerful types of units,
+so defeating all the enemies with limited observation range
+is challenging, demanding proper cooperation strategies and
+micro-control of movement and attack. We choose the hard
+map 5m vs 6m to evaluate TEM. TEM controls 5 Marines to
+fight with 6 enemy Marines.
+The baselines include MAPPO, MADDPG, Full Com-
+munication (FC) and Randomly-stop Communication (RC).
+MAPPO is the CTDE backbone we are using in the follow-
+ing experiments, which is proven to have state-of-the-art per-
+formance on several MARL cooperative benchmarks [Yu et
+al., 2021]. MADDPG is another classic CTDE approach for
+multi-agent cooperation tasks [Lowe et al., 2020]. FC and
+RC are two special cases of TEM. We keep the communica-
+tion protocol the same, but disable the decoder in the mes-
+sage module, instead, the agents choose the communication
+targets by pre-defined rules. In FC, the agent will keep ran-
+domly choosing someone to communicate with, to extend the
+message chain until no one is available. In RC, the agent will
+randomly stop the message chain by a probability p, or keep
+forwarding to a random one.
+We run the experiments over 6 seeds. For each seed, we
+compute the win rate over 32 test games after each training
+iteration as shown in Fig. 4.TEM gets the highest win rate
+over the baselines. FC and RC perform worse than MAPPO
+benchmark. One possible reason is that targeted communica-
+tion by TEM could improve cooperation while random com-
+munication by FC and RC may bring redundant information
+for decision-making. The win rate of baseline MADDPG re-
+mains zero, showing that it is hard to defeat an army with
+more units and MADDPG fails to learn such a strategy.
+Figure 5: Reward for Predator Prey (PP) during training, the shaded
+regions represent the 95% confidence intervals.
+5.2
+Predator Prey
+In the Predator Prey (PP) task, N predators try to chase, sur-
+round and finally capture M preys, as shown in Fig 1. The
+predators are the agents to be trained and the preys flee in the
+opposite direction of the closest predator at a faster speed fol-
+lowing pre-defined rules. So the predators have to be grouped
+automatically and cooperate to surround each prey, and it is
+impossible for one predator to capture a prey itself. In prac-
+tice, we set N as 7 and M as 3, denoted as 7-3 scenario.
+Different from the PP task in some previous works, here,
+the agents can only partially observe the teammates and tar-
+gets. The rewards are the sum of the agents’ negative dis-
+tances to their closest preys or landmarks. In addition, the
+agents are penalized for collisions with other agents.
+The baselines include MAPPO, I2C, MADDPG, DDPG,
+FC and RC. I2C proposes an individual communication
+mechanism [Ding et al., 2020]. DDPG is a classic deep re-
+inforcement learning algorithm for continuous control [Lill-
+icrap et al., 2019]. We apply DDPG independently to each
+agent as a baseline without considering cooperation.
+As shown in Fig 5, while other baselines gradually con-
+verge at the last episodes, TEM keeps raising the rewards and
+improves the final reward by 17.2% compared with MAPPO.
+5.3
+Cooperative Navigation
+In the Cooperative Navigation (CN) task, N agents try to
+occupy N stationary landmarks separately, as shown in Fig
+7. The positions of landmarks and agents are randomly ini-
+tialized. The best strategy is that each agent has a different
+target from the beginning through communication instead of
+rescheduling when collisions happen because of choosing the
+same target. In practice, we set N as 7, denoted as 7-7 sce-
+nario. The baselines and reward settings are the same as PP.
+We compare TEM with the baselines on the training perfor-
+mance Fig 6. We can see that TEM converges to the highest
+Figure 6: Reward for Cooperative Navigation (CN) during training,
+the shaded regions represent the 95% confidence intervals.
+
+WinRatefor5mvs6m
+1.0
+TEM
+MAPPO
+0.8
+MADDPG
+RC
+0.6
+FC
+Rate
+Win
+0.4
+0.2
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Step
+1e7RewardforPredatorPrey(PP)
+TEM
+-25
+MAPPO
+I2C
+-30
+DDPG
+MADDPG
+Reward
+RC
+-35
+FC
+-40
+-45
+-50
+2
+3
+4
+5
+Step
+1e7RewardforCooperative Navigation (CN)
+-40
+TEM
+MAPPO
+-50
+I2C
+DDPG
+MADDPG
+Reward
+-60
+RC
+FC
+-70
+-80
+-90
+3
+5
+Step
+1e7Figure 7: Comparison between (a) TEM and (b) MAPPO on the
+same environment of CN. Five frames are illustrated. Green lines
+are the trajectories and pink lines are message chains.
+reward than all the baselines. FC and RC are only slightly bet-
+ter than MAPPO, suggesting that the communication actions
+m_a learned by TEM are targeted, and the message chain
+brings helpful information to the ones that really need it.
+We compare the illustrations on CN between TEM and
+MAPPO in Fig 7. In (a), the TEM agents Agent 1 and Agent 4
+notice Landmark 1 by communication (pink message chain).
+Thus each agent moves straight forward to the corresponding
+landmark. While in (b), the MAPPO agents miss Landmark
+1, so for Agent 4, there will be nowhere to go. Agent 4 first
+tries to scramble with Agent 1 but fails, then turns to Agent
+2. Agent 2 is forced to leave to avoid collision and turns to
+Agent 3. We can see that communication brought by TEM
+can improve cooperation and reduce internal strife.
+5.4
+Scalability of TEM
+We further examine the scalability of TEM on PP task in Ta-
+ble 1. We take average episode rewards (R), successful cap-
+ture times (S), collision times (C) as the metrics. For R and
+S, the performance is better when the values are higher, while
+for C, the performance is better when the values are lower.
+Note that the existing MARL communication approaches are
+not scalable due to the modeling of each agent, the base-
+lines are the transferred MAPPO and the specifically trained
+MAPPO. We directly transfer the learned model from the 7-
+3 scenario to 9-3 and 3-1 scenarios without further training.
+For 9-3 scenario, two new agents are included, and it will be
+easier to capture the preys. But more agents also increase the
+risks of collision, so the cooperation mode could be different
+and the agents need to communicate to suit the new scenario.
+For TEM, the average episode rewards rise from -40.5 to -
+17.9, and the gain is 55.8%, while for MAPPO, the gain of
+rewards is 52.3%. TEM does not only perform better after
+transferring, but also gains more. For 3-1 scenario, both the
+numbers of the agents and preys change. The results show
+that TEM still keeps a better performance on all the metrics.
+Moreover, it shows that after TEM learns how to commu-
+nicate in a complex scenario, it can successfully transfer to
+simple ones.
+We also train MAPPO from scratch specifically on 9-3
+and 3-1 (denoted as MAPPO (learned)), and the performance
+of transferred TEM (trained on 7-3) is close to MAPPO
+(learned) on 9-3 without training. But the transferred TEM
+works worse on 3-1, and we suggest that cooperation by com-
+munication may not play an essential role in such a simple
+TEM (7-3)
+MAPPO (7-3)
+MAPPO (learned)
+TEM (finetuned)
+7-3
+R
+-40.5±4.7
+-44.9±4.3
+-
+-
+S
+61.6±18.3
+49.0±16.5
+-
+-
+C
+1.4±0.6
+12.6±2.6
+-
+-
+3-1
+R
+-10.6±4.7
+-12.7±5.9
+-7.52±2.6
+-7.0±2.8
+S
+18.7±14.0
+13.5±12.7
+36.5±13.2
+31.7±13.4
+C
+1.2±1.2
+3.6±2.7
+1.8±1.8
+0.8 ±0.6
+9-3
+R
+-17.9±5.0
+-21.3±7.5
+-17.1± 8.2
+-14.0 ±2.6
+S
+107.2±23.6
+69.3±32.0
+109.5±18.7
+127.9±5.9
+C
+16.2±1.8
+30.6±6.7
+7.2±1.9
+5.4±1.6
+Table 1: Scalability of TEM on PP. R: average episode rewards, S:
+successful capture times, C: collision times. TEM (7-3) and MAPPO
+(7-3) are trained on the scenario 7-3: 7 agents to capture 3 preys,
+and tested on ten random environments on 7-3, 3-1, 9-3 scenarios.
+MAPPO (learned) is specifically trained from scratch on the corre-
+sponding test environments. TEM (finetuned) is the TEM model
+trained on 7-3 and tuned on the corresponding test environments.
+environment. We further finetune TEM (7-3) on the new sce-
+narios and the finetuned models even outperform the specially
+learned MAPPO.
+Similar experiments are conducted on CN as shown in Ta-
+ble 2. TEM keeps the scalability when transferred from 7-
+7 scenario to 6-6 and 9-9, and outperforms the transferred
+MAPPO. Surprisingly, the transferred TEM even outper-
+forms the MAPPO trained from scratch (denoted as MAPPO
+(learned)) on most metrics. It suggests that CN requires more
+communication to coordinate the agents to explore the land-
+marks at the corner. And the results also show that the com-
+munication pattern learned from 7-7 still works well in other
+scenarios. Similarly, the finetuned TEM gets even better per-
+formance.
+TEM (7-7)
+MAPPO (7-7)
+MAPPO (learned)
+TEM (finetuned)
+7-7
+R
+-38.8±15.1
+-46.6±14.8
+-
+-
+S
+35.8± 6.2
+23.3±9.0
+-
+-
+C
+2.8±0.3
+4.2±0.2
+-
+-
+6-6
+R
+-39.8±5.3
+-45.0±8.0
+-43.6±10.1
+-36.7 ±5.2
+S
+35.3±6.2
+20.0±5.9
+19.3±6.8
+36.1±5.9
+C
+7.2±0.4
+8.4±0.4
+4.8±0.4
+4.8 ±0.2
+9-9
+R
+-45.8±23.9
+-57.5±25.3
+-50.8±12.9
+-41.1 ±11.4
+S
+39.4±4.9
+26.6±8.7
+29.0±6.6
+38.9±5.2
+C
+9.0±0.3
+28.8±0.4
+10.8±0.2
+8.0±0.2
+Table 2: Scalability of TEM on CN. TEM (7-7) and MAPPO (7-7)
+are trained on the scenario 7-7: 7 agents to occupy 7 landmarks, and
+tested on ten random environments on 7-3, 6-6, 9-9 scenarios.
+6
+Conclusions
+To tackle the scalability problem of MARL communication,
+this paper proposes a novel framework Transformer-based
+Email Mechanism (TEM). The agents adopt local communi-
+cation to send and forward messages like emails to form mes-
+sage chains, which set up bridges among partial-observation
+ranges. We introduce Transformer to encode and decode the
+message chain to choose the next receiver selectively. Em-
+pirical results in diverse multi-agent cooperative tasks show
+that our method outperforms the baselines. Furthermore, we
+can directly apply TEM to a new environment with a different
+number of agents without retraining. Better performance than
+the baselines when zero-shot transferring shows the scalabil-
+ity of TEM. Based on TEM, communication for hundreds of
+agents and further tailored message generation can be devel-
+oped, which may be an important step for MARL applications
+to real-world tasks.
+
+Agent 1
+Agent 1
+Agent 4
+Agent 4
+Agent 2
+Agent 3
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf,len=649
+page_content='Scalable Communication for Multi-Agent Reinforcement Learning  via Transformer-Based Email Mechanism  Xudong Guo , Daming Shi , Wenhui Fan  Department of Automation, Tsinghua University  {gxd20, shidm18}@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='cn, fanwenhui@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='cn  Abstract  Communication can impressively improve co-  operation in multi-agent reinforcement learning  (MARL), especially for partially-observed tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  However, existing works either broadcast the mes-  sages leading to information redundancy, or learn  targeted communication by modeling all the other  agents as targets, which is not scalable when the  number of agents varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' In this work, to tackle  the scalability problem of MARL communication  for partially-observed tasks, we propose a novel  framework Transformer-based Email Mecha-  nism (TEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The agents adopt local communica-  tion to send messages only to the ones that can  be observed without modeling all the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' In-  spired by human cooperation with email forward-  ing, we design message chains to forward informa-  tion to cooperate with the agents outside the ob-  servation range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' We introduce Transformer to en-  code and decode the message chain to choose the  next receiver selectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Empirically, TEM outper-  forms the baselines on multiple cooperative MARL  benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' When the number of agents varies,  TEM maintains superior performance without fur-  ther training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  1  Introduction  Multi-agent reinforcement learning (MARL) has achieved re-  markable success in many complex challenges, especially in  game playing [OpenAI et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  MARL shows great potential to solve cooperative multi-agent  real-world tasks, such as autonomous vehicle teams [Shalev-  Shwartz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2016], robotics control [Kober et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2013]  and intelligent traffic control [Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' However,  some essential obstacles still exist for MARL to reach satis-  factory performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' When training the MARL algorithms,  the agents keep updating their policies and causing dynam-  ics in the environment, which may hinder the model con-  vergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Worse still, in most cooperative multi-agent tasks,  agents can only observe part of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Partial ob-  servability and non-stationarity make it harder to success-  fully cooperate, even though some works employ centralized  Figure 1:  Message chain formed by email forwarding in the  Transformer-based Email Mechanism (TEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The agents (circles)  are trying to surround and capture the target (square).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The dotted  circle is the observation range for the agent with the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  The black lines are message chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' ⟨·⟩ demotes concatenating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The  cross denotes not sending or forwarding after receiver selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The  agents a, b and c indirectly cooperate by sending (ma) and forward-  ing (⟨ma, mb⟩) messages to capture the target T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' As they are not in  the same observation range, forwarding like emails is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  training and decentralized execution (CTDE) paradigm to im-  port a critic to coordinate the whole team [Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  Lowe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Rashid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  Inspired by the ways how humans and animals cooperate,  communication is introduced to share information between  agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  Some works broadcast the messages to all the  other agents [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Sukhbaatar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  Foerster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2016], and other recent works try to learn  targeted peer-to-peer communication to reduce the commu-  nication bandwidth [Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Jiang and Lu, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Attention mechanism from Transformer  [Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2017] is also employed to learn the commu-  nication [Jiang and Lu, 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' However, the existing meth-  ods rely on modeling every teammate in the environment by  ID to decide whether to communicate, which will bring huge  computational overhead when the number of agents is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  As the modeling network is trained by a specific amount of  IDs, the learned communication mechanism is not scalable  to reuse when the number of agents changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' In fact, the  agent cannot know the state of an agent outside the observa-  tion range, and cannot judge whether the information is useful  for it, so it is unreasonable to directly share information with  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='01919v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='MA]  5 Jan 2023  d  e  区  T  ar  Csuch an agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' For example, for applications to autonomous  vehicles, only the vehicles nearby are worth communicating  with to avoid collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Thus, global communication with  all vehicles is unnecessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Moreover, communication with  other vehicles should adapt to different numbers of agents as  the traffic situation varies a lot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  In this work, to tackle this new problem - the scalability  of MARL communication, we propose a scalable multi-agent  communication mechanism via Transformer-based Email  Mechanism (TEM) to tackle the abovementioned challenges  as shown in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' We adopt local communication to send  messages only to the agents in the observation range, with-  out modeling all the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The agent will decide whom to  communicate with by its own intention and by observing the  agents in the range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Thus, no matter how the overall num-  ber of the agents changes, the learned communication mech-  anism is scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' To better utilize the key information and  indirectly cooperate with the agents outside the range, we de-  sign a new communication mechanism like email forward-  ing to form a message chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The agents can send and for-  ward the messages so that the chain connects agents from  different ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' For example, the agent a in Fig 1 would  like to surround and capture the target T with other agents,  thus the agent a may send a message to the agent b instead of  d, though d is the nearest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Then the agent b can forward the  message together with the information from itself to c, so that  a, b and c can cooperate for the same goal though there is no  direct communication between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Similarly, in our daily  life, cooperation in a big company or organization relies on  such forwarding emails to share information, as it is always  hard to directly find the exact contact in another department.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  To suit the unfixed length of the message chain and ensure  the communication mechanism is scalable, we design a new  message network and employ Transformer to encode and de-  code the sequence of messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Furthermore, augmented by  the attention mechanism in the Transformer, the communi-  cation is selective by modeling the correlation between the  messages and the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The message network is inde-  pendent and can be plugged into any CTDE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' What’s  more, we design a loss to guide the agent to estimate the im-  pact of the message on other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Note that we do not  introduce the broadcast mechanism from email to keep the  communication efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  For evaluation, we test TEM on three partial-observation  cooperative MARL benchmarks: the Starcraft multi-agent  challenge (SMAC) [Samvelyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2019], Predator Prey  (PP) [Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2019] and Cooperative Navigation (CN)  [Lowe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2020], where TEM reaches better performance  than the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' We also evaluate the scalability of TEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  Without extra training, TEM can suit both situations where  the number of agents increases and decreases, and still out-  performs the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  2  Related Works  Learning how to communicate is a popular research domain  in multi-agent reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Researches in this do-  main focus mainly on cooperative scenarios, where agents  could communicate with each other explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' In the early  works, RIAL and DIAL [Foerster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2016] are designed  to learn communication, where messages are delivered from  one timestep to the next timestep in a broadcast way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Comm-  Net [Sukhbaatar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2016] proposes a hidden layer as com-  munication and allows the agents to communicate repeatedly  in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' IC3Net [Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2018] brings in the gat-  ing mechanism to control communication based on Comm-  Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Both of BiC-Net [Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2017] and ATOC [Jiang  and Lu, 2018] implement the communication layer as bidi-  rectional RNN, which inputs the observations of all agents  and outputs the action or the integrated thought of each agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  However, these methods either broadcast the messages or rely  on a centralized communication layer, which is high-cost and  not stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Communication should not only serve as an inte-  gration of information, instead, the agents should share infor-  mation selectively through peer-to-peer communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  To avoid broadcasting messages, recent works try to de-  sign more intelligent communication mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  CTDE  paradigms are also imported to implement decentralized com-  munication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  Some works are based on QMIX [Rashid et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2018]: VBC [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2019] proposes a request-  reply mechanism and a variance-based regularizer to elimi-  nate the noisy components in messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' NDQ [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=',  2020] learns nearly decomposable value functions with com-  munication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' TMC [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2020] maximizes the mu-  tual information between the decentralized Q functions and  the communication messages while minimizing the entropy  of messages between agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' MAIC [Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2022] al-  lows each agent to learn to generate incentive messages by  modeling every teammate and bias other agents’ value func-  tions directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Some are based on another CTDE framework  MADDPG [Lowe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2020]: TarMAC [Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2019]  proposes a targeted communication behavior via a signature-  based soft attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Besides the message, the  sender broadcasts a key used by the receivers to gauge the  message’s relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' DAACMP [Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2020] adds a  double attention mechanism in the actor and critic network  respectively to select and process the important messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  I2C [Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2020] learns a prior net via causal inference  for peer-to-peer communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The influence of one agent  on another is inferred via the joint action-value function and  quantified to label the necessity of peer-to-peer communica-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Nevertheless, the methods above need to model every  other agent in the environment to achieve individual commu-  nication, which is not scalable and practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  To the best of our knowledge, none of the existing MARL  communication methods considers the scalability of the com-  munication mechanism and the forwarding protocol inspired  by email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  3  Background  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='1  Policy Gradient (PG) Reinforcement Learning  Policy Gradient (PG) reinforcement learning has the advan-  tage of learning a policy network explicitly, in contrast to  value-based reinforcement learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' PG methods  optimize the policy parameter θ to maximize its objective  function J(θ) = ES(Vπθ(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' However, due to the variance  of environments, it is hard to choose a subtle learning rate in  reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' To resolve this problem and ensure  the safe optimization of policy learning, the Trust Region Pol-  icy Optimization (TRPO) [Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2015] increases  the constraint of the parameter difference between policy up-  dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Such constraint ensures the parameter changes in a  small range, so that the collapse of value can be avoided and  the policy can learn monotonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The parameter update of  TRPO is θk+1 = arg maxθL(θk, θ) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' ¯DKL(θ||θk) ≤ δ,  where L(θk, θ) = E[ πθ(a|s)  πθk (a|s)Aπθk (s, a)] is the approxima-  tion of the original policy gradient object J(θ) within the  constraint of KL divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  Based on TRPO, a simplified version Proximal Policy Op-  timization [Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2017] is carried out, maintaining  the motivation to constrain the learning step while more ef-  ficient and easy to be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The object function of  PPO can be written as:  L(s, a, θk, θ) =min[ πθ(a|s)  πθk(a|s)Aπθk (s, a),  clip( πθ(a|s)  πθk(a|s)), 1 − ϵ, 1 + ϵ)Aπθk (s, a)],  (1)  which forces the ratio of πθ(a|s)  πθk (a|s) to locate in the interval (1−  ϵ, 1 + ϵ), so that the new θ is not too far away from old θk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='2  MAPPO Algorithm  Multi-agent PPO (MAPPO) introduces PPO into the multi-  agent scenario [Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' MAPPO mainly considers  decentralized partially observable Markov decision processes  (DEC-POMDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' In an environment with n agents, s ∈ S de-  notes the state of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The agent i only has a  local observation of environment oi = O(s) and chooses its  action based on its observation and policy ai = πi(ai|oi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  The joint action A = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', an) denotes the set of actions of  all agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Then, the environment transits its state based on  the transition probability P(s′|s, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' In MARL, all the agents  will get rewards based on the transition of state and their ac-  tions (or more likely joint action) ri = R(s, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Each agent is  supposed to get a higher accumulated reward �  t rt  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' There-  fore, the agents optimize their policy to maximize the dis-  count accumulated reward J(θ) = Eat,st[�  t γtR(st, at)],  where γ ∈ (0, 1] is the discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  MAPPO utilizes parameter sharing within homogeneous  agents, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', homogeneous agents share the same set of net-  work structure and parameters during training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  MAPPO is also a CTDE framework, namely, each PPO agent  maintains an actor network πθ to learn the policy and a critic  network Vφ(s) to learn the value function, where θ and φ are  the parameters of policy network and value network, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The value function requires the global state and only  works during training procedures to reduce variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' In our  work, we take MAPPO as our baseline and backbone, and  add TEM as the communication mechanism into MAPPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  4  Methods  In this section, we introduce the detailed structure and de-  sign of TEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Before each action decision-making, the agents  Figure 2: Workflow of TEM during one time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' A denotes the  actor network, C denotes the critic network, E denotes the environ-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' One training step has three phases: communication, action  and learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The execution only includes the first two phases and  the critic will not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  communicate with each other following the designed proto-  col, sharing the key information efficiently and selectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  We design a message module based on Transformer to en-  code the messages received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' At the same time, the module  is able to decide whether to communicate and whom to com-  municate with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The message module works together with the  original action decision module from MAPPO, to form the  actor network in the CTDE structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The workflow of TEM  is illustrated in Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' We design an independent loss to en-  courage the message module to maximize the messages’ im-  pact on other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The whole model has the scalability to  transfer from one scenario to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' As the message mod-  ule is parallel to the action module, our model can be plugged  into any CTDE structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='1  Communication Protocol Design  We design a communication protocol following the way how  humans communicate by email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The information flow is like  a forwarding chain: the chain starts with an agent with key in-  formation to share, and the following agents merge their own  information into the chain and then forward the new message  to the next agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The chain ends when the final agent finds  the message useless for others, or there are no more potential  communication objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  When designing the communication protocol, we mainly  consider the following questions: (1) Whether to communi-  cate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' (2) Whom to communicate with?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' (3) What to commu-  nicate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' (4) How to utilize the messages?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  (1) Whether to communicate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' As shown in Fig 2, in ev-  ery step of execution and training, the first stage is commu-  nication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' When the communication stage is done, the actor  networks for each agent will make the action decisions by  the observations and messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' In the communication stage,  each agent has the chance to decide whether to start a new  chain and send a message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' And the agents who receive mes-  sages can decide whether to continue forwarding the mes-  sages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Multiple message chains are allowed and the informa-  tion from different chains is merged if there is a shared node  for the chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  Communication !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content="  Action  Critic & Actor  Learning  01'  A  a1  m_a1 =  2  m1  02  A  a2  m_a2  m2  E  +R  V  03  A  m_a3  04  A  Execution  TrainingFigure 3: Network structure of TEM." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' (a) Actor network of agent i, including an action network and a message network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Emb denotes the  embedding network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' (b) Encoder module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' (c) Decoder module, where m_dec0  i = o_fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  (2) Whom to communicate with?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  We think that for  partial-observation (PO) problems, communication with all  the agents is not reasonable and effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The direct commu-  nication with the agent outside the observation range may not  bring helpful information as the sender does not even know  the receiver’s state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Therefore, we do not model all the other  agents to decide whether to communicate with them like in  some previous works [Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  Instead, when the agent i chooses communication objects, we  only consider the agents in the observation range Oi , and in  our experiments, Oi includes the nearest several agents of the  agent i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' By training the message module, the agent can pre-  dict the impact of the message on other agents, and is more  likely to choose the one with the highest impact to communi-  cate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  We combine the two decisions (1) and (2) into one commu-  nication action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The communication actions of agent i m_ai  include not sending at all m_ai = 0, and sending to one agent  j in the observation range m_ai = j, (j ∈ Oi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Namely, we  have:  P(m_ai = 0) +  �  j∈Oi  (P(m_ai = j)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  (2)  This way, the agent can decide when and whom to communi-  cate by one action, simplifying the modeling and learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  (3) What to communicate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' To keep the information from  the head nodes in the chain, and merge the information from  different chains, every agent maintains a message buffer to  store the messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' In practice, the message buffer is imple-  mented as a queue, with a fixed storage length, but can flexi-  bly push in and pop out elements as the communication goes  (First Input First Output, FIFO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' We use m_bi to denote all  the messages inside the agent i ’s message buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' When send-  ing the new message, the agent i merges its own observation  into the chain, then the message chain expands to ⟨m_bi, oi⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  Here, the operation ⟨·⟩ demotes pushing into the queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The  buffer is clear when every step starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  (4) How to utilize the messages?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Instead of some previ-  ous works [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2022], we do not  think the messages directly influence the value estimation of  other agents is the natural way of communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The infor-  mation exchange should be separated from the information  utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' And the final effects of the messages should be  determined by the receiver instead of the sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Thus, in our  model, messages are taken as a counterpart of the observa-  tion, serving as part of the inputs of the actor network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='2  Network Design  The schematics of the network design in our model are shown  as Fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Each agent has an actor network to observe the envi-  ronment and communicate with other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The actor net-  work of the agent i will output the action to interact with the  environment ai, the action to communicate m_ai (whether  to communicate and whom to communicate with), and the  corresponding message to be sent mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' To better utilize the  history information and get a smoother action sequence, an  RNN is employed in the actor network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Thus the agent i also  keeps a hidden state ht  i, and updates it every time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  The actor network consists of two sub-networks, the ac-  tion network and the message network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The action network  mainly concentrates on the task itself and tries to get bet-  ter rewards by outputting reasonable actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The message  network concentrates on the communication to share infor-  mation with other agents instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The two sub-networks ex-  change the representation feature of the observations o_fi and  that of the messages m_fi to merge the information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  In the action network, o_fi is learned by a multi-layer  perceptron (MLP), and then the action network concatenates  o_fi and m_fi to input into the RNN together with the hidden  state from the last time step ht−1  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Another MLP, in the end,  processes the output of the RNN to generate the final action  ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  On the other hand, the messages from other agents like  mj · · · mk are stored in the message buffer, like the email  inbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The embedding layer (we implement it as a full con-  nected (FC) layer by practice) converts the messages to fit  Message Buffer  FC  FC  FC  Action  Network  Message  Emb  Self-Attention  Network  MLP  m_enci  Softmax  o_fi  Encoder  × Ne  m_fi  MLP  FC  Attention  cat  m_fi  Network  o_f i  Decoder  Encoder Φ  RNN  m_deci  MLP  m_bi  Decoder  MLP  i  MLPthe input dimensions of the encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Ne sequential encoder  modules and Nd sequential decoder modules are followed by  the embedding layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The output of encoder modules m_fi  serves as the representation of all the messages in the buffer,  with the key information emphasized by the attention mecha-  nism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The decoder modules further combine the information  from both of m_fi and o_fi to get the output m_deci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Fi-  nally, one MLP produces the communication decision m_ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  For each encoder module, it takes in m_encne−1  i  from the  embedding layer or the last encoder, then generates m_encne  i  as the input for the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The transformer in the mod-  ule can model the sequential information and is flexible to  fit message chains with different lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Also, the attention  mechanism will help the agent to pick out the key informa-  tion from the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' ne implies the position of the layer in  the encoder sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' To prevent gradient vanishing, the en-  coder module employs the residual connections to link the  self-attention mechanism and the MLP [Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  The structure of the self-attention mechanism is the same as  the attention network in the decoder while k, q and v are gen-  erated from the same input m_encne−1  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  In the decoder module, the first m_dec0  i is the representa-  tion of the observations o_fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' In the attention network, full  connected layers generates key k and query q by m_decnd−1  i  and m_fi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Also, the third FC layer generates  value v from m_decnd−1  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' k and q are used for calculating  the weights α of the value v as Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  α = Softmax(exp(qkT  √dk  )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  (3)  In fact, the weight α learns the correlations between the  m_decnd−1  i  and m_fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' By multiplying v and α, then we get  the weighted representation of m_decnd−1  i  from the ending  FC layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' With a similar structure of the residual connections  and MLP, we get m_decnd  i  as the input for the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='3  Loss Function Design  The communication among the agents during a collaborative  task aims to share the key information that one believes is  useful for some specific other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' So the learning of the  message network is driven by the impact of the message to be  sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  As the communication will not change either the action of  other agents or the loss of the action network, an indepen-  dent loss to model the influence of the messages on other  agents’ actions is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  We denote the communication  loss as L(i)  m (θ), where θ is the parameters of the actor net-  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The action of an agent j is sampled from the categorical  distribution P(aj|oj, m_bj) learned by the action network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  Then, when considering the new message from the agent i mi,  we can estimate the distribution P(aj|oj, ⟨m_bj, mi⟩) as the  consequence of the communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Kullback-Leibler (KL)  divergence is widely used to measure the discrepancy be-  tween these two conditional probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Thus,  the causal effect Γ(i)  j  of the message from agent i on agent j  can be defined as:  Γ(i)  j  = DKL (P(aj|oj, ⟨m_bj, mi⟩)||P(aj|oj, m_bj)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' (4)  By considering all the possible agents to send the message to  in the observation range, we can get the expectation of the  causal effect of the message EΓ(i)(θ) by Equation 5:  EΓ(i)(θ) =  �  j∈Oi  �  Pθ(m_ai = j|oi, m_bi)Γ(i)  j  �  ,  (5)  where Oi denotes the observation range of the agent i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The  communication decision of agent i is sampled form the cat-  egorical distribution Pθ(m_ai|oi, m_bi) learned by the mes-  sage network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Pθ denotes that the gradient of this item should  be propagated when training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  The expectation EΓ(i)(θ) represents the overall effect the  message mi can bring to the whole system, which we should  maximize in the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  However, communication  should also be sparse and efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  If we do not control  the communication times by the external guidance, the agents  will tend to send as many messages as possible to get higher  EΓ(i)(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  Therefore, we also designed another item for  communication loss to reduce the communication overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  When the agent i chooses not to send the message to any  agents in the observation range for most of the times, the  probability Pθ(m_ai = 0|oi, m_bi) should be relatively high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  So we need to maximize this probability at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  So far, we can get the final communication loss L(i)  m (θ) by  the following equation and maximize it when training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  L(i)  m (θ) = EΓ(i)(θ) + δPθ(m_ai = 0|oi, m_bi),  (6)  where δ is the weight of the communication reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  The loss of the action network L(i)  a (θ) is defined followed  by Equation 1 in MAPPO as:  L(i)  a (θ) =min(r(i)  θ A(i)  πθold , clip(r(i)  θ , 1 − ϵ, 1 + ϵ)A(i)  πθold ),  (7)  where r(i)  θ  =  πθ(a(i)|o(i))  πθold(a(i)|o(i)), A(i)  πθold is the advantage function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  What’s more, to encourage more exploration when train-  ing, we adopt an entropy loss L(i)  e (θ) as [Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2021]:  L(i)  e (θ) = S(πθ(oi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  (8)  We can get the overall loss function for the actor network  when training:  L(θ) =  n  �  i=1  �  L(i)  a (θ) + λmL(i)  m (θ) + λeL(i)  e (θ)  �  ,  (9)  where n is the number of the agents, and λm, λe are the coef-  ficients to weight the corresponding losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  The critic network is trained to minimize the loss function  L(φ) =  n  �  i=1  (max[(Vφ(s(i)) − R)2,  (clip(Vφ(s(i)), Vφold(s(i)) − ϵ, Vφold(s(i)) + ϵ) − R)2]),  (10)  where R is the discounted accumulated reword.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  Figure 4: Test win rate for the SMAC map 5m vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' 6m, the shaded  regions represent the 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' FC: Full Communi-  cation, RC: Randomly-stop Communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  5  Experiments  We evaluate the performance of TEM on three widely-used  partially-observed multi-agent cooperative tasks: the Star-  craft multi-agent challenge (SMAC), Predator Prey (PP) and  Cooperative Navigation (CN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' We compare the training pro-  cess of TEM with the baselines and analyze the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  We test the scalability of TEM to scenarios with different  numbers of agents and targets when zero-shot transferring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='1  StarCraft II Micromanagement Benchmark  In the SMAC task, N units controlled by the algorithm try  to kill all the M enemies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' There are usually more enemies  than agents, or the enemies are more powerful types of units,  so defeating all the enemies with limited observation range  is challenging, demanding proper cooperation strategies and  micro-control of movement and attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' We choose the hard  map 5m vs 6m to evaluate TEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' TEM controls 5 Marines to  fight with 6 enemy Marines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  The baselines include MAPPO, MADDPG, Full Com-  munication (FC) and Randomly-stop Communication (RC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  MAPPO is the CTDE backbone we are using in the follow-  ing experiments, which is proven to have state-of-the-art per-  formance on several MARL cooperative benchmarks [Yu et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' MADDPG is another classic CTDE approach for  multi-agent cooperation tasks [Lowe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' FC and  RC are two special cases of TEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' We keep the communica-  tion protocol the same, but disable the decoder in the mes-  sage module, instead, the agents choose the communication  targets by pre-defined rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' In FC, the agent will keep ran-  domly choosing someone to communicate with, to extend the  message chain until no one is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' In RC, the agent will  randomly stop the message chain by a probability p, or keep  forwarding to a random one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  We run the experiments over 6 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' For each seed, we  compute the win rate over 32 test games after each training  iteration as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='TEM gets the highest win rate  over the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' FC and RC perform worse than MAPPO  benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' One possible reason is that targeted communica-  tion by TEM could improve cooperation while random com-  munication by FC and RC may bring redundant information  for decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The win rate of baseline MADDPG re-  mains zero, showing that it is hard to defeat an army with  more units and MADDPG fails to learn such a strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  Figure 5: Reward for Predator Prey (PP) during training, the shaded  regions represent the 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='2  Predator Prey  In the Predator Prey (PP) task, N predators try to chase, sur-  round and finally capture M preys, as shown in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The  predators are the agents to be trained and the preys flee in the  opposite direction of the closest predator at a faster speed fol-  lowing pre-defined rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' So the predators have to be grouped  automatically and cooperate to surround each prey, and it is  impossible for one predator to capture a prey itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' In prac-  tice, we set N as 7 and M as 3, denoted as 7-3 scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  Different from the PP task in some previous works, here,  the agents can only partially observe the teammates and tar-  gets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The rewards are the sum of the agents’ negative dis-  tances to their closest preys or landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' In addition, the  agents are penalized for collisions with other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  The baselines include MAPPO, I2C, MADDPG, DDPG,  FC and RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' I2C proposes an individual communication  mechanism [Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' DDPG is a classic deep re-  inforcement learning algorithm for continuous control [Lill-  icrap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' We apply DDPG independently to each  agent as a baseline without considering cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  As shown in Fig 5, while other baselines gradually con-  verge at the last episodes, TEM keeps raising the rewards and  improves the final reward by 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='2% compared with MAPPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='3  Cooperative Navigation  In the Cooperative Navigation (CN) task, N agents try to  occupy N stationary landmarks separately, as shown in Fig  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The positions of landmarks and agents are randomly ini-  tialized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The best strategy is that each agent has a different  target from the beginning through communication instead of  rescheduling when collisions happen because of choosing the  same target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' In practice, we set N as 7, denoted as 7-7 sce-  nario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The baselines and reward settings are the same as PP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  We compare TEM with the baselines on the training perfor-  mance Fig 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' We can see that TEM converges to the highest  Figure 6: Reward for Cooperative Navigation (CN) during training,  the shaded regions represent the 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  WinRatefor5mvs6m  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='0  TEM  MAPPO  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='8  MADDPG  RC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='6  FC  Rate  Win  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='0  Step  1e7RewardforPredatorPrey(PP)  TEM  25  MAPPO  I2C  30  DDPG  MADDPG  Reward  RC  35  FC  40  45  50  2  3  4  5  Step  1e7RewardforCooperative Navigation (CN)  40  TEM  MAPPO  50  I2C  DDPG  MADDPG  Reward  60  RC  FC  70  80  90  3  5  Step  1e7Figure 7: Comparison between (a) TEM and (b) MAPPO on the  same environment of CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Five frames are illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Green lines  are the trajectories and pink lines are message chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  reward than all the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' FC and RC are only slightly bet-  ter than MAPPO, suggesting that the communication actions  m_a learned by TEM are targeted, and the message chain  brings helpful information to the ones that really need it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  We compare the illustrations on CN between TEM and  MAPPO in Fig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' In (a), the TEM agents Agent 1 and Agent 4  notice Landmark 1 by communication (pink message chain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  Thus each agent moves straight forward to the corresponding  landmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' While in (b), the MAPPO agents miss Landmark  1, so for Agent 4, there will be nowhere to go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Agent 4 first  tries to scramble with Agent 1 but fails, then turns to Agent  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Agent 2 is forced to leave to avoid collision and turns to  Agent 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' We can see that communication brought by TEM  can improve cooperation and reduce internal strife.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='4  Scalability of TEM  We further examine the scalability of TEM on PP task in Ta-  ble 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' We take average episode rewards (R), successful cap-  ture times (S), collision times (C) as the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' For R and  S, the performance is better when the values are higher, while  for C, the performance is better when the values are lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  Note that the existing MARL communication approaches are  not scalable due to the modeling of each agent, the base-  lines are the transferred MAPPO and the specifically trained  MAPPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' We directly transfer the learned model from the 7-  3 scenario to 9-3 and 3-1 scenarios without further training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  For 9-3 scenario, two new agents are included, and it will be  easier to capture the preys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' But more agents also increase the  risks of collision, so the cooperation mode could be different  and the agents need to communicate to suit the new scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  For TEM, the average episode rewards rise from -40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='5 to -  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='9, and the gain is 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='8%, while for MAPPO, the gain of  rewards is 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' TEM does not only perform better after  transferring, but also gains more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' For 3-1 scenario, both the  numbers of the agents and preys change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The results show  that TEM still keeps a better performance on all the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  Moreover, it shows that after TEM learns how to commu-  nicate in a complex scenario, it can successfully transfer to  simple ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  We also train MAPPO from scratch specifically on 9-3  and 3-1 (denoted as MAPPO (learned)), and the performance  of transferred TEM (trained on 7-3) is close to MAPPO  (learned) on 9-3 without training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' But the transferred TEM  works worse on 3-1, and we suggest that cooperation by com-  munication may not play an essential role in such a simple  TEM (7-3)  MAPPO (7-3)  MAPPO (learned)  TEM (finetuned)  7-3  R  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='5±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='7  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='9±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='3 S  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='6±18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='3  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='0±16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='5 C  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='6  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='6±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='6 3-1  R  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='6±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='7  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='7±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='9  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='52±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='6  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='0±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='8  S  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
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+page_content='6  Table 1: Scalability of TEM on PP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' R: average episode rewards, S:  successful capture times, C: collision times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' TEM (7-3) and MAPPO  (7-3) are trained on the scenario 7-3: 7 agents to capture 3 preys,  and tested on ten random environments on 7-3, 3-1, 9-3 scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  MAPPO (learned) is specifically trained from scratch on the corre-  sponding test environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' TEM (finetuned) is the TEM model  trained on 7-3 and tuned on the corresponding test environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' We further finetune TEM (7-3) on the new sce-  narios and the finetuned models even outperform the specially  learned MAPPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  Similar experiments are conducted on CN as shown in Ta-  ble 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' TEM keeps the scalability when transferred from 7-  7 scenario to 6-6 and 9-9, and outperforms the transferred  MAPPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Surprisingly, the transferred TEM even outper-  forms the MAPPO trained from scratch (denoted as MAPPO  (learned)) on most metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' It suggests that CN requires more  communication to coordinate the agents to explore the land-  marks at the corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' And the results also show that the com-  munication pattern learned from 7-7 still works well in other  scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Similarly, the finetuned TEM gets even better per-  formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  TEM (7-7)  MAPPO (7-7)  MAPPO (learned)  TEM (finetuned)  7-7  R  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='8±15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='1  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='6±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='8 S  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='8± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='2  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
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+page_content='9  C  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
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+page_content='3  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
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+page_content='9  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='1 ±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='4  S  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='4±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='9  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='6±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='7  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='0±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='6  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='9±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='2  C  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='3  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='4  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='2  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='2  Table 2: Scalability of TEM on CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' TEM (7-7) and MAPPO (7-7)  are trained on the scenario 7-7: 7 agents to occupy 7 landmarks, and  tested on ten random environments on 7-3, 6-6, 9-9 scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content='  6  Conclusions  To tackle the scalability problem of MARL communication,  this paper proposes a novel framework Transformer-based  Email Mechanism (TEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' The agents adopt local communi-  cation to send and forward messages like emails to form mes-  sage chains, which set up bridges among partial-observation  ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' We introduce Transformer to encode and decode the  message chain to choose the next receiver selectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
+page_content=' Em-  pirical results in diverse multi-agent cooperative tasks show  that our method outperforms the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQf9f42/content/2301.01919v1.pdf'}
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf,len=977
+page_content='Photon Bose-Einstein condensation and lasing in semiconductor cavities  Aurelian Loirette-Pelous and Jean-Jacques Greffet  Universit´e Paris-Saclay, Institut d’Optique Graduate School,  CNRS, Laboratoire Charles Fabry, 91127, Palaiseau, France  (Dated: January 31, 2023)  Photon Bose-Einstein condensation and photon thermalisation have been largely studied with  molecular gain media in optical cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Their observation with semiconductors has remained elusive  despite a large body of experimental results and a very well established theoretical framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' We  use this theoretical framework as a convenient platform to revisit photon Bose-Einstein condensation  in the driven dissipative regime and compare with the lasing regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' We discuss the thermalisation  figures of merit and the different experimental procedures to asses thermalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' We compare the  definitions of lasing and condensation thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Finally, we explore the fluctuations of the system  and their relation to the different regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  INTRODUCTION  In  2010,  experiments  by  Klaers  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  [1,  2]  identified and demonstrated Bose-Einstein condensation  of photons, a new light emission regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  While this  regime share with lasing the macroscopic occupation of  one mode, cavity photons are in near-thermodynamic  equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  As a direct consequence, cavity modes  occupation follow a Bose-Einstein (BE) distribution and  condensation is forced in the lowest energy cavity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  At first glance, Bose-Einstein condensation (BEC)  with photons seems to be impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  On the one  hand, lasers are usually thought to operate far from  equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  On the other hand,  in the so-called  blackbody radiation, equilibrium between photons is  reached due to walls acting as a reservoir, but the null  chemical potential precludes condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Actually, a  suitable gain material such as pumped dyes molecules or  semiconductors can act as a reservoir providing a photon  chemical potential [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Thermalization of the photon gas  with such a reservoir is made possible with a high-Q  cavity, when the number of absorption-emission cycles  made by a photon before leaving the cavity becomes  large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Furthermore, the cavity introduces a band gap  in the photon dispersion relation so that a lowest energy  state can be defined for a given band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' These ingredients  are sufficient to ensure BE condensation of photons at  room temperature in the weak coupling regime [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In the last decade, the pioneering experiments [1, 2]  in a dye-filled microcavity triggered a large amount of  works in similar devices in order to understand further  this new regime and its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' An important issue  has been to clarify the similarities and differences with  the lasing regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' While the overall crossover from the  standard out-of-equilibrium lasing phase to the BEC one  has been shown to be quite smooth [4], some features  of BEC have appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  At equilibrium, the emission  spectrum follows a BE distribution, and condensation  occurs into the lowest energy cavity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  When  thermalization breaks down, major spectral alterations  have been observed, ranging from deformation of the  thermal tail [1] to lasing in excited modes and multimode  lasing [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Early experiments investigating the second-  order coherence in the BEC regime evidenced large  fluctuations g(2)(0) = 2 even far above the condensation  threshold [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' This thermal behaviour suggests a closest  resemblance of a photon BEC to a pumped blackbody  than to a standard laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  As a consequence, first  order temporal coherence is also delayed to above-  threshold excitation [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In recent years, the question  about the difference between BEC and lasing has been  renewed due to the emergence of nanostructured cavity  mirrors enabling to realize complex potentials for light  [10–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Indeed,  in these systems,  controlling the  thermalization enables, for example, the study of vortices  formation and annihilation [15–17], or to envision analog  simulation with synchronized arrays of out-of-equilibrium  condensates [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Still, in the quest for these new  applications, we observe that several aspects of the  problem have been overlooked so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We list several  of them in the next paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We first note that BE condensation of photons  has  been  observed  in  dye-filled  microcavities  [2,  6,  21]  and  plasmonic  nanoparticles  arrays  [22],  and  erbium–ytterbium  co-doped  fiber  cavities  [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Alternatively, semiconductors have received much less  attention up to now, in spite of being a very common and  versatile active medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In particular, photon BEC in  semiconductor-based devices is not fully recognized yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  This is surprising in many respects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' On the experimental  one, spectral signatures hinting at thermalization and  BEC of photons has been observed early in a VCSEL  designed for polariton physics [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  More recently,  similar features have been observed in a commercial  VCSEL [26], suggesting that BEC (or near-equilibrium  BEC) of photons could be more common than it is  usually thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  A high absorption/emission cycles  number before cavity loss has also  been  reported  in a quantum-well photonic crystal laser [27], while  not interpreted as BEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' On the theoretical side, the  possibility of a chemical potential for photons has been  historically demonstrated on a semiconductor example  [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Furthermore, simple and accurate models of gain  and lasing in semiconductors are available so that this  system is a very good playground to explore the physics  of photon Bose-Einstein condensation and lasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='13073v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='quant-gas]  30 Jan 2023  2  Second,  finding  a  clear  signature  of  photon  thermalization  is  not  an  obvious  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  On  the  experimental side,  the analysis of emission spectra  is often compared with a Bose-Einstein function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  On  the theoretical side, a dimensionless number quantifying  the  degree  of  thermalization  has  been  introduced  theoretically by some authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  A simple connection  between these two approaches is still lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Third, the connection between lasing and condensation  is not fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  While a clear threshold  is observed in both cases, its exact positions differs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  This may impact the interpretation of the observed  phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Hence, there is a need to compare the  definitions of thresholds from laser and from equilibrium  BE condensates physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Fourth,  intensity  correlations  are  often  used  to  distinguish coherent light from stochastic light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  It is  interesting to revisit condensation and lasing by studying  fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' While many results have been reported, the  role of the degree of thermalization and the role of the  β-factor of the cavity have not been fully discussed so  that it is difficult to draw final conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In  this  paper,  we  take  advantage  of  the  well-  developped formalism to describe gain in semiconductors  to analyse all these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In the next section, we  present a simple unified theory of equilibrium and non-  equilibrium condensation of photons in a semiconductor-  based cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' While similar to the pioneering model by  Kirton and Keeling [28, 29] for dye-filled microcavities,  we show that our model provides a straightforward  interpretation of the photons chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We  then  derive  a  generalized  BE  distribution  in  the  driven-dissipative regime and exhibit a dimensionless  number that characterizes quantitatively the degree of  thermalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We discuss some of its properties and  clarify the connection with other dimensionless numbers  such as cooperativity, Knudsen number and optical  thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In this new framework, we show how to  revisit some lasing features such as gain clamping and  inversion, and discuss the selection of the lasing mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  An extended definition of the equilibrium condensation  threshold is also introduced for nonequilibrium systems,  and compared to the standard lasing threshold definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In Section 3, we discuss several observables to evaluate  to which extent a device is thermalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Equipped  with the explicit form of the degree of thermalization  introduced in the previous section, we can revisit the  typical experimental situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In particular, we show  that the most common practice, consisting in studying  the emission spectrum, should be used with caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We finally focus on the second-order coherence in  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We tackle the thermalisation issue by  calculating analytically the intensity autocorrelation  function g(2)(0) as a function of the β-factor and the  degree of thermalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  EQUILIBRIUM AND NON-EQUILIBRIUM  CONDENSATION OF PHOTONS IN A  SEMICONDUCTOR-BASED CAVITY  In this section, we first summarize basic forms of  the emission and absorption rate in a semiconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We then use this formalism to recover the equilibrium  number of photons per mode in a lossless cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We  finally compare this case to the one of a lossy cavity with  gain operating in the so-called driven-dissipative regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  This approach enables us to discuss in a very simple  framework (i) the thermalization regime, introducing a  degree of thermalization in a very systematic way, (ii)  the connection between condensation and lasing and (iii)  the definitions of their respective threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Model of semiconductor gain medium in a  cavity  Throughout this paper, we will focus on a piece of  semiconductor placed in a cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We assume finite  extension of the cavity so that photonics modes are  spectrally discretized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' We index them with l = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='nc  corresponding to increasing energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The various  particle exchange pathways between the gain medium,  the modes and the environment are shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' 1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In the cavity, photons in the l-th mode can be created  or annihilated by the gain medium at the rates Rl  em  for spontaneous emission, Rl  emN l for stimulated emission  and Rl  absN l for absorption, where N l is the number of  photons in the mode l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Alternatively, radiative cavity  losses occurs at the rate κlN l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In the semiconductor,  excited electrons are created at the rate Rin through  pumping (indistinctly electrical or optical) Conversely,  relaxation can occur through the above-depicted emission  in the cavity modes, through spontaneous emission into  vacuum modes at the rate Rvac  em , or through non-radiative  relaxation pathways (for example Auger effect) at the  rate Rnr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In contrast with dye molecules, explicit forms of Rl  em  and Rl  abs can be derived for semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Here  we focus on an intrinsic direct bandgap semiconductor,  indifferently  2  or  3-dimensional,  and  follow  usual  approximations [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  As sketched on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  1 (b),  the conduction and heavy-hole valence band [31] are  described by the isotropic dispersion Ec(k) and Ev(k)  respectively, were k stands for the wavevector modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Assuming that only vertical interband transitions are  possible, a transition involving a photon in the mode l  with energy El requires an electron and a hole with the  same wavevector ⃗kl so that Ec(kl) − Ev(kl) = El.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' We  also assume that the ground cavity mode energy is higher  than the gap energy E0 > Egap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Interestingly, conduction electrons and valence holes  close to the gap edges can be well described as free  particles with an effective mass m∗  c/v, which leads to  the simple parabolic band model Ec/v(k) = E0  c/v ±  3  Gain medium  Cavity  mode 𝑙  𝑅𝑖𝑛  𝜅𝑙𝑁𝑙  𝑅𝑎𝑏𝑠  𝑙  𝑁𝑙  𝑅𝑒𝑚  𝑙  𝑁𝑙  𝑅𝑒𝑚  𝑙  𝑅𝑒𝑚  𝑣𝑎𝑐  Pumping  Cavity loss  Absorption  Stim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  emission  Spont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  emission  Spont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  emission into  vacuum modes  a)  𝑅𝑛𝑟  Non-radiative  relaxation  0  k l  k=|k|  E 0  v  E 0  c  Ec(k l)  Ev(k l)  E  E l  E gap  Ec(k)=E 0  c +  2k 2  2m *  c  Ev(k)=E 0  v  2k 2  2m *  v  b)  0  1  2  1 Occupation  probability  E  c  v  fFD(E,T,  c)  fFD(E,T,  v)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Scheme of the system and notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Panel (a): flux  of particles between the gain medium, the cavity, and the  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Panel (b): semiconductor band structure as a  function of the wavevector modulus (left) and distribution of  the electrons in each band (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' See the main text for a  detailed description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  ℏ2k2  2m∗  c/v with E0  c/v the energy minimum/maximum of the  conduction/valence band and ℏ the Planck constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Analytical expressions for El(kl) can be derived, as well  as for the density of states in each band ρc/v(k) and  the joint density of state ρJ(k) associated to the vertical  transitions [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Next,  we  assume  that  the  bands  are  in  local  thermodynamic equilibrium characterized by a Fermi-  Dirac distribution with a common temperature T and  local chemical potentials µc and µv, the so-called quasi-  Fermi levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In the case of electrical pumping, we have  µc − µv = eV where V is the applied voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  As  the voltage increases, µc increases (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' µv decreases)  from the Fermi-level, so that their difference µc − µv is  controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' It is also possible to define quasi-Fermi levels  under optical pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In this context, the spontaneous emission, stimulated  emission and absorption rates for the mode l can be  written respectively as [30]:  Rl  em = glfF D(Ec(kl), T, µc)[1 − fF D(Ev(kl), T, µv)]  (1)  and  Rl  abs = glfF D(Ev(kl), T, µv)[1−fF D(Ec(kl), T, µc)], (2)  where gl is a pumping-independent transition rate  and fF D(E, T, µ) = 1/[exp ( E−µ  kBT ) + 1] is the Fermi-  Dirac distribution with E the electron or hole energy,  kB the Boltzmann constant and µ a quasi-Fermi levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The microscopic expression of gl is given in Appendix  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' The right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  (1) expresses that  emission is proportional to the probability of finding an  electron at the right energy in the conduction band and  a corresponding hole in the valence band, and conversely  for absorption in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Finally, we define the fraction of spontaneous emission  into the mode l through the generalized β-factor:  βl =  Rl  em  Rvac  em + �  j Rj  em  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  (3)  In a single cavity mode context, this dimensionless  number  characterizes  the  emission  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  A  macroscopic laser corresponds to β → 0 while a nanolaser  corresponds to β → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Indeed due to large (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  small) mode volume, a macroscopic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' nano-) laser  is characterized by a low (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  high) Purcell factor,  so that spontaneous emission into the numerous vacuum  modes (the mode l) is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Reducing the volume  further tends to reduce the cavity mode number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Still the  cavity modes spacing and number can also be adjusted  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  through engineering of the mirrors curvature for  Fabry-Perot-like cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Hence the quantities Rvac  em and  �  l Rl  em can be partially tuned independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Photon BEC in a lossless cavity with gain  Bose-Einstein  condensation  is  a  property  of  an  ensemble  of  bosons  in  thermodynamic  equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Quantitatively, thermodynamic equilibrium means that  a state at energy E is occupied according to a Bose-  Einstein distribution 1/[exp( E−µ  kBT ) − 1] where µ is the  chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Condensation may occur when the  chemical potential approaches the ground state energy  µ → E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In  the  blackbody  radiation,  photons  reach  a  thermodynamic equilibrium due to walls acting as a  reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  This equilibrium is characterized by a null  photon chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Remarkably, Wurfel showed  that it is possible to introduce a photon chemical  potential when dealing with stationary systems with gain  [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We reproduce here the reasoning for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We  4  start by assuming a perfectly lossless cavity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' κl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In the steady-state regime, the balance between the  spontaneous and stimulated emission processes and the  absorption in the l-th photonic mode yields:  Rl  em + Rl  emN l = Rl  absN l,  (4)  where N l is the number of photons in the l-th mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' It  is readily seen that the photon number only depends on  the ratio between the absorption rate and the emission  rate Rl  abs/Rl  em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Given Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' (1),(2) which assumes that  the gain medium is in local thermodynamic equilibrium,  simple algebra allows to recover the Van Roosbroeck-  Shockley relation [32]:  Rl  abs  Rlem  = exp  �El − µ  kBT  �  ,  (5)  where µ = µc − µv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' From Eqs (4) and (5), it follows  that the photon number in the mode l is given by:  N l =  1  exp  � El−µ  kBT  �  − 1  ,  (6)  namely a Bose-Einstein distribution with temperature  T and a chemical potential defined as the quasi-Fermi  levels splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In the absence of pumping, the chemical  potential is null and we recover the blackbody radiation  distribution with the temperature of the semiconductor  at equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Finally, beyond the semiconductor model used here,  we emphasize the key role of local thermodynamic  equilibrium in each band under pumping to derive this  result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Indeed, this appears as a sufficient condition on  the gain medium to reach photons BEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In particular,  this explains why eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  (5) can be written similarly  for dyes molecules in terms of emission and absorption  cross sections, a formula known as the Kennard-Stepanov  relation [33–35] (sometimes also called the Neporent-  McCumber relation,  see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  [36] and references  therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  To summarize, the number of photons in  a non-lossy cavity filled with a gain medium in local  thermodynamic equilibrium can be described by a Bose-  Einstein distribution with a non-zero chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The driven-dissipative regime of a lossy cavity  with gain: Lasing or BEC ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We now consider a cavity coupled to the environment  through the loss rates κl > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Such a system composed  of a gain medium and a cavity with radiative losses is  usually considered to be a laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  A natural question  then arises: what is the difference between Bose-Einstein  condensation and lasing ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We repeat the analysis of the previous section using  the same assumptions and notations, now accounting  for cavity losses so that the system is in the driven-  dissipative regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The balance equation (4) becomes  Rl  em +Rl  emN l = (Rl  abs +κl)N l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' The steady-state photon  number in the mode l can then be cast in the form [30]:  N l =  Rl  em  κl − (Rlem − Rl  abs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  (7)  In this last equation, the quantity Rl  em −Rl  abs is better  known as the net gain rate of the mode l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Hence, this  simple model recovers that the mode l starts to lase as the  net gain compensates the radiative losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' So far, we have  isolated a mode and computed its occupation number by  expressing the balance between gain and losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  This  approach is at first glance at odds with the study of  the population of different modes in an equilibrium  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Nevertheless, we now cast this laser equation  in a form that mimicks Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Upon factorization by  Rl  em and inserting the relation (5) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' (7), we find the  alternative form [37]:  N l =  1  exp  �  El−µ  kBT  �  [1 + Kln(T, µ)] − 1  ,  (8)  where  Kl  n(T, µ) =  κl  Rl  abs(T, µ)  (9)  is a dimensionless number often called Knudsen  number in the context of transport phenomena and  Boltzmann equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' The Knudsen number is given by  the ratio of the absorption time  1  Rl  abs by a characteristic  time of the cavity, the residence time of a photon in  the cavity  1  κl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Hence, in the regime where a photon  undergoes a large number of absorption and emission  cycles during the residence time, the Knudsen number  is small and the distribution (8) approaches the BE  distribution of a non-lossy cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In other words, the  large number of absorption and emission events enables  the photons to thermalize with the semiconductor acting  as a reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The Knudsen number appears to be  the natural quantity that quantifies how thermalized is  a mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Importantly, note that a Knudsen number is  associated to each mode, it is not a global quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' We  stress that some modes may be thermalized while others  are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  As a conclusion of this section, it is clear from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  (8) that Bose-Einstein condensation of photons is a  particular regime of lasing, in which (i) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' (5) is satisfied  for the gain medium and (ii) the Knudsen number is small  for all the modes to ensure that they are all thermalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In the remaining of this work, we will use ”lasing” to refer  indistinctly to Bose-Einstein condensation or standard  out-of-equilibrium lasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In addition, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' (8) provides  an alternative point of view to interpret lasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Indeed,  5  while Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' (7) provides a good description of single mode  lasing in a system with significant losses and gain, we  anticipate that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' (8) will be more suited to the study  of multimode phenomena in the thermalized regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Knudsen number, thermalization degree,  optical thickness, cooperativity and photon number  at transparency  In the last section, we have introduced the Knudsen  number Kl  n of a mode l as the absorption time divided  by the residence time in the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  It takes small  values in the thermalized regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Its inverse, that we  note Dl, was called thermalization degree in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' [6] or  thermalization coefficient in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Its key role in  photon Bose-Einstein condensation had been suggested  [1] and identified [39] in early papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Here, we have  shown how it appears naturally from laser rates equation  in the context of an equilibrium distribution perturbed  by the introduction of cavity losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Let us now discuss  alternative physical interpretations of the thermalization  degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' We first note that it can be viewed as the effective  cavity length Ll = c/κl divided by the absorption mean  free path ll  abs = c/Rl  abs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  With this point of view,  which is often used to discriminate between diffusive  regime and ballistic regime in transport phenomena, we  identify the degree of thermalization with the optical  thickness Ll/ll  abs = Dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Second, we remind that the  optical thickness is proportional to the cooperativity  C(Na).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' This quantity had been initially introduced to  characterize the absorption of a photon by an ensemble of  Na atoms in a cavity in the context of non-linear optics in  a cavity [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' It is currently used as a measure of the light-  matter interaction in cavity quantum electrodynamics  (CQED) [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Finally, the thermalization degree has been  interpreted historically in laser physics as the photon  number at transparency [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Here this follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  (7) when Rl  em = Rl  abs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Interestingly, this suggests to  reinterpret some experiments featuring a high photon  number at transparency as Bose-Einstein condensation  of photons, see for example Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' [27] for a semiconductor  laser in a photonic crystal cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Lasing mode in the BEC picture  In the previous sections, we showed that the laser  equation (8) giving the mode photon number has the  structure of a Bose-Einstein distribution apart from a  correction term given by 1 + Kl  n(T, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Hence, we can  revisit the lasing transition in terms of Bose-Einstein  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We start with the laser point of view given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In this framework, lasing in the mode l occurs as the gain  rate saturates when it approaches the loss rate (Rl  em −  Rl  abs) → κl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' This is called gain clamping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In addition,  finite losses require positive gain, that is, population  inversion of the corresponding transition [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We now place in the perspective of the generalized  Bose-Einstein distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We start by writing (8) in  a slightly different form [37]:  N l =  1  exp  � El−µl  eff (T,µ)  kBT  �  − 1  ,  (10)  where  we  have  introduced  an  effective  chemical  potential µl  eff(T, µ) = µ − kBT log[1 + Kl  n(T, µ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Here,  we stress that this form enables to use the Bose-  Einstein distribution which is an equilibrium concept  in the nonequilibrium driven-dissipative regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The  effective chemical potential is composed of a term µ  which accounts for the gain and a term −kBT log[1+Kn]  which accounts for the losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' The usual condition for  Bose-Einstein condensation in the mode l is then directly  generalized as:  µl  eff(T, µ) → El.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  (11)  Here, the increase of the pump power is interpreted as  increasing the quasi-Fermi levels splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Therefore µ  converges toward a fixed value µclp defined as the solution  of the implicit equation:  µclp − kBT log[1 + Kl  n(T, µclp)] = El.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  (12)  This saturation of µ corresponds to gain clamping  in the BEC point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In this last equation,  the correction term is always negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Hence, the  quasi-Fermi levels splitting must exceed the transition  energy El to trigger lasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  This corresponds to  population inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  It highlights the importance to  distinguish between the quasi-Fermi levels splitting and  the effective chemical potential, since only the latter can  be interpreted as the photon chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We now focus on a multimode system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' The usual laser  textbook picture is the following [30, 44, 45]: the gain  curve is taken to be a bell-shaped function of frequency,  while the frequency dependence of the mirrors losses is  neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Lasing is thus expected to occur in the cavity  mode with largest gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  This picture is at odds with  the one of ideal Bose-Einstein condensation, which is  expected to occur in the ground cavity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We now revisit this issue using the Bose-Einstein  picture given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In the present multimode  situation, each mode l defines a different clamping  value of the quasi-Fermi levels splitting, that we note  µl  clp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Single mode lasing takes place in the mode with  the smallest µl  clp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  To gain further insight, we assume  Kl  n(µl  clp) ≈ Kl  n(El).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The clamped quasi-Fermi levels  splitting of each mode l is simply given by:  µl  clp ≈ El + kBT log[1 + Kl  n(T, El)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  (13)  6  Interestingly,  this expression is composed of two  competing terms: on one hand, the mode energy favors  lasing in low energy modes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  on the other hand, it  depends on the Knudsen number and favors lasing  in highly thermalized modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Therefore, without the  second contribution coming from the cavity losses, we  would recover the usual condensation on the ground  mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In practice, lasing in a mode above the ground  mode is thus the signature of a system in which the  modes have very different thermalization degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' This  discussion highlights that thermalization is primarily a  modal property and not a system property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Indeed, as  explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' II B, thermalization occurs between a  mode and the reservoir, rather than between modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Finally, we note that some authors used lasing in  the ground mode versus an excited mode as a criterion  to distinguish between BE condensation and out-of-  equilibrium lasing [38, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' While lasing in an excited  mode is indeed a signature of nonequilibrium operation,  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' (13) shows that condensation in the ground mode is  only the signature of a Knudsen number slowly varying  from one mode to another, regardless of its absolute  amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Condensation versus lasing threshold  In the previous section, we showed how to interpret  the  mode  selected  for  lasing  within  a  generalized  Bose-Einstein condensation approach,  stressing that  condensation and lasing are two faces of the same coin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  As a next step, it is natural to compare the definitions  used for lasing threshold and for condensation threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We first remind the lasing threshold definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Many  different criteria can be used to characterize lasing [47,  48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Here, we consider the widely used condition based  on an input/output curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' On Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' 2 (a), the number  of photons N j in the cavity is plotted as a function of  the injection rate of excited carriers, that we note Rin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  On a linear scale, N j turns suddenly from sublinear to  linear on a small pumping range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' The threshold is defined  as the input rate of excited carriers Rin,LAS when the  linear slope is continued down to 0 output rate (see the  dashed blue line on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' 2 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' This input rate is equal  to the value of the losses, evaluated at gain clamping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Indeed, close to clamping, stimulated emission funnel  all additional photons in the lasing mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The losses  are due to different mechanisms: the leakage through  non-lasing cavity modes with rate �  l̸=j κlN l(µj  clp), the  emission into vacuum modes (Rvac  em (µj  clp)) and other non-  radiative charge carrier relaxation processes (Rnr(µj  clp)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The lasing threshold is thus given by:  Rin,LAS = Rnr(µj  clp)+Rvac  em (µj  clp)+  �  l̸=j  κlN l(µj  clp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' (14)  We now focus on the condensation threshold definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In the literature on BEC in thermodynamic equilibrium,  the BEC threshold is defined by the equality between  the total number of particles and the number of particles  in the excited states in the condensed phase [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' First,  note that in photons BEC experiments, the number N l  of photons in a mode l cannot be measured directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Still,  the driven-dissipative regime enables to derive it from the  measured flux κlN l and the knowledge of the loss rate κl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Second, note that in essence, this definition relies on the  same idea as for a laser: beyond threshold, all additional  photons will go to the condensed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' As shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  2 (b), the condensation threshold is extracted graphically  in a similar fashion as for the laser threshold when  plotting the number of photons in the condensed mode  versus the total number of photons in the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' The  total number of photons in the cavity at threshold is then  given by the sum of non-condensing modes population at  clamping, namely �  l̸=j N l(µj  clp) = N tot  BEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Still, this  procedure differs from the lasing threshold definition, as  it is based on a number of photons in a cavity and not  on a comparison of fluxes of input carriers and emitted  photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In particular, the nonradiative losses and the  radiative losses into vacuum modes are not taken into  account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Hence, the BEC definition leads to a smaller  value of the threshold for the quasi-Fermi levels splitting  than the lasing condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The difference is not very  large when the β factor is close to 1 but may be very  large when emission into vacuum modes dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' This  is illustrated in Figure 2 (c) where it is seen that the  thresholds can differ by orders of magnitude (in term of  photons in the lasing mode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  To conclude, the choice of using the BEC or laser  threshold has to be conducted carefully, as they can  take very different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  To guide this choice, one  should note that for the lasing threshold, both the  input and emitted power must be monitored, while  the emitted power spectrum is sufficient to determine  the condensation one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  As already encountered in  the previous sections, this suggests that other than  making a real difference between BEC and lasing, the  ”condensation” point of view is a framework suited to the  study of the multimode character of the system, while the  ”lasing” one rather focus on its driven-dissipative aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  EXPERIMENTAL ASSESSMENT OF  THERMALIZATION  In the previous section, we made a clear distinction  between BEC and lasing using the thermalization degree  of the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' However, the thermalization degree cannot  be measured directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Indeed, it is proportionnal to the  absorption rate Rl  abs, but only the net absorption rate  Rl  abs − Rl  em is given by a transmission measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In this section, we aim at finding observable quantities  that depend sharply on the thermalization degree,  enabling its assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' We first analyze the emission  spectrum under homogenenous pumping, which is the  most common experimental practice,  and find that  7  Rin,LAS/ j  Rin/ j  0  N j  LAS  N j  a)  Laser input-ouput curve  (Rin  Rin,LAS)/ j fit  N tot  BEC  N tot  0  N j  BEC  N j  b)  Photon-Photon curve  (N tot  N tot  BEC) fit  10  1  101  103  105  107  109  Rin/ 0  10  1  101  103  105  107  109  N 0  c)  R vac  em /  l  j  lN l =0  0 =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='5×10 2  R vac  em /  l  j  lN l =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='4×102  0 =10 4  R vac  em /  l  j  lN l =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='5×106  0 =10 8  Condensation threshold  Laser threshold  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Panel (a): schematic input-output laser curve (red  line) on a linear scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' The dashed blue line is a linear fit of  the laser curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Its intersection with the N j = 0 axis defines  the laser pumping threshold Rin,LAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The corresponding  lasing mode photon number at threshold is noted N j  LAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Panel (b): schematic photon-photon curve of a multimode  driven-dissipative BEC condensing in the mode j (red line)  on a linear scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  N tot = �  l N l is the total number of  photons in the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The dashed blue line is a linear fit  of the BEC curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Its intersection with the N j = 0 axis  defines the BEC threshold N tot  BEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' The corresponding lasing  mode photon number at threshold is noted N j  BEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Panel  (c): comparison of the lasing mode photon number at laser  and BEC thresholds, on input-output curves corresponding to  different rates of spontaneous emission into vacuum modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' A  constant cavity modes spacing is assumed so that their energy  reads El = E0(1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='001 × l), with E0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='271 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' κl and gl  are assumed constant over the modes, with a ratio gl/κl = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  This enforces lasing in the ground mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Non-radiative losses  are neglected Rnr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  To help considering the value of  Rvac  em (µj  clp)/ �  l̸=j κlN l(µj  clp), the corresponding value of β0  is given in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Other parameters are compiled in  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  this method may not be reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We then discuss  spectral and spatial measurements under inhomogeneous  pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' We finally discuss the influence of band-filling  on the thermalization degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Spectrum analysis  In an ideally thermalized system, we saw in Section  II B that the mode occupation follows a Bose-Einstein  distribution N l = 1/[exp( El−µ  kBT ) − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' At low occupation  numbers,  the classical regime is recovered,  namely,  the BE distribution reduces to a Maxwell-Boltzmann  distribution N l  ≈ exp(− El−µ  kBT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Hence, a common  practice to prove thermalization consists in looking for  a linear decay on a semilogarithmic plot of the spectrum  [24, 26, 50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Here, we compare this approach with the  characterization based on the Knudsen number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In the classical regime, the generalized BE distribution  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' (8) becomes:  N l ≈  exp(− El−µ  kBT )  1 + Kln  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  (15)  It is readily seen that an exponential decay of the  cavity photons spectrum is observed in two cases: (i) the  Knudsen number of all the modes is much lower than  1, and (ii) the Knudsen number is constant over the  modes, whatever its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In the second case, despite  an exponential behaviour of the spectrum, the Knudsen  number may take values ≳ 1 indicating a non thermalized  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Beyond the classical regime, it is noteworthy that this  issue persists in the quantum degenerate regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Indeed,  according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' (10), the generalized BE distribution  with constant Knudsen number Kn simplifies in an  equilibrium BE distribution with the effective chemical  potential µeff  = µ − kBT log[1 + Kn] [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  All in  all, it means that spectrum analysis with homogeneous  pumping in order to quantify the thermalization may  not be reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In particular, we note in Appendix B  that devices featuring a large, planar and homogeneously  pumped cavity are likely to feature a nearly constant  Knudsen number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' This may explain the BE-like spectra  observed in optically [24, 25] and electrically [26] pumped  large area VCSELs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Inhomogeneous pumping  An interesting signature of thermalization can be  observed when using an inhomogeneous pumping with  a beam or injection area much smaller than the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Indeed, the pumped part of the gain medium emits  photons isotropically through spontaneous emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  These photons can be reabsorbed efficiently everywhere  in a thermalized system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  As a consequence,  the  gain is homogeneous in the cavity despite a localized  pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  To describe this effect, it is necessary to  include additional rate equations describing locally the  gain medium population [38, 39, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  While this goes  far beyond the scope of the present work, we give a  hint of the complexity of this case by writing how the  8  photon occupation number is modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The balance  equation (4) with the losses κl for a mode l has  to be integrated over the gain medium volume (also  called active volume) Vact, namely  �  Vact d3⃗r  �  Rl  em(⃗r ) +  Rl  em(⃗r )N l�  = N l �  Vact d3⃗r  �  Rl  abs(⃗r ) + κl/Vact  �  where the  rates are now defined locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In particular, the local  Knudsen number is Kl  n(⃗r) = κl/[Rl  abs(⃗r )Vact].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The  photon number in the mode l then becomes:  N l =  1  exp(  El  kBT )  �  exp( µ(⃗r )  kBT )  �l  �  1 +  �  Kln(⃗r )  �l�  − 1  ,  (16)  where ⟨A(⃗r )⟩l =  �  Vact d3⃗r Rl  abs(⃗r )A(⃗r )/  �  Vact d3⃗r Rl  abs(⃗r )  is a spatial average weighted and normalized by the  absorption rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  While the global distribution still  appears as a generalized BE distribution, additional  complexity is brought by the spatial average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In  particular,  the  weighting  by  the  local  absorption  now gives a modal dependence to the quasi-Fermi  levels splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  An enhanced sensitivity to imperfect  thermalization is thus expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Experimentally, it was  reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  [1] a departure from the ideal BE  distribution of high energy modes occupation, while the  thermalization degree was tuned down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Given the small  extension of the optical pump used compared to the  large extension of these high energy modes, this is in  good qualitative agreement with our considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' A  similar observation has also been made in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' [22] for  plasmon-polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Beside spectrum analysis, we eventually mention two  other types of measurements that reveal efficiently  the thermalization of the system with inhomogeneous  pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' The first consists in measuring the size of the  condensate as a function of the size of the pumping beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  When the system is well thermalized, the condensate  size is invariant, while it follows the size of the spot in  the opposite case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' This type of measurement has been  reported [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In the same fashion, the spatial position  of the condensate in a trap can be compared with the  position of the pump beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' As the pump is moved away  from the center of the trap, the longer condensation keeps  occurring in the center, the higher the thermalization  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' This measurement has been reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' [1, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Thermalization and saturation at high pumping  In  the  last  subsection,  we  discussed  how  the  thermalization  of  a  system  can  be  probed  with  inhomogeneous pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Noteworthy, this has been  done as if the thermalization of a mode was a general  quantity, independent on the pumping strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Here,  we discuss how the thermalization evolves as the system  is driven toward the degenerate regime through strong  pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The key issue is simple:  thermalization is  ensured by absorption and reemission;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' if the gain medium  is highly pumped and approaches saturation, absorption  is reduced and hence thermalization decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='35  (eV)  10  1  100  101  102  103  D l  gl/ l =1000  gl/ l =100  gl/ l =10  gl/ l =2  =E l  ( clp,D l  clp)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Variation of the thermalization degree Dl of a  mode l as a function of the quasi-Fermi levels splitting µ,  for various ratio gl/κl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' The colored dots indicate clamping as  defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' (12), at the quasi-Fermi levels splitting µclp  and the thermalization degree Dl  clp = Dl(µclp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' The vertical  dark dashed line indicates transparency, namely µ = El.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' The  energy of the mode is El = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='271 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Other parameters are  compiled in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' (2) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' (9), the dependence of the  thermalization degree on the quasi-Fermi levels (that is  pumping) reads:  Dl = gl  κl fF D(Ev(kl), T, µv)[1−fF D(Ec(kl), T, µc)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' (17)  On Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' 3, we show the evolution of the thermalization  degree of a mode l as a function of the quasi-Fermi  levels splitting µ [54] for various gl/κl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  At low  pumping, filling of the conduction band (and accordingly  depletion of the valence band) is negligible so that Dl =  gl/κl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' When increasing the quasi-Fermi levels splitting,  the thermalization degree decreases significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  At  clamping (colored dot), the fall is about a multiplication  factor 1/5 at high gl/κl, and more than 1/10 at low gl/κl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In the first case, corresponding to a well thermalized  mode, clamping occurs right over transparency (dark  dashed vertical line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In a two-level system, transparency  corresponds to an occupation probability of 1/2 of the  upper and lower level, so that the product of the levels  occupation is 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Here, the slightly different value is  due to the asymmetry of the bands of our semiconductor  model (see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In the low mode thermalization  case, a large inversion population is needed for lasing,  that occurs well above transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The occupation  probability of the conduction band is then much greater  than 1/2, and conversely for the valence band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Hence,  the degree of thermalization is significantly decreased  compared to the near-equilibrium case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In  summary,  reliable  assessments  of  the  system  thermalization should be made in the degenerate regime  9  due to this dependence of the thermalization degree  dependence on pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  INTENSITY FLUCTUATIONS: ARE THEY  A BEC SIGNATURE ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In the previous part, we showed that the spectrum is a  quantity that can reveal the thermalization of the system,  but which needs to be analyzed and probed with care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In  this section, we investigate the intensity fluctuations as  an alternative observable to distinguish between the BEC  and the out-of-equilibrium laser regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Context  In the textbook picture of out-of-equilibrium lasing,  coherence sets up right at the lasing threshold [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Above threshold, the intensity fluctuations are ruled  by Poissonian statistics resulting in a second order  correlation function at zero-time delay g(2)(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' On  the contrary, earlier works on intensity fluctuations in the  BEC regime predicted [55] and then measured [8] super-  Poissonian statistics for light well-above condensation  threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  This thermal regime,  characterized by  g(2)(0) = 2, was found to extend deeply in the condensed  phase, before the crossover to the usual Poissonian light  was recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  This ask the question whether large  fluctuations are a signature of BEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  While the picture described in the last paragraph  suggests studying the fluctuations according to the  thermalization degree,  other parameter have to be  taken  into  account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In  Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  [8,  55],  it  has  been  pointed  out  that  the  reservoir  size  has  an  important influence on fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' For large reservoirs,  the  gain  medium  can  be  loosely  thought  as  an  infinite reservoir, recovering grand-canonical ensemble  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' The large condensed mode photon number  fluctuations, comparable to its mean value even above  condensation threshold, are then identified to the so-  called grand-canonical fluctuation catastrophe [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' On  the contrary, fluctuations become limited when the  reservoir excitations number is smaller than the mean  photon number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Besides the role of the volume, it has been shown that  the β-factor has a strong influence on the fluctuations  for micro- and nano-lasers [57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' While macroscopic  lasers with low β−factor show the usual steep crossover  between thermal and Poissonian light at lasing threshold,  in high-β devices the crossover is slow and occurs well  above threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Knowing that the perfect equilibrium  approach of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' [55] assumed a β = 1 cavity, this rather  suggests that intensity fluctuations could be independent  on the system thermalization, at least in the nanolaser  limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In summary, assessing the role of thermalization on  fluctuations requires to carefully control both the effect of  the size of the reservoir and the β-factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' With that many  degrees of freedom, it is a theoretical challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Methods to calculate the photon number distribution  like master equations for the lasing mode photon number  [29, 55, 59, 60] or stochastic rate equations [61–64] can  hardly been solved numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In the next subsections,  we proceed to a simpler investigation, focusing only on  the second-order coherence at zero-time delay g(2)(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We calculate this quantity by studying the small photon  number deviations over the steady state in the Langevin  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Interestingly, we note that for this quantity,  this approach showed good agreement with a more  rigorous stochastic rate equations model [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Thus,  this allows to get accurate and analytical insights for an  observable easily accessible experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Second-order coherence at zero delay time  We base our investigation on the dynamical evolution  equation of the photon number N j(t) in the cavity mode  j that is lasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' For simplicity, we neglect the influence of  other cavity modes on the system dynamics, and of non-  radiative losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Hereafter, we omit the mode superscript  N j = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Following the notations of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' 1 (a) for the  various exchange pathways between the cavity, the gain  medium and the environment, the dynamical evolution  equation for N(t) is given by:  dN  dt = −κN +[Rem(Ne)−Rabs(Ne)]N +Rem(Ne), (18)  where Ne(t) is the number of excited electrons in the  gain medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We switched from the variables µc and  µv to Ne for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In semiconductors gain media,  the excited electrons dynamics is usually commensurable  with the one of the cavity photons [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Hence the  corresponding evolution equation of Ne(t) must be taken  into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' 1, it yields:  dNe  dt  = Rin −[Rem(Ne)−Rabs(Ne)]N − Rem(Ne)  β  , (19)  where we used that Rvac  em = (1/β − 1)Rem as follows  from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In the following, β will be assumed to be  independent on pumping, as usual in laser physics [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  All in all, these 2-coupled rate equations correspond to  a standard class-B model broadly used to describe the  dynamics of most semiconductor single mode lasers [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We now note the steady-state solutions of equations  (18),(19) as Nss, Ne,ss, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' We also introduce  the small deviations δN(t), δNe(t), with |δN| ≪ Nss  and |δNe| ≪ Ne,ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We then linearize Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  (18),(19)  to first order in these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' The noise due to the  quantization of the emission, absorption, pumping and  loss process is finally added to each equation through  the respective stochastic terms Fp, Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We obtain the  following coupled Langevin equations:  10  dδN  dt  = −γppδN + γpeδNe + Fp  (20)  and  dδNe  dt  = −γepδN − γeeδNe + Fe,  (21)  where  we  have  defined  the  short-hands  γpp  =  −[Rem(Ne,ss) − Rabs(Ne,ss)] + κ, γpe = Nss∂Ne[Rem −  Rabs](Ne,ss) + ∂NeRem(Ne,ss), γep  =  [Rem(Ne,ss) −  Rabs(Ne,ss)],  γee  =  Nss∂Ne[Rem − Rabs](Ne,ss) +  (1/β)∂NeRem(Ne,ss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In addition, the stochastic terms  verify the usual correlations properties ⟨Fx(t1)Fy(t2)⟩ =  2Sxyδ(t1 − t2) with x, y  ∈  (p, e) [30],  where the  expressions of the Sxy are given in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In this linearized Langevin approach, the second-order  intensity correlation g(2)(0) = ⟨N(0)[N(0) − 1]⟩/N 2  ss is  given by:  g(2)(0) = 1 −  1  Nss  + ⟨δN(0)2⟩  N 2ss  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  (22)  A detailed expression is then obtained by Fourier  transforming (20),(21), so that the problem can be  reformulated into a matricial form which is easy to invert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The full result, well-known in the literature [30, 61, 65],  is given in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Simple asymptotic expressions  can be written for limiting values of some parameters, as  discussed in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Results  We first focus on the usual macroscopic laser limit  β → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' From the full expression in Appendix D, simple  algebra show that the second-order coherence at zero  delay time reduces to:  g(2)(0) = 1 +  1  1 +  � Nss  NLAS  �2 ,  (23)  where NLAS is the photon number at lasing threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The coherence threshold is defined at g(2)(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='5  corresponding to N = NLAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Going straight to the point,  this equality between the coherence and laser thresholds  does not allow for a distinction between standard laser  and photons BEC in this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Indeed here the crossover  from thermal to Poissonian statistics always occurs at  lasing threshold, regardless of the thermalization degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We now focus on the opposite, ”nanolaser” limit β →  1, where most of the spontaneous emission goes into the  single cavity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' The second-order coherence at zero  delay time now follows the asymptotic behaviour [55, 57,  58, 63, 65]:  g(2)(0) = 1 +  1  1 +  � Nss  NCO  �2 ,  (24)  where the coherence threshold is now given by NCO ≈  �  Rem(Nss=∞)  ∂Ne[Rem−Rabs](Nss=∞)  �1/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' It is seen that NCO presents  no explicit dependence on the thermalization degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Coming back to our initial question, we conclude that  the study of intensity fluctuations through the second-  order coherence at zero delay time does not provide a  mean to distinguish between the out-of-equilibrium laser  and the BE condensation regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Finally, we discuss the consequence of this conclusion  on the grand canonical fluctuation catastrophe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In the  nanolaser limit, the coherence threshold is not given  by the laser nor the BEC threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' In particular, for  realistic parameter values, the coherence threshold is  shifted to much stronger pumping values than the laser  threshold [57, 58, 65] and the BEC threshold [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Hence,  there is a lasing/BEC regime with large fluctuations  between these two thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' It is possible to attribute  this regime to grand canonical fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Indeed,  it has been shown [8] that the coherence threshold  square N 2  CO corresponds to an effective number of excited  carriers in the gain medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Therefore, in the range  between the condensation and the coherence thresholds,  the gain medium is large compared to the photon gas and  can be considered to be an infinite reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Remarkably,  we find that the concept of grand canonical fluctuations  is not restricted to equilibrium BE condensation but can  be extended to non-equilibrium systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  CONCLUSION  To summarize, we have explored the photon Bose-  Einstein condensate regime for semiconductors in a  cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Owing to the explicit form of the gain for  semiconductors and the extensive body of knowledge for  semiconductors lasers, this system is a very convenient  playground which provides a theoretical framework to  discuss both lasing and condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Using the  Van Roosbroek-Schockley relation, we have shown that  the photon Bose-Einstein condensation in the driven  dissipative regime is a particular case of the lasing regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The theoretical framework also enables to compare the  definitions of threshold used either for condensation  or for lasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  A Knudsen number emerges naturally  from the analysis to characterize thermalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We  have discussed its close connection with other quantities  introduced in different contexts such as thermalisation  degree, optical thickness and cooperativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Equipped  with  this  theoretical  figure  of  merit  to  quantify  thermalization, we have analysed different experimental  procedures to assess thermalization and put forward their  strengths and limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Finally, we have explored  the connection between the intensity fluctuations and  11  the emission regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Large fluctuations are a priori  expected to be a signature of the grand canonical regime  typical of the equilibrium condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' However, using  a Langevin analytical model of the fluctuations in the  driven-dissipative regime, we showed that the coherence  threshold does not depend on the thermalization degree,  both for large and small β-factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In this paper, we have explored the stationary regime  of a single BEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' The semiconductor platform appears to  be a very fruitful playground to study BEC physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' An  interesting direction for future work is to revisit in the  BEC regime recent results obtained with semiconductor  cavities such as topological lasers [66–68], chiral emission  [69], nonlinearities [70] including superfluidity [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' The  platform is also well suited to further explore the  dynamical behavior of BEC [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Also, the analysis of  the fluctuations has revealed an interesting regime for  micro and nanolasers above the lasing threshold and  below the coherence threshold which can be viewed as  grand-canonical fluctuations in non-equilibrium systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  This calls for more detailed studies of this phenomenon  in the framework of open systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' It may provide new  experimental platforms for the study of nonequilibrium  statistical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  We are grateful to Gian Luca Lippi for helpful  discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  This work is supported by the French  National Agency (ANR) (ANR-17-CE24-0046).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  acknowledges the support of Institut Universitaire de  France (IUF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Appendix A: Microscopic expression of gl  According to the Fermi golden rule, it is shown that gl  can be cast into the form [30]:  gl = 2π  ℏ [ℏΩ]2ρJVactΓl,  (A1)  where ℏΩ is the projected light-matter coupling  Hamiltonian between a single vertical transition and a  plane wave, Vact is the volume of the gain medium (also  called active medium) and Γl is the overlap integral  between the gain medium and the cavity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The  spatial structure of the mode electric field is thus fully  contained in Γl which is thus mode dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Appendix B: Thermalization degree in devices  featuring a large, planar and homogeneously  pumped cavity  Plugging Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  (2) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  (9), the thermalization  degree has the form:  Dl = gl  κl fF D(Ev(kl), T, µv)[1 − fF D(Ec(kl), T, µc)],  (B1)  that is a pump-independent term gl/κl and a pump  dependent term corresponding to the product of the  Fermi-Dirac distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  According to the Appendix A, the pump-independent  term  reads  gl/κl  =  2π  ℏ [ℏΩ]2ρJVactΓl/κl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In  semiconductor devices featuring a large and planar  cavity, the measurable spectrum typically extends over  ∼ 40 meV due to detection angle limitation and high  refractive index material [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The variations of the  transition matrix element Ωl, the joint density of states  ρJ and the mirror loss rate κl are negligible in this range  [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Also, as the in-plane part of the modes is nearly a  plane wave, the overlap integral Γl is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Hence,  the pump-independent part of the thermalization degree  gl/κl should thus be constant over the modes to a good  approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We now focus on the pump dependent term, which  describes the saturation of the absorption through  pumping (see Section III C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Rigorously speaking, this  term has always some dependence on the modes, since  saturation of the absorption is greater for low energy  modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Still, it is showed on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  4 that the impact  of this dependence is almost unnoticeable on the emitted  spectrum for values of g/κ as low as ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Noting that  lasing is prevented if g/κ < 1 [73], the range over which  the absorption saturation term has significant influence is  quite narrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' As a conclusion, devices featuring a large,  planar and homogeneously pumped cavity are expected  to be well described by a constant Knudsen number over  the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='50  E (eV)  100  101  102  103  104  105  106  Normalized occupation  Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' BE dis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' (E,  =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='3073eV,g/ =4)  Fit by BE dis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' (E,  fit =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='2709eV)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Best fit of a generalized BE distribution ( Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' (8))  by an ideal BE distribution ( Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' (6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' The fit parameter  of the ideal BE distribution is the quasi-Fermi levels splitting  µfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  gl and κl are assumed constant over all modes with  g/κ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  All curves are normalized to 1 at E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='50 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Other parameters are compiled in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  12  Appendix C: Model parameters values in figures  We take parameter values representative of the VCSEL  used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The semiconductor gain medium  consists of InGaAs quantum wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' We take Egap = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='215  eV, m∗  c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='059 × me, m∗  v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='37 × me where me  is the electron mass [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  The hole mass corresponds  to the valence band heavy-hole mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Contribution of  transitions with other valence bands is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  We  assume room temperature operation T = 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Appendix D: Full expression of g(2)(0)  According to the treatment and notations of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  IV B, the full expression of the second-order intensity  correlations at zero-time delay writes [30, 61, 65]:  g(2)(0) = 1 −  1  Nss  + γ2  peSee + γpeγee(2Sp e) + (γpeγep + γppγee + γ2  ee)Spp  (γpp + γee)(γpeγep + γppγee)N 2ss  ,  (D1)  where 2See = 2Spp = 2Rem[Nss + 1] and 2Sep =  −Rem[2Nss + 1] + Nss[Rem − Rabs].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  This expression can be expanded more explicitely as  g(2)(0) = 1 −  1  Nss  +  �  1  1 + Nss/Nβ + N 2ss/N 2  CO  �  ×  �  1  1 + N 2ss/N 2  LAS  �  ×  �  1 +  N 2  ss  N 2  LAS  + Nss  Nβ  +  � Nss  NCO  �2�  ∂NeRem  ∂Ne[Rem − Rabs] +  Nss  N 2  LAS  + κRem  g  ��  (D2)  where Nβ ≈  �  1  β − 1  �−1  Rem(Nss=∞)  ∂NeRem(Nss=∞) and NCO ≈  �  Rem(Nss=∞)  ∂Ne[Rem−Rabs](Nss=∞)  �1/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  In the limit β → 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  β → 0), the first (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' second) term between brackets  dominates while the product of the other terms ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' The  term 1/Nss is always negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
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+page_content=' Vewinger, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
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+page_content=' Schmitt, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
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+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
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+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
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+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Trichet,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Smith,  and  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
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+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
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+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Oulton, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Mintert, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Nyman, Learning  the fuzzy phases of small photonic condensates, Physical  Review Letters 126, 150602 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
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+page_content=' Damm, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Dung, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Vewinger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Klaers,  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Weitz, Observation of grand-canonical number  statistics in a photon Bose-Einstein condensate, Physical  Review Letters 112, 030401 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
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+page_content=' Damm, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Dung, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Wahl, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Vewinger,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Klaers, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Weitz, Spontaneous symmetry breaking  and phase coherence of a photon bose-einstein condensate  coupled to a reservoir, Physical Review Letters 116,  033604 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
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+page_content='  Dung,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Kurtscheid,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
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+page_content='  Yacomotti, Spontaneous mirror-symmetry breaking in  coupled photonic-crystal nanolasers, Nature Photonics 9,  311 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  [71] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Keijsers, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Geng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Peters, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Wouters, and  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Rodriguez, Steady-state superfluidity of light in a  tunable cavity at room temperature, arXiv preprint  arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='13463 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  [72] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Gladilin and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Wouters, Classical field model for  arrays of photon condensates, Physical Review A 101,  043814 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  [73] This  follows  from  the  gain  clamping  condition  [Rl  em(µclp) − Rl  abs(µclp)]  =  κl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' The left hand side  of this expression is bounded by the full inversion value  of the gain [Rl  em(µclp → ∞) − Rl  abs(µclp → ∞)] = gl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content='  Losses that exceeds this bounds prevents gain clamping  to occur, and thus lasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
+page_content=' Hence, the condition for lasing  reformulates in gl/κl > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFPT4oBgHgl3EQfYjSJ/content/2301.13073v1.pdf'}
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+IMAGE AND VIDEO COMPRESSION OF FLUID FLOW DATA
+A PREPRINT
+Vishal Anatharaman, Jason Feldkamp, Kai Fukami∗, Kunihiko Taira
+Department of Mechanical and Aerospace Engineering,
+University of California, Los Angeles, CA 90095, USA
+Corresponding author: kfukami1@g.ucla.edu
+January 3, 2023
+ABSTRACT
+We study the compression of spatial and temporal features in fluid flow data using multimedia com-
+pression techniques. The efficacy of spatial compression techniques, including JPEG and JPEG2000
+(JP2), and spatio-temporal video compression techniques, namely H.264, H.265, and AV1, in limiting
+the introduction of compression artifacts and preserving underlying flow physics are considered for
+laminar periodic wake around a cylinder, two-dimensional turbulence, and turbulent channel flow.
+These compression techniques significantly compress flow data while maintaining dominant flow
+features with negligible error. AV1 and H.265 compressions present the best performance across a
+variety of canonical flow regimes and outperform traditional techniques such as proper orthogonal
+decomposition in some cases. These image and video compression algorithms are flexible, scalable,
+and generalizable holding potential for a wide range of applications in fluid dynamics in the context
+of data storage and transfer.
+1
+Introduction
+High-fidelity simulations and experiments within the field of fluid dynamics produce exceedingly large amounts of
+data. As the need for higher fidelity simulations and advanced experimental resources expands, storage and transfer
+requirements for spatio-temporal data from simulations, become a major challenge. To address this issue, spatio-
+temporal redundancies or repeated dominant flow features can be exploited by a variety of compression techniques
+to alleviate memory constraints for fluid flow data storage. A variety of compression techniques, including modal
+analysis [1, 2, 3], sub-sampling and local re-simulation [4], and deep learning [5, 6, 7, 8] have been considered in an
+effort to reduce the size of fluid flow data. Although effective, these techniques can be application-specific and struggle
+to achieve substantial compression ratios without introducing undesirable compression artifacts such as discontinuities
+or deletions of flow features.
+In comparison, multimedia compression techniques are general and simple to use, and have benefited from demand
+for the modern technologies of high-resolution video streaming [9, 10, 11] and video-conferencing [12, 13, 14].
+These compression techniques are classified into two groups: lossless compression and lossy compression [15]. With
+lossless techniques, the data retrieved from or reconstructed from the compressed state is identical to that preceding
+the application of a compression algorithm. Hence, this is preferred for archival purposes and used for medical
+imaging [16] and technical drawings [17]. In contrast, processed data with lossy techniques do not necessarily match the
+original data, enabling a significant data-size reduction in the compressed state. Since this may introduce compression
+artifacts such as discontinuities in image data or the loss of high spatial frequency information, it is suitable for natural
+images such as photographs in applications where imperceptible loss may be acceptable [18]. We consider here the
+impacts of such losses on fluid mechanics simulation data to assess the costs of applying lossy techniques. In 2003,
+Schmalzl [19] considered multimedia data compression for fluid flows with an example of Rayleigh-Bénard convection.
+With multimedia compression technologies having undergone significantly advances in the last two decades, we reassess
+image and video compressions with modern algorithms for applications to fluid flow data.
+Lossy techniques of interest typically involve frequency-domain transformation, filtering, and entropy coding as
+components in the compression process. The development of the discrete cosine transform (DCT) [20, 21] has played a
+arXiv:2301.00078v1  [physics.flu-dyn]  31 Dec 2022
+
+A PREPRINT - JANUARY 3, 2023
+Original data
+Compressed data
+Reconstruction
+…
+Spatial 
+Compression
+Temporal 
+Compression
+t
+Original 
+Data
+Compressed 
+Data
+…
+!(#, %, &)
+!′(#, %, &)
+Original data
+Reconstruction
+(a)
+(b)
+Figure 1: (a) Spatial compression: an example velocity field of flow over a cylinder q(x) is represented as a grayscale
+image, encoded using an image-based technique to a compressed form, and reconstructed as ˜q(x) using a decoder.
+(b) Spatio-temporal compression: multiple snapshots of this flow field data q(x, t) are represented as a grayscale video
+and are compressed to ˜q(x, t) with both spatial and temporal techniques.
+crucial role in image compression, and is the basis of Joint Photographic Experts Group (JPEG) [22]. The emergence of
+JPEG enabled efficient image compression in a wide range of communities and it became a generally accepted format
+for digital images. After the development of DCT, wavelet transforms began to be utilized for image compression
+in such algorithms as JPEG2000 (JP2) [23], which achieves better compression than the DCT of JPEG as a result of
+multi-scale properties of wavelets.
+In tandem with the growth of image compression techniques, advancement in video compression technologies followed
+suit since video data can be characterized as a time series of image frames. Generally, these time frames include both
+spatial and temporal redundancies. In fact, we often see the similarities (redundancies) between temporally adjacent
+frames or spatially adjacent pixels. Video compression algorithms are designed to remove such redundancies and
+obtain a compact form of the original information. Current video compression technologies are generally based on the
+DCT [24]. Although other candidates including fractal compression [25, 26], matching pursuit [27], and discrete wavelet
+transform (DWT) have been investigated as the subject of some studies, these are still not used in practical products.
+Moving Picture Experts Group (MPEG) series have been traditionally used for video compression of high-definition
+television [28, 29, 30]. H.2xx series was then developed and they have achieved significant compression compared to
+the conventional MPEGs [31, 32]. Especially in the recent versions such as H.264 and H.265, motion compensation,
+quantization, and entropy coding are applied for efficient video compression. More recently, AOMedia Video 1 (AV1),
+an open, royalty-free video coding format, was released in 2018, achieving enhanced compression compared to the
+aforementioned techniques [33, 34].
+To meet the demand for these image and video compression tools, significant investment and research have produced
+compression techniques of impressive efficiency and usability in addition to free video encoders [35] to promote
+widespread accessibility. As such, leveraging these multimedia-inspired compression techniques should also be of
+particular interest to the fluid dynamics community given the massive scale of data produced, stored, and transferred.
+2
+
+Original Data
+Compressed Data
+Reconstruction
+nxm
+< nxm
+nxm
+01100111 01101111
+11.
+Encoding
+Decoding
+(qn1
+g(x, y)gfa,tgfr,t2A PREPRINT - JANUARY 3, 2023
+A standardization on one or more multimedia compression formats for storing fluid flow data in a compressed
+representation can yield dividends in research output by allowing greater access to high-fidelity fluid flow data sets and
+by removing memory constraints as a barrier to entry.
+This paper investigates the effectiveness of these image and video compression techniques on fluid flow data. Spatial
+image compression techniques, such as JPEG and JP2, alongside spatio-temporal video compression techniques,
+namely H.264, H.265, and AV1, are examined for various flow fields, including laminar cylinder flow, two-dimensional
+turbulence, and turbulent channel flow. Field variables from simulation data, such as streamwise velocity and vorticity,
+are represented as grayscale images, and multiple snapshots are packaged into a video. These videos are then encoded
+into a compressed form using the aforementioned multimedia compression methods. Modern techniques can compress
+flow data well below 10% of the original file size with negligible error and preserve the underlying physics of the flow.
+Although this paper focuses on applications to canonical fluid flows, the flexibility and scalability of these algorithms
+suggest an expansive potential within this field.
+Compression is a process in which data is compressed (encoded) into a representation that uses less data, and
+decompressed (decoded) into identical data in the case of lossless compression or nearly-identical data in the case
+of lossy compression. Through this procedure, a compression method reduces bits of the original data q(x, t) by
+eliminating statistical redundancies that may be contained within temporally adjacent frames and spatially adjacent
+pixels. In general, a data compression algorithm is referred to as an encoder φ while one that performs the decompression
+is called a decoder ψ,
+γ(x, t) = φ(q(x, t)),
+q(x, t) ≈ ˜q(x, t) = ψ(γ(x, t)),
+(1)
+where γ(x, t) is the compressed data corresponding to the original data q(x, t). Depending on the extent of compression,
+the data, and a choice of encoder/decoder, the reconstruction ˜q(x, t) generally includes some amount of error.
+The data compression process is illustrated in figure 1 for both image and video compressions. Figure 1(a) depicts a lossy
+spatial image compression technique, involving quantization of the image data in a compressed space and producing
+a reconstruction in the image space showing the operations of JPEG and JP2. Figure 1(b) provides a visualization
+of a spatio-temporal compression technique, exploiting a redundant block of a frame that remains consistent across
+subsequent frames, similar to H.264, H.265, and AV1. As these algorithms originated in the multimedia industry, they
+are optimized for human viewers and involve the removal of high-frequency components in the data and down-sampling
+of the color spectrum such that the eyes cannot easily distinguish compressed data from the original data. For the
+purposes of this study, we only consider grayscale images and videos, which are comprised only of a single-component
+field data matrix, denoted as ˜q(x). This is in contrast to full-color data, which requires red, green, and blue components,
+and is unnecessary for the current analysis as we are interested in considering field variables individually. Herein, we
+consider the application of five compression techniques on grayscale images and videos. The encoding schemes, which
+package the data into a compressed binary form, are detailed in what follows.
+2
+Compression techniques
+2.1
+Image Compression
+2.1.1
+JPEG
+Let us first describe JPEG, which is a standard lossy spatial compression used for encoding image data based on the
+discrete cosine transform (DCT). An example of a JPEG compression process with a vorticity field of two-dimensional
+decaying isotropic turbulence is presented in figure 2. The images are partitioned into 8 × 8 blocks in a left-to-right,
+top-to-bottom scan. Pixel values within blocks are quantized to values of [−128, 127] from [0, 255]. The forward DCT
+is individually performed at each block and outputs compressed data. The DCT for 8 × 8 blocks is mathematically
+expressed as
+F(kx, ky) = 1
+4C(kx)C(ky)
+�
+7
+�
+ix=0
+7
+�
+iy=0
+f(ix, iy) cos
+�(2ix + 1)kxπ
+16
+�
+cos
+�(2iy + 1)kyπ
+16
+��
+,
+(2)
+f(ix, iy) = 1
+4
+�
+7
+�
+kx=0
+7
+�
+ky=0
+C(kx)C(ky)F(kx, ky) cos
+�(2ix + 1)kxπ
+16
+�
+cos
+�(2iy + 1)kyπ
+16
+��
+,
+(3)
+where
+C(k) =
+�1/
+√
+2
+for k = 0
+1
+otherwise.
+(4)
+3
+
+A PREPRINT - JANUARY 3, 2023
+Input
+8×8 block
+DCT for 8×8 block
+Reconstructed image
+(Keeping 8.31% of the DCT coefficients)
+Figure 2: JPEG compression process with an example of two-dimensional isotropic turbulent vorticity.
+Here, F(kx, ky) denotes the DCT coefficient corresponding to the horizontal wavelength kx and vertical wavelength ky
+and f(ix, iy) describes the pixel value at the location corresponding to ix and iy. In other words, the forward DCT takes
+as input a discrete signal of 64 points and produces coefficients for a linear combination of 64 unique basis signals, each
+denoting a specific spatial wavelength. Most of the spatial domain information is concentrated across lower wavelength
+because of slow spatial variation from one pixel to the next in image data. This quality permits lossy quantization,
+which refers to constant values in a quantization table Q(kx, ky) with 64 elements. The DCT coefficient is normalized
+by a constant Q(kx, ky) in an element-wise manner,
+F Q(kx, ky) = ⌊F(kx, ky)
+Q(kx, ky)⌋
+(5)
+where F Q(kx, ky) is a normalized coefficient and the operation ⌊·⌋ denotes rounding to the nearest integer. Quantization
+tables are provided by the Joint Photographics Experts Group. Note that dividing the DCT coefficients by values in the
+quantization table reduces high-wavenumber coefficients to 0, which permits efficient entropy coding (explained later)
+to perform the cutoff at high frequencies [36]. The resulting quantized DCT coefficients form a matrix of size 8 × 8
+with low-wavenumber components generally located in the top-left of the matrix and high-wavenumber coefficients at
+the bottom-right, as a consequence of the similar distribution of spatial modes to which these coefficients correspond.
+Subsequently, quantized coefficients are ordered from low to high wavenumbers.
+To reduce the data size, entropy coding [37, 38], a lossless method of compressing bitstreams with redundancies, is then
+performed for the output of DCT. The idea of entropy coding is used not only for JPEG but also other image/video
+compression techniques such as JP2, H.2xx series, and AV1. To express the encoding-based data compression, let us
+consider a message of DAEBCBACBBBC (12 characters). Since this message includes five different characters, it
+needs to prepare 3 bits to convert these characters to bits or binary digits representation. Here, we use the following
+conversion table,
+A
+B
+C
+D
+E
+000
+001
+010
+011
+100
+With this table, the message is expressed as
+D
+A
+E
+B
+C
+B
+A
+C
+B
+B
+B
+C
+011
+000
+100
+001
+010
+001
+000
+010
+001
+001
+001
+010
+As shown, the number of bits is 36. The idea of the encoding-based compression is to prepare an adaptive conversion
+table assigning a shorter bit length for characters that appear in a high probability and a longer bit length for characters
+that barely appear. For example, the following adaptive table can be used:
+A
+B
+C
+D
+E
+110
+0
+10
+1110
+1111
+4
+
+A PREPRINT - JANUARY 3, 2023
+With this new table, the message can be expressed as
+D
+A
+E
+B
+C
+B
+A
+C
+B
+B
+B
+C
+1110
+110
+1111
+0
+10
+0
+110
+10
+0
+0
+0
+10
+The current table can save the total number of bits for the message from 36 to 25. Modern data compression techniques
+efficiently find such an adaptive conversion table for saving image and video sizes.
+The presence of a better adaptive conversion can be proven with the source coding theorem [39]. For any data, the
+expected code length should satisfy the relationship,
+Eβ∼P [l(d(β))] ≥ Eβ∼P [− logb(P(β))],
+(6)
+where l is the number of symbols in a message, d is the coding function, b is the number of symbols in a table, and P is
+the probability of the original symbol. An entropy coding method attempts to approach the lower bound. For JPEG
+compression, Huffman coding [40] is used to determine an adaptable table composed of the estimated probability of
+occurrence for each possible value. Huffman coding uses binary trees [41] for efficient encoding.
+2.1.2
+JPEG2000 (JP2)
+JPEG2000 (JP2) is a successor to JPEG. JP2 operates using a similar four-step process to JPEG, involving image
+partitioning, frequency-domain transformation, quantization, and entropy coding. In contrast to JPEG, JP2 introduces
+more advanced dynamic tiling algorithms based on variable-sized macroblocks. This makes use of the discrete wavelet
+transform (DWT), and performs additional preprocessing prior to entropy coding. Tiling in this context refers to the
+partitioning of the source image into several non-overlapping rectangular blocks, each of which is processed distinctly.
+Whereas JPEG restricts tile sizes to 8 × 8, tiles in JP2 can be of arbitrary size up to the image dimensions. The DWT is
+applied to each tile in a manner similar to DCT, decomposing a signal into a linear combination of wavelet functions.
+The coefficients in this linear combination correspond to a specific wavelet basis function in the signal. A wavelet can
+be defined as a scale and shift of a basis wavelet. Child wavelets [42] are generally considered for DWT, given by
+ψg,r(s) =
+1
+2g/2 ψ
+�s − 2gr
+2g
+�
+(7)
+where g is a scaling factor, r is a shift factor, and s corresponds to the index of the one-dimensional representation of an
+image. In other words, the flow field snapshot is converted to a one-dimensional representation and the independent
+variable upon which this one-dimensional signal f(s). The DWT coefficient given a wavelet of the preceding definition
+is then
+Fg,r =
+� ∞
+−∞
+f(s)ψg,rds
+(8)
+where f(s) is a one-dimensional signal. The signal can be reconstructed through the summation of the product of each
+coefficient with the corresponding wavelet. In a discrete interpretation, this is written as
+f(s) =
+∞
+�
+g=−∞
+∞
+�
+r=−∞
+Fg,rψg,r.
+(9)
+The summation bounds for both the calculation of the coefficients and reconstruction of the signal can be set to finite
+values and can still produce lossless reconstructions assuming that the wavelets contain the maximum and minimum
+wavelengths within the source image.
+An example of the DWT for the vorticity field of two-dimensional decaying turbulence is presented in figure 3. The
+DWT can be applied recursively to one-dimensional signals to produce higher fidelity representations of data. As such,
+successive high-pass filters are applied on down-sampled images, producing a higher fidelity representation of spatial
+frequencies in the source image. The DWT can be extended to higher dimensions by applying the one-dimensional
+DWT on rows and columns. The recursive application of the DWT produces 2n distinct filtered images where n is the
+number of times the DWT is applied.
+Similar to JPEG compression, the DWT coefficients are quantized following the transformation on the wavespace,
+F Q
+a,b = sign(Fa,b)⌊|Fa,b|
+∆b
+⌋,
+(10)
+5
+
+A PREPRINT - JANUARY 3, 2023
+(b)
+(c)
+(d)
+(e)
+(g)
+(f)
+(a)
+0
+50
+100
+-50
+-100
+Figure 3: An example of the two-level discrete wavelet transform for two-dimensional homogeneous turbulent vorticity
+field, used in JP2 compression. High-pass filtering yields three large images. Low-pass filtering and downscaling are
+then performed, producing the three small images. (a) The final approximation image. (b, e) Vertical, (c, f) horizontal,
+and (d, g) diagonal coefficients of the second (b − d) and the first levels (e − g) are shown.
+where ∆b is defined as the quantization step. DWT coefficients within the range (−∆b, ∆b) are quantized to 0.
+Following quantization, the coefficients are processed in preparation for entropy coding. Arithmetic coding [43] is
+used for entropy coding in JPEG2000. While Huffman coding separates the original data into component symbols and
+replaces each with a code in a table, Arithmetic coding encodes the entire message into a single number represented
+with an arbitrary-precision fraction pa, where 0 ≤ pa < 1.
+2.2
+Video Compression
+2.2.1
+H.264 (AVC)
+The H.264 video compression includes a multi-step process, consisting generally of prediction, transformation (a set
+of frequency-domain representation and quantization in image compression), and entropy encoding on the encoder
+side. A similar process for file reconstruction is performed on the decoder side. Prediction in video compression
+amounts to an operation to remove redundancies in the given signal. H.264 supports a range of prediction options
+6
+
+500
+450
+400
+350
+300
+250
+200
+150
+100
+50
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200160
+140
+120
+100
+80
+60
+40
+20
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200160
+140
+120
+100
+80
+60
+40
+20
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200300
+250
+200
+150
+100
+50
+50
+100
+150
+200
+250
+300
+350
+400300
+250
+200
+150
+100
+50
+50
+100
+150
+200
+250
+300
+350
+400100
+ 50
+0
+-50
+-100A PREPRINT - JANUARY 3, 2023
+Frame 1
+MC Frame 2
+Frame 2 MC Residual
+Figure 4: Motion compensation (MC), used in temporal compression of H.264, H.265, and AV1, is exemplified by the
+calculation of a motion vector field, describing the translation of pixels between successive frames. As an example, a
+streamwise velocity field u of three-dimensional turbulent channel flow is considered. Subtracting this field from the
+original image yields a residual image, which is stored.
+such as intra-prediction used for data within the current frame, inter-prediction for motion compensation, and multiple
+block-size-based predictions. An accurate prediction implies that the residual contains very little information, amounting
+to good compression performance.
+Video data is first partitioned into macroblocks of dimension 16 × 16 pixels. A prediction of the current macroblock
+is formed using 4 × 4 and 16 × 16-sized blocks in the case of intra-frame prediction, referring to predictions from
+surrounding blocks within the same frame. A range of block sizes from 4 × 4 to 16 × 16 are also considered in the
+case of inter-frame prediction, referring to predictions from previously coded frames. Macroblock prediction is further
+discretized into intra-prediction with neighboring blocks in the current frame, blocks in a previously coded frame, and
+blocks from up to two previously coded frames.
+In intra-prediction, the size of prediction macroblocks can be three cases: 16 × 16, 8 × 8, or 4 × 4 pixels [31]. The
+choice of block size is made primarily based on prediction efficiency. One of several prediction modes where each
+prediction mode indicates a direction in which to extrapolate pixel values or to average across all pixels. In the case of
+inter-prediction, the reference frame, where the prediction block is situated, is chosen from several previously decoded
+frames. As shown in figure 4, a motion vector is then obtained for the current macroblock, based on the offset from the
+prediction block or from previously coded motion vectors. These motion vectors are optionally weighted to account for
+temporal proximity between frames, and are sent into the data stream. Generally, a deblocking filter is further applied to
+each frame to store in a decoded format for subsequent inter-frame predictions in smoothing the sharp edges caused by
+the use of block coding [44].
+Transformation involves the conversion of blocks to frequency-domain representations and quantization of coefficients
+corresponding to high wavelength data. The DCT step is given by the transformation of block X by matrix A into DCT
+coefficients Y for each macroblock. For the case of 4 × 4 blocks, these matrices X, Y ∈ R4×4 are expressed as
+Y = AXAT =
+�
+��
+a
+a
+a
+a
+b
+c
+−c
+−b
+a
+−a
+−a
+a
+c
+−b
+b
+−c
+�
+�� X
+�
+��
+a
+b
+a
+c
+a
+c
+−a
+−b
+a
+−c
+−a
+b
+a
+−b
+a
+−c
+�
+�� ,
+(11)
+where a = 1/2, b =
+�
+1/2 cos(π/8), c =
+�
+1/2 cos(3π/8). The rows of A are orthonormal. To calculate equation 11
+on a practical processor, the approximation for b and c is required. This is achieved with a fixed-point approximation,
+which is equivalent to scaling each row of A by 2.5 and rounding to the nearest integer. A core transform Cf4, which
+scales each term of A by 2.5 and rounds to the nearest integer, and a scaling matrix Sf4, which restores norms of row of
+Cf4 to 1 by scaling, are respectively defined as,
+Cf4 =
+�
+��
+c1
+c1
+c1
+c1
+c2
+c1
+−c1
+−c2
+c1
+−c1
+−c1
+c1
+c1
+−c2
+c2
+−c1
+�
+�� ,
+Sf4 =
+�
+��
+s1
+s2
+s1
+s2
+s2
+s3
+s2
+s3
+s1
+s2
+s1
+s2
+s2
+s3
+s2
+s3
+�
+�� ,
+(12)
+where c1 = 1, c2 = 2, s1 = 1/4, s2 = 1/(2
+√
+10), and s3 = 1/10. Matrix Y is then determined as
+Y = [Cf4XCT
+f4]Sf4.
+(13)
+Similar DCT approximations are specified for other block sizes, involving Cf8, Sf8, and others [31]. A quantization
+mechanism similar to JPEG is applied, with a quantization table specified for various block sizes. The quantized DCT
+coefficients are then traversed in an oscillating, “zig-zag” manner from low- to high-wavenumber components. Entropy
+coding is finally applied to the output of DCT.
+7
+
+Frame 1
+MC Frame 2
+Frame 2 MC ResidualA PREPRINT - JANUARY 3, 2023
+2.2.2
+H.265 (HEVC)
+H.265 was released in 2013 as the successor to H.264. While the fundamental architecture is unchanged from H.264,
+H.265 makes use of coding tree units which are similar to macroblocks but expand the range of possible dimensions,
+with variable dimensions selected by the encoder, allowing coding tree units to be divided into sub-blocks. Predictions
+and reconstructions are performed on coding tree units and supported sub-block sizes range from 64 × 64 to 4 × 4
+pixels. Motion vectors are predicted based on those of adjacent units or blocks in the case of intra prediction, or previous
+encoded frames in the case of inter prediction. As in H.264, the DCT is performed at the coding tree block level, and
+the resultant coefficients are subjected to scalar quantization and entropy coding. Instead of deblocking filter in H.264,
+a sample adaptive offset filter is applied within the prediction loop to improve the quality of the compressed data [45].
+2.2.3
+AV1
+AV1 was released in 2018 in an effort to replace the H.2XX series of video compression algorithms. In AV1, much
+of the fundamental architecture from H.2XX is maintained. Frames are partitioned using a 4-way partition tree with
+dimensions ranging from 128 × 128 to 4 × 4. AV1 supports 56 directional spatial modes for intra-frame prediction with
+finer angle variations than that provided by H.2XX. AV1 extends the number of reference frames that any given frame
+can use to perform predictions to seven references for inter-frame prediction, thus enabling more accurate encoding
+of data with rich temporal characteristics. Motion vector field formation is improved by expanding the spatial search
+domain for vector candidates and through the utilization of a temporal motion field estimation system. AV1 also
+extends frequency-domain transform algorithms to include the asymmetric discrete sine transform with a richer set
+of transform kernels for varying block sizes. Entropy coding and scalar quantization are also used as well as H.2XX
+compression algorithms. To perform deblocking, a constrained directional enhancement filter and loop restoration
+filters are applied [33]. These filters are able to effectively remove artifacts without causing blurring, compared to
+conventional deblocking filters [46].
+3
+Flow Fields
+Let us apply the compression techniques presented above to representative fluid flow data sets from numerical
+simulations. We describe herein the problem setup of the example flow fields we analyze and the simulation approach
+used to generate them.
+3.1
+Two-dimensional laminar cylinder wake
+Bluff body flow forms a large class of problems, such as the vortex shedding around a cylinder. We first apply the
+compression techniques to the two-dimensional cylinder wake obtained by direct numerical simulation (DNS) [47, 48].
+The governing equations are the incompressible Navier–Stokes equations,
+∇ · u = 0,
+(14)
+∂u
+∂t + u · ∇u = −∇p +
+1
+ReD
+∇2u,
+(15)
+where u and p are the non-dimensionalized velocity vector and pressure, respectively.
+All variables are non-
+dimensionalized with the fluid density ρ, the uniform velocity U∞, and the cylinder diameter D. The Reynolds
+number is defined as ReD = U∞D/ν = 100 with ν being the kinematic viscosity. We consider five nested lev-
+els of multi-domains with the finest grid level covering (x, y) = [−1, 15] × [−8, 18] and the largest domain being
+(x, y) = [−5, 75] × [−40, 40]. The time step for DNS is ∆t = 2.50 × 10−3. We extract the domain around a cylinder
+body over (x∗, y∗)/D = [−0.7, 15] × [−5, 5] with (Nx, Ny) = (500, 300) and (∆x, ∆y) = (0.0314, 0.0333). The
+flow exhibits vortex shedding with a single period with 21 snapshots. For the compression analysis, 160 temporal
+snapshots of grayscale images of the streamwise velocity field u are considered.
+3.2
+Two-dimensional decaying homogeneous isotropic turbulence
+To examine the data compression performance by the present techniques for more complex turbulent flows, we also
+consider a two-dimensional decaying homogeneous isotropic turbulence. This time-varying flow can be regarded
+as a canonical fluid flow example for a broad range of turbulent flows. The data set is prepared by DNS using
+the two-dimensional vorticity transport equation [49]. We set the initial Reynolds number Re0 ≡ u∗l∗
+0/ν = 80.4,
+where u∗ is the characteristic velocity obtained by the square root of the spatially averaged initial kinetic energy,
+l∗
+0 = [2u2(t0)/ω2(t0)]1/2 is the initial integral length, and ν is the kinematic viscosity. The computational domain and
+8
+
+A PREPRINT - JANUARY 3, 2023
+the numbers of grid points are set to Lx = Ly = 1 and Nx = Ny = 128, respectively. We use 1000 snapshots in an
+eddy turn-overtime of t ∈ [2, 6] with a time interval of ∆t = 0.004. For the data compression analysis, 128 × 128 grid
+with grayscale contours of the vorticity field ω are used.
+3.3
+Turbulent channel flow
+To further demonstrate the present data compression techniques, we also examine turbulent channel flow at a friction
+Reynolds number of Reτ = uτδ/ν = 180, where uτ is the friction velocity, δ is the half-width of the channel, and ν
+is the kinematic viscosity. This flow involves a broader range of spatio-temporal flow scales and fewer redundancies
+compared to the previous two examples. The data sets are prepared by a three-dimensional DNS [50, 51], numerically
+solving the incompressible Navier–Stokes equations. The present DNS has been validated by comparison with spectral
+DNS data of Moser et al [52]. The streamwise, wall-normal, and spanwise spatial coordinates are denoted by x, y, and z,
+respectively. The size of the computational domain and the number of grid points here are (Lx, Ly, Lz) = (4πδ, 2δ, 2πδ)
+and (Nx, Ny, Nz) = (256, 96, 256), respectively. The grids in the streamwise and spanwise directions are taken to be
+uniform, while that in the y direction is a non-uniform grid. The no-slip boundary condition is imposed on the walls and
+a periodic boundary condition is applied to the x and z directions. The flow is driven by a constant pressure gradient. In
+what follows, we denote wall-unit quantities with the superscript +.
+For the present study, an x − z cross-sectional streamwise velocity u at y+ = 13.2 is analyzed, where representative
+streak structures are present [53]. Fifty temporal snapshots of a 256 × 256 spatial grid over t+ ∈ [0, 63] are formatted
+into grayscale data and are used for compression assessment.
+4
+Results
+For image compression, matrices of flow field data corresponding to specific temporal snapshots are represented as
+uncompressed grayscale images. These images are compressed using JPEG and JP2 encoders. For video compression,
+the matrices of flow field data are represented as grayscale images and concatenated into uncompressed, grayscale
+videos. This raw video file is used as the input to the H.264, H.265, and AV1 encoders. To obtain control over the
+outputted file size for the purposes of this analysis, a two-pass encoding scheme is considered. In the two-pass encoding,
+the encoder runs twice. The first run is used to collect some information and statistics such as how many bytes would
+be needed for data compression and the second run performs the actual encoding. These two processes enable the use
+of the information collected in the first run to achieve enhanced compression.
+This study uses FFmpeg [35], a free and open-source software consisting of various libraries for handling video, audio,
+and other multimedia files. These libraries can be easily used from the command line ffmpeg. Default encoding
+settings for the relevant FFMpeg library are used to maintain consistency across all tests.
+4.1
+Image compression
+To establish a baseline performance for comparison, individual flow snapshots are compressed using singular value
+decomposition (SVD). Individual snapshots are decomposed into left and right singular vector matrices U and V T , and
+a diagonal matrix Σ containing the singular values. Snapshots are reconstructed by retaining the r leading modes. Here,
+the compression ratio η in the SVD context is then defined as
+η = r(m + n + 1)
+mn
+,
+(16)
+where m and n are the snapshot dimensions in the horizontal and vertical directions and a value of η = 1 corresponds
+to the original, uncompressed snapshot. In the present study, the snapshot dimensions for each flow example are set
+as (m, n) = (500, 300) for cylinder wake, (128, 128) for two-dimensional decaying turbulence, and (256, 256) for
+turbulent channel flow, respectively.
+Let us compare the image compression techniques with the cylinder wake example. As for the data attribute, we use
+a streamwise velocity u. The compressed wake fields and the spatial absolute error distributions with η ≈ 0.05 are
+presented in figure 5. The L2 error norm of reconstruction ε = ||fRef − fComp||2/||fRef||2, where fRef and fComp
+are respectively the reference and compressed flow fields, is also shown underneath the decoded fields. The SVD
+produces negligible error for the entire flow field, although slight discontinuities are observed in the wake region. By
+comparison, JPEG compression introduces some compression artifacts including discontinuities of grayscale contours
+and granulated vortical structures. This is due to fixed-size areas upon which the DCT is performed and elementary
+anti-blocking features. This indicates that compression based on the fixed block of 8 × 8 pixels is not appropriate for
+flow fields that include fine-scale spatial variations. In contrast, the compressed flow field with the JP2 algorithm retains
+9
+
+A PREPRINT - JANUARY 3, 2023
+JPEG
+JPEG2000
+SVD
+{ε, η} = {0.0176, 0.0548}
+{ε, η} = {0.00370, 0.0507}
+{ε, η} = {0.00670, 0.0532}
+Compressed flow field
+Spatial error distribution
+0
+50
+100
+-50
+-100
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+Figure 5: Comparison of image compression techniques for cylinder wake at ReD = 100. A streamwise velocity field
+u is considered. The L2 error norm of the reconstruction ε and the compression ratio η are shown underneath each flow
+field contour. Spatial absolute error distribution for each compression technique is also presented.
+JPEG
+JPEG2000
+SVD
+{ε, η} = {0.0350, 0.282}
+{ε, η} = {0.0233, 0.270}
+{ε, η} = {0.00820, 0.282}
+0
+50
+100
+-50
+-100
+Figure 6: Comparison of image compression techniques for two-dimensional decaying turbulence. A vorticity field ω is
+considered. The L2 error norm of the reconstruction ε and the compression ratio η are shown underneath each flow
+field contour.
+wake features even while achieving significant data compression, with the L2 error of 0.00370. These results support
+the effectiveness of the adaptive block size of DWT in JP2 compression for bluff body wake data sets.
+Next, we apply the image compression techniques to two-dimensional decaying homogeneous isotropic turbulence,
+as shown in figure 6. We use a vorticity field ω as a quantity of interest and compare the compression results with
+η ≈ 0.280. Similar to the cylinder example, the SVD can provide a smooth field and small error for the entire flow field.
+Although SVD can achieve a reasonable compression for the laminar cylinder wake and two-dimensional turbulence
+that are mainly composed of large vortical structures, we discuss later how the presence of fine-scale turbulent structures
+alters the compression performance. For this two-dimensional turbulence, the effect of 8 × 8 pixel blocks can be clearly
+observed in JPEG compression. Such pixelized artifacts on the flow field can be mitigated by using the JP2 compression
+technique, analogous to the observation with the cylinder example.
+The limitation of the SVD and the efficacy of the DWT-based process in the JP2 algorithm are further emphasized in
+the example of more complex turbulence. Here, the compression techniques are applied to a streamwise velocity u
+10
+
+JPEG
+JPEG2000
+SVD
+(8, n) = {0.0176, 0.0548)
+(8, n) = {0.00370, 0.0507
+(s, n) = {0.00670, 0.0532)
+(8, n) = (0.00490, 0.136)
+(8, n) = (0.00310, 0.120)
+(8, n) = (0.00180, 0.112)100
+ 50
+0
+-50
+-100JPEG
+JPEG2000
+SVD
+ = 0.0350
+8= 0.0233
+8 = 0.00820
+n = 0.282
+n = 0.270
+n = 0.282
+8 = 0.0168
+8 = 0.0134
+8= 0.00310
+n = 0.511
+n = 0.452
+n = 0.502100
+ 50
+0
+-50
+-100A PREPRINT - JANUARY 3, 2023
+{ε, η} = {0.129, 0.0281}
+{ε, η} = {0.0265, 0.234}
+{ε, η} = {0.0903, 0.0234}
+{ε, η} = {0.235, 0.0235}
+{ε, η} = {0.0122, 0.240}
+{ε, η} = {0.0258 0.235}
+JPEG
+SVD
+JPEG2000
+{ε, η} = {0.129, 0.0281}
+{ε, η} = {0.0265, 0.234}
+{ε, η} = {0.0903, 0.0234}
+{ε, η} = {0.235, 0.0235}
+{ε, η} = {0.0122, 0.240}
+{ε, η} = {0.0258 0.235}
+JPEG
+SVD
+JPEG2000
+{ε, η} = {0.129, 0.0281}
+{ε, η} = {0.0265, 0.234}
+{ε, η} = {0.0903, 0.0234}
+{ε, η} = {0.235, 0.0235}
+{ε, η} = {0.0122, 0.240}
+{ε, η} = {0.0258 0.235}
+JPEG
+SVD
+JPEG2000
+JPEG
+JPEG2000
+SVD
+{ε, η} = {0.129, 0.0281}
+{ε, η} = {0.0903, 0.0234}
+{ε, η} = {0.235, 0.0235}
+0
+50
+100
+-50
+-100
+Figure 7: Comparison of image compression techniques for turbulent channel flow at Reτ = 180. A streamwise
+velocity field u is considered. The L2 error norm of the reconstruction ε and the compression ratio η are shown
+underneath each flow field contour.
+(a)
+(b)
+: JPEG
+: JP2
+: SVD
+: Cylinder
+: 2D turbulence
+: Channel
+SVD
+JPEG
+JP2
+SVD
+JP2
+JPEG
+SVD
+JPEG
+JP2
+(c)
+(d)
+(e)
+Figure 8: Relationship between (a) the L2 error norm ε, (b) SSIM, and image compression ratio η. Zoom-in view of
+η-SSIM curve for (c) cylinder wake, (d) two-dimensional turbulence, and (e) turbulent channel flow.
+of the three-dimensional channel flow. The compression results with η ≈ 0.025 are compared in figure 7. As shown,
+the SVD-based compression cannot retain the important features of the streaks. Compared to SVD, JPEG provides a
+better reconstruction although it also introduces discontinuities that obscure small spatial length scales in the flow field.
+Surprisingly, JP2 produces non-negligible artifacts and maintains an L2 error norm less than half that of SVD. The
+channel flow field at this low η remains nearly indistinguishable from the uncompressed flow field, also preserving the
+streak spacing of the reference DNS field [53, 54]. These observations suggest the effectiveness of the JP2 algorithm
+for image compression of complex fluid flow data.
+Building on these assessments, the L2 error between compressed and uncompressed flow fields is evaluated across
+compression ratios, as shown in figure 8(a). The error is averaged over all temporal snapshots of each flow example. In
+general, all compression algorithms produce an asymptotically decaying L2 error. JPEG introduces appreciable error at
+low η, in the same order as SVD compression. It is worth pointing out that JP2 performs especially well at low η while
+SVD compression produces the lowest L2 error for high η for all flow fields.
+11
+
+100
+ 50
+0
+-50
+-100100
+10
+-2
+10
+10
+0.0
+0.2
+0.4
+0.6
+0.8
+n1.00
+0.75
+SSIM
+0.50
+L
+0.25
+0.00
+0.0
+0.2
+0.4
+0.6
+0.8
+n1.005
+1.000
+EE
+0.995
+SSIM
+0.990
+0.985
+0.980
+0.0
+0.2
+0.4
+0.6
+0.8
+n1.000
+0.975
+SSIM
+0.950
+0.925
+0.900
+0.0
+0.2
+0.4
+0.6
+0.8
+n1.00
+SSIM
+0.99
+0.98
+0.97
+0.0
+0.2
+0.4
+0.6
+0.8
+nA PREPRINT - JANUARY 3, 2023
+(a)
+(d)
+(b)
+(e)
+(c)
+(f)
+ηLow = 0.0299
+ηMed = 0.0913
+ηHigh = 0.237
+ηLow = 0.295
+ηMed = 0.539
+ηHigh = 0.885
+ηLow = 0.215
+ηMed = 0.505
+ηHigh = 0.932
+ηLow = 0.0299
+ηMed = 0.0913
+ηHigh = 0.237
+ηLow = 0.0381
+ηMed = 0.0880
+ηHigh = 0.803
+ηLow = 0.0381
+ηMed = 0.0880
+ηHigh = 0.803
+Figure 9: Kinetic energy spectra for two-dimensional decaying homogeneous isotropic turbulence using (a) JPEG and
+(d) JP2. (b, e) Streamwise and (c, f) spanwise kinetic energy spectrum of three-dimensional turbulent channel flow
+compressed with (b, c) JPEG and (e, f) JP2.
+As an additional metric for quantifying the error introduced by each compression method, the localized structural
+similarity index (SSIM) [55] is computed between compressed and uncompressed flow fields. SSIM can capture spatial
+correlation around pixels and is less sensitive against a pixel-wise error caused by translation and rotational difference
+compared to the L2 error. Hence, SSIM is suited for the image and video-based compression analysis. The SSIM χ is
+defined as
+χ = l(ix, iy)c(ix, iy)s(ix, iy)
+(17)
+where
+l(ix, iy) = 2µxµy + C1
+µ2x + µ2y + C1
+,
+c(ix, iy) = 2σxσy + C2
+σ2x + σ2y + C2
+,
+s(ix, iy) = σxy + C3
+σxσy + C3
+(18)
+with µx and σx defined as the mean and standard deviation of ix respectively, σxy being the covariance of ix and iy,
+and c1, c2, and c3 being constants to stabilize division. We set {C1, C2, C3} = {0.16, 1.44, 0.72} following Wang et
+al. [55]. The resultant value lies between 0, representing no similarity, and 1, representing an identical image. The
+relationship between the image compression ratio and the L2 error is depicted in figure 8(b) Generally, JP2 and SVD
+produce a negligible decrease in the SSIM at low compression ratios and asymptotically approach an SSIM value of 1
+at higher compression ratios. The SSIM value of the cylinder flow field with JPEG compression applied decays by
+approximately 10%, as a result of significant discontinuities produced by JPEG.
+We also present the zoom-in view of the relationship between SSIM and the compression ratio η for each flow, in
+figures 8(c) − (e). Similar to the observation in the η − ε curves in figure 8(a), SVD and JP2 provide high SSIM scores
+compared to JPEG. Especially at excessive compression (low η) of three-dimensional turbulent channel flow, JP2 can
+provide better reconstructions than the other two cases. Although scalar metrics such as the L2 error ε and SSIM are
+useful, we note that monitoring not only scalar values but also decoded flow fields is important in assessing how vortical
+structures can be retained through data compression because the influence of local structures are averaged.
+We are additionally interested in whether finer structures in flow images can still be retained through the present
+compression process. To examine this aspect, we consider the kinetic energy spectrum of both two- and three-
+dimensional turbulence examples, as summarized in figure 9. The kinetic energy spectrum E(k) for two-dimensional
+decaying turbulence is
+E(k) = 1
+2(uiui),
+(19)
+12
+
+Uncompressed
+JPEG LoW
+JPEG Medium
+JPEG High
+10~5
+E
+10-10
+101
+102
+103
+k-Uncompressed
+JP2 LoW
+JP2 Medium
+JP2 High
+10-5
+E
+10-10
+101
+102
+103
+k100
+10-4
+10-6
+10-1
+10-2100
+10-4
+10-6
+10~2
+10-1100
+10-4
+10-6
+10-1
+10~2100
+10-4
+10-6
+10-1
+102A PREPRINT - JANUARY 3, 2023
+{ε, η} = {0.0411, 0.0380}
+{ε, η} = {0.00470, 0.148}
+H.265
+{ε, η} = {0.0132, 0.0196}
+{ε, η} = {0.00450, 0.142}
+AV1
+{ε, η} = {0.0171, 0.0309}
+POD
+{ε, η} = {0.00200, 0.130}
+{ε, η} = {0.0578, 0.0300}
+{ε, η} = {0.00770, 0.157}
+H.264
+{ε, η} = {0.0411, 0.0380}
+{ε, η} = {0.00470, 0.148}
+H.265
+{ε, η} = {0.0132, 0.0196}
+{ε, η} = {0.00450, 0.142}
+AV1
+{ε, η} = {0.0171, 0.0309}
+POD
+{ε, η} = {0.00200, 0.130}
+{ε, η} = {0.0578, 0.0300}
+{ε, η} = {0.00770, 0.157}
+H.264
+0
+50
+100
+-50
+-100
+H.264
+{ε, η} = {0.0578, 0.0300}
+H.265
+{ε, η} = {0.0411, 0.0380}
+{ε, η} = {0.0132, 0.0196}
+{ε, η} = {0.0171, 0.0309}
+AV1
+POD
+Figure 10: Comparison of video compression techniques applied on a streamwise velocity field u of cylinder wake
+at ReD = 100, compressed using H.264, H.265, AV1, and POD compression algorithms. The L2 error norm of the
+reconstruction ε and the compression ratio η are shown underneath each flow field contour.
+where ui are the components of the fluctuating velocity and the overbar denotes an averaging operation in space and
+time. For three-dimensional turbulent channel flow, the one-dimensional streamwise and spanwise spectra is evaluated
+Euu(k+
+x ; y+) = ˆu∗ˆu
+z,t, Euu(k+
+z ; y+) = ˆu∗ˆu
+x,t,
+(20)
+where (·)∗ represents the complex conjugate and ˆ(·) denotes the one-dimensional Fourier transformed variable. Here,
+we compare three compression ratios, denoted as low, medium, and high, for each turbulent flow.
+JP2 demonstrates a strong adherence to the kinetic energy spectrum of the uncompressed flow field in both the x
+and z directions while JPEG compression at low η introduces non-negligible errors at higher wave numbers. This is
+a consequence of the quantization step of JPEG compression that removes high wavelength scales from the image.
+Considering the overestimation of Euu(k+
+x ) as seen in figure 9 when using JPEG, this is likely caused by the absence
+of a deblocking filter, producing more high wavelength artifacts in the image than what exists in the uncompressed
+data. Similarly, the underestimation of E(k) by JP2 can be attributed to adaptive block sizes that produce a lower
+peak signal-to-noise ratio, indicative of lower quality. In general, JP2 is more adept at preserving high-wavenumber
+structures.
+4.2
+Video compression
+From the perspective of information, fluid flows are inherently temporally-redundant — as such, video compression
+algorithms that perform temporal compression are a powerful tool, achieving compression performance that outperforms
+the previously analyzed image-based techniques. This section assesses the capabilities of video compression techniques
+such as H.264, H.265, and AV1 compression algorithms for time-varying fluid flow data. Additionally, proper orthogonal
+decomposition (POD) compression [56] is considered to compare this familiar method of compression within the
+fluid dynamics community with those analyzed herein [3, 57]. POD is used to decompose a matrix of vectorized,
+temporally evolving flow field data into a set of basis modes and eigenvalues that contain coherent flow structures
+and can be used for flow field reconstruction. Formally, a flow field q(x, t) − q(x) can be represented as �n
+j=1 ajφj
+where aj is the temporal coefficient for mode φj. The value of aj is the inner product between the mode φj and the
+mean-subtracted flow field, q(x, t) − q(x). This modal representation can be truncated to r modes, such that the flow
+field is approximated by �r
+j=1 ajφj. This study uses the snapshot POD method [58] for comparison to the other video
+13
+
+100
+ 50
+0
+-50
+-100A PREPRINT - JANUARY 3, 2023
+H.264
+ε = 0.0377
+η =0.0256
+ε = 0.00660
+η =0.222
+ε = 0.0227
+η =0.0324
+ε = 0.00580
+η = 0.230
+ε = 0.0175
+η = 0.0207
+ε = 0.00480
+η = 0.206
+ε = 0.0170
+η = 0.0209
+ε = 0.00300
+η = 0.201
+H.265
+AV1
+POD
+H.264
+ε = 0.0377
+η =0.0256
+ε = 0.00660
+η =0.222
+ε = 0.0227
+η =0.0324
+ε = 0.00580
+η = 0.230
+ε = 0.0175
+η = 0.0207
+ε = 0.00480
+η = 0.206
+ε = 0.0170
+η = 0.0209
+ε = 0.00300
+η = 0.201
+H.265
+AV1
+POD
+H.264
+ε = 0.0377
+η =0.0256
+ε = 0.00660
+η =0.222
+ε = 0.0227
+η =0.0324
+ε = 0.00580
+η = 0.230
+ε = 0.0175
+η = 0.0207
+ε = 0.00480
+η = 0.206
+ε = 0.0170
+η = 0.0209
+ε = 0.00300
+η = 0.201
+H.265
+AV1
+POD
+H.264
+ε = 0.0377
+η =0.0256
+ε = 0.00660
+η =0.222
+ε = 0.0227
+η =0.0324
+ε = 0.00580
+η = 0.230
+ε = 0.0175
+η = 0.0207
+ε = 0.00480
+η = 0.206
+ε = 0.0170
+η = 0.0209
+ε = 0.00300
+η = 0.201
+H.265
+AV1
+POD
+0
+50
+100
+-50
+-100
+H.264
+{ε, η} = {0.0377, 0.0256}
+H.265
+{ε, η} = {0.0227, 0.0324}
+{ε, η} = {0.0175, 0.0207}
+{ε, η} = {0.0170, 0.0209}
+AV1
+POD
+Figure 11: Comparison of video compression techniques applied on two-dimensional isotropic turbulent vorticity field,
+compressed using H.264, H.265, AV1, and POD compression algorithms. The L2 error norm of the reconstruction ε
+and the compression ratio η are shown underneath each flow field contour.
+{ε, η} = {0.231, 0.147}
+{ε, η} = {0.0890, 0.403}
+{ε, η} = {0.151, 0.135}
+{ε, η} = {0.0832, 0.185}
+{ε, η} = {0.0588, 0.367}
+{ε, η} = {0.0502, 0.385}
+{ε, η} = {0.255, 0.157}
+{ε, η} = {0.139, 0.412}
+H.264
+H.265
+AV1
+POD
+{ε, η} = {0.231, 0.147}
+{ε, η} = {0.0890, 0.403}
+{ε, η} = {0.151, 0.135}
+{ε, η} = {0.0832, 0.185}
+{ε, η} = {0.0588, 0.367}
+{ε, η} = {0.0502, 0.385}
+{ε, η} = {0.255, 0.157}
+{ε, η} = {0.139, 0.412}
+H.264
+H.265
+AV1
+POD
+0
+50
+100
+-50
+-100
+{ε, η} = {0.231, 0.147}
+{ε, η} = {0.0890, 0.403}
+{ε, η} = {0.151, 0.135}
+{ε, η} = {0.0832, 0.185}
+{ε, η} = {0.0588, 0.367}
+{ε, η} = {0.0502, 0.385}
+{ε, η} = {0.255, 0.157}
+{ε, η} = {0.139, 0.412}
+H.264
+H.265
+AV1
+POD
+H.264
+{ε, η} = {0.231, 0.147}
+H.265
+{ε, η} = {0.0832, 0.185}
+{ε, η} = {0.151, 0.135}
+{ε, η} = {0.255, 0.157}
+AV1
+POD
+Figure 12: Comparison of video compression techniques applied on a streamwise velocity field u of three-dimensional
+turbulent channel flow at Reτ = 180, compressed using H.264, H.265, AV1, and POD compression algorithms. The
+L2 error norm of the reconstruction ε and the compression ratio η are shown underneath each flow field contour.
+compression techniques. The compression ratio for a reconstructed flow field containing r modes is evaluated as
+η = r(m + n) + m
+n(m + n) + m,
+(21)
+where m is the total number of pixels in the flow field and n is the total number of flow snapshots.
+The results of video compression for laminar cylinder wake at ReD = 100 are shown in figure 10. Here, we compare
+the decoded streamwise velocity field u with η ≈ 0.02 − 0.03. POD compression introduces negligible error, likely
+as a result of the temporally redundant nature of periodic wake and the larger coherent modal structures that POD
+is able to extract. By comparison, H.264 compression produces significant artifacts at low η. H.264 struggles likely
+because inter-frame prediction candidates are chosen from a shallow time range. We also observe that H.265 fails to
+improve in terms of error level over H.264 for the cylinder wake. This is due to the employment of a similar inter-frame
+prediction and selection algorithm to that of H.264. Compared to these H.2XX series, the AV1 algorithm provides much
+14
+
+100
+ 50
+0
+-50
+-100100
+ 50
+0
+-50
+-100A PREPRINT - JANUARY 3, 2023
+(a)
+(b)
+: H264
+: H265
+: POD
+: Cylinder
+: 2D turb.
+: Channel
+POD
+AV1
+(c)
+(d)
+: AV1
+H265 H264
+POD
+AV1
+H265
+H264
+Figure 13: Relationship between (a) the L2 error norm ε, (b) SSIM, and video compression ratio η. Zoom-in view of
+η-SSIM curve for (c) cylinder wake and (d) two-dimensional isotropic turbulence.
+better compression, achieving a lower L2 error than that achieved by H.264 and H.265. This highlights the enhanced
+capability of AV1 to compress laminar and temporally redundant flow fields.
+We next examine the video compression techniques for two-dimensional decaying homogeneous isotropic turbulence,
+as summarized in figure 11. The flow fields are compared for the compression ratios of η ≈ 0.02 − 0.03. Similar to the
+cylinder case, POD compression provides a reasonable reconstruction, likely because large-scale vortical structures
+are dominant at this particular time. It is, however, easily anticipated that the error of this time-varying flow relies
+on the presence of a range of length scales, as the small length scales disappear with the progress of the decay over
+time [59, 60]. The dependence of the compression performance over time for decaying flow will be examined later.
+While H.2xx compression techniques provide a reasonable reconstruction, AV1 provides better compression without
+suffering from pixelized artifacts. These results suggest the powerful capabilities of novel deblocking filters for fluid
+flow applications.
+The video compression techniques are also applied to the x−z sectional streamwise velocity field u of three-dimensional
+turbulent channel flow at Reτ = 180, as depicted in figure 12. The compressed flow fields are compared for η ≈ 0.150.
+In contrast to the other flow examples, POD compression produces significant visible artifacts and a high error value
+for turbulent channel flow because of a complex temporal evolution of the flow field. POD requires a greater number
+of modes for adequate reconstruction [61, 62, 63]. Although H.265 improves over H.264 significantly for turbulent
+channel flow, this still produces few observable discontinuities. This is likely caused by adaptive tiling in macroblocks
+for prediction procedures, allowing lower η with similar flow field representation. AV1 exceeds the performance of
+H.264 and H.265 consistently and POD compression on turbulent channel flow. Flow fields compressed using AV1 are
+indistinguishable from uncompressed flow fields at high η.
+The L2 error and SSIM are evaluated across a range of compression ratios for each type of flow field, as presented in
+figure 13. In general, all video compression algorithms produce asymptotically decaying L2 error values with increasing
+η. AV1 performs well at low η, especially for the cylinder wake. Additionally, all compression algorithms perform
+well for two-dimensional turbulence, likely as a result of the flow field snapshots holding slow changes from one
+frame to the next due to the decaying nature of the flow. Moreover, as observed with samples at various compression
+ratios, the L2 error for all algorithms plateau at non-negligible values for the cylinder wake flow field. SSIM values
+15
+
+n
+10
+-2
+10
+3
+10
+0.0
+0.2
+0.4
+0.6
+0.8
+n1.0
+0.8
+4
+SSIM
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+n1.000
+SSIM
+0.995
+0.990
+0.985
+0.0
+0.2
+0.4
+0.6
+0.8
+n1.000
+SSIM
+0.995
+0.990
+0.985
+0.0
+0.2
+0.4
+0.6
+0.8
+nA PREPRINT - JANUARY 3, 2023
+(b)
+(c)
+(a)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(b)
+(c)
+(a)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(b)
+(c)
+(a)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(b)
+(c)
+(a)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(a)
+(b)
+(c)
+Figure 14: L2 error norm ε of vorticity field ω for two-dimensional decaying homogeneous isotropic turbulence over
+time. (a) H.264, (b) H.265, and (c) AV1. (i) and (ii) in each case are chosen due to their employment in inter-frame
+prediction in each algorithm.
+generally diverge from the asymptotic limit at low η. The exceptional cases include cylinder wake and two-dimensional
+turbulence compressed using AV1, which introduces negligible error at low η. AV1 outperforms H.264 and H.265 on
+turbulent channel flow as well, due to the improved blocking techniques.
+In addition, the time evolution of the L2 error is examined to gain insight into the performance of video compression
+algorithms for individual snapshots. The temporal evolution of the L2 error norm ε for two-dimensional decaying
+turbulence is shown in figure 14. H.2xx compression techniques exhibit repeated temporal structures in its L2 error
+evolution, likely as a consequence of inter-frame prediction selecting frames to make predictions from at relatively
+similar intervals. AV1 compression provides a distinctive reduction in the L2 error over time for medium and low η,
+indicating improved accuracy as snapshots begin to show redundancies due to the vortex field decaying and exhibiting
+similar large-scale coherent structures from one snapshot to the next. We also observe that H.264 produces a high error
+at low η for early flow field snapshots. This relates to the time-varying flow nature of the present decaying turbulence,
+as mentioned above. The presence of finer structures at the high Taylor Reynolds number Reλ(t) portion of the flow
+likely causes the difficulty in compressing vortical flow data.
+We also examine the L2 error norm ε and the flow fields over time for turbulent channel flow, as depicted in figure 15.
+H.264 generally produces a larger L2 error compared to the other techniques, as we also observed with the visual
+assessments in figure 12. With low η of H.264 compression, the error decreases over time, likely as a result of a later
+snapshot being selected for inter-frame prediction. Compared to H.264, H.265 provides better compression over time.
+Similar to the observation with H.264, the L2 error significantly varies over time at a low η. This is likely due to the
+inter-frame selection of an early frame from which further predictions were made. AV1 produces a negligible error at a
+high η while the errors increase as η decreases.
+We are also interested in the performance of video compression algorithms in preserving high wavenumber structures in
+the compressed state. The general performance of each compression algorithm with regard to kinetic energy spectra of
+each flow field is investigated, as shown in figure 16. H.264 performs well for two-dimensional turbulence, but produces
+a noticeable error at all η in both the stream- and spanwise directions of the kinetic energy spectrum of turbulent channel
+flow. A similar divergence from the expected data can be observed at low η in the spanwise direction as well. H.265
+performs comparatively well for two-dimensional decaying isotropic turbulence, and for turbulent channel flow in the
+spanwise direction. However, it produces a non-negligible error at high wavenumbers when compressed at low η in the
+spanwise direction. This indicates an over-representation of high-wavenumber components due to blocking as a result
+of the adaptive subblock sizes of H.265. Generally, AV1 is the best-suited algorithm for preserving spatial frequency
+information, particularly at high wavenumbers, for both two and three-dimensional turbulent flow fields. At higher η,
+the energy contents at each wavenumber are almost indistinguishable.
+16
+
+0.012
+AV1 Low
+AV1 Medium
+Error
+0.01
+AV1 High
+0.008
+0.006
+0.004
+0.002
+0
+2
+3
+4
+5
+6
+t10
+0
+10
+w
+2
+10
+10
+3
+0.0
+0.2
+0.4
+0.6
+0.8
+n0.015
+H265 L0W
+^H265 Medium
+H265 High
+0.01
+0.005
+2
+3
+4
+5
+6
+t0.025
+H264 L0w
+_H264 Medium
+0.02
+ H264 High
+0.015
+0.01
+0.005
+2
+3
+4
+5
+6
+tA PREPRINT - JANUARY 3, 2023
+(b)
+(c)
+(a)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(b)
+(c)
+(a)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(b)
+(c)
+(a)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(a)
+(b)
+(c)
+Figure 15: L2 error norm ε of streamwise velocity field u for turbulent channel flow over time. (a) H.264, (b) H.265,
+and (c) AV1. (i) and (ii) in each case are chosen due to their employment in inter-frame prediction in each algorithm.
+At last, we investigate whether the video compression techniques can preserve the temporal evolution of complex
+turbulent flows. Here, let us examine the temporal two-point correlation for three-dimensional channel flow compressed
+using all three video compression algorithms. The temporal two-point correlation coefficient at a given t+ is defined
+as R+
+uu(t+)/R+
+uu(0) [64, 65] and is depicted in figure 17. The assessment of temporal two-point correlation provides
+insight into the relations of flow snapshots to preceding snapshots.
+Consistent with the insights gained from the kinetic energy spectrum, H.264 compression at a η exhibits disagreement
+with the reference curve at t+ values between 5 and 30, and above 50. This is indicative of a de-correlation of the
+velocity field and is likely a result of poor performance in capturing high-wavenumber information. Except for this
+particular case, all compression algorithms generally perform well, with temporal two-point correlation coefficients
+closely following that of the uncompressed flow field. These results suggest that these novel video compression
+techniques capture the spatio-temporal redundancies well even for complex turbulent flows and also significantly reduce
+data size while preserving their physics.
+5
+Conclusion
+We compressed flow field data from canonical flow examples using a number of widely-available multimedia com-
+pression techniques. The performance of the JPEG and JP2 spatial image compression techniques and the H.264,
+H.265, and AV1 spatio-temporal video compression techniques were considered for simulated laminar cylinder flow,
+decaying isotropic turbulence, and turbulent channel flow. Streamwise velocity and vorticity field data were represented
+as grayscale images and videos, and were compressed using the aforementioned techniques.
+All techniques, with the exception of JPEG, were shown to compress flow data below 10% of the original file size while
+introducing negligible error and preserving underlying flow physics. AV1 and H.265 compression were shown to have
+the best performance across a variety of flow regimes. The spatial error distributions were concentrated on the cylinder
+surface and directly behind the cylinder for the streamwise velocity data compression and in the vortex shedding wake
+for the vorticity data. Turbulence statistics in the form of kinetic energy spectra were preserved under compression for
+all methods except JPEG.
+17
+
+0.14
+0.12
+0.1
+0.08
+0.06
+0.04
+0.02
+20
+40
+0
+6010
+0
+10
+w
+2
+10
+10
+3
+0.0
+0.2
+0.4
+0.6
+0.8
+n0.4
+0.3
+0.2
+20
+40
+601.2
+0.8
+0.6
+0.4
+0.2
+20
+40
+60A PREPRINT - JANUARY 3, 2023
+(d)
+(g)
+(e)
+(h)
+(f)
+(i)
+(a)
+(b)
+(c)
+ηLow = 0.0640
+ηMed = 0.585
+ηHigh = 0.880
+ηLow = 0.0724
+ηMed = 0.595
+ηHigh = 0.976
+ηLow = 0.0587
+ηMed = 0.562
+ηHigh = 0.898
+ηLow = 0.119
+ηMed = 0.367
+ηHigh = 0.776
+ηLow = 0.0872
+ηMed = 0.403
+ηHigh = 0.817
+ηLow = 0.0872
+ηMed = 0.403
+ηHigh = 0.817
+ηLow = 0.119
+ηMed = 0.367
+ηHigh = 0.776
+ηLow = 0.187
+ηMed = 0.385
+ηHigh = 0.767
+ηLow = 0.187
+ηMed = 0.385
+ηHigh = 0.767
+Figure 16: Kinetic energy spectra E(k) for (a, d, g) two-dimensional decaying homogeneous isotropic turbulence
+using H.264, H.265, and AV1. (b, e, h) Streamwise Euu(k+
+x ) and (c, f, i) spanwise kinetic energy spectra Euu(k+
+z ) of
+three-dimensional turbulent channel flow.
+For single snapshots of data represented as an image, JP2 compression was shown to far outperform JPEG compression,
+with a tolerable increase in computational complexity. For multiple temporal snapshots of data represented as a
+video, the choice of compression method becomes more nuanced. JP2 compression was shown to achieve the lowest
+compression error as temporal compression adds slight error to the data. The AV1 algorithm maximizes η at the expense
+of computational complexity and non-negligible encoding time. This algorithm is new and emerging from the research
+environment, so future optimizations could bring this encoding time to a manageable level. The H.265 algorithm
+provided excellent compression performance at a fast encoding time, and appears as a promising algorithm for current
+fluid dynamics applications. H.264 provided acceptable compression performance, but was largely triumphed by the
+AV1 and H.265 algorithms.
+We have shown that modern multimedia compression algorithms provide robust performance in a variety of fluid flow
+applications. The implementation of these techniques becomes especially pertinent as simulations within computational
+fluid dynamics become exceedingly data-intensive, a trend that decreases the accessibility to high-fidelity models. These
+methods are free, easily accessible, regularly updated and supported, and provide flexible and scalable compression
+performance. As such, the implementation of these compression techniques has exciting potential across the fluid
+dynamics community for data storage and transfer with minimal loss.
+18
+
+-Uncompressed
+1H265 L0W
+^H265 Medium
+H265 High
+10~5
+10-10
+101
+102
+103
+k-Uncompressed
+AV1 Low
+AV1 Medium
+·AV1 High
+10~5
+10-10
+101
+102
+103
+k100
+10-4
+10-2
+10-1100
+10-4
+10-2
+10-1
+Y100
+10-4
+10-2
+10-1100
+10-4
+10-2
+10-1-Uncompressed
+H264 LoW
+H264 Medium
+H264 High
+10~5
+10-10
+101
+102
+103
+k100
+10-4
+10-6
+10-2
+10-1100
+10
+10-4
+10-6
+10-2
+10-1A PREPRINT - JANUARY 3, 2023
+(a)
+(b)
+(c)
+ηLow = 0.187
+ηMed = 0.385
+ηHigh = 0.767
+ηLow = 0.0872
+ηMed = 0.403
+ηHigh = 0.817
+ηLow = 0.119
+ηMed = 0.367
+ηHigh = 0.776
+Figure 17: Normalized temporal two-point correlation coefficients Ruu(t+)/Ruu(0) for three-dimensional turbulent
+channel flow using H.264, H.265, and AV1.
+JPEG
+JP2
+H.264
+H.265
+AV1
+u, Cylinder Flow
+0.58
+2.27
+0.94
+2.85
+61.21
+ω, Cylinder Flow
+0.59
+1.53
+0.92
+2.39
+43.83
+u, Channel Flow
+0.29
+1.42
+0.45
+1.96
+49.59
+Table 1: Encoding time (s) for different compression algorithms and flow regimes, compressed at 100 KB/s bitrate.
+Acknowledgements
+KT acknowledges the support from the US Army Research Office (W911NF-21-1-0060), the US Air Force Office
+of Scientific Research (FA9550-21-1-0178), and the US Department of Defense Vannevar Bush Faculty Fellowship
+(N00014-22-1-2798). We also thank Professor Koji Fukagata (Keio University) for sharing his DNS code.
+Appendix: Encoding time
+The increased performance of new compression algorithms comes at a cost; non-negligible increases in computational
+complexity should be considered when implementing these algorithms. In fact, in a paper from 2000 on compressing
+three-dimensional flows with the JPEG and JP2 algorithms [66], the added complexity of the JP2 algorithm caused
+JPEG to be recommended over JP2, despite losing clear performance benefits. The recommendation of the present
+study reverses that statement. As such, it is important to quantify the encoding time of these algorithms at the time of
+writing this study.
+The decoding time is observed to be negligibly small for all compression codecs; thus, this appendix focuses on
+encoding. The streamwise velocity and vorticity data are encoded for both the laminar cylinder flow and turbulent
+channel flow cases at the same bitrate (100 KB/s) for all compression algorithms and the encoding time is measured.
+The encoding is performed with a 2.5GHz i7 Intel Core processor and 8 GB RAM. The results are summarized in
+table 1. Encoding time per frame is observed to be larger for the turbulent channel flow than the laminar cylinder flow,
+indicating that the algorithms struggle to encode multiscale turbulent flow data. Across encoding algorithms, JPEG and
+H.264 compression are the fastest, a testament to the maturity and low complexity of these methods. JP2 and H.265
+encoding are generally several times slower, but still relatively fast, justifying their added compression performance.
+AV1 is observed to be far slower in encoding than the other methods: over 100 times slower than JPEG and H.264, and
+over 25 times slower than JP2 and H.265. This severe encoding time increase limits the practicality of implementing this
+algorithm in large-scale applications, and perhaps justifies the use of H.265 over AV1. As the algorithm was released
+only a few years prior to the writing of this paper, advances in computing and algorithm development could increase its
+practicality in the near future.
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+19
+
+-Uncompressed
+H264 L0W
+0.8
+H264 Medium
+H264 High
+0.6
+0.4
+0.2
+0
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+0
+40
+60-Uncompressed
+H265 L0W
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+H265 Medium
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+AV1 Low
+0.8
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf,len=1593
+page_content='IMAGE AND VIDEO COMPRESSION OF FLUID FLOW DATA  A PREPRINT  Vishal Anatharaman, Jason Feldkamp, Kai Fukami∗, Kunihiko Taira  Department of Mechanical and Aerospace Engineering,  University of California, Los Angeles, CA 90095, USA  Corresponding author: kfukami1@g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='edu  January 3, 2023  ABSTRACT  We study the compression of spatial and temporal features in fluid flow data using multimedia com-  pression techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The efficacy of spatial compression techniques, including JPEG and JPEG2000  (JP2), and spatio-temporal video compression techniques, namely H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265, and AV1, in limiting  the introduction of compression artifacts and preserving underlying flow physics are considered for  laminar periodic wake around a cylinder, two-dimensional turbulence, and turbulent channel flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  These compression techniques significantly compress flow data while maintaining dominant flow  features with negligible error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' AV1 and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265 compressions present the best performance across a  variety of canonical flow regimes and outperform traditional techniques such as proper orthogonal  decomposition in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' These image and video compression algorithms are flexible, scalable,  and generalizable holding potential for a wide range of applications in fluid dynamics in the context  of data storage and transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  1  Introduction  High-fidelity simulations and experiments within the field of fluid dynamics produce exceedingly large amounts of  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' As the need for higher fidelity simulations and advanced experimental resources expands, storage and transfer  requirements for spatio-temporal data from simulations, become a major challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' To address this issue, spatio-  temporal redundancies or repeated dominant flow features can be exploited by a variety of compression techniques  to alleviate memory constraints for fluid flow data storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' A variety of compression techniques, including modal  analysis [1, 2, 3], sub-sampling and local re-simulation [4], and deep learning [5, 6, 7, 8] have been considered in an  effort to reduce the size of fluid flow data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Although effective, these techniques can be application-specific and struggle  to achieve substantial compression ratios without introducing undesirable compression artifacts such as discontinuities  or deletions of flow features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  In comparison, multimedia compression techniques are general and simple to use, and have benefited from demand  for the modern technologies of high-resolution video streaming [9, 10, 11] and video-conferencing [12, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  These compression techniques are classified into two groups: lossless compression and lossy compression [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' With  lossless techniques, the data retrieved from or reconstructed from the compressed state is identical to that preceding  the application of a compression algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Hence, this is preferred for archival purposes and used for medical  imaging [16] and technical drawings [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' In contrast, processed data with lossy techniques do not necessarily match the  original data, enabling a significant data-size reduction in the compressed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Since this may introduce compression  artifacts such as discontinuities in image data or the loss of high spatial frequency information, it is suitable for natural  images such as photographs in applications where imperceptible loss may be acceptable [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' We consider here the  impacts of such losses on fluid mechanics simulation data to assess the costs of applying lossy techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' In 2003,  Schmalzl [19] considered multimedia data compression for fluid flows with an example of Rayleigh-Bénard convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  With multimedia compression technologies having undergone significantly advances in the last two decades, we reassess  image and video compressions with modern algorithms for applications to fluid flow data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Lossy techniques of interest typically involve frequency-domain transformation, filtering, and entropy coding as  components in the compression process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The development of the discrete cosine transform (DCT) [20, 21] has played a  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00078v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='flu-dyn]  31 Dec 2022  A PREPRINT - JANUARY 3, 2023  Original data  Compressed data  Reconstruction  …  Spatial  Compression  Temporal  Compression  t  Original  Data  Compressed  Data  …  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' (#, %, &)  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='′(#, %, &)  Original data  Reconstruction  (a)  (b)  Figure 1: (a) Spatial compression: an example velocity field of flow over a cylinder q(x) is represented as a grayscale  image, encoded using an image-based technique to a compressed form, and reconstructed as ˜q(x) using a decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  (b) Spatio-temporal compression: multiple snapshots of this flow field data q(x, t) are represented as a grayscale video  and are compressed to ˜q(x, t) with both spatial and temporal techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  crucial role in image compression, and is the basis of Joint Photographic Experts Group (JPEG) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The emergence of  JPEG enabled efficient image compression in a wide range of communities and it became a generally accepted format  for digital images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' After the development of DCT, wavelet transforms began to be utilized for image compression  in such algorithms as JPEG2000 (JP2) [23], which achieves better compression than the DCT of JPEG as a result of  multi-scale properties of wavelets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  In tandem with the growth of image compression techniques, advancement in video compression technologies followed  suit since video data can be characterized as a time series of image frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Generally, these time frames include both  spatial and temporal redundancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' In fact, we often see the similarities (redundancies) between temporally adjacent  frames or spatially adjacent pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Video compression algorithms are designed to remove such redundancies and  obtain a compact form of the original information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Current video compression technologies are generally based on the  DCT [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Although other candidates including fractal compression [25, 26], matching pursuit [27], and discrete wavelet  transform (DWT) have been investigated as the subject of some studies, these are still not used in practical products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Moving Picture Experts Group (MPEG) series have been traditionally used for video compression of high-definition  television [28, 29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2xx series was then developed and they have achieved significant compression compared to  the conventional MPEGs [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Especially in the recent versions such as H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264 and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265, motion compensation,  quantization, and entropy coding are applied for efficient video compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' More recently, AOMedia Video 1 (AV1),  an open, royalty-free video coding format, was released in 2018, achieving enhanced compression compared to the  aforementioned techniques [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  To meet the demand for these image and video compression tools, significant investment and research have produced  compression techniques of impressive efficiency and usability in addition to free video encoders [35] to promote  widespread accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' As such, leveraging these multimedia-inspired compression techniques should also be of  particular interest to the fluid dynamics community given the massive scale of data produced, stored, and transferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  2  Original Data  Compressed Data  Reconstruction  nxm  < nxm  nxm  01100111 01101111  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Encoding  Decoding  (qn1  g(x, y)gfa,tgfr,t2A PREPRINT - JANUARY 3, 2023  A standardization on one or more multimedia compression formats for storing fluid flow data in a compressed  representation can yield dividends in research output by allowing greater access to high-fidelity fluid flow data sets and  by removing memory constraints as a barrier to entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  This paper investigates the effectiveness of these image and video compression techniques on fluid flow data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Spatial  image compression techniques, such as JPEG and JP2, alongside spatio-temporal video compression techniques,  namely H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265, and AV1, are examined for various flow fields, including laminar cylinder flow, two-dimensional  turbulence, and turbulent channel flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Field variables from simulation data, such as streamwise velocity and vorticity,  are represented as grayscale images, and multiple snapshots are packaged into a video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' These videos are then encoded  into a compressed form using the aforementioned multimedia compression methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Modern techniques can compress  flow data well below 10% of the original file size with negligible error and preserve the underlying physics of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Although this paper focuses on applications to canonical fluid flows, the flexibility and scalability of these algorithms  suggest an expansive potential within this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Compression is a process in which data is compressed (encoded) into a representation that uses less data, and  decompressed (decoded) into identical data in the case of lossless compression or nearly-identical data in the case  of lossy compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Through this procedure, a compression method reduces bits of the original data q(x, t) by  eliminating statistical redundancies that may be contained within temporally adjacent frames and spatially adjacent  pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' In general, a data compression algorithm is referred to as an encoder φ while one that performs the decompression  is called a decoder ψ,  γ(x, t) = φ(q(x, t)),  q(x, t) ≈ ˜q(x, t) = ψ(γ(x, t)),  (1)  where γ(x, t) is the compressed data corresponding to the original data q(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Depending on the extent of compression,  the data, and a choice of encoder/decoder, the reconstruction ˜q(x, t) generally includes some amount of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  The data compression process is illustrated in figure 1 for both image and video compressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Figure 1(a) depicts a lossy  spatial image compression technique, involving quantization of the image data in a compressed space and producing  a reconstruction in the image space showing the operations of JPEG and JP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Figure 1(b) provides a visualization  of a spatio-temporal compression technique, exploiting a redundant block of a frame that remains consistent across  subsequent frames, similar to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265, and AV1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' As these algorithms originated in the multimedia industry, they  are optimized for human viewers and involve the removal of high-frequency components in the data and down-sampling  of the color spectrum such that the eyes cannot easily distinguish compressed data from the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' For the  purposes of this study, we only consider grayscale images and videos, which are comprised only of a single-component  field data matrix, denoted as ˜q(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This is in contrast to full-color data, which requires red, green, and blue components,  and is unnecessary for the current analysis as we are interested in considering field variables individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Herein, we  consider the application of five compression techniques on grayscale images and videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The encoding schemes, which  package the data into a compressed binary form, are detailed in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  2  Compression techniques  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='1  Image Compression  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='1  JPEG  Let us first describe JPEG, which is a standard lossy spatial compression used for encoding image data based on the  discrete cosine transform (DCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' An example of a JPEG compression process with a vorticity field of two-dimensional  decaying isotropic turbulence is presented in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The images are partitioned into 8 × 8 blocks in a left-to-right,  top-to-bottom scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Pixel values within blocks are quantized to values of [−128, 127] from [0, 255].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The forward DCT  is individually performed at each block and outputs compressed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The DCT for 8 × 8 blocks is mathematically  expressed as  F(kx, ky) = 1  4C(kx)C(ky)  �  7  �  ix=0  7  �  iy=0  f(ix, iy) cos  �(2ix + 1)kxπ  16  �  cos  �(2iy + 1)kyπ  16  ��  ,  (2)  f(ix, iy) = 1  4  �  7  �  kx=0  7  �  ky=0  C(kx)C(ky)F(kx, ky) cos  �(2ix + 1)kxπ  16  �  cos  �(2iy + 1)kyπ  16  ��  ,  (3)  where  C(k) =  �1/  √  2  for k = 0  1  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  (4)  3  A PREPRINT - JANUARY 3, 2023  Input  8×8 block  DCT for 8×8 block  Reconstructed image  (Keeping 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='31% of the DCT coefficients)  Figure 2: JPEG compression process with an example of two-dimensional isotropic turbulent vorticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Here, F(kx, ky) denotes the DCT coefficient corresponding to the horizontal wavelength kx and vertical wavelength ky  and f(ix, iy) describes the pixel value at the location corresponding to ix and iy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' In other words, the forward DCT takes  as input a discrete signal of 64 points and produces coefficients for a linear combination of 64 unique basis signals, each  denoting a specific spatial wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Most of the spatial domain information is concentrated across lower wavelength  because of slow spatial variation from one pixel to the next in image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This quality permits lossy quantization,  which refers to constant values in a quantization table Q(kx, ky) with 64 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The DCT coefficient is normalized  by a constant Q(kx, ky) in an element-wise manner,  F Q(kx, ky) = ⌊F(kx, ky)  Q(kx, ky)⌋  (5)  where F Q(kx, ky) is a normalized coefficient and the operation ⌊·⌋ denotes rounding to the nearest integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Quantization  tables are provided by the Joint Photographics Experts Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Note that dividing the DCT coefficients by values in the  quantization table reduces high-wavenumber coefficients to 0, which permits efficient entropy coding (explained later)  to perform the cutoff at high frequencies [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The resulting quantized DCT coefficients form a matrix of size 8 × 8  with low-wavenumber components generally located in the top-left of the matrix and high-wavenumber coefficients at  the bottom-right, as a consequence of the similar distribution of spatial modes to which these coefficients correspond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Subsequently, quantized coefficients are ordered from low to high wavenumbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  To reduce the data size, entropy coding [37, 38], a lossless method of compressing bitstreams with redundancies, is then  performed for the output of DCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The idea of entropy coding is used not only for JPEG but also other image/video  compression techniques such as JP2, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2xx series, and AV1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' To express the encoding-based data compression, let us  consider a message of DAEBCBACBBBC (12 characters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Since this message includes five different characters, it  needs to prepare 3 bits to convert these characters to bits or binary digits representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Here, we use the following  conversion table,  A  B  C  D  E  000  001  010  011  100  With this table, the message is expressed as  D  A  E  B  C  B  A  C  B  B  B  C  011  000  100  001  010  001  000  010  001  001  001  010  As shown, the number of bits is 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The idea of the encoding-based compression is to prepare an adaptive conversion  table assigning a shorter bit length for characters that appear in a high probability and a longer bit length for characters  that barely appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' For example, the following adaptive table can be used:  A  B  C  D  E  110  0  10  1110  1111  4  A PREPRINT - JANUARY 3, 2023  With this new table, the message can be expressed as  D  A  E  B  C  B  A  C  B  B  B  C  1110  110  1111  0  10  0  110  10  0  0  0  10  The current table can save the total number of bits for the message from 36 to 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Modern data compression techniques  efficiently find such an adaptive conversion table for saving image and video sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  The presence of a better adaptive conversion can be proven with the source coding theorem [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' For any data, the  expected code length should satisfy the relationship,  Eβ∼P [l(d(β))] ≥ Eβ∼P [− logb(P(β))],  (6)  where l is the number of symbols in a message, d is the coding function, b is the number of symbols in a table, and P is  the probability of the original symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' An entropy coding method attempts to approach the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' For JPEG  compression, Huffman coding [40] is used to determine an adaptable table composed of the estimated probability of  occurrence for each possible value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Huffman coding uses binary trees [41] for efficient encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2  JPEG2000 (JP2)  JPEG2000 (JP2) is a successor to JPEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' JP2 operates using a similar four-step process to JPEG, involving image  partitioning, frequency-domain transformation, quantization, and entropy coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' In contrast to JPEG, JP2 introduces  more advanced dynamic tiling algorithms based on variable-sized macroblocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This makes use of the discrete wavelet  transform (DWT), and performs additional preprocessing prior to entropy coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Tiling in this context refers to the  partitioning of the source image into several non-overlapping rectangular blocks, each of which is processed distinctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Whereas JPEG restricts tile sizes to 8 × 8, tiles in JP2 can be of arbitrary size up to the image dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The DWT is  applied to each tile in a manner similar to DCT, decomposing a signal into a linear combination of wavelet functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  The coefficients in this linear combination correspond to a specific wavelet basis function in the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' A wavelet can  be defined as a scale and shift of a basis wavelet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Child wavelets [42] are generally considered for DWT, given by  ψg,r(s) =  1  2g/2 ψ  �s − 2gr  2g  �  (7)  where g is a scaling factor, r is a shift factor, and s corresponds to the index of the one-dimensional representation of an  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' In other words, the flow field snapshot is converted to a one-dimensional representation and the independent  variable upon which this one-dimensional signal f(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The DWT coefficient given a wavelet of the preceding definition  is then  Fg,r =  � ∞  −∞  f(s)ψg,rds  (8)  where f(s) is a one-dimensional signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The signal can be reconstructed through the summation of the product of each  coefficient with the corresponding wavelet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' In a discrete interpretation, this is written as  f(s) =  ∞  �  g=−∞  ∞  �  r=−∞  Fg,rψg,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  (9)  The summation bounds for both the calculation of the coefficients and reconstruction of the signal can be set to finite  values and can still produce lossless reconstructions assuming that the wavelets contain the maximum and minimum  wavelengths within the source image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  An example of the DWT for the vorticity field of two-dimensional decaying turbulence is presented in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The  DWT can be applied recursively to one-dimensional signals to produce higher fidelity representations of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' As such,  successive high-pass filters are applied on down-sampled images, producing a higher fidelity representation of spatial  frequencies in the source image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The DWT can be extended to higher dimensions by applying the one-dimensional  DWT on rows and columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The recursive application of the DWT produces 2n distinct filtered images where n is the  number of times the DWT is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Similar to JPEG compression, the DWT coefficients are quantized following the transformation on the wavespace,  F Q  a,b = sign(Fa,b)⌊|Fa,b|  ∆b  ⌋,  (10)  5  A PREPRINT - JANUARY 3, 2023  (b)  (c)  (d)  (e)  (g)  (f)  (a)  0  50  100  50  100  Figure 3: An example of the two-level discrete wavelet transform for two-dimensional homogeneous turbulent vorticity  field, used in JP2 compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' High-pass filtering yields three large images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Low-pass filtering and downscaling are  then performed, producing the three small images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' (a) The final approximation image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' (b, e) Vertical, (c, f) horizontal,  and (d, g) diagonal coefficients of the second (b − d) and the first levels (e − g) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  where ∆b is defined as the quantization step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' DWT coefficients within the range (−∆b, ∆b) are quantized to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Following quantization, the coefficients are processed in preparation for entropy coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Arithmetic coding [43] is  used for entropy coding in JPEG2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' While Huffman coding separates the original data into component symbols and  replaces each with a code in a table, Arithmetic coding encodes the entire message into a single number represented  with an arbitrary-precision fraction pa, where 0 ≤ pa < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2  Video Compression  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='1  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264 (AVC)  The H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264 video compression includes a multi-step process, consisting generally of prediction, transformation (a set  of frequency-domain representation and quantization in image compression), and entropy encoding on the encoder  side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' A similar process for file reconstruction is performed on the decoder side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Prediction in video compression  amounts to an operation to remove redundancies in the given signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264 supports a range of prediction options  6  500  450  400  350  300  250  200  150  100  50  20  40  60  80  100  120  140  160  180  200160  140  120  100  80  60  40  20  20  40  60  80  100  120  140  160  180  200160  140  120  100  80  60  40  20  20  40  60  80  100  120  140  160  180  200300  250  200  150  100  50  50  100  150  200  250  300  350  400300  250  200  150  100  50  50  100  150  200  250  300  350  400100  50  0  50  100A PREPRINT - JANUARY 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' 2023  Frame 1  MC Frame 2  Frame 2 MC Residual  Figure 4: Motion compensation (MC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' used in temporal compression of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265, and AV1, is exemplified by the  calculation of a motion vector field, describing the translation of pixels between successive frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' As an example, a  streamwise velocity field u of three-dimensional turbulent channel flow is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Subtracting this field from the  original image yields a residual image, which is stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  such as intra-prediction used for data within the current frame, inter-prediction for motion compensation, and multiple  block-size-based predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' An accurate prediction implies that the residual contains very little information, amounting  to good compression performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Video data is first partitioned into macroblocks of dimension 16 × 16 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' A prediction of the current macroblock  is formed using 4 × 4 and 16 × 16-sized blocks in the case of intra-frame prediction, referring to predictions from  surrounding blocks within the same frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' A range of block sizes from 4 × 4 to 16 × 16 are also considered in the  case of inter-frame prediction, referring to predictions from previously coded frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Macroblock prediction is further  discretized into intra-prediction with neighboring blocks in the current frame, blocks in a previously coded frame, and  blocks from up to two previously coded frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  In intra-prediction, the size of prediction macroblocks can be three cases: 16 × 16, 8 × 8, or 4 × 4 pixels [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The  choice of block size is made primarily based on prediction efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' One of several prediction modes where each  prediction mode indicates a direction in which to extrapolate pixel values or to average across all pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' In the case of  inter-prediction, the reference frame, where the prediction block is situated, is chosen from several previously decoded  frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' As shown in figure 4, a motion vector is then obtained for the current macroblock, based on the offset from the  prediction block or from previously coded motion vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' These motion vectors are optionally weighted to account for  temporal proximity between frames, and are sent into the data stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Generally, a deblocking filter is further applied to  each frame to store in a decoded format for subsequent inter-frame predictions in smoothing the sharp edges caused by  the use of block coding [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Transformation involves the conversion of blocks to frequency-domain representations and quantization of coefficients  corresponding to high wavelength data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The DCT step is given by the transformation of block X by matrix A into DCT  coefficients Y for each macroblock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' For the case of 4 × 4 blocks, these matrices X, Y ∈ R4×4 are expressed as  Y = AXAT =  �  ��  a  a  a  a  b  c  −c  −b  a  −a  −a  a  c  −b  b  −c  �  �� X  �  ��  a  b  a  c  a  c  −a  −b  a  −c  −a  b  a  −b  a  −c  �  �� ,  (11)  where a = 1/2, b =  �  1/2 cos(π/8), c =  �  1/2 cos(3π/8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The rows of A are orthonormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' To calculate equation 11  on a practical processor, the approximation for b and c is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This is achieved with a fixed-point approximation,  which is equivalent to scaling each row of A by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='5 and rounding to the nearest integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' A core transform Cf4, which  scales each term of A by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='5 and rounds to the nearest integer, and a scaling matrix Sf4, which restores norms of row of  Cf4 to 1 by scaling, are respectively defined as,  Cf4 =  �  ��  c1  c1  c1  c1  c2  c1  −c1  −c2  c1  −c1  −c1  c1  c1  −c2  c2  −c1  �  �� ,  Sf4 =  �  ��  s1  s2  s1  s2  s2  s3  s2  s3  s1  s2  s1  s2  s2  s3  s2  s3  �  �� ,  (12)  where c1 = 1, c2 = 2, s1 = 1/4, s2 = 1/(2  √  10), and s3 = 1/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Matrix Y is then determined as  Y = [Cf4XCT  f4]Sf4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  (13)  Similar DCT approximations are specified for other block sizes, involving Cf8, Sf8, and others [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' A quantization  mechanism similar to JPEG is applied, with a quantization table specified for various block sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The quantized DCT  coefficients are then traversed in an oscillating, “zig-zag” manner from low- to high-wavenumber components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Entropy  coding is finally applied to the output of DCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  7  Frame 1  MC Frame 2  Frame 2 MC ResidualA PREPRINT - JANUARY 3, 2023  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265 (HEVC)  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265 was released in 2013 as the successor to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' While the fundamental architecture is unchanged from H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265 makes use of coding tree units which are similar to macroblocks but expand the range of possible dimensions,  with variable dimensions selected by the encoder, allowing coding tree units to be divided into sub-blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Predictions  and reconstructions are performed on coding tree units and supported sub-block sizes range from 64 × 64 to 4 × 4  pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Motion vectors are predicted based on those of adjacent units or blocks in the case of intra prediction, or previous  encoded frames in the case of inter prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' As in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264, the DCT is performed at the coding tree block level, and  the resultant coefficients are subjected to scalar quantization and entropy coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Instead of deblocking filter in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264,  a sample adaptive offset filter is applied within the prediction loop to improve the quality of the compressed data [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='3  AV1  AV1 was released in 2018 in an effort to replace the H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2XX series of video compression algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' In AV1, much  of the fundamental architecture from H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2XX is maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Frames are partitioned using a 4-way partition tree with  dimensions ranging from 128 × 128 to 4 × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' AV1 supports 56 directional spatial modes for intra-frame prediction with  finer angle variations than that provided by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2XX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' AV1 extends the number of reference frames that any given frame  can use to perform predictions to seven references for inter-frame prediction, thus enabling more accurate encoding  of data with rich temporal characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Motion vector field formation is improved by expanding the spatial search  domain for vector candidates and through the utilization of a temporal motion field estimation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' AV1 also  extends frequency-domain transform algorithms to include the asymmetric discrete sine transform with a richer set  of transform kernels for varying block sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Entropy coding and scalar quantization are also used as well as H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2XX  compression algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' To perform deblocking, a constrained directional enhancement filter and loop restoration  filters are applied [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' These filters are able to effectively remove artifacts without causing blurring, compared to  conventional deblocking filters [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  3  Flow Fields  Let us apply the compression techniques presented above to representative fluid flow data sets from numerical  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' We describe herein the problem setup of the example flow fields we analyze and the simulation approach  used to generate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='1  Two-dimensional laminar cylinder wake  Bluff body flow forms a large class of problems, such as the vortex shedding around a cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' We first apply the  compression techniques to the two-dimensional cylinder wake obtained by direct numerical simulation (DNS) [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  The governing equations are the incompressible Navier–Stokes equations,  ∇ · u = 0,  (14)  ∂u  ∂t + u · ∇u = −∇p +  1  ReD  ∇2u,  (15)  where u and p are the non-dimensionalized velocity vector and pressure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  All variables are non-  dimensionalized with the fluid density ρ, the uniform velocity U∞, and the cylinder diameter D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The Reynolds  number is defined as ReD = U∞D/ν = 100 with ν being the kinematic viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' We consider five nested lev-  els of multi-domains with the finest grid level covering (x, y) = [−1, 15] × [−8, 18] and the largest domain being  (x, y) = [−5, 75] × [−40, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The time step for DNS is ∆t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='50 × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' We extract the domain around a cylinder  body over (x∗, y∗)/D = [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='7, 15] × [−5, 5] with (Nx, Ny) = (500, 300) and (∆x, ∆y) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0314, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0333).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The  flow exhibits vortex shedding with a single period with 21 snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' For the compression analysis, 160 temporal  snapshots of grayscale images of the streamwise velocity field u are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2  Two-dimensional decaying homogeneous isotropic turbulence  To examine the data compression performance by the present techniques for more complex turbulent flows, we also  consider a two-dimensional decaying homogeneous isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This time-varying flow can be regarded  as a canonical fluid flow example for a broad range of turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The data set is prepared by DNS using  the two-dimensional vorticity transport equation [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' We set the initial Reynolds number Re0 ≡ u∗l∗  0/ν = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='4,  where u∗ is the characteristic velocity obtained by the square root of the spatially averaged initial kinetic energy,  l∗  0 = [2u2(t0)/ω2(t0)]1/2 is the initial integral length, and ν is the kinematic viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The computational domain and  8  A PREPRINT - JANUARY 3, 2023  the numbers of grid points are set to Lx = Ly = 1 and Nx = Ny = 128, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' We use 1000 snapshots in an  eddy turn-overtime of t ∈ [2, 6] with a time interval of ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' For the data compression analysis, 128 × 128 grid  with grayscale contours of the vorticity field ω are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='3  Turbulent channel flow  To further demonstrate the present data compression techniques, we also examine turbulent channel flow at a friction  Reynolds number of Reτ = uτδ/ν = 180, where uτ is the friction velocity, δ is the half-width of the channel, and ν  is the kinematic viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This flow involves a broader range of spatio-temporal flow scales and fewer redundancies  compared to the previous two examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The data sets are prepared by a three-dimensional DNS [50, 51], numerically  solving the incompressible Navier–Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The present DNS has been validated by comparison with spectral  DNS data of Moser et al [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The streamwise, wall-normal, and spanwise spatial coordinates are denoted by x, y, and z,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The size of the computational domain and the number of grid points here are (Lx, Ly, Lz) = (4πδ, 2δ, 2πδ)  and (Nx, Ny, Nz) = (256, 96, 256), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The grids in the streamwise and spanwise directions are taken to be  uniform, while that in the y direction is a non-uniform grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The no-slip boundary condition is imposed on the walls and  a periodic boundary condition is applied to the x and z directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The flow is driven by a constant pressure gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' In  what follows, we denote wall-unit quantities with the superscript +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  For the present study, an x − z cross-sectional streamwise velocity u at y+ = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2 is analyzed, where representative  streak structures are present [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Fifty temporal snapshots of a 256 × 256 spatial grid over t+ ∈ [0, 63] are formatted  into grayscale data and are used for compression assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  4  Results  For image compression, matrices of flow field data corresponding to specific temporal snapshots are represented as  uncompressed grayscale images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' These images are compressed using JPEG and JP2 encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' For video compression,  the matrices of flow field data are represented as grayscale images and concatenated into uncompressed, grayscale  videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This raw video file is used as the input to the H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265, and AV1 encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' To obtain control over the  outputted file size for the purposes of this analysis, a two-pass encoding scheme is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' In the two-pass encoding,  the encoder runs twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The first run is used to collect some information and statistics such as how many bytes would  be needed for data compression and the second run performs the actual encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' These two processes enable the use  of the information collected in the first run to achieve enhanced compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  This study uses FFmpeg [35], a free and open-source software consisting of various libraries for handling video, audio,  and other multimedia files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' These libraries can be easily used from the command line ffmpeg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Default encoding  settings for the relevant FFMpeg library are used to maintain consistency across all tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='1  Image compression  To establish a baseline performance for comparison, individual flow snapshots are compressed using singular value  decomposition (SVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Individual snapshots are decomposed into left and right singular vector matrices U and V T , and  a diagonal matrix Σ containing the singular values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Snapshots are reconstructed by retaining the r leading modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Here,  the compression ratio η in the SVD context is then defined as  η = r(m + n + 1)  mn  ,  (16)  where m and n are the snapshot dimensions in the horizontal and vertical directions and a value of η = 1 corresponds  to the original, uncompressed snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' In the present study, the snapshot dimensions for each flow example are set  as (m, n) = (500, 300) for cylinder wake, (128, 128) for two-dimensional decaying turbulence, and (256, 256) for  turbulent channel flow, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Let us compare the image compression techniques with the cylinder wake example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' As for the data attribute, we use  a streamwise velocity u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The compressed wake fields and the spatial absolute error distributions with η ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='05 are  presented in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The L2 error norm of reconstruction ε = ||fRef − fComp||2/||fRef||2, where fRef and fComp  are respectively the reference and compressed flow fields, is also shown underneath the decoded fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The SVD  produces negligible error for the entire flow field, although slight discontinuities are observed in the wake region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' By  comparison, JPEG compression introduces some compression artifacts including discontinuities of grayscale contours  and granulated vortical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This is due to fixed-size areas upon which the DCT is performed and elementary  anti-blocking features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This indicates that compression based on the fixed block of 8 × 8 pixels is not appropriate for  flow fields that include fine-scale spatial variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' In contrast, the compressed flow field with the JP2 algorithm retains  9  A PREPRINT - JANUARY 3, 2023  JPEG  JPEG2000  SVD  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0176, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0548}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00370, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0507}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00670, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0532}  Compressed flow field  Spatial error distribution  0  50  100  50  100  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='05  Figure 5: Comparison of image compression techniques for cylinder wake at ReD = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' A streamwise velocity field  u is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The L2 error norm of the reconstruction ε and the compression ratio η are shown underneath each flow  field contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Spatial absolute error distribution for each compression technique is also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  JPEG  JPEG2000  SVD  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0350, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='282}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0233, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='270}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00820, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='282}  0  50  100  50  100  Figure 6: Comparison of image compression techniques for two-dimensional decaying turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' A vorticity field ω is  considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The L2 error norm of the reconstruction ε and the compression ratio η are shown underneath each flow  field contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  wake features even while achieving significant data compression, with the L2 error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' These results support  the effectiveness of the adaptive block size of DWT in JP2 compression for bluff body wake data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Next, we apply the image compression techniques to two-dimensional decaying homogeneous isotropic turbulence,  as shown in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' We use a vorticity field ω as a quantity of interest and compare the compression results with  η ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Similar to the cylinder example, the SVD can provide a smooth field and small error for the entire flow field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Although SVD can achieve a reasonable compression for the laminar cylinder wake and two-dimensional turbulence  that are mainly composed of large vortical structures, we discuss later how the presence of fine-scale turbulent structures  alters the compression performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' For this two-dimensional turbulence, the effect of 8 × 8 pixel blocks can be clearly  observed in JPEG compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Such pixelized artifacts on the flow field can be mitigated by using the JP2 compression  technique, analogous to the observation with the cylinder example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  The limitation of the SVD and the efficacy of the DWT-based process in the JP2 algorithm are further emphasized in  the example of more complex turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Here, the compression techniques are applied to a streamwise velocity u  10  JPEG  JPEG2000  SVD  (8, n) = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0176, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0548)  (8, n) = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00370, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0507  (s, n) = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00670, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0532)  (8, n) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00490, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='136)  (8, n) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00310, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='120)  (8, n) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00180, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='112)100  50  0  50  100JPEG  JPEG2000  SVD  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0350  8= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0233  8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00820  n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='282  n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='270  n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='282  8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0168  8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0134  8= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00310  n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='511  n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='452  n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='502100  50  0  50  100A PREPRINT - JANUARY 3, 2023  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='129, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0281}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0265, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='234}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0903, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0234}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='235, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0235}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0122, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='240}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0258 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='235}  JPEG  SVD  JPEG2000  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='129, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0281}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0265, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='234}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0903, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0234}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='235, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0235}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0122, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='240}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0258 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='235}  JPEG  SVD  JPEG2000  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='129, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0281}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0265, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='234}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0903, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0234}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='235, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0235}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0122, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='240}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0258 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='235}  JPEG  SVD  JPEG2000  JPEG  JPEG2000  SVD  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='129, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0281}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0903, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0234}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='235, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0235}  0  50  100  50  100  Figure 7: Comparison of image compression techniques for turbulent channel flow at Reτ = 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' A streamwise  velocity field u is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The L2 error norm of the reconstruction ε and the compression ratio η are shown  underneath each flow field contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  (a)  (b)  : JPEG  : JP2  : SVD  : Cylinder  : 2D turbulence  : Channel  SVD  JPEG  JP2  SVD  JP2  JPEG  SVD  JPEG  JP2  (c)  (d)  (e)  Figure 8: Relationship between (a) the L2 error norm ε, (b) SSIM, and image compression ratio η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Zoom-in view of  η-SSIM curve for (c) cylinder wake, (d) two-dimensional turbulence, and (e) turbulent channel flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  of the three-dimensional channel flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The compression results with η ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='025 are compared in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' As shown,  the SVD-based compression cannot retain the important features of the streaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Compared to SVD, JPEG provides a  better reconstruction although it also introduces discontinuities that obscure small spatial length scales in the flow field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Surprisingly, JP2 produces non-negligible artifacts and maintains an L2 error norm less than half that of SVD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The  channel flow field at this low η remains nearly indistinguishable from the uncompressed flow field, also preserving the  streak spacing of the reference DNS field [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' These observations suggest the effectiveness of the JP2 algorithm  for image compression of complex fluid flow data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Building on these assessments, the L2 error between compressed and uncompressed flow fields is evaluated across  compression ratios, as shown in figure 8(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The error is averaged over all temporal snapshots of each flow example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' In  general, all compression algorithms produce an asymptotically decaying L2 error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' JPEG introduces appreciable error at  low η, in the same order as SVD compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' It is worth pointing out that JP2 performs especially well at low η while  SVD compression produces the lowest L2 error for high η for all flow fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  11  100  50  0  50  100100  10  2  10  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='8  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='75  SSIM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='50  L  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='8  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='005  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='000  EE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='995  SSIM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='990  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='985  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='980  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='8  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='975  SSIM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='950  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='925  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='900  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='8  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00  SSIM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='97  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='8  nA PREPRINT - JANUARY 3, 2023  (a)  (d)  (b)  (e)  (c)  (f)  ηLow = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0299  ηMed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0913  ηHigh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='237  ηLow = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='295  ηMed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='539  ηHigh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='885  ηLow = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='215  ηMed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='505  ηHigh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='932  ηLow = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0299  ηMed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0913  ηHigh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='237  ηLow = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0381  ηMed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0880  ηHigh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='803  ηLow = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0381  ηMed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0880  ηHigh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='803  Figure 9: Kinetic energy spectra for two-dimensional decaying homogeneous isotropic turbulence using (a) JPEG and  (d) JP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' (b, e) Streamwise and (c, f) spanwise kinetic energy spectrum of three-dimensional turbulent channel flow  compressed with (b, c) JPEG and (e, f) JP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  As an additional metric for quantifying the error introduced by each compression method, the localized structural  similarity index (SSIM) [55] is computed between compressed and uncompressed flow fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' SSIM can capture spatial  correlation around pixels and is less sensitive against a pixel-wise error caused by translation and rotational difference  compared to the L2 error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Hence, SSIM is suited for the image and video-based compression analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The SSIM χ is  defined as  χ = l(ix, iy)c(ix, iy)s(ix, iy)  (17)  where  l(ix, iy) = 2µxµy + C1  µ2x + µ2y + C1  ,  c(ix, iy) = 2σxσy + C2  σ2x + σ2y + C2  ,  s(ix, iy) = σxy + C3  σxσy + C3  (18)  with µx and σx defined as the mean and standard deviation of ix respectively, σxy being the covariance of ix and iy,  and c1, c2, and c3 being constants to stabilize division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' We set {C1, C2, C3} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='16, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='44, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='72} following Wang et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The resultant value lies between 0, representing no similarity, and 1, representing an identical image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The  relationship between the image compression ratio and the L2 error is depicted in figure 8(b) Generally, JP2 and SVD  produce a negligible decrease in the SSIM at low compression ratios and asymptotically approach an SSIM value of 1  at higher compression ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The SSIM value of the cylinder flow field with JPEG compression applied decays by  approximately 10%, as a result of significant discontinuities produced by JPEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  We also present the zoom-in view of the relationship between SSIM and the compression ratio η for each flow, in  figures 8(c) − (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Similar to the observation in the η − ε curves in figure 8(a), SVD and JP2 provide high SSIM scores  compared to JPEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Especially at excessive compression (low η) of three-dimensional turbulent channel flow, JP2 can  provide better reconstructions than the other two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Although scalar metrics such as the L2 error ε and SSIM are  useful, we note that monitoring not only scalar values but also decoded flow fields is important in assessing how vortical  structures can be retained through data compression because the influence of local structures are averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  We are additionally interested in whether finer structures in flow images can still be retained through the present  compression process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' To examine this aspect, we consider the kinetic energy spectrum of both two- and three-  dimensional turbulence examples, as summarized in figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The kinetic energy spectrum E(k) for two-dimensional  decaying turbulence is  E(k) = 1  2(uiui),  (19)  12  Uncompressed  JPEG LoW  JPEG Medium  JPEG High  10~5  E  10-10  101  102  103  k-Uncompressed  JP2 LoW  JP2 Medium  JP2 High  10-5  E  10-10  101  102  103  k100  10-4  10-6  10-1  10-2100  10-4  10-6  10~2  10-1100  10-4  10-6  10-1  10~2100  10-4  10-6  10-1  102A PREPRINT - JANUARY 3, 2023  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0411, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0380}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00470, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='148}  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0132, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0196}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00450, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='142}  AV1  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0171, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0309}  POD  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00200, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='130}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0578, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0300}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00770, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='157}  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0411, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0380}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00470, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='148}  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0132, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0196}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00450, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='142}  AV1  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0171, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0309}  POD  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00200, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='130}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0578, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0300}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00770, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='157}  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264  0  50  100  50  100  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0578, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0300}  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0411, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0380}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0132, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0196}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0171, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0309}  AV1  POD  Figure 10: Comparison of video compression techniques applied on a streamwise velocity field u of cylinder wake  at ReD = 100, compressed using H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265, AV1, and POD compression algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The L2 error norm of the  reconstruction ε and the compression ratio η are shown underneath each flow field contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  where ui are the components of the fluctuating velocity and the overbar denotes an averaging operation in space and  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' For three-dimensional turbulent channel flow, the one-dimensional streamwise and spanwise spectra is evaluated  Euu(k+  x ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' y+) = ˆu∗ˆu  z,t, Euu(k+  z ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' y+) = ˆu∗ˆu  x,t,  (20)  where (·)∗ represents the complex conjugate and ˆ(·) denotes the one-dimensional Fourier transformed variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Here,  we compare three compression ratios, denoted as low, medium, and high, for each turbulent flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  JP2 demonstrates a strong adherence to the kinetic energy spectrum of the uncompressed flow field in both the x  and z directions while JPEG compression at low η introduces non-negligible errors at higher wave numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This is  a consequence of the quantization step of JPEG compression that removes high wavelength scales from the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Considering the overestimation of Euu(k+  x ) as seen in figure 9 when using JPEG, this is likely caused by the absence  of a deblocking filter, producing more high wavelength artifacts in the image than what exists in the uncompressed  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Similarly, the underestimation of E(k) by JP2 can be attributed to adaptive block sizes that produce a lower  peak signal-to-noise ratio, indicative of lower quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' In general, JP2 is more adept at preserving high-wavenumber  structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2  Video compression  From the perspective of information, fluid flows are inherently temporally-redundant — as such, video compression  algorithms that perform temporal compression are a powerful tool, achieving compression performance that outperforms  the previously analyzed image-based techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This section assesses the capabilities of video compression techniques  such as H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265, and AV1 compression algorithms for time-varying fluid flow data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Additionally, proper orthogonal  decomposition (POD) compression [56] is considered to compare this familiar method of compression within the  fluid dynamics community with those analyzed herein [3, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' POD is used to decompose a matrix of vectorized,  temporally evolving flow field data into a set of basis modes and eigenvalues that contain coherent flow structures  and can be used for flow field reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Formally, a flow field q(x, t) − q(x) can be represented as �n  j=1 ajφj  where aj is the temporal coefficient for mode φj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The value of aj is the inner product between the mode φj and the  mean-subtracted flow field, q(x, t) − q(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This modal representation can be truncated to r modes, such that the flow  field is approximated by �r  j=1 ajφj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This study uses the snapshot POD method [58] for comparison to the other video  13  100  50  0  50  100A PREPRINT - JANUARY 3, 2023  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0377  η =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0256  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00660  η =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='222  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0227  η =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0324  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00580  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='230  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0175  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0207  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00480  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='206  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0170  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0209  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00300  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='201  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265  AV1  POD  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0377  η =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0256  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00660  η =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='222  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0227  η =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0324  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00580  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='230  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0175  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0207  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00480  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='206  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0170  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0209  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00300  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='201  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265  AV1  POD  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0377  η =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0256  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00660  η =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='222  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0227  η =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0324  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00580  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='230  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0175  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0207  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00480  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='206  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0170  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0209  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00300  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='201  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265  AV1  POD  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0377  η =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0256  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00660  η =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='222  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0227  η =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0324  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00580  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='230  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0175  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0207  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00480  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='206  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0170  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0209  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='00300  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='201  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265  AV1  POD  0  50  100  50  100  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0377, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0256}  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0227, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0324}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0175, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0207}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0170, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0209}  AV1  POD  Figure 11: Comparison of video compression techniques applied on two-dimensional isotropic turbulent vorticity field,  compressed using H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265, AV1, and POD compression algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The L2 error norm of the reconstruction ε  and the compression ratio η are shown underneath each flow field contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='231, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='147}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0890, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='403}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='151, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='135}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0832, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='185}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0588, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='367}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0502, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='385}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='255, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='157}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='139, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='412}  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265  AV1  POD  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='231, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='147}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0890, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='403}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='151, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='135}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0832, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='185}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0588, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='367}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0502, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='385}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='255, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='157}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='139, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='412}  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265  AV1  POD  0  50  100  50  100  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='231, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='147}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0890, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='403}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='151, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='135}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0832, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='185}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0588, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='367}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0502, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='385}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='255, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='157}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='139, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='412}  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265  AV1  POD  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='231, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='147}  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0832, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='185}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='151, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='135}  {ε, η} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='255, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='157}  AV1  POD  Figure 12: Comparison of video compression techniques applied on a streamwise velocity field u of three-dimensional  turbulent channel flow at Reτ = 180, compressed using H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265, AV1, and POD compression algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The  L2 error norm of the reconstruction ε and the compression ratio η are shown underneath each flow field contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  compression techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The compression ratio for a reconstructed flow field containing r modes is evaluated as  η = r(m + n) + m  n(m + n) + m,  (21)  where m is the total number of pixels in the flow field and n is the total number of flow snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  The results of video compression for laminar cylinder wake at ReD = 100 are shown in figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Here, we compare  the decoded streamwise velocity field u with η ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='02 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' POD compression introduces negligible error, likely  as a result of the temporally redundant nature of periodic wake and the larger coherent modal structures that POD  is able to extract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' By comparison, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264 compression produces significant artifacts at low η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264 struggles likely  because inter-frame prediction candidates are chosen from a shallow time range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' We also observe that H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265 fails to  improve in terms of error level over H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264 for the cylinder wake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This is due to the employment of a similar inter-frame  prediction and selection algorithm to that of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Compared to these H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2XX series, the AV1 algorithm provides much  14  100  50  0  50  100100  50  0  50  100A PREPRINT - JANUARY 3, 2023  (a)  (b)  : H264  : H265  : POD  : Cylinder  : 2D turb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  : Channel  POD  AV1  (c)  (d)  : AV1  H265 H264  POD  AV1  H265  H264  Figure 13: Relationship between (a) the L2 error norm ε, (b) SSIM, and video compression ratio η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Zoom-in view of  η-SSIM curve for (c) cylinder wake and (d) two-dimensional isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  better compression, achieving a lower L2 error than that achieved by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264 and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This highlights the enhanced  capability of AV1 to compress laminar and temporally redundant flow fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  We next examine the video compression techniques for two-dimensional decaying homogeneous isotropic turbulence,  as summarized in figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The flow fields are compared for the compression ratios of η ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='02 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Similar to the  cylinder case, POD compression provides a reasonable reconstruction, likely because large-scale vortical structures  are dominant at this particular time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' It is, however, easily anticipated that the error of this time-varying flow relies  on the presence of a range of length scales, as the small length scales disappear with the progress of the decay over  time [59, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The dependence of the compression performance over time for decaying flow will be examined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  While H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2xx compression techniques provide a reasonable reconstruction, AV1 provides better compression without  suffering from pixelized artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' These results suggest the powerful capabilities of novel deblocking filters for fluid  flow applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  The video compression techniques are also applied to the x−z sectional streamwise velocity field u of three-dimensional  turbulent channel flow at Reτ = 180, as depicted in figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The compressed flow fields are compared for η ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  In contrast to the other flow examples, POD compression produces significant visible artifacts and a high error value  for turbulent channel flow because of a complex temporal evolution of the flow field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' POD requires a greater number  of modes for adequate reconstruction [61, 62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Although H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265 improves over H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264 significantly for turbulent  channel flow, this still produces few observable discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This is likely caused by adaptive tiling in macroblocks  for prediction procedures, allowing lower η with similar flow field representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' AV1 exceeds the performance of  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264 and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265 consistently and POD compression on turbulent channel flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Flow fields compressed using AV1 are  indistinguishable from uncompressed flow fields at high η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  The L2 error and SSIM are evaluated across a range of compression ratios for each type of flow field, as presented in  figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' In general, all video compression algorithms produce asymptotically decaying L2 error values with increasing  η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' AV1 performs well at low η, especially for the cylinder wake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Additionally, all compression algorithms perform  well for two-dimensional turbulence, likely as a result of the flow field snapshots holding slow changes from one  frame to the next due to the decaying nature of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Moreover, as observed with samples at various compression  ratios, the L2 error for all algorithms plateau at non-negligible values for the cylinder wake flow field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' SSIM values  15  n  10  2  10  3  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='8  n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='8  4  SSIM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
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+page_content='990  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
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+page_content='000  SSIM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
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+page_content='8  nA PREPRINT - JANUARY 3, 2023  (b)  (c)  (a)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (b)  (c)  (a)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (b)  (c)  (a)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (b)  (c)  (a)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (a)  (b)  (c)  Figure 14: L2 error norm ε of vorticity field ω for two-dimensional decaying homogeneous isotropic turbulence over  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' (a) H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264, (b) H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265, and (c) AV1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' (i) and (ii) in each case are chosen due to their employment in inter-frame  prediction in each algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  generally diverge from the asymptotic limit at low η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The exceptional cases include cylinder wake and two-dimensional  turbulence compressed using AV1, which introduces negligible error at low η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' AV1 outperforms H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264 and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265 on  turbulent channel flow as well, due to the improved blocking techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  In addition, the time evolution of the L2 error is examined to gain insight into the performance of video compression  algorithms for individual snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The temporal evolution of the L2 error norm ε for two-dimensional decaying  turbulence is shown in figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2xx compression techniques exhibit repeated temporal structures in its L2 error  evolution, likely as a consequence of inter-frame prediction selecting frames to make predictions from at relatively  similar intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' AV1 compression provides a distinctive reduction in the L2 error over time for medium and low η,  indicating improved accuracy as snapshots begin to show redundancies due to the vortex field decaying and exhibiting  similar large-scale coherent structures from one snapshot to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' We also observe that H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264 produces a high error  at low η for early flow field snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This relates to the time-varying flow nature of the present decaying turbulence,  as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The presence of finer structures at the high Taylor Reynolds number Reλ(t) portion of the flow  likely causes the difficulty in compressing vortical flow data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  We also examine the L2 error norm ε and the flow fields over time for turbulent channel flow, as depicted in figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264 generally produces a larger L2 error compared to the other techniques, as we also observed with the visual  assessments in figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' With low η of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264 compression, the error decreases over time, likely as a result of a later  snapshot being selected for inter-frame prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Compared to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265 provides better compression over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Similar to the observation with H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264, the L2 error significantly varies over time at a low η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This is likely due to the  inter-frame selection of an early frame from which further predictions were made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' AV1 produces a negligible error at a  high η while the errors increase as η decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  We are also interested in the performance of video compression algorithms in preserving high wavenumber structures in  the compressed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The general performance of each compression algorithm with regard to kinetic energy spectra of  each flow field is investigated, as shown in figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264 performs well for two-dimensional turbulence, but produces  a noticeable error at all η in both the stream- and spanwise directions of the kinetic energy spectrum of turbulent channel  flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' A similar divergence from the expected data can be observed at low η in the spanwise direction as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265  performs comparatively well for two-dimensional decaying isotropic turbulence, and for turbulent channel flow in the  spanwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' However, it produces a non-negligible error at high wavenumbers when compressed at low η in the  spanwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This indicates an over-representation of high-wavenumber components due to blocking as a result  of the adaptive subblock sizes of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Generally, AV1 is the best-suited algorithm for preserving spatial frequency  information, particularly at high wavenumbers, for both two and three-dimensional turbulent flow fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' At higher η,  the energy contents at each wavenumber are almost indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='012  AV1 Low  AV1 Medium  Error  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='01  AV1 High  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='002  0  2  3  4  5  6  t10  0  10  w  2  10  10  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='8  n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='015  H265 L0W  ^H265 Medium  H265 High  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='005  2  3  4  5  6  t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='025  H264 L0w  _H264 Medium  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='02  H264 High  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='005  2  3  4  5  6  tA PREPRINT - JANUARY 3, 2023  (b)  (c)  (a)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (b)  (c)  (a)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (b)  (c)  (a)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (i)  (ii)  (a)  (b)  (c)  Figure 15: L2 error norm ε of streamwise velocity field u for turbulent channel flow over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' (a) H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264, (b) H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265,  and (c) AV1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' (i) and (ii) in each case are chosen due to their employment in inter-frame prediction in each algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  At last, we investigate whether the video compression techniques can preserve the temporal evolution of complex  turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Here, let us examine the temporal two-point correlation for three-dimensional channel flow compressed  using all three video compression algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The temporal two-point correlation coefficient at a given t+ is defined  as R+  uu(t+)/R+  uu(0) [64, 65] and is depicted in figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The assessment of temporal two-point correlation provides  insight into the relations of flow snapshots to preceding snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Consistent with the insights gained from the kinetic energy spectrum, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264 compression at a η exhibits disagreement  with the reference curve at t+ values between 5 and 30, and above 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This is indicative of a de-correlation of the  velocity field and is likely a result of poor performance in capturing high-wavenumber information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Except for this  particular case, all compression algorithms generally perform well, with temporal two-point correlation coefficients  closely following that of the uncompressed flow field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' These results suggest that these novel video compression  techniques capture the spatio-temporal redundancies well even for complex turbulent flows and also significantly reduce  data size while preserving their physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  5  Conclusion  We compressed flow field data from canonical flow examples using a number of widely-available multimedia com-  pression techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The performance of the JPEG and JP2 spatial image compression techniques and the H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265, and AV1 spatio-temporal video compression techniques were considered for simulated laminar cylinder flow,  decaying isotropic turbulence, and turbulent channel flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Streamwise velocity and vorticity field data were represented  as grayscale images and videos, and were compressed using the aforementioned techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  All techniques, with the exception of JPEG, were shown to compress flow data below 10% of the original file size while  introducing negligible error and preserving underlying flow physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' AV1 and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265 compression were shown to have  the best performance across a variety of flow regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The spatial error distributions were concentrated on the cylinder  surface and directly behind the cylinder for the streamwise velocity data compression and in the vortex shedding wake  for the vorticity data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Turbulence statistics in the form of kinetic energy spectra were preserved under compression for  all methods except JPEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
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+page_content='585  ηHigh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='880  ηLow = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0724  ηMed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='595  ηHigh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='976  ηLow = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0587  ηMed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='562  ηHigh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='898  ηLow = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='119  ηMed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
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+page_content='776  ηLow = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0872  ηMed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='403  ηHigh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='817  ηLow = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0872  ηMed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='403  ηHigh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='817  ηLow = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='119  ηMed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='367  ηHigh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='776  ηLow = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='187  ηMed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='385  ηHigh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='767  ηLow = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='187  ηMed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='385  ηHigh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='767  Figure 16: Kinetic energy spectra E(k) for (a, d, g) two-dimensional decaying homogeneous isotropic turbulence  using H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265, and AV1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' (b, e, h) Streamwise Euu(k+  x ) and (c, f, i) spanwise kinetic energy spectra Euu(k+  z ) of  three-dimensional turbulent channel flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  For single snapshots of data represented as an image, JP2 compression was shown to far outperform JPEG compression,  with a tolerable increase in computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' For multiple temporal snapshots of data represented as a  video, the choice of compression method becomes more nuanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' JP2 compression was shown to achieve the lowest  compression error as temporal compression adds slight error to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The AV1 algorithm maximizes η at the expense  of computational complexity and non-negligible encoding time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This algorithm is new and emerging from the research  environment, so future optimizations could bring this encoding time to a manageable level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265 algorithm  provided excellent compression performance at a fast encoding time, and appears as a promising algorithm for current  fluid dynamics applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264 provided acceptable compression performance, but was largely triumphed by the  AV1 and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265 algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  We have shown that modern multimedia compression algorithms provide robust performance in a variety of fluid flow  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The implementation of these techniques becomes especially pertinent as simulations within computational  fluid dynamics become exceedingly data-intensive, a trend that decreases the accessibility to high-fidelity models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' These  methods are free, easily accessible, regularly updated and supported, and provide flexible and scalable compression  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' As such, the implementation of these compression techniques has exciting potential across the fluid  dynamics community for data storage and transfer with minimal loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  18  Uncompressed  1H265 L0W  ^H265 Medium  H265 High  10~5  10-10  101  102  103  k-Uncompressed  AV1 Low  AV1 Medium  AV1 High  10~5  10-10  101  102  103  k100  10-4  10-2  10-1100  10-4  10-2  10-1  Y100  10-4  10-2  10-1100  10-4  10-2  10-1-Uncompressed  H264 LoW  H264 Medium  H264 High  10~5  10-10  101  102  103  k100  10-4  10-6  10-2  10-1100  10  10-4  10-6  10-2  10-1A PREPRINT - JANUARY 3, 2023  (a)  (b)  (c)  ηLow = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='187  ηMed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='385  ηHigh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='767  ηLow = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='0872  ηMed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='403  ηHigh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='817  ηLow = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='119  ηMed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='367  ηHigh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='776  Figure 17: Normalized temporal two-point correlation coefficients Ruu(t+)/Ruu(0) for three-dimensional turbulent  channel flow using H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265, and AV1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  JPEG  JP2  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265  AV1  u, Cylinder Flow  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='58  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='94  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='85  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='21  ω, Cylinder Flow  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='59  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='92  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='39  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='83  u, Channel Flow  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='29  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='42  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='96  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='59  Table 1: Encoding time (s) for different compression algorithms and flow regimes, compressed at 100 KB/s bitrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Acknowledgements  KT acknowledges the support from the US Army Research Office (W911NF-21-1-0060), the US Air Force Office  of Scientific Research (FA9550-21-1-0178), and the US Department of Defense Vannevar Bush Faculty Fellowship  (N00014-22-1-2798).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' We also thank Professor Koji Fukagata (Keio University) for sharing his DNS code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Appendix: Encoding time  The increased performance of new compression algorithms comes at a cost;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' non-negligible increases in computational  complexity should be considered when implementing these algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' In fact, in a paper from 2000 on compressing  three-dimensional flows with the JPEG and JP2 algorithms [66], the added complexity of the JP2 algorithm caused  JPEG to be recommended over JP2, despite losing clear performance benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The recommendation of the present  study reverses that statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' As such, it is important to quantify the encoding time of these algorithms at the time of  writing this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  The decoding time is observed to be negligibly small for all compression codecs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' thus, this appendix focuses on  encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The streamwise velocity and vorticity data are encoded for both the laminar cylinder flow and turbulent  channel flow cases at the same bitrate (100 KB/s) for all compression algorithms and the encoding time is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  The encoding is performed with a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='5GHz i7 Intel Core processor and 8 GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' The results are summarized in  table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Encoding time per frame is observed to be larger for the turbulent channel flow than the laminar cylinder flow,  indicating that the algorithms struggle to encode multiscale turbulent flow data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Across encoding algorithms, JPEG and  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264 compression are the fastest, a testament to the maturity and low complexity of these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' JP2 and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265  encoding are generally several times slower, but still relatively fast, justifying their added compression performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  AV1 is observed to be far slower in encoding than the other methods: over 100 times slower than JPEG and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='264, and  over 25 times slower than JP2 and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' This severe encoding time increase limits the practicality of implementing this  algorithm in large-scale applications, and perhaps justifies the use of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='265 over AV1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' As the algorithm was released  only a few years prior to the writing of this paper, advances in computing and algorithm development could increase its  practicality in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
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+page_content='  19  Uncompressed  H264 L0W  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='8  H264 Medium  H264 High  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2  0  20  0  40  60-Uncompressed  H265 L0W  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='8  H265 Medium  H265 High  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2  0  20  0  40  60-Uncompressed  AV1 Low  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='8  AV1 Medium  AV1 High  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='2  0  0  20  40  60A PREPRINT - JANUARY 3, 2023  [2] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
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+page_content=' Modal analysis of fluid flows: An overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
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+page_content='  [4] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Wu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Zaki, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Meneveau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Data compression for turbulence databases using spatiotemporal subsampling  and local resimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
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+page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Liang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Lu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' A novel in situ compression method for CFD data  based on generative adversarial network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
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+page_content='  [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Glaws, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' King, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Sprague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Deep learning for in situ data compression of large turbulent flow simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
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+page_content='  [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
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+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' Tan, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfR_ex/content/2301.00078v1.pdf'}
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diff --git a/D9E1T4oBgHgl3EQfWgQs/content/tmp_files/2301.03114v1.pdf.txt b/D9E1T4oBgHgl3EQfWgQs/content/tmp_files/2301.03114v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d7006d5f94e1e9bcbda043134d35fd9f46a4493f
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@@ -0,0 +1,12941 @@
+Lepton-Nucleus Scattering Measurements for
+Neutrino Interactions and Oscillations
+by
+Afroditi Papadopoulou
+B.S., National Kapodistrian University of Athens (2016)
+Submitted to the Department of Physics
+in partial fulfillment of the requirements for the degree of
+Doctor of Philosophy
+at the
+MASSACHUSETTS INSTITUTE OF TECHNOLOGY
+May 2022
+© Massachusetts Institute of Technology 2022. All rights reserved.
+Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+Department of Physics
+May 12, 2022
+Certified by. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+Or Hen
+Associate Professor
+Thesis Supervisor
+Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+Deepto Chakrabarty
+Associate Department Head of Physics, MIT
+arXiv:2301.03114v1  [hep-ex]  8 Jan 2023
+
+2
+
+Lepton-Nucleus Scattering Measurements for
+Neutrino Interactions and Oscillations
+by
+Afroditi Papadopoulou
+B.S., National Kapodistrian University of Athens (2016)
+Submitted to the Department of Physics
+on May 12, 2022, in partial fulfillment of the
+requirements for the degree of
+Doctor of Philosophy
+Abstract
+Currently running and forthcoming precision neutrino oscillation experiments aim
+to unambiguously determine the neutrino mass ordering, the charge-parity violating
+phase in the lepton sector and the possible existence of physics Beyond the Standard
+Model. To have an understanding of all the effects necessary for the success of these
+experiments, lepton-nucleus interactions must be modeled in unprecedented detail.
+With this thesis, expertise in both neutrino and electron cross-section modeling and
+analysis was leveraged in order to make fundamental and critical improvements to
+our understanding of these interactions. The outlined work takes a significant step
+towards this high-precision measurement era with three complementary approaches.
+Cross sections are reported using neutrino data sets from the MicroBooNE liquid
+argon time projection chamber detector at Fermi National Laboratory, as well as
+electron scattering data from the CLAS detector at Thomas Jefferson National Labo-
+ratory. Furthermore, the modeling development of the commonly used GENIE event
+generator is presented.
+3
+
+ Neutrino Cross Sections
+Phys. Rev. Lett. 125, 201803 (2020)
+·Event Generator
+·Electron
+Modeling
+Cross Sections
+Phys. Rev. D 103, 113003 (2021)
+Nature 599, 565-570 (2021)Thesis Supervisor: Or Hen
+Title: Associate Professor
+4
+
+Acknowledgments
+First and foremost, my eternal gratitude goes to my advisor, Prof. Or Hen. Or has
+provided me with all the resources necessary to proceed in my academic career. I am
+also grateful to Prof. Lawrence Weinstein. Their guidance has been crucial in my
+formation as a physicist.
+I’m grateful to all the members of our group for their invaluable support and to
+all the MIT graduate students in my year. Many thanks to all the members of the
+MicroBooNE, “Electrons-For-Neutrinos”, CLAS, GENIE, and GlueX collaborations
+for all their input and guidance. I’m also grateful to my entire family and all my
+friends for their continuous support and encouragement.
+5
+
+6
+
+Contents
+1
+Introduction
+11
+1.1
+Neutrinos In The Standard Model . . . . . . . . . . . . . . . . . . . .
+11
+1.2
+Neutrino Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+1.3
+Long-Baseline Accelerator-Based Neutrino
+Oscillation Experiments
+. . . . . . . . . . . . . . . . . . . . . . . . .
+20
+1.4
+Neutrino-Nucleus Interactions . . . . . . . . . . . . . . . . . . . . . .
+24
+1.5
+Connections To Electron Scattering . . . . . . . . . . . . . . . . . . .
+28
+1.6
+Thesis Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+31
+2
+The MicroBooNE Experiment At Fermi National Laboratory
+33
+2.1
+Fermilab Neutrino Beamlines
+. . . . . . . . . . . . . . . . . . . . . .
+33
+2.2
+The MicroBooNE Detector . . . . . . . . . . . . . . . . . . . . . . . .
+36
+2.3
+Liquid Argon Time Projection Chambers . . . . . . . . . . . . . . . .
+38
+2.4
+Scintillation Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+42
+2.5
+Hardware And Software Triggers
+. . . . . . . . . . . . . . . . . . . .
+44
+2.6
+Optical Event Reconstruction . . . . . . . . . . . . . . . . . . . . . .
+46
+2.7
+TPC Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . . .
+47
+2.8
+Pandora Reconstruction Framework . . . . . . . . . . . . . . . . . . .
+48
+2.9
+Cosmic Overlay Simulation . . . . . . . . . . . . . . . . . . . . . . . .
+52
+3
+MicroBooNE Quasielastic-like Cross-Section Results
+[Phys. Rev. Lett. 125, 201803 (2020)]
+55
+7
+
+3.1
+First Measurement of Differential Charged Current Quasielastic-like
+Scattering Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . .
+55
+3.1.1
+Quasielastic-like Neutrino Data Analysis . . . . . . . . . . . .
+55
+3.1.2
+Quasielastic-like Cross-Section Results
+. . . . . . . . . . . . .
+63
+3.1.3
+Quasielastic-like Cross-Section Analysis Conclusions . . . . . .
+68
+3.2
+First Multidimensional Measurement Of Kinematic Imbalance Cross
+Sections On Argon . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+68
+3.2.1
+Kinematic Imbalance Neutrino Data Analysis
+. . . . . . . . .
+68
+3.2.2
+Cross-Section Extraction Technique . . . . . . . . . . . . . . .
+74
+3.2.3
+Event Generator Modeling And Configurations . . . . . . . . .
+78
+3.2.4
+Kinematic Imbalance Differential Cross-Section Results . . . .
+79
+3.2.5
+Kinematic Imbalance Cross-Section Analysis Conclusions . . .
+87
+3.3
+Prospects With Future Neutrino Experiments
+. . . . . . . . . . . . .
+87
+4
+Inclusive Electron Scattering And The GENIE Event Generator
+[Phys. Rev. D 103, 113003 (2021)]
+91
+4.1
+Electron-Nucleus Modeling Development . . . . . . . . . . . . . . . .
+91
+4.2
+Inclusive Electron Scattering Data Comparisons . . . . . . . . . . . .
+101
+4.3
+Implications For Neutrinos . . . . . . . . . . . . . . . . . . . . . . . .
+109
+5
+The CLAS Electrons-For-Neutrinos Experiment At Thomas Jeffer-
+son National Laboratory
+115
+5.1
+Continuous Electron Beam Accelerator Facility . . . . . . . . . . . . .
+115
+5.2
+The CEBAF Large Acceptance Spectrometer . . . . . . . . . . . . . .
+116
+5.3
+Toroidal Magnet
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+117
+5.4
+Drift Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+117
+5.5
+Electromagnetic Calorimeter . . . . . . . . . . . . . . . . . . . . . . .
+119
+5.6
+Cherenkov Counters
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+120
+5.7
+Time-Of-Flight Detector . . . . . . . . . . . . . . . . . . . . . . . . .
+121
+5.8
+E2a Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+122
+8
+
+6
+Electrons-For-Neutrinos Results
+[Nature 599, 565–570 (2021)]
+125
+6.1
+Electron Data Mining Analysis
+. . . . . . . . . . . . . . . . . . . . .
+125
+6.2
+Incident Energy Reconstruction Results . . . . . . . . . . . . . . . . .
+139
+6.3
+Kinematic Imbalance Results
+. . . . . . . . . . . . . . . . . . . . . .
+144
+6.4
+Electrons-For-Neutrinos Conclusions
+. . . . . . . . . . . . . . . . . .
+153
+6.5
+Prospects With CLAS12 . . . . . . . . . . . . . . . . . . . . . . . . .
+154
+7
+Summary
+157
+8
+Appendices
+159
+8.1
+Total Struck Nucleon Momentum Derivation . . . . . . . . . . . . . .
+159
+8.2
+Wiener SVD Regularization Technique . . . . . . . . . . . . . . . . .
+163
+8.3
+Electrons-For-Neutrinos Fiducials . . . . . . . . . . . . . . . . . . . .
+171
+List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+180
+List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+199
+References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+200
+9
+
+10
+
+Chapter 1
+Introduction
+1.1
+Neutrinos In The Standard Model
+The Standard Model (SM) of particle physics was developed to describe the parti-
+cles that are considered to be fundamental and their interactions [1]. Though an
+extremely accurate theory, we are already aware of particles exhibiting behaviors not
+predicted within the scope of the SM. These peculiar particles are called neutrinos,
+which travel enormous distances before they interact with matter, exclusively via
+weak interactions. The recent realization of neutrino oscillations was a historic dis-
+covery [2, 3]. The SM had been incredibly successful, yet it requires neutrinos to
+be massless. The new observations clearly showed that neutrinos oscillate between
+different identities. This behavior is driven by their non-zero masses and is indicative
+of new physics Beyond the Standard Model.
+Though very challenging to detect them, the study of neutrinos is a promising
+venue towards the generalisation of the SM. The latter, initially developed more than
+40 years ago, introduced the notion of fundamental particles. Those particles are
+either the building blocks of matter referred to as fermions with semi-integer spins, or
+mediators of the interactions referred to as bosons with integer spins. The collection
+of the fundamental particles and the carriers is shown in figure 1-1.
+Prior to the SM, the electroweak theory incorporated neutrinos (𝜈𝑒 , 𝜈𝜇, 𝜈𝜏) as
+left-handed partners of the three families of charged leptons (e, 𝜇, 𝜏) and the corre-
+11
+
+Figure 1-1: The Standard Model of particle physics illustrating the three generations
+of fermions, the gauge bosons, and the scalar Higgs boson.
+Figure adapted from
+ScienceAlert [4].
+sponding anti-neutrinos as right-handed partners, which is illustrated in figure 1-2.
+The experimental verification that neutrinos (anti-neutrinos) are left-handed (right-
+handed) was established with the Goldhaber experiment [5].
+Figure 1-2: Graphic illustrating the left-handed (right-handed) nature of neutrinos
+(anti-neutrinos) via the orientation of the momentum-spin vectors. Figure adapted
+from [6].
+In the context of the SM, neutrinos are assumed to be massless and their individ-
+ual lepton number is conserved. Under the assumption that neutrinos are massless,
+only the left-handed component of the Dirac spinor interacts via the weak force and
+12
+
+FERMIONS
+(matter particles)
+BOSONS (force carriers)
+C
+t
+H
+QUARKS
+up
+charm
+top
+gluon
+Higgs boson
+6
+down
+strange
+bottom
+photon
+e
+LEPTONS
+electron
+muon
+tau
+Z boson
+Ve
+VT
+W
+electron
+muon
+tau
+w boson
+neutrino
+neutrino
+neutrinoMomentum
+Spin
+Momentum
+Spinright-handed components are completely absent. Prior to the discovery of neutrino
+oscillations, there was no experimental evidence of violation of the individual lepton
+number. A typical example of such searches is the muon- and electron-number violat-
+ing decay 𝜇± → 𝑒± + 𝛾, where only upper limits have been placed on the branching
+ratio [7]. Yet, the experimental discovery of neutrino oscillations established that
+the conservation of individual lepton number is not universal. That raises a wealth
+of questions related to the possible right-handed nature and the non-zero mass of
+neutrinos.
+If neutrinos are massive, helicity is not exactly conserved. A massive neutrino
+can have its spin and momentum anti-aligned, and that would correspond to a left-
+handed neutrino. However, in the reference frame of an observer travelling faster than
+the neutrino itself, the neutrino spin and momentum can be aligned, and that would
+correspond to a right-handed particle. On the other hand, if neutrinos are massless,
+they must travel at the speed of light. Since there exists no reference frame travelling
+faster than that, the neutrino helicity cannot change.
+At the same time, the neutrino masses are significantly smaller compared to other
+particles. The latest measurements by KATRIN indicate that electrons are 𝒪(106)
+more massive than neutrinos [8].
+Massless particles require no extra terms to be
+added in the SM Lagrangian. On the other hand, the tiny neutrino masses constitute
+a riddle that requires fine tuning of the additional Lagrangian terms. Instead, many
+theorists argue in favor of a fundamental reason why the neutrino masses are so small.
+In order to accommodate a non-zero neutrino mass, there are two potential ap-
+proaches. The first one includes new particles, namely Dirac neutrinos, and the second
+introduces a new particle type, namely Majorana neutrinos [9].
+A Dirac neutrino can acquire mass via its coupling to the Higgs field. Those parti-
+cles that interact with the Higgs field change helicity and, thus, left-handed particles
+become right-handed and vice versa. Experimental results to-date indicate that in-
+teracting neutrinos are left-handed [5]. A potential extension of the SM includes such
+right-handed neutrinos that obtain mass via the Higgs field. However, these neutri-
+nos do not have electroweak charge, thus they interact only via the mixing with the
+13
+
+left-handed counterpart. If indeed neutrinos are Dirac particles and obtain their mass
+due to the coupling with the Higgs field, their masses should be comparable to the
+other fermions predicted by the SM. However, the neutrino interaction coupling can
+be tuned to retrieve masses that are comparable with the experimental observations.
+Another possibility would be the introduction of Majorana particles. With this
+approach, there is no distinction between neutrinos and anti-neutrinos, with the latter
+being plausible since neutrinos are electrically neutral.
+Massive neutrinos can be
+accommodated in this extension of the SM. Earlier the argument that an observer
+traveling at the speed of light might observe a flip of the neutrino helicity was used.
+However, if neutrinos are their own anti-particles, there is no helicity change within
+this massive neutrino hypothesis.
+The simplest SM extension with Majorana-like
+particles is the type 1 seesaw mechanism [10].
+In this framework, a left-handed
+neutrino interacts with the Higgs boson and a really heavy right-handed neutrino is
+briefly produced. The latter further interacts with the Higgs field to produce a light
+left-handed Majorana neutrino.
+1.2
+Neutrino Oscillations
+Neutrinos are produced in specific flavor eigenstates, namely 𝜈𝑒 , 𝜈𝜇 or 𝜈𝜏.
+The
+neutrino mass eigenstates can be expressed as a superposition of flavor eigenstates
+and vice versa. The flavor eigenstates evolve as a function of the distance that neu-
+trinos travel, the neutrino energy and the neutrino mass differences squared. Such
+an evolution is referred to as “neutrino oscillation” and is a natural consequence of
+the fact that a neutrino flavor state is composed of multiple mass eigestates. For
+simplicity, the derivation of the formalism for the two-neutrino oscillation case with
+masses 𝑚𝑖 for i ∈ {1,2} is outlined here. The neutrino wavefunction is treated as a
+quantum-mechanical plane wave 𝜓 that evolves in time and space as:
+𝜓(𝐿) = 𝜓(0) · 𝑒𝑖·𝑝·𝑥,
+(1.1)
+with 𝑝·𝑥 = 𝐸𝑡−⃗𝑝·⃗𝑥 corresponding to the Lorentz-invariant phase and with (𝐸,⃗𝑝 )
+14
+
+being the energy and the three-momentum, respectively. For a particle of mass 𝑚,
+its momentum 𝑝 can be obtained in the highly relativistic limit where 𝐸 ≫ 𝑚 via a
+Taylor expansion as:
+𝑝 =
+√
+𝐸2 − 𝑚2 ≈ 𝐸 − 𝑚2/2𝐸.
+(1.2)
+If a neutrino travels some distance 𝐿 in vacuum, a phase shift is introduced in its
+wavefunction given by equation 1.1, and equation 1.2 becomes:
+𝜈𝑖(𝐿) = 𝜈𝑖(0) · 𝑒−𝑖·𝑚2
+𝑖 ·𝐿/2𝐸
+(1.3)
+where all the constants have been absorbed in the global phase 𝜈𝑖(0). It is impor-
+tant to stress the fact that these expressions are accurate for plane waves propagat-
+ing in vacuum. Additional phase shifts have to be introduced when neutrinos travel
+through a high density material.
+In the two-neutrino case, the flavor eigenstates 𝜈𝑒 and 𝜈𝜇 can be expressed as a
+linear superposition of the mass eigenstates 𝜈1 and 𝜈2:
+𝜈𝑒 = 𝑐𝑜𝑠𝜃 · 𝜈1 + 𝑠𝑖𝑛𝜃 · 𝜈2,
+𝜈𝜇 = −𝑠𝑖𝑛𝜃 · 𝜈1 + 𝑐𝑜𝑠𝜃 · 𝜈2,
+(1.4)
+where 𝜃 is the mixing angle between the two states. Consider now a pure beam
+of electron neutrinos produced at the source, effectively at a distance of 𝐿 = 0. The
+wavefunction evolves following the formalism in equation 1.3,
+𝜈𝑒(𝐿) = 𝑐𝑜𝑠𝜃 · 𝑒−𝑖·𝑚2
+1·𝐿/2𝐸 · 𝜈1(0) + 𝑠𝑖𝑛𝜃 · 𝑒−𝑖·𝑚2
+2·𝐿/2𝐸 · 𝜈2(0).
+(1.5)
+Given that in our detectors the products of weak interactions are reconstructed
+15
+
+based on the neutrino flavor, equation 1.5 needs to be rewritten using the flavor basis,
+𝜈𝑒(𝐿) = [𝑐𝑜𝑠2𝜃 · 𝑒−𝑖·𝑚2
+1·𝐿/2𝐸 + 𝑠𝑖𝑛2𝜃 · 𝑒−𝑖·𝑚2
+2·𝐿/2𝐸] · 𝜈𝑒(0)
+−𝑠𝑖𝑛𝜃 · 𝑐𝑜𝑠𝜃 · [𝑒𝑖·𝑚2
+1·𝐿/2𝐸 − 𝑒𝑖·𝑚2
+2·𝐿/2𝐸] · 𝜈𝜇(0)
+(1.6)
+The probability of detecting a neutrino of a given flavor is obtained via the square
+of the amplitude,
+𝑃𝜈𝑒→𝜈𝑒 = ⟨𝜈𝑒(𝐿)|𝜈𝑒(𝐿)⟩ = 1 − 𝑠𝑖𝑛2(2𝜃)𝑠𝑖𝑛2(∆𝑚2𝐿
+4𝐸
+),
+(1.7)
+𝑃𝜈𝑒→𝜈𝜇 = ⟨𝜈𝑒(𝐿)|𝜈𝜇(𝐿)⟩ = 𝑠𝑖𝑛2(2𝜃)𝑠𝑖𝑛2(∆𝑚2𝐿
+4𝐸
+),
+(1.8)
+where ∆𝑚2 = 𝑚2
+2 − 𝑚2
+1 is the neutrino mass difference squared. Provided that
+𝜃 ̸= 0 or 𝜋/2 and ∆𝑚2 ̸= 0 for oscillations to take place, the neutrino beam evolves
+as a function of 𝐿/𝐸. The amplitude of this oscillation is given by 𝑠𝑖𝑛2(2𝜃). The
+wavelength, expressed in commonly used units, is obtained by
+1.27∆𝑚2 [𝑒𝑉 2]𝐿 [𝑘𝑚]
+𝐸 [𝐺𝑒𝑉 ]
+.
+(1.9)
+The experimental observation of neutrino oscillations serves as proof of evidence
+that the initial state indeed undergoes a phase shift. The explanation for such a shift is
+that at least some neutrinos have non-zero mass and that the transformation between
+the mass and the flavor eigenstates involves a non-zero mixing angle. Equation 1.8
+gives the neutrino oscillation probability for the simplified case of only two types of
+neutrinos. The full three-flavor probability for 𝜈𝜇 → 𝜈𝑒 oscillation in vacuum is given
+by
+𝑃𝜈𝜇→𝜈𝑒(𝐸, 𝐿)
+≈
+𝐴 sin2 ∆𝑚2
+13𝐿
+4𝐸
+(1.10)
+−𝐵 cos
+(︂∆𝑚2
+13𝐿
+4𝐸
++ 𝛿𝐶𝑃
+)︂
+sin ∆𝑚2
+13𝐿
+4𝐸
+,
+16
+
+where ∆𝑚2
+13 = 𝑚2
+1 − 𝑚2
+3 is the neutrino mass difference squared that determines
+the oscillation wavelength as a function of 𝐿/𝐸 and 𝛿𝐶𝑃 is the charge-parity (CP)
+symmetry violating phase [11–13]. The coefficients 𝐴 and 𝐵 depend primarily on the
+neutrino oscillation mixing angles,
+𝐴 = sin2 𝜃23 sin2 2𝜃13, 𝐵 = −sin 2𝜃12 sin 2𝜃23
+2 sin 𝜃13
+sin ∆𝑚2
+21𝐿
+4𝐸
+sin2 2𝜃13.
+(1.11)
+The different flavor neutrinos (labelled 𝜈𝑒, 𝜈𝜇 and 𝜈𝜏) are linear combinations of
+the different mass neutrinos labelled 1, 2, 3.
+The outlined neutrino mixing can be generalized to N neutrino eigenstates. The
+probability of finding an 𝛼 flavor eigenstate neutrino in a flavor eigenstate 𝛽, after
+traveling some distance 𝐿, is given by:
+𝑃𝛼→𝛽 = 𝛿𝛼𝛽 − 4
+∑︁
+𝑗>𝑖
+𝑅𝑒[𝑈 *
+𝛼𝑖𝑈𝛽𝑖𝑈𝛼𝑖𝑈 *
+𝛽𝑗]𝑠𝑖𝑛2
+(︂[︂1.27 𝐺𝑒𝑉
+𝑒𝑉 2 𝑘𝑚
+]︂ ∆𝑚2
+𝑗𝑖𝐿
+𝐸
+)︂
++2
+∑︁
+𝑗>𝑖
+𝐼𝑚[𝑈 *
+𝛼𝑖𝑈𝛽𝑖𝑈𝛼𝑖𝑈 *
+𝛽𝑗]𝑠𝑖𝑛
+(︂[︂2.54 𝐺𝑒𝑉
+𝑒𝑉 2 𝑘𝑚
+]︂ ∆𝑚2
+𝑗𝑖𝐿
+𝐸
+)︂
+(1.12)
+where U is the N×N unitary neutrino mixing matrix and ∆𝑚2
+𝑗𝑖 = 𝑚2
+𝑗 − 𝑚2
+𝑖 [14].
+The corresponding antineutrino oscillation probability can be obtained by replacing
+𝑈 → 𝑈 †.
+Within the widely accepted neutrino model, there exist three active neutrinos [15],
+resulting into two squared mass splittings, ∆𝑚2
+21 and ∆𝑚2
+32. The corresponding 3 × 3
+mixing matrix, referred to as the Pontecorvo-Maki-Nakagawa-Sakata (PNMS) matrix,
+relates the mass eigenstates (𝜈1, 𝜈2, 𝜈3) to the flavor eigenstates (𝜈𝑒, 𝜈𝜇, 𝜈𝜏):
+⎛
+⎜
+⎜
+⎜
+⎜
+⎜
+⎜
+⎝
+𝜈𝑒
+𝜈𝜇
+𝜈𝜏
+⎞
+⎟
+⎟
+⎟
+⎟
+⎟
+⎟
+⎠
+=
+⎛
+⎜
+⎜
+⎜
+⎜
+⎜
+⎜
+⎝
+𝑈𝑒1
+𝑈𝑒2
+𝑈𝑒3
+𝑈𝜇1
+𝑈𝜇2
+𝑈𝜇3
+𝑈𝜏1
+𝑈𝜏2
+𝑈𝜏3
+⎞
+⎟
+⎟
+⎟
+⎟
+⎟
+⎟
+⎠
+·
+⎛
+⎜
+⎜
+⎜
+⎜
+⎜
+⎜
+⎝
+𝜈1
+𝜈2
+𝜈3
+⎞
+⎟
+⎟
+⎟
+⎟
+⎟
+⎟
+⎠
+(1.13)
+17
+
+The mixing of the mass-flavor eigenstates is parametrised with three mixing angles
+(𝜃12, 𝜃23, 𝜃13) and the 𝛿𝐶𝑃 violating phase. The best measured values of the these
+angular parameters to date are obtained from global fits as documented in the latest
+Review of Particle Physics edition [1] and listed in table 1.1.
+Table 1.1: Summary table of the 3𝜈 oscillation parameters. The 1𝜎 intervals for both
+the case normal are inverse ordering are shown. Table adapted from [1].
+Normal Ordering
+Inverted Ordering
+𝑠𝑖𝑛2𝜃12
+0.320+0.020
+−0.016
+0.320+0.020
+−0.016
+𝑠𝑖𝑛2𝜃23
+0.547+0.020
+−0.030
+0.551+0.018
+−0.030
+𝑠𝑖𝑛2𝜃13
+(2.160+0.083
+−0.069) · 10−2
+(2.220+0.074
+−0.076) · 10−2
+∆𝑚2
+21
+(7.55+0.02
+−0.16) · 10−5 𝑒𝑉 2
+(7.55+0.02
+−0.16) · 10−5 𝑒𝑉 2
+∆𝑚2
+32
+(2.42+0.03
+−0.03) · 10−3 𝑒𝑉 2
+(−2.50+0.04
+−0.03) · 10−3 𝑒𝑉 2
+The 𝛿𝐶𝑃 phase and the neutrino mass hierarchy (𝑚2 < 𝑚3 or the other way
+around) are still not determined. A potential way to resolve these open questions is
+via the Mikheyev-Smirnov-Wolfenstein (MSW) effect [16]. In the presence of matter,
+the neutrino wavefunction propagation is modified. When a neutrino travels through
+a dense material, the energy-momentum relationship is affected via coherent interac-
+tions with matter particles. Given that regular matter on Earth contains electrons,
+but not muons or taus, charged current interactions with the medium only affect the
+electron neutrino propagation. This interaction flavor dependence results in mea-
+surable changes in the contribution of the different flavors. Thus, the MSW effect
+is exploited to perform long-baseline neutrino experiments. The Deep Underground
+Neutrino Experiment (DUNE) is a forthcoming multi-billion-dollar international ex-
+periment aiming to resolve the aforementioned open questions. To successfully achieve
+that goal, DUNE will utilize an intense muon beam, with a near detector located at
+Fermi National Laboratory IL and a far detector ≈ 1300 km away from the neutrino
+source at Sanford Underground Research Facility SD.
+18
+
+Before DUNE starts taking data, a number of other neutrino experiments will
+have already provided constraints for 𝛿𝐶𝑃 and the mass hierarchy [17]. For these
+results to be obtained, different neutrino oscillation experiments utilize a number of
+neutrino sources covering a wide range of energies.
+Reactor neutrino experiments exploit the large electron anti-neutrino fluxes pro-
+duced in nuclear reactors by 𝛽 decays of heavy nuclei (nuclear fissions of 235𝑈, 238𝑈,
+239𝑃𝑢, 241𝑃𝑢). The typical energy scale of reactor 𝜈𝑒’s is a few MeVs.
+Atmospheric neutrino experiments take advantage of neutrinos produced by cos-
+mic rays.
+The latter interact with the upper layers of the atmosphere producing
+a large flux of pions and kaons.
+These decay in the atmosphere into muons and
+muon neutrinos. These muons might decay into electrons, electron anti-neutrinos,
+and muon neutrinos before they reach the Earth. Atmospheric neutrino experiments
+aim to detect these muon neutrinos. The energy of these neutrinos spans a really
+wide range up to 𝒪(100) GeV. In some cases, like in IceCube [18], even neutrinos
+with PeV energies can be detected. Such neutrinos cover distances between 𝒪(10) km
+- for neutrinos produced in the upper layers of the atmosphere directly above the de-
+tector - to 𝒪(104) km - for netrinos that are produced on the other side of the Earth
+and travel through the core. Atmospheric experiments provide the current best limit
+of ∆𝑚2
+32 = 2.56+0.13
+−0.11 × 10−3𝑒𝑉 2 [1].
+Solar neutrino experiments detect the electron neutrinos produced in the Sun core
+due to nuclear fusion processes. Solar neutrino experiments are designed to detect
+the 𝜈𝑒’s produced via these processes. The currently best limit corresponds to an
+extremely small value of ∆𝑚2
+21 = 7.37+0.59
+−0.44 × 10−5𝑒𝑉 2 [1], much smaller than the
+∆𝑚2
+32 splitting mentioned above.
+Accelerator-based experiments use muon neutrino beams produced via the decay of
+primarily pions produced when a proton beam hits a heavy target. Such experiments
+are further classified into appearance and disappearance experiments. The former
+search for electron neutrinos oscillated from the initial muon neutrino beam. The
+latter look for the reduction of muon neutrino interactions due to oscillations.
+Different experiments are designed to be sensitive to different values of ∆𝑚2 by
+19
+
+choosing the appropriate 𝐿/𝐸 ratio. Building on equation 1.9, the value of ∆𝑚2 for
+which
+∆𝑚2𝐿
+2𝐸
+≃ 1
+(1.14)
+corresponds to the ∆𝑚2 sensitivity of a given experiment.
+Neutrino oscillation experiments are further classified depending on the average
+value of this 𝐿/𝐸 ratio. Using equation 1.14, short baseline experiments with 𝐿/𝐸 ≲ 1
+km/GeV are sensitive to ∆𝑚2 ≳ 1 𝑒𝑉 2. Long baseline experiments where 𝐿/𝐸 ≲ 103
+km/GeV are sensitive to ∆𝑚2 ≳ 10−3 𝑒𝑉 2.
+1.3
+Long-Baseline Accelerator-Based Neutrino
+Oscillation Experiments
+Figure 1-3: The experimental setup of an accelerator-based long-baseline neutrino
+oscillation experiment.
+Long-baseline accelerator-based experiments consist of a near detector (ND) posi-
+tioned close to the neutrino source, and a far detector (FD), which is positioned close
+to the oscillation maximum„ as shown in figure 1-3. The number of interactions of
+20
+
+Laboratory 1
+Laboratory 2
+Neutrino
+detector
+Neutrino
+sourceflavor 𝛼 in the near detector (𝑁 𝛼
+𝑁𝐷) is obtained as
+𝑁 𝑁𝐷
+𝛼
+(𝐸𝑟𝑒𝑐𝑜) =
+∫︁
+Φ𝑁𝐷
+𝛼
+(𝐸𝑡𝑟𝑢𝑒)𝜎𝛼(𝐸𝑡𝑟𝑢𝑒)𝜖𝛼(𝐸𝑡𝑟𝑢𝑒)𝑓 𝑁𝐷(𝐸𝑡𝑟𝑢𝑒, 𝐸𝑟𝑒𝑐𝑜)𝑑𝐸𝑡𝑟𝑢𝑒
+(1.15)
+where Φ𝑁𝐷
+𝛼
+(𝐸𝑡𝑟𝑢𝑒) is the neutrino flux of flavor 𝛼 close to the source, 𝜎𝛼(𝐸𝑡𝑟𝑢𝑒)
+is the cross section for a given flavor 𝛼, 𝜖𝛼(𝐸𝑡𝑟𝑢𝑒) is the reconstruction efficiency for
+flavor 𝛼, and 𝑓 𝑁𝐷(𝐸𝑡𝑟𝑢𝑒, 𝐸𝑟𝑒𝑐𝑜) is the detector response function, describing how 𝐸𝑡𝑟𝑢𝑒
+is mapped to 𝐸𝑟𝑒𝑐𝑜 in the near detector. For 𝜈𝛼 → 𝜈𝛽 oscillations, the number of
+events in the far detector can be obtained as
+𝑁 𝐹𝐷
+𝛼→𝛽(𝐸𝑟𝑒𝑐𝑜) =
+∫︁
+Φ𝐹𝐷
+𝛼 (𝐸𝑡𝑟𝑢𝑒)𝑃𝛼→𝛽(𝐸𝑡𝑟𝑢𝑒)𝜎𝛽(𝐸𝑡𝑟𝑢𝑒)𝜖𝛽(𝐸𝑡𝑟𝑢𝑒)𝑓 𝐹𝐷(𝐸𝑡𝑟𝑢𝑒, 𝐸𝑟𝑒𝑐𝑜)𝑑𝐸𝑡𝑟𝑢𝑒
+(1.16)
+where 𝑁 𝐹𝐷
+𝛼→𝛽(𝐸𝑟𝑒𝑐𝑜) is the number of 𝛽 flavor interactions, Φ𝐹𝐷
+𝛼 (𝐸𝑡𝑟𝑢𝑒) is the neu-
+trino flux of flavor 𝛼, 𝑃𝛼→𝛽(𝐸𝑡𝑟𝑢𝑒) is the oscillation probability for 𝛼 → 𝛽, 𝜎𝛽(𝐸𝑡𝑟𝑢𝑒)
+is the cross section for flavor 𝛽, 𝜖𝛽(𝐸𝑡𝑟𝑢𝑒) is the reconstruction efficiency for flavor
+𝛽, and 𝑓 𝐹𝐷(𝐸𝑡𝑟𝑢𝑒, 𝐸𝑟𝑒𝑐𝑜) is the detector response function, describing how 𝐸𝑡𝑟𝑢𝑒 is
+reconstructed in 𝐸𝑟𝑒𝑐𝑜.
+Accelerator-produced neutrino beams predominantly contain muon neutrinos [19].
+Therefore, long-baseline accelerator-based neutrino oscillation experiments focus on
+muon neutrino disappearance and electron neutrino appearance studies. The former
+are sensitive to the oscillation parameters 𝜃23 and ∆𝑚2
+23, while the latter have sensi-
+tivity to 𝜃13 and 𝛿𝐶𝑃 [20]. At a distance 𝐿 from the neutrino production point, some
+muon neutrinos will oscillate to electron neutrinos, resulting in fluxes of approximately
+Φ𝐹𝐷
+𝑒
+(𝐸𝑡𝑟𝑢𝑒)
+∝
+𝑃𝜈𝜇→𝜈𝑒(𝐸𝑡𝑟𝑢𝑒, 𝐿) Φ𝑁𝐷
+𝜇
+(𝐸𝑡𝑟𝑢𝑒),
+(1.17)
+Φ𝐹𝐷
+𝜇 (𝐸𝑡𝑟𝑢𝑒)
+∝
+[︀
+1 − 𝑃𝜈𝜇→𝜈𝑒(𝐸𝑡𝑟𝑢𝑒, 𝐿)
+]︀
+Φ𝑁𝐷
+𝜇
+(𝐸𝑡𝑟𝑢𝑒),
+where the proportionality constant depends on the experiment geometry, that can
+21
+
+be affected by the different experimental acceptances at the ND and FD locations,
+and 𝑃𝜈𝜇→𝜈𝑒 is the electron neutrino appearance probability.
+𝜈𝜇 → 𝜈𝑒 oscillations
+are thus observed by measuring the neutrino fluxes Φ𝐹𝐷
+𝑒
+(𝐸𝑡𝑟𝑢𝑒) and Φ𝐹𝐷
+𝜇 (𝐸𝑡𝑟𝑢𝑒) as
+a function of energy and distance. The three-flavor oscillation equations are similar
+but include additional terms. Charge-parity (CP) symmetry violation in the leptonic
+sector would add a phase 𝛿𝐶𝑃 to the three-flavor oscillation with an opposite sign for
+neutrinos and anti-neutrinos [21,22].
+Therefore, the precision to which oscillation parameters can be determined exper-
+imentally depends on our ability to extract Φ𝐹𝐷
+𝛼 (𝐸) from 𝑁 𝐹𝐷
+𝛼
+(𝐸𝑟𝑒𝑐𝑜), as can be seen
+in equation 1.16 and is graphically illustrated in figure 1-4.
+Figure 1-4: Neutrino energy spectra reconstruction depends on our ability to model
+the interaction of neutrinos with atomic nuclei and the propagation of particles
+through the atomic nucleus.
+This flow chart shows the process, starting with an
+oscillated far-detector incident-energy spectrum (green), differentiating the physical
+neutrino interactions (green arrows) from the experimental analysis (blue arrows),
+and ending up with an inferred incident-energy spectrum that hopefully matches the
+actual one.
+While experimental effects are generally understood and can be minimized us-
+22
+
+PHYSICS PROCESS
+n
+10
+Oscillated v Flux
+8
+元
+6
+4
+元°
+2
+0
+p
+ particles
+E[GeV]
+EXPERIMENTALANALYSISing improved detectors, nuclear effects are irreducible and must be accounted for
+using theoretical models, typically implemented in neutrino event generators. Thus,
+the experimental sensitivity is largely determined by the accuracy of the theoreti-
+cal models used to calculate the interaction cross sections and the accuracy of the
+energy reconstruction.
+The available models have many free parameters that are
+poorly constrained and are “tuned” by each neutrino experiment. Current oscillation
+experiments report significant systematic uncertainties due to these interaction mod-
+els [22–25]. Simulations further show that energy reconstruction errors can lead to
+significant biases in extracting 𝛿𝐶𝑃 at DUNE [26]. There is a robust theoretical effort
+to improve these models [27–29].
+Since there are no mono-energetic high-energy neutrino beams, these models can-
+not be tested for individual neutrino energies. Instead, experiments tune models to
+their near-detector data, where the unoscillated flux Φ𝑁𝐷
+𝛼
+(𝐸𝑡𝑟𝑢𝑒) is calculated from
+hadronic reaction rates [30–32]. While highly informative, such integrated constraints
+are insufficient to ensure that the models are correct for each value of 𝐸𝑡𝑟𝑢𝑒. Thus, for
+precision measurements using a broad-energy neutrino beam, the degree to which the
+near-detector data alone can constrain models is unclear for a number of reasons. In
+some cases, the near and far detectors use different target nuclei, which demands the
+cross-section extrapolation between the two targets. At the same time, the fluxes at
+the two detectors are not identical due to their distance from the source, and the fact
+that neutrinos oscillate. The near detector measures neutrino interactions originat-
+ing from pion decays in a long pipe with a length of 𝒪(50 m), while the far detector
+measures neutrinos from a much smaller solid angle. Thus, the particle acceptance is
+different between the two detectors. Furthermore, the selection efficiencies 𝜖𝛼(𝐸𝑡𝑟𝑢𝑒)
+in equations 1.15 and 1.16, both for the signal and the background events at the two
+detectors, are model dependent and different.
+Yet another challenge for neutrino oscillation experiments is the accuracy of the
+true neutrino energy reconstruction that enters the oscillation probability formula
+shown in equation 1.12. The neutrino energy is reconstructed using the measured en-
+ergy deposition of the final state particles. However, energy losses due to the energy
+23
+
+of neutral particles, particles below thresholds, and inactive detector regions have to
+be taken into account. Inevitably, assumptions about these effects have to be made
+based on the underlying modeling choices and on the detector capabilities. This pro-
+cedure introduces systematic uncertainties that might limit the precise reconstruction
+of the true neutrino energy. Thus, the success of forthcoming neutrino experiments
+like DUNE relies on the accurate identification and reconstruction of all particles
+produced. Hence, tracking detectors with low detection thresholds are key elements.
+Furthermore, neutrinos traveling through matter experience a potential due to
+the coherent elastic scattering with electrons and nucleons. Coherent scattering takes
+place when a neutrino wavefunction interacts with matter as a whole. The implication
+of this behavior is that neutrinos and antineutrinos are affected in different ways, due
+to the lack of positrons in regular matter. This effect can mimic a CP violating picture
+with 𝑃(𝜈𝛼 → 𝜈𝛽) ̸= 𝑃(¯𝜈𝛼 → ¯𝜈𝛽), though it contains no fundamental information
+related to the matter-antimatter asymmetry. Therefore, accounting for matter effects
+in oscillation experiments that aim to extract the oscillation parameters is crucial.
+All these open questions need to be addressed in order to ensure the success of
+forthcoming high-precision neutrino oscillation experiments. Hence, this thesis pro-
+gresses in that direction by improving the understanding of lepton-nucleus interactions
+described in sections 1.4 and 1.5.
+1.4
+Neutrino-Nucleus Interactions
+Neutrino-nucleus interactions are extremely complicated processes that become even
+more complex on heavier nuclei, such as on argon, the target nucleus used in liquid
+argon time projection chambers. These interactions are approximated by deploying
+a two-step approach.
+The first step involves the primary neutrino interaction with a single nucleon or
+a pair of nucleons with the nucleons being treated as quasi-free objects. There are
+four main types of neutrino scattering processes that play a major role in the energy
+spectrum of neutrino experiments like MicroBooNE and DUNE, namely quasielastic
+24
+
+(QE), meson exchange currents (MEC), resonant production (RES), and deep inelas-
+tic scattering (DIS). These four interaction types and the relevant outgoing particles
+are shown in figure 1-5.
+Figure 1-5: The four main interaction processes for neutrino-nucleus scattering events.
+Each one of those processes dominates in different energy ranges, as illustrated in
+figure 1-6. At neutrino energies below ≈ 1 GeV, QE interactions are the ones that
+dominate. With this process, leptons scatter off, and liberate, a single nucleon from
+the target nucleus. In the region around 𝒪(1 GeV), a set of nuclear forces include
+the exchange of virtual mesons between two (or more) nucleons, which is referred to
+as 2 particle-2 hole effects (2p2h). Such an effect has major contributions from MEC
+events that lead to the emission of two nucleons from the primary neutrino interaction
+point. For interaction energies greater than the ∆ baryon mass of 1232 MeV, RES
+interactions become energetically allowed. Such processes are the dominant ones in
+the energy range between 1-4 GeV. With such an interaction type, the struck nucleon
+is brought into an excited state that is called a baryon resonance. The resonance
+deexcites and that leads to the emission of a single nucleon and a single pion in the
+final state. At even higher energies starting at ≈ 4 GeV, the incoming neutrino scatters
+off a quark in the nucleus. Such an energy transfer results in the production of multiple
+hadrons in the final state. Interference terms across the different interaction processes
+need to be added coherently at the quantum-mechanical amplitude level. However,
+commonly used neutrino event generators, such as GENIE [33], do not include such
+interference terms between the different reaction modes. In other words, the total
+cross section is obtained by adding the individual cross sections 𝜎𝑖(𝐸) incoherently.
+For neutrino scattering events, interaction processes can take place in the form of
+either charged current (CC) or neutral current (NC) interactions. The former includes
+25
+
+O
+MEC
+RES
+DISFigure 1-6: Neutrino cross-section evolution as a function of the neutrino energy
+illustrating the energy range where each one of the four main processes dominates.
+Figure adapted from [34].
+the exchange of charged W bosons, and the latter of neutral Z bosons respectively.
+The corresponding Feyman diagrams are shown in figure 1-7.
+Figure 1-7: Feyman diagrams illustrating charged current (CC) and neutral current
+(NC) processes.
+Beyond the possible interactions and processes mentioned above, neutrino-nucleon
+interactions become even more sophisticated due to the fact that the interaction takes
+place in a dense nuclear medium, such as an argon nucleus. These complex nuclear
+effects can be further classified into initial and final state interactions.
+Initial state interactions are associated with the nucleon-nucleon correlations in
+the target nucleus. The nucleons inside the target nucleus are not at rest. Their
+26
+
+1
+TOTAL
+DIS
+cross
+RES
+>
+10-1
+1
+10
+102
+E, (GeV)CC
+NC
+μ
+Vu
+-
+-
+-
+-
+W
+1
+Z
+-
+-
+-
+-
+-
+V
+n
+p
+p
+pFigure 1-8: Nucleon momentum distribution options available in commonly used
+neutrino event generators. Figure adapted from [35].
+momentum distribution can be approximated in a number of ways, typical examples
+of which would be a Bodek-Richie Fermi Gas [36], a Local Fermi Gas [37], or an
+Effective Spectral Function [38], as illustrated in figure 1-8. The precise behavior of
+the Fermi motion is unknown, and it results in the smearing of the reconstructed
+value of the true neutrino energy.
+Figure 1-9: Schematic illustration the wealth of possible final state interactions that
+the final state hadrons can undergo. Figure adapted from [39].
+27
+
+ntranuclearMedium
+HadronAbsorption
+n
+n元
+ChargeExchange
+CCQEInteraction
+Pion Absorption
+b元
+p
+(In)elastic
+Scattering
+PionProduction
+ElasticScattering
+p元8
+Probability / GeV
+- Bodek-Ritchie Fermi Gas
+6
+- Local Fermi Gas
+- Effective Spectral Function
+2
+0.2
+0.4
+0.6
+Momentum [GeV]In the second part of the factorization process, after the primary neutrino-nucleus
+interaction, the outgoing nucleons can undergo a wealth of final state interactions
+while transversing the nuclear medium, before they exit the nucleus. Such re-interactions
+might result in processes such as the emission of further hadrons, the absorption of the
+initially emitted hadrons, charge exchange processes, and/or acceleration/deceleration
+of the primary hadrons, as illustrated in figure 1-9.
+All the effects described above have a direct impact on the ability to accurately
+reconstruct the energy of the incoming neutrino using the properties of the final state
+particles.
+1.5
+Connections To Electron Scattering
+Neutrinos and electrons interact with atomic nuclei by exchanging intermediate vector
+bosons, a massive 𝑊 ± or 𝑍 for the neutrino and a massless photon for the electron.
+Electrons interact via a vector current 𝑗𝜇
+𝐸𝑀 = ¯𝑢𝛾𝜇𝑢 and neutrinos interact via vector
+and axial-vector 𝑗𝜇
+𝐶𝐶 = ¯𝑢𝛾𝜇(1 − 𝛾5)𝑢 −𝑖𝑔𝑊
+2
+√
+2 currents.
+Fundamental considerations give an inclusive (𝑒, 𝑒′) electron-nucleon elastic scat-
+tering cross section that depends on only two structure functions [40],
+𝑑2𝜎𝑒
+𝑑𝑥𝑑𝑄2 = 4𝜋𝛼2
+𝑄4
+[︂1 − 𝑦
+𝑥
+𝐹 𝑒
+2 (𝑥, 𝑄2) + 𝑦2𝐹 𝑒
+1 (𝑥, 𝑄2)
+]︂
+.
+(1.18)
+Here 𝐹 𝑒
+1 and 𝐹 𝑒
+2 are the standard electromagnetic vector structure functions, 𝑄2 =
+q2 − 𝜈2 is the squared momentum transfer and q and 𝜈 are the three-momentum
+and energy transfers, 𝑥 = 𝑄2/(2𝑚𝜈) is the Bjorken scaling variable, 𝑚 is the nucleon
+mass, 𝑦 = 𝜈/𝐸𝑒 is the electron fractional energy loss, and 𝛼 is the fine structure
+constant. This formula shows the simplest case where 𝑄2 ≫ 𝑚2.
+The corresponding inclusive charged-current (CC) (𝜈, 𝑙±) neutrino-nucleon elastic
+cross section has a similar form, where 𝑙± is the outgoing charged lepton [41]. The
+vector part of the current is subject to the same fundamental considerations as above,
+28
+
+but the axial-vector part of the current does not conserve parity. This leads to a third,
+axial, structure function,
+𝑑2𝜎𝜈
+𝑑𝑥𝑑𝑄2 =𝐺2
+𝐹
+2𝜋
+[︂1 − 𝑦
+𝑥
+𝐹 𝜈
+2 (𝑥, 𝑄2) + 𝑦2𝐹 𝜈
+1 (𝑥, 𝑄2)
+−𝑦(1 − 𝑦/2)𝐹 𝜈
+3 (𝑥, 𝑄2)
+]︀
+.
+(1.19)
+Here 𝐹 𝜈
+1 and 𝐹 𝜈
+2 are the parity-conserving neutrino-nucleus vector structure functions,
+𝐹 𝜈
+3 is the axial structure function, and 𝐺𝐹 is the Fermi constant. The vector form
+factors, 𝐹 𝜈
+1 and 𝐹 𝜈
+2 , have both vector-vector and axial-axial contributions.
+These simple equations are very similar for electron-nucleus scattering. In the limit
+of electron-nucleon elastic scattering (𝑥 = 1), the two structure functions reduce to the
+Dirac and Pauli form factors, which are linear combinations of the electric 𝐺𝐸(𝑄2) and
+magnetic 𝐺𝑀(𝑄2) form factors. Neutrino-nucleon elastic scattering has an additional
+axial form factor. In the simplest case where a lepton scatters quasielastically from
+a nucleon in the nucleus and the nucleon does not reinteract as it leaves the nucleus
+shown in figure 1-10, the lepton-nucleus cross section is the integral over all initial
+state nucleons,
+𝑑𝜎
+𝑑𝐸𝑑Ω =
+∫︁
+pi
+∫︁
+𝐸𝑏
+𝑑3pi𝑑𝐸𝑏𝐾𝑆(pi, 𝐸𝑏)𝑑𝜎𝑓𝑟𝑒𝑒
+𝑑Ω
+𝛿3(q − pf − pr)𝛿(𝜔 − 𝐸𝑏 − 𝑇𝑓 − 𝑇𝑟),
+(1.20)
+where pi and pf = q + pi are the initial and final momenta of the struck nucleon
+in the absense of any reinteraction, pr = −pi is the momentum of the recoil 𝐴 − 1
+nucleus, 𝐸𝑏 is the nucleon binding energy, 𝑆(pi, 𝐸𝑏) is the probability of finding a
+nucleon in the nucleus with momentum pi and binding energy 𝐸𝑏, 𝑇𝑓 and 𝑇𝑟 are
+the kinetic energies of the final state nucleon and 𝐴 − 1 system, 𝑑𝜎𝑓𝑟𝑒𝑒/𝑑Ω is the
+lepton-bound nucleon elastic cross section, and 𝐾 is a known kinematic factor.
+This simple picture is complicated by nucleon reinteractions which change the
+29
+
+Figure 1-10: Quasielastic lepton-nucleus scattering where the outgoing nucleon does
+not reinteract as it leaves the nucleus.
+overlap integral between the initial and final states, and thus the cross section. The
+latter further changes the momentum and angle of the outgoing nucleon.
+Thus,
+to calculate even the simplest type of lepton-nucleus interaction, the momentum and
+binding energy distribution of all nucleons in the nucleus need to be known, as well as
+how the outgoing nucleon wave function is distorted by the nucleon-nucleus potential,
+and how the outgoing nucleon kinematics is changed by final state interactions.
+Electron-nucleus scattering is much easier to understand than neutrino-nucleus
+scattering for three reasons: (a) electron beams have a single, well-known, energy; (b)
+electron experiments typically have low statistical uncertainty because electron beams
+have high flux, and electron-nucleus cross sections are far higher than their neutrino
+counterparts; and (c) electron cross sections are purely vector. The strength of the
+interaction is very different: 4𝜋𝛼2/𝑄4 for electrons versus 𝐺2
+𝐹/(2𝜋) for neutrinos,
+where the factor of 1/𝑄4 in the electron cross section is due to the exchanged boson
+mass (i.e. massless photon) in the propagator. When compensating for the factor
+of 1/𝑄4, the shapes of the electron- and neutrino-nucleus cross sections are very
+similar [42]. Nuclear medium effects such as nucleon motion, binding energy, two-
+body currents, and final state interactions are similar or identical. Therefore electron-
+nucleus scattering can be used to constrain models of neutrino-nucleus scattering. Any
+model which fails to accurately describe electron-nucleus (vector-vector) scattering
+data cannot be used with confidence to simulate neutrino-nucleus (vector-vector +
+axial-axial + vector-axial) interactions. Thus, models of the neutrino-nucleus cross
+30
+
+Final StateMuon
+FinalStateProtonsection must be able to describe the more limited electron-nucleus cross section.
+1.6
+Thesis Scope
+As detailed in section 1.3, the success of future precision neutrino oscillation experi-
+ments depends on an unprecedented understanding of neutrino-nucleus interactions.
+Insufficient knowledge of either the energy reconstruction or the cross section will limit
+the experimental precision. In this thesis, the expertise in both neutrino and electron
+cross-section modeling and analysis is leveraged in order to alleviate this insufficient
+knowledge using three complementary approaches, illustrated in figure 1-11.
+Figure 1-11: Graphic illustration of the three complementary approaches used in this
+thesis to improve our understanding of lepton-nucleus interactions.
+Namely, neutrino scattering data sets from the MicroBooNE detector at Fermi
+National Laboratory were analysed (chapter 2). The first measurements of exclusive
+cross sections with a single proton and no pions detected in the final state were
+reported. These results were used to identify regions where modeling improvements
+are required and specific nuclear effects can be studied in detail (chapter 3). The
+exact same event topology was investigated using electron scattering data sets. For
+the connection across neutrinos and electrons to be established, significant modeling
+improvements took place to ensure a consistent modeling across the two particle
+species in the commonly used GENIE event generator (chapter 4). Building on those
+improvements, the “Electrons-For-Neutrinos” analysis reported on the first use of wide
+31
+
+ Neutrino Cross Sections
+Phys. Rev. Lett. 125, 201803 (2020)
+·Event Generator
+·Electron
+Modeling
+Cross Sections
+Phys. Rev. D 103, 113003 (2021)
+Nature 599, 565-570 (2021)phase-space electron scattering data sets from the CLAS detector at Thomas Jefferson
+National Laboratory (chapter 5). The analysis identified significant shortcomings in
+our lepton-nucleus interaction understanding by reporting cross sections as a function
+of energy reconstruction methods and testing the validity of models commonly used
+in neutrino oscillation analyses (chapter 6).
+32
+
+Chapter 2
+The MicroBooNE Experiment At
+Fermi National Laboratory
+2.1
+Fermilab Neutrino Beamlines
+Fermi National Laboratory (Fermilab) takes advantage of artificially produced neu-
+trino beams in order to study neutrino oscillations. The advantage of that choice is
+the better control over the energy spectrum range and the neutrino flavor content.
+Thus, the spectrum can be tuned so that, given the distance of the detector from the
+source, the experiment is located at an oscillation maximum or minimum, depending
+on the objective of the intended measurement [43]. On top of that, a narrower energy
+spectrum can be obtained via the off-axis technique [44].
+The neutrino beam production includes a well-defined procedure that is outlined
+below for the Fermilab beamlines. The Fermilab linac is used to accelerate a proton
+beam up to a kinetic energy of 400 MeV. Then the Booster synchrotron is used to
+further accelerate the protons up to a kinetic energy of 8 GeV.
+For the Booster Neutrino Beam (BNB), a kicker extracts the protons in a single
+turn and those protons are redirected to the relevant target hall. For all the other
+experiments, the protons are redirected to the Main Injector. The latter is the high-
+est energy US-based accelerator facility and the source of neutrinos from the Main
+Injector (NuMI), the source of muons for the muon campus experiments, and the
+33
+
+forthcoming high intensity DUNE neutrino beam. Figure 2-1 illustrates the series of
+the relevant accelerator components and the different beams.
+Figure 2-1: The Fermilab accelerator complex showing the accelerator components
+and the different beams. Figure adapted from [45].
+The BNB takes advantage of the 8 GeV protons from the Booster synchrotron.
+These protons collide with a beryllium target which is located within a pulsed elec-
+tromagnet called horn. Such collisions lead to the production of mesons, mainly 𝜋± ,
+𝐾±, and 𝐾0. The channel that dominates is p + Be → 𝜋+ + X, with X corresponding
+to all the other hadrons produced out of the interaction. When operating in neutrino
+mode, the horn focuses the positively charged particles. The focused particle beam
+then enters the decay pipe via a concrete-made collimator with a length of ≈ 2 m. In
+the decay pipe, some of the particles decay, a process that results in the production
+of neutrinos primarily via the channel 𝜋+ → 𝜇+ + 𝜈𝜇. The massive particles are
+stopped using a beam absorber made of steel and concrete. On the other hand, neu-
+trinos transverse the absorber and, at the end of this process, a neutrino-dominated
+beam is obtained. A graphic illustration of the process is shown in figure 2-2.
+Apart from the desired 𝜈𝜇 beam, neutrinos can also be produced via the decay of
+anti-muons coming out of the proton-target collision and the channel 𝜇+ → 𝑒+ +𝜈𝑒 +
+¯𝜈𝜇 [46]. Such interactions are the main sources of intrinsic 𝜈𝑒 contamination to the
+main 𝜈𝜇 beam. Similar hadron decays result in further contamination of the 𝜈𝜇 beam.
+The main source of the ¯𝜈𝜇 contamination comes from 𝜋−’s which are not separated
+34
+
+Main Injector
+Recycler Ring
+Low-Energy
+Neutrino
+Experiments
+High-Energy
+Neutrino
+Experiments
+Muon
+Delivery
+Booster
+Ring
+Fixed-Target
+Experiments,
+Test Beam
+Facility
+Linac
+Muon
+lon Source
+ExperimentsFigure 2-2: The ingredients for a neutrino beam include the accelerated protons,
+the target, the magnetic horn, the decay pipe, and the absorbers. Figure adapted
+from [45].
+from the main beam by the horn. Figure 2-3 illustrates the BNB energy spectrum
+while operated in neutrino mode.
+Figure 2-3: The BNB neutrino flux prediction through the MicroBooNE detector for
+𝜈𝜇, ¯𝜈𝜇, 𝜈𝑒, and ¯𝜈𝑒. A TPC volume with dimensions 2.56 m × 2.33 m × 10.37 m is
+assumed. Figure adapted from [46].
+The Fermilab-based NuMI neutrino beamline is used by the MINOS, Miner𝜈a,
+ArgoNeuT, NO𝜈A, and MINOS+ experiments [47–53]. On top of that, the Mini-
+35
+
+Neutrino BeamRecipe
+Rock,
+BeamAbsorber
+DecayPipe
+Neutrinos
+MagneticFocusing
+Horn
+Muons,
+Neutrinos
+Target
+PionmKaons
+Protons2
+10
+9
+v/POT/GeV/cm
+MicroBooNE Simulation Preliminary
+10-11
+10-12
+10
+.13
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+4.5
+5.0
+Energy (GeV)BooNE and MicroBooNE experiments use the beamline for off-axis studies. In this
+case, neutrinos are produced via the collision of a 120 GeV beam on a graphite tar-
+get. Due to the large angle with respect to the NuMI beam dump, MicroBooNE
+collects a significant number of low-energy neutrinos originating from kaons decay-
+ing at rest. Using the off-axis NuMI beamline, MicroBooNE records events with a
+narrower energy spectrum and with a higher electron-neutrino contribution.
+2.2
+The MicroBooNE Detector
+The Micro Booster Neutrino Experiment (MicroBooNE) at Fermilab was proposed
+to succeed MiniBooNE in order to resolve the Low Energy Excess (LEE) [54]. It is
+located ≈ 20 m away from the MiniBooNE detector, thus establishing almost the same
+𝐿/𝐸 ratio. MicroBooNE uses the same beamline as MiniBooNE, therefore the two
+experiments share similar flux uncertainties. At the core of the MicroBooNE detector,
+there exists a Liquid Argon Time Projection Chamber (LArTPC). This detector
+technology offers high spacial and momentum resolution, allowing an unprecedented
+energy reconstruction accuracy and a precise particle identification.
+Some neutrinos from the two beamlines enter the MicroBooNE LArTPC and
+interact with the argon nuclei. Out of these interactions, charged and neutral particles
+are produced.
+When the charged particles transverse the liquid argon, the argon
+atoms get excited and they further ionize the medium, a process that results in
+the emission of ionization electrons.
+In the presence of a strong electric field of
+273 V/cm, the ionization electrons drift towards the anode plane, where three wire
+planes are located.
+Apart from the ionization electrons, scintillation light is also
+produced and is collected by 32 photomultiplier tubes (PMTs), which are located
+behind the anode plane. Both the TPC and the light collection system are embedded
+withing a cylindrical 170 ton liquid argon cryostat. Figure 2-4 shows the size of the
+detector.
+The readout electronics are embedded withing the liquid argon in order to sig-
+nificantly reduce the electronic noise. The analogue-to-digital conversion (ADC) and
+36
+
+Figure 2-4: Schematic illustration of the MicroBooNE detector and its dimensions.
+Figure adapted from [55].
+PMT electronics are located outside the cryostat.
+Given the baseline of ≈ 470 m, neutrinos in the two beamlines need to be charac-
+terized by two angles in the MicroBooNE coordinate system. As shown in figure 2-5,
+𝜃 is the angle with respect to the z-direction along the beam direction.
+Further-
+more, 𝜑 defines the orientation with respect to the XY-plane, orthogonal to the beam
+direction.
+Figure 2-5: The MicroBooNE coordinate system.
+A Cosmic Ray Tagger (CRT) system shown in figure 2-6 was installed in 2017 [56]
+to improve on the rejection of cosmics, which constitute the greatest source of back-
+grounds on MicroBooNE. This detector sub-system consists of 73 scintillating modules
+made of interleaved layers of scintillating plastic strips situated on the top, bottom,
+37
+
+2.6 m
+2.3 m
+10.4 mY
+A
+Z
+Xand two sides parallel to the neutrino beam [56]. Based on simulation predictions
+from CORSIKA [57] and GEANT [58], an estimated coverage of 85% is obtained.
+The CRT installation aimed to improve on the identification and rejection of the
+dominant cosmic background. Cosmic muons transversing the CRT result in the pro-
+duction of scintillation light that can be reconstructed as hits on the CRT channels.
+Such hits allow the identification of the cosmic-induced muon tracks with a time pre-
+cision of ≈ 100 ns. The latter offers a complementary way to resolve the x-direction
+ambiguity in the TPC reconstruction.
+Figure 2-6: The design of CRT planes as part of the MicroBooNE detector. Simula-
+tion of cosmic rays crossing the CRT, the brown lines represent possible cosmic ray
+trajectories. There are four CRT planes: top plane, bottom plane, pipe side plane
+and feedthrough side plane. The beam direction is along the z axis. Figure adapted
+from [56].
+2.3
+Liquid Argon Time Projection Chambers
+The TPC technology was introduced in the 1970s by David Nygren [59]. Carlo Rubbia
+designed a LArTPC in 1977 using the same TPC principles as Nygren, but with liquid
+argon instead of gas [60]. Figure 2-7 shows the working principle of a LArTPC. The
+cuboid volume of the LArTPC is filled with ultra pure liquid argon. The presence
+38
+
+Top
+Top
+Pipe Side
+Top
+5.4m
+Pipe Side
+Top
+PipeSide
+Bottom
+.4m
+Extra
+Feedthrough
+Side
+Bottom
+1.4ml
+Botton
+Pipe Side
+Bottom
+Cryostat(wrapping TPC)
+LArTF Pit Ground
+xof a high-voltage cathode on one side of the detector and a grounded anode on the
+other side create a homogeneous electric field.
+Figure 2-7: Working principle of a LArTPC detector. Figure adapted from [61].
+When a neutrino interaction takes place in the TPC, charged particles are pro-
+duced in the final state. Along their propagation path in LAr, such particles excite
+and ionize the argon nuclei, a process that results in the emission of ionization elec-
+trons. In the presence of a strong electric field, these ionization electrons drift towards
+the anode plane. To ensure an electron drift time of 𝒪(ms) before recombination ef-
+fects take place, a very small 𝑂2 contamination of 10 parts per trillion has to be
+guaranteed.
+On the anode plane, there exist three wire planes with a 3 mm spacing where the
+clouds of the arriving ionization electrons create signals. The first two planes corre-
+spond to induction planes and are oriented ±60𝑜 with respect to the vertical axis. In
+order to obtain 3D views of the particle trajectories, at least two planes with different
+orientations are required. The third collection plane removes the ambiguities due to
+dead wires. On top of that, the calorimetric and tracking abilities are improved. Bias
+voltages are applied on the wire planes so that the two induction planes satisfy the
+transparency condition outlined in [62,63]. The condition requires that the drifting
+39
+
+Sense Wires
+V
+Vwire plane waveforms
+Liquid Argon TPC
+Charged Particles
+Cathode
+Plane
+>0
+Edrift
+Y wire plane waveformselectrons pass the two induction planes and are fully captured on the third collection
+plane. The drifting charge induces a bipolar signal on the two induction planes and is
+collected on the third one, where a unipolar signal is produced, as shown in figure 2-8
+(left). The signal area is proportional to the ionization.
+Figure 2-8: (Left) bipolar (U and V induction planes) and unipolar (Y collection
+plane) signal induction on the three MicroBooNE planes. Figure adapted from [63].
+(Right) schematic view of the wire planes. The vertical collection Y wires are shown
+in pink, the induction U wires, angled at +60𝑜 are shown in blue and the induction
+V wires, angled at -60𝑜, are shown in green. Figure adapted from [64].
+The MicroBooNE field cage has a height of 2.3 m, a width of 2.6 m and a length
+along the beam direction of 10.4 m. The liquid argon is kept at a pressure of 1.2 atm,
+a boiling temperature of 89 K, and a resulting density of 1.38 g/cm3. The TPC is
+merged into the liquid argon. The active TPC volume is 86 tons. The cathode is kept
+at -70 kV and the anode is grounded, which results into a homogeneous electric field of
+273 V/cm. That translates into a drift velocity of 1.14 m/ms. Thus, the MicroBooNE
+readout window is 2.3 ms. The induction planes U and V are biased at -110 V and
+the collection V plane is biased at 230 V. The induction planes include 2400 wires and
+the collection plane consists of 3456 wires. The distance between both the different
+planes and the plane wires is 3 mm. Figure 2-8 (right) shows the schematic view of
+the wire planes.
+Within the liquid argon, the wire signals are fed into a front end ASIC. Inter-
+mediate amplifiers further amplify the signal and pass it to the feed-through. Then,
+outside the cryogenic environment, the signal is digitized by readout modules with a
+frequency of 16 Hz. The next step is to downsample the signal to lower frequencies
+40
+
+0.15
+[e-
+sn
+U plane
+0.1
+V plane
+Y plane
+0.05
+0
+-0.05
+50
+-40
+-30
+-20
+-10
+0
+10
+Time [us]of 2 Hz [65].
+There exist three 1.6 ms signal readout windows for each event. These frames
+are further truncated to the range between -0.4 ms to 2.7 ms. The hardware-defined
+trigger 0 time is obtained from the accelerator division. Given the 2.3 ms MicroBooNE
+drift time, the extra buffer of 0.4 ms makes sure that there is enough time to isolate
+the neutrino interaction from the cosmic rays that arrive close to the neutrino beam
+trigger time.
+Figure 2-9: Entry/exit points of cosmic muon tracks with a signal from a muon
+counter located outside of the cryostat. In the absence of space charge effects, the
+points should be located along the TPC boundaries indicated by the dashed lines.
+Figure adapted from [66].
+The ion drift velocity in liquid argon is ≈ 5 mm/s, orders of magnitude smaller
+than the electron one. Thus, the argon ions result in the build-up of positive charge in
+the LArTPC for minutes. On top of that, the continuous interaction of cosmic rays in
+the TPC at ≈ 5 kHz also results in a continuous build-up of positively charged argon
+ions. The existence of this positive charge leads to a distortion of the homogeneous
+electric field within the TPC, as shown in figure 2-9. This distortion is referred to as
+41
+
+MicroBooNE
+y100
+50
+0
+-50
+-100
+_150
+0
+50
+100
+150
+200
+250
+x [cm]“space charge effect” (SCE) [66], which leads to a displacement of the reconstructed
+signal ionization source by up to 𝒪(10 cm).
+2.4
+Scintillation Light
+The scintillation light production and propagation is almost instantaneous and of the
+order of 𝒪(ns). Thus, collecting the emitted scintillation light is crucial to identify
+the time that the event took place and to determine the x-position along the drift
+direction in the TPC. When charged particles transverse the liquid argon, scintillation
+light is produced that results in the deposition of ≈ 104 𝛾 / MeV. This light can be
+produced via two mechanisms, shown in figure 2-10.
+Figure 2-10: The production of scintillation light in liquid argon. A charged particle
+can either excite or ionise the argon. Figure adapted from [6].
+The first mechanism is the self-trapped exciton luminescence. In that case, charged
+particles transverse the liquid argon and they leave some of the argon atoms in an
+excited state called excitons. These states are molecules with another argon atom
+with a short lifetime and are called dimers or excimers. Roughly 65% of them are in
+a singlet state 1Σ𝑢 and the remaining ones are in a triplet state 3Σ𝑢.
+With the second mechanism, the charged particles ionize the argon atoms and
+42
+
+quenching by impurity
+excited state
+singlet
+65%
+scintillation light
+35%
+Ar
+excimer states
+50%
+50%
+absorption by impurity
+ionisation
+recombination
+tripletthat results in the creation of free electrons.
+These electrons recombine with the
+positive argon ions, a process that also creates excited dimers. With this mechanism,
+the probability of creating either singlets or triplets is equal.
+The singlet states result in the emission of fast scintillation light with a decay
+time of ≈ 6 ns. The triplet states result in a slow component and in a decay time of
+≈ 1.5 𝜇s. Both the fast and the slow components have a peak wavelength at 128 nm
+in the Vacuum Ultra-Violet (VUV) region. Both states have an energy minimum
+which is equivalent to a distance between the atoms of ≈ 2.8 Å [67]. The liquid argon
+inter-atom separation is ≈ 4 Å, which is greater than the one that corresponds to
+the excimer energy minimum. Thus, liquid argon is transparent to its own light and,
+therefore, the light can be detected over long distances.
+The second mechanism is related to the recombination luminescence and relies on
+the free electron and argon ion local density. Thus, recombination effects are stronger
+for particles with higher energy deposition per unit length. This implies a particle-
+type dependence of the energy deposition in the detector to the light-equivalent pro-
+duction. That also means that the scintillation light yield depends on the choice for
+the electric field. A strong electric field will reduce the recombination effects and,
+therefore, the recombination-induced luminescence. The amount of scintillation light
+can be further reduced due to impurities that result into quenching and absorption
+effects. With that in mind, the liquid argon purity in MicroBooNE is monitored to
+ensure an absorption length greater than the 2.5 m-long TPC width.
+The MicroBooNE light detection system consists of 32 8-inch Hamamatsu cryo-
+genic PMTs located behind the anode plane, shown in figure 2-11. A Tetra-Phenyl
+Butadiene (TPB) coated acrylic plate is located in front of the PMT. The TPB con-
+verts the 128 nm light argon scintillation light to 425 nm visible light. The tube is
+surrounded by mu-metal to shield the electronics against any magnetic fields. The
+PMTs operate at a voltage of 1300 V and have a gain of 107 [68]. The PMT signal is
+sent to a splitter board, where it gets pre-amplified and digitized. The signals have a
+60 ns rise time, much smaller than the spill windows for the BNB (1.6 𝜇s) and NuMI
+(10 𝜇s) beams.
+43
+
+Figure 2-11: The MicroBooNE light collection system with the 32 PMTs. Figure
+adapted from [69].
+2.5
+Hardware And Software Triggers
+Even when accelerator-induced neutrino beams transverse the MicroBooNE detec-
+tor, it is highly unlikely that any neutrino will interact due to the extremely small
+neutrino-nucleus cross sections. Each BNB bunch includes ≈ 1.2 × 1012 protons and
+the TPC surface area is ≈ 60 × 103 cm2. That yields a neutrino interaction every ≈
+500 bunches [19]. In order to significantly reduce the amount of data that is stored on
+tape, a minimal amount of optical activity is required. Regardless, background events
+with light activity within the beam spill arrival time are still the ones that dominate.
+Figure 2-12 shows the differences between two such events [70]. The first one (left)
+corresponds to pure cosmic activity that arrives in coincidence with the beam spill.
+The second one (right) includes a neutrino interaction of interest, although there is
+still a significant cosmic contamination.
+Cosmic-induced background events are accounted for by recording events when
+the neutrino beam is off. That leads to the existence of two trigger types. The first
+one, referred to as “Beam On”, corresponds to data samples recorded when the beam
+is on and the light system yielded a signal. The second one, referred to as “Beam Off”
+or “External BNB” (ExtBNB), uses the same configuration as in the Beam On case,
+but those data sets are recorded when the beam is off.
+The MicroBooNE readout system is triggered with the arrival of a signal [64]. The
+signal might originate from the BNB/NuMI accelerator clocks. Alternatively, there
+44
+
+Figure 2-12: (Left) cosmic-induced event that was stored because of the coincidence
+of a 1.6 𝜇s accelerator BNB signal and light detected by the PMTs. (Right) neutrino-
+induced event where the light was coming from a neutrino interaction. Figure adapted
+from [70].
+is a function generator in the trigger rack producing pulses at a fixed frequency. The
+latter is used to record Beam Off events. The trigger board sends a signal to all the
+readout crates to start recording data. On top of that, the trigger keeps track of the
+trigger type and the time that the signal was received. The accelerator-based signals
+are produced in couples. The former (early signal) vetos Beam Off triggers just before
+the beam triggers. That process aims to avoid any trigger overlap that might result
+in the reduction of the exposure to the beam. The latter signal is the one used to
+trigger the readout. The MicroBooNE TPC readout is completely unbiased, thus all
+the time ticks are stored and the readout is not zero-suppressed. The PMT readout
+is biased though, with data being stored during specific time intervals determined by
+the “discriminators”.
+There exist two PMT data-taking configurations that differ by their duration and
+by the suppression level. The duration is frequently expressed in tick units, with each
+tick corresponding to 15.625 ns. For the BNB triggering, the beam discriminator
+starts the data recording simultaneously for all PMTs. That happens by replicating
+the trigger signal and by redirecting it to all the PMT boards. The duration of this
+beam window is 23.4 𝜇s or 1500 optical ticks. On average, the neutrino arrival time
+is ≈ 4 𝜇s after the opening of the beam window. The cosmic discriminators span a
+range outside the 23.4 𝜇s window. The ultimate goal of the cosmic discriminators is to
+45
+
+"Cosmic Rays Only"
+"Neutrino+CosmicRays
+A
+1.6μs beam
+2.3 ms drift time
+1.6 μs beam
+2.3 ms drift time
+spill time
+spill timesuppress the amount of data that is recorded over a long time interval, once the signal
+for a triggered event has arrived. The duration of the cosmic window is 4.8 ms and
+spans the range of [-1.6,+3.2] ms. Only waveforms with more than 130 ADC counts
+are stored. That number corresponds to ≈ 7 photo-electrons (PE). The accelerator
+signals are meticulously timed so that neutrinos originating from the 1.6 𝜇s BNB spill
+arrive in MicroBooNE during this time window of 23.4 𝜇s. When the beamline is fully
+operational, that takes place ≈ 5 times/s. The majority of the triggered events based
+on the hardware-driven signal do not include neutrino interactions. Thus, a software
+trigger is further applied to determine whether an event is a neutrino candidate or
+not. This trigger uses PMT optical waveforms and searches for light activity within
+the 1.6 𝜇s BNB beam-spill. That action takes place after the TPC data has been sent
+to the DAQ crates.
+2.6
+Optical Event Reconstruction
+In order to perform a MicroBooNE data analysis, the low-level PMT signals have
+to be converted to high-level data products.
+That is achieved by using the raw
+PMT waveforms as an input quantity and, based on these, flashes are reconstructed.
+Such flashes are indicative of optical activity in the TPC recorded by multiple PMTs
+simultaneously. This light activity is collected by the 32 8-inch PMTs oriented as
+shown in figure 2-13. The neutrino-induced light activity results in a higher PE yield
+compared to the cosmic-induced one, as shown in figure 2-14.
+Figure 2-13: Optically reconstructed flash object recorded by the MicroBooNE PMT
+light collection system. The dark orange regions represent a higher PE yield. Figure
+adapted from [64].
+The PMTs have a low (≈ 20 ADC/PE) and a high (≈ 2 ADC/PE) gain read-
+46
+
+80
+OHO
+CH1
+CH7
+CHB
+CH13
+CH14
+CH19
+CH20
+CH25
+CH26
+60
+40
+20
+CH3
+GHO
+OLHO
+CH15
+CH16
+CH21
+CH22
+CH28
+0
+CH4
+20
+CHG
+LLHO
+CH12
+CH17
+CH18
+CH23
+CH24
+40
+CH3
+60
+80
+-500
+400
+-300
+200
+-100
+100
+200
+300
+400
+500out [64]. The optical reconstruction merges the two streams into a corrected waveform
+which corrects saturated high-gain pulses based on the information obtained from the
+low-gain pulses. Depending on the discriminator type, a different baseline estimation
+is used.
+The cosmic discriminator uses a constant baseline, called pedestal.
+The
+beam discriminator uses a time dependent baseline estimate. This is a more accurate
+estimation and addresses potential overestimations or underestimations in the signal
+baseline. Once the pedestal has been identified, the waveform ADC counts are used.
+Those pulses above threshold are identified and propagated to the next stage for the
+“hit” reconstruction. These pulses result in the creation of data products based on this
+optical hit reconstruction. Each optical hit refers to optical PMT activity, namely
+the number of produced PEs and the event time. The optical hits are clustered and
+the PMT PE production is summed in order to reconstruct flashes [71].
+Figure 2-14: Neutrino-induced tracks (black) are matched to the corresponding light
+signals collected by PMTs (red circles) and are clearly separated from the cosmic-
+induced ones (dimmed color tracks). Figure adapted from [72].
+2.7
+TPC Signal Processing
+While drifting towards the anode plane, the ionization electrons repel each other.
+That results in a diffused signal arriving on the anode plane and the level of the
+diffusion depends on the position that the interaction took place.
+Those drifting
+electrons result in the induction of current on the neighboring wires.
+Therefore,
+waveforms produced on a specific wire might have an effect on those produced on a
+47
+
+neighboring wire. The current induced on the wires is amplified and shaped by the
+ASICs located within the liquid argon.
+The objective of the noise filtering and signal processing on MicroBooNE is to
+convert those raw digitized waveforms into the number of ionization electrons passing
+through a specific wire plane at a given time [73,74]. For that to be achieved, the first
+step is the application of noise filters to remove the external noise and the electronics-
+induced one. The major sources of external noise originate from the TPC drift high-
+voltage power supply and the low-voltage regulators for the front-end ASICs [63].
+Then, the application of a deconvolution of the digitized TPC wire signals follows.
+That takes place in two dimensions, with the first one being over time and the second
+one being the effect across multiple wires. A region of interest (ROI) is identified
+based on the deconvoluted charge distribution. The ionization charge is obtained
+with a linear baseline subtraction within the start/end bins of the ROI window. At
+the final step, the processed signals accounting for the number of electrons on a given
+wire at a certain time are used as input to the MicroBooNE event reconstruction.
+The signals are calibrated before the conversion to deposited energy is performed.
+Figure 2-15 illustrates the necessity of signal processing before any reconstruction is
+enabled.
+2.8
+Pandora Reconstruction Framework
+The processed signals from the previous stage are used as the input for higher level re-
+construction of objects such as vertices, tracks, and showers. In order to reconstruct
+objects in the TPC, the Pandora reconstruction framework is deployed [75]. Pan-
+dora uses pattern-recognition algorithms, along with the use of multiple algorithms
+completing specific tasks for a given topology. The waveforms obtained with 4.8 ms
+window are used as an input to the reconstruction framework.
+The first step includes fitting the processed waveforms with Gaussian distributions
+to each peak. This fitting process results into the creation of a 2D hit. Then, Pandora-
+Cosmic is a track-focused selection that aims to tag the cosmic muons. The selection
+48
+
+Figure 2-15: Candidate neutrino event display from MicroBooNE data on one of
+the induction planes.
+(a) The raw waveform image.
+(b) The image after noise-
+filtering. (c) The image after 2D deconvolution. The image quality near the neutrino
+interaction vertex significantly improves after the 2D deconvolution and the latter
+leads to improvements in the pattern recognition. Figure adapted from [74].
+results in the creation of a cosmic-free hit collection. PandoraNu identifies neutrino
+interaction vertices and uses them to reconstruct tracks and showers originating from
+the vertex. A parent neutrino particle is defined and the reconstructed objects are
+added as daughter particles. Creating a “slice” is the first step that Pandora performs.
+A slice is defined as a collection of reconstructed particles originating from the same
+interaction. For the creation of the slices, the PandoraCosmic algorithm is first ran
+over all the hits in order to identify the cosmic-induced muon tracks and the associated
+𝛿-rays and Michel electrons under a cosmic hypothesis. The obvious cosmic activity
+of through-going muon tracks is identified based on the geometric information. The
+remaining hits are used by the PandoraNu algorithm and objects are reconstructed
+under the neutrino hypothesis, creating a slice. Each slice is reconstructed using both
+49
+
+Raw
+A.U.
+AfterNoiseFiltering
+After2-DDeconvolution
+750(a)
+40000
+(b)
+(c)
+MicroBooNE
+30000
+600
+20000
+10000
+[sr
+450
+m
+Time
+0
+300
+-10000
+-20000
+150
+-30000
+40000
+0
+0
+20
+40
+60
+80
+0
+20
+40
+60
+80
+0
+20
+40
+60
+80
+Wire[3mmspacing]hypotheses. For the interactions to be reconstructed in a three-dimensional space,
+Pandora requires information from at least two wire planes [75]. The 2D hits are
+clustered on each wire plane and for each slice. A collection of 3D candidate vertices
+is produced by identifying locations that project to the same points of the 2D clus-
+ters. All the candidate vertices are propagated into a Support Vector Machine (SVM)
+selection and the candidate with the highest score is isolated. The cluster matching
+algorithms are ran around this candidate vertex on each plane and are compared to
+improve the matching of the reconstructed objects [75]. With this process, a collec-
+tion of reconstructed Particle Flow Particles (PFParticles) is constructed. Such a
+PFParticle is created by combining 2D cluster objects on the three wire planes. Each
+one of those PFParticles is associated with a vertex location and has a collection of 3D
+points, which contain the charge details from the relevant 2D hits. These 3D points
+are referred to as SpacePoints. The reconstructed PFParticles in the neutrino slice
+are organized with a hierarchical structure based on parent-daughter assocciations,
+as shown in figure 2-16.
+Figure 2-16: Illustration of the hierarchical structure of particles reconstructed for a
+simulated charged current 𝜈𝜇 event in MicroBooNE. The interaction includes a muon,
+proton and charged pion in the visible final state. Figure adapted from [75].
+A candidate neutrino PFParticle is assigned at the very top of the hierarchy
+by the PandoraNu algorithm. That candidate neutrino will be having at least one
+50
+
+Track (p), daughter of primary p
+Track (p),primary daughter ofvμ
+Shower (e+),daughterofprimary
+Track (π+),primary daughter of vμ
+w,wireposition
+Parent Vμ
+interactionvertex
+Track (μ), primary daughter of vμ
+x, drift positiondaughter PFParticle. These daughter particles are assigned a score that classifies
+them as either-track-like or shower-like objects. A Support Vector Machine uses the
+collection of hits to determine the nature of reconstructed object. Track-like objects
+have a score close to 1 and shower-like objects score closer to 0. Based on that score,
+a shower- or a track-like data product is constructed for each particle.
+In the case of a track-like classification, Pandora uses a linear fit, described in detail
+in [75], and returns the direction and the position of each point across the particle
+trajectory in 3D. For each one of the points, the charge deposition dQ/dx and the
+residual range - the distance from the end of the track - are stored. This approach
+allows accurate measurements of dx that might include deflections and displacements
+due to space charge effects. For track-like objects, dQ/dx is converted to dE/dx using
+the inverse Modified Box model [76], as shown in figure 2-17. The advantage of this
+model is that the non-linear dependence of the local density of ions is taken into
+consideration.
+Figure 2-17: Illustration of the measured dQ/dx vs dE/dx distribution with the mod-
+ified recombination model in the MC simulation with the ArgoNeuT parametrization.
+Figure adapted from [76].
+51
+
+MicroBooNE Simulation
+dQ/dx (e/cm)
+200X103
+180
+100(
+160
+140
+800
+120
+600
+100
+80
+400
+60
+40
+200
+20
+T
+Modified box with ArgoNeuT parametrization
+2
+4
+6
+10
+12
+14161820
+0
+8
+dE/dx (MeV/cm)For shower-like objects, Pandora creates a 3D cone along the hit collection with
+a fixed 3D orientation, solid angle and length. The shower energy is obtained and
+calibrated using the same techniques as the ones used on the 𝜋0 reconstruction [77].
+Furthermore, showers are fitted with a Kalman filter [78]. With this fit, hits that are
+longitudinally or transversely displaced from the main shower cone are removed. This
+fitting process returns a track-like object. Therefore, the calorimetric tools mentioned
+in the previous paragraph become available for shower-like objects too.
+2.9
+Cosmic Overlay Simulation
+For the purposes of the oscillation and cross section analyses on MicroBooNE, neutrino-
+induced interactions in our detector need to be induced. However, MicroBooNE is
+a surface detector dominated by the cosmic activity. In order to simulate this cos-
+mic contamination as accurately as possible, real cosmic events collected with the
+unbiased trigger are used. This trigger stores events outside the beam-related trig-
+ger windows and does not demand the existence of any optical activity in a specific
+part of the detector. Such events are overlaid on top of GENIE simulated neutrino
+interactions [79]. These resulting samples are referred to as “cosmic overlays” and an
+example event display is shown in figure 2-18.
+Figure 2-18: MicroBooNE event display of an event in a cosmic overlay sample. A
+simulated neutrino event (orange box) is overlaid on top of cosmic events (blue box).
+52
+
+μBooNE
+BNBRun1Simulation
+5cm
+Run 7021Subrun 857Event 42856For the simulated part of the cosmic overlays, the reconstructed information is
+“backtracked” to the underlying truth-level information. That is achieved by associ-
+ating the hits on each plane to the GEANT particles that resulted in the production
+of these hits. Thus, it is feasible to relate reconstructed PFParrticles to the under-
+lying simulated interaction products. It is further possible to identify the amount of
+charge originating from the cosmic part of the overlay samples. Reconstructed track-
+and shower-like objects are matched to a simulated object when they have more hits
+in common than any of the other simulated particles or cosmic tracks in a given event.
+Motivated by the analyzer’s involvement in the development and validation of
+these samples, the analyses presented in sections 3.1 and 3.2 were the first ones
+to adopt the aforementioned overlay technique.
+Due to the success in accurately
+describing the cosmic background, this technique is currently used as the default
+simulation option across the MicroBooNE collaboration.
+53
+
+54
+
+Chapter 3
+MicroBooNE Quasielastic-like
+Cross-Section Results
+[Phys. Rev. Lett. 125, 201803 (2020)]
+3.1
+First Measurement of Differential Charged Cur-
+rent Quasielastic-like Scattering Cross Sections
+3.1.1
+Quasielastic-like Neutrino Data Analysis
+As outlined in section 1.4, understanding the interaction of neutrinos with argon nuclei
+is of particular importance as a growing number of neutrino oscillation experiments
+employ liquid argon time projector chamber (LArTPC) neutrino detectors. Exper-
+imentally, the energy of interacting neutrinos is determined from the measured mo-
+menta of particles that are emitted following the neutrino interaction in the detector.
+Many accelerator-based oscillation studies focus on measurements of charged-current
+(CC) neutrino-nucleon quasielastic (QE) scattering interactions [80–89], where the
+neutrino removes a single intact nucleon from the nucleus without producing any
+additional particles.
+This choice is guided by the fact that CCQE reactions can
+be reasonably well approximated as two-body interactions, and their experimental
+signature of a correlated muon-proton pair is relatively straightforward to measure.
+Therefore, precise measurements of CCQE processes are expected to allow precise
+reconstruction of neutrino energies with discovery-level accuracy [90].
+55
+
+A working definition for identifying CCQE interactions in experimental measure-
+ments requires the identification of a neutrino interaction vertex with an outgoing
+lepton, exactly one outgoing proton, and no additional particles. These events are
+referred to herein as “CCQE-like”.
+This definition can include contributions from
+non-CCQE interactions that lead to the production of additional particles that are
+absent from the final state due to nuclear effects, such as pion absorption, or have
+momenta that are below the experimental detection threshold. Pre-existing data on
+neutrino CCQE-like interactions came from experiments using various energies and
+target nuclei [91]. These primarily included measurements of CCQE-like muon neu-
+trino (𝜈𝜇) cross sections for interactions where a muon and no pions were detected,
+with [86–89] and without [80–85] requiring the additional detection of a proton in
+the final state. While most relevant for LArTPC based oscillation experiments, no
+measurements of CCQE-like cross sections on argon with the detection of a proton in
+the final state existed until 2021.
+This analysis presents the first measurement of exclusive CCQE-like neutrino-
+argon interaction cross sections, measured using the MicroBooNE LArTPC [92]. Our
+data serve as the first study of exclusive CCQE-like differential cross sections on ar-
+gon as well as a benchmark for theoretical models of 𝜈𝜇-Ar interactions, which are
+key for performing a precise extraction of oscillation parameters by future LArTPC
+oscillation experiments. We focused on a specific subset of CCQE-like interactions, de-
+noted here as CC1p0𝜋, where the contribution of CCQE interactions is enhanced [55].
+These include charged-current 𝜈𝜇-Ar scattering events with a detected muon and ex-
+actly one proton, with momenta greater than 100 MeV/𝑐 and 300 MeV/𝑐, respectively.
+The measured muon-proton pairs were required to be co-planar with small missing
+transverse momentum and minimal residual activity near the interaction vertex that
+was not associated with the measured muon or proton. For these CC1p0𝜋 events,
+the flux-integrated 𝜈𝜇-Ar total and differential cross sections in muon and proton mo-
+mentum and angle were extracted. The relevant cross sections were further reported
+as a function of the calorimetric measured energy and the reconstructed momentum
+transfer.
+56
+
+The measurement used data from the MicroBooNE LArTPC detector [93], which
+is the first of a series of LArTPCs to be used for precision oscillation measure-
+ments [94–99]. As described in chapter 2, the MicroBooNE detector has an active
+mass of 85 tons and is located along the Booster Neutrino Beam (BNB) at Fermilab,
+463 m downstream from the target. The BNB energy spectrum extends to 2 GeV and
+peaks around 0.7 GeV [19]. A neutrino is detected by its interaction with an argon
+nucleus in the LArTPC. The secondary charged particles produced in the interac-
+tion travel through the liquid argon, leaving a trail of ionization electrons that drift
+horizontally and transverse to the neutrino beam direction in an electric field of 273
+V/cm, to a system of three anode wire planes located 2.5 m from the cathode plane
+detailed in section 2.3. The Pandora tracking package [75] described in section 2.8 is
+used to form individual particle tracks from the measured ionization signals. Particle
+momenta are determined from the measured track length for protons and multiple
+Coulomb scattering pattern for muons [100].
+The analysis presented here was performed on data collected from the BNB beam,
+with an exposure of 4.59 × 1019 protons on target (POT). At nominal running con-
+ditions, one neutrino interaction is expected in ≈ 500 BNB beam spills. The trigger,
+based on the scintillation light detected by the 32 photomultiplier tubes (PMTs),
+increases the fraction of recorded spills with a neutrino interaction to ≈ 10%. Appli-
+cation of additional software selection further rejects background events, mostly from
+cosmic muons, to provide a sample that contains a neutrino interaction in ≈ 15% of
+the selected spills [74,101]. A CCQE-like event selection, further cosmic rejection and
+neutrino-induced background rejection cuts, described in detail in [55], are applied.
+Muon-proton pair candidates are identified by requiring two tracks with a common
+vertex and an energy deposition profile consistent with a proton and a muon [102].
+Further cuts on the track pair opening angle (|∆𝜃𝜇,𝑝 − 90∘| < 55∘) and the muon and
+proton track lengths (𝑙𝜇 > 𝑙𝑝) reduce the cosmic background rate to less than 1% [55].
+The selected CC1p0𝜋 event definition includes events with any number of protons
+with momenta below 300 MeV/𝑐, neutrons at any momenta, and charged pions with
+momentum lower than 70 MeV/𝑐. The minimal proton momentum requirement of
+57
+
+300 MeV/𝑐 is guided by its stopping range in LAr and corresponds to five wire pitches
+in the TPC, to ensure an efficient particle identification.
+To avoid contributions from cosmic tracks, our CC1p0𝜋 selection considered only
+pairs of tracks with a fully-contained proton candidate, and a fully or partially con-
+tained muon candidate in the fiducial volume of the MicroBooNE detector.
+The
+fiducial volume is defined by 3 < 𝑥 < 253 cm, -110 < 𝑦 < 110 cm, and 5 < 𝑧 <
+1031 cm. The 𝑥 axis points along the negative drift direction with 0 cm placed at
+the anode plane, 𝑦 points vertically upward with 0 cm at the center of the detector,
+and 𝑧 points along the direction of the beam, with 0 cm at the upstream edge of the
+detector. Tracks are fully contained if both the start point and end point are within
+this volume, and partially contained if only the start point is within this volume.
+We limited our analysis to a phase space region where the detector response to
+our signal is well understood and its effective detection efficiency is higher than 2.5%.
+This corresponds to 0.1 < 𝑝𝜇 < 1.5 GeV/𝑐, 0.3 < 𝑝𝑝 < 1.0 GeV/𝑐, −0.75 < cos 𝜃𝜇 <
+0.95, and cos 𝜃𝑝 > 0.15. Additional kinematical selections were used to enhance the
+contribution of CCQE interactions in our CC1p0𝜋 sample. These include requiring
+that the measured muon-proton pairs be coplanar (|∆𝜑𝜇,𝑝−180∘| < 35∘) relative to the
+beam axis, have small missing transverse momentum relative to the beam direction
+(𝑝𝑇 = |⃗𝑝 𝜇
+𝑇 + ⃗𝑝 𝑝
+𝑇| < 350 MeV/𝑐), and have a small energy deposition around the
+interaction vertex that is not associated with the muon or proton tracks. This event
+selection results in a CCQE dominated sample, where table 3.1 shows the fractional
+contribution for each interaction channel. Figure 3-1 shows the relevant interaction
+breakdown for the entire sample of selected events as a function of cos𝜃𝜇. The same
+nominal MC sample was also used to compute the purity.
+After the application of the event selection requirement on the data sample, we
+retained 410 CC1p0𝜋 candidate events. It is estimated that our CC1p0𝜋 CCQE-
+like event selection purity equals ≈ 84% [55], with ≈ 81% of the measured events
+originating from an underlying CCQE interaction as defined by the GENIE event
+generator. The efficiency for detecting CC1p0𝜋 events, out of all generated CC1p0𝜋
+with an interaction vertex within our fiducial volume, was estimated using our Monte
+58
+
+Carlo (MC) simulation and equals ≈ 20% [55]. We note that this efficiency includes
+acceptance effects, as the typical LArTPC efficiency for reconstructing a contained
+high-momentum proton or muon track is grater than ≈ 90% [75].
+Table 3.1: Interaction breakdown after the application of our selection cuts.
+Interaction
+Mode
+Fractional
+Contribution (%)
+QE
+81.1
+MEC
+10.9
+RES
+6.6
+DIS
+1.4
+0.6
+0.4
+0.2 0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+cos(
+)
+0
+500
+1000
+1500
+2000
+2500
+3000
+Event Count
+QE
+MEC
+RES
+DIS
+Figure 3-1: Interaction breakdown of the cos𝜃𝜇 plot illustrating the dominance of
+CCQE interactions after the application of our selection cuts.
+Single differential cross sections are reported in measured proton and muon kine-
+matics. The differential cross section is given by:
+d𝜎
+d𝑋𝑛
+=
+𝑁 on
+𝑛 − 𝑁 off
+𝑛 − 𝐵𝑛
+𝜖𝑛 · Φ𝜈 · 𝑁target · ∆𝑝
+𝑛,
+(3.1)
+59
+
+where 𝑋 stands for the kinematical variable that the cross section is differential
+in and 𝑛 marks the cross-section bin. In each bin 𝑛, 𝑁 on
+𝑛 is the number of measured
+events when the beam is on, 𝑁 off
+𝑛 is the number of measured events when the beam is
+off and cosmic-induced background events are collected, 𝐵𝑛 is the non-CC1p0𝜋 beam-
+related background estimated from MC, 𝑁target is the number of scattering nuclei, Φ𝜈
+is the integrated incoming neutrino flux, ∆𝜇
+𝑛 and ∆𝑝
+𝑛 are the differential bin widths,
+and 𝜖𝑛 is the effective particle detection efficiency. We note that the high cos(𝜃𝜇)
+bin has large beam-related background corresponding 𝐵𝑛 in equation 3.1, which is
+estimated using the GENIE v2.12.2 based MC simulation and is presented in figure 3-
+2.
+0.6
+0.4
+0.2 0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+cos(
+)
+0
+50
+100
+150
+200
+250
+300
+350
+NonCC1p Bkg Event Count
+QE
+MEC
+RES
+DIS
+Figure 3-2: Interaction breakdown of the cos𝜃𝜇 plot illustrating the dominance of RES
+interactions after the application of our selection cuts for the non-CC1p0𝜋 background
+part of the MC sample.
+As the detection efficiency is a multidimensional function of the interaction vertex
+and the particle momentum and direction, the data were binned in three-dimensional
+momentum, in-plane, and out-of-place angle bins with the effective detection efficiency
+calculated for each such bin separately and integrated over the interaction vertex in
+the detector. The efficiency was extracted based on simulation and is defined as the
+60
+
+ratio of the number of reconstructed CC1p0𝜋 events to the number of true generated
+CC1p0𝜋 events with a vertex inside our fiducial volume in bin n. This procedure
+accounts for bin migration effects such that cross-sections are obtained as a function
+of true kinematical variables, as opposed to experimentally reconstructed ones.
+The proton and muon efficiencies were extracted independently of each other, such
+that, when the cross-section is differential in muon kinematics, the proton kinematics
+is integrated over and vise versa. This is done due to the limited data and simulation
+statistics and is justified since the proton and muon efficiencies are largely indepen-
+dent in the region of interest. The effect of residual correlations is accounted for in the
+systematic uncertainties. We further note that the missing transverse momentum re-
+quirement increases the sensitivity of our efficiency corrections to the meson exchange
+current (MEC) and final state interaction (FSI) models used in our simulations. We
+accounted for the model sensitivity in our systematic studies detailed below.
+The neutrino flux was predicted using the flux simulation of the MiniBooNE col-
+laboration that used the same beamline [82]. We accounted for the small distance
+between MiniBooNE and MicroBooNE. Neutrino cross section modeling uncertainties
+were estimated using the GENIE framework of event reweighting [103,104] with its
+standard reweighting parameters. For both cross section and flux systematics, we
+use a multisim technique [105], which consists of generating many MC replicas, each
+one called a “universe”, where model parameters are varied within their uncertainties.
+Each universe represents a different reweighting. The simultaneous reweighting of all
+model parameters allows the correct treatment of their correlations.
+A different model is followed for detector model systematic uncertainties, that
+are dominated by individual detector parameters. Unisim samples [105] were gener-
+ated, where one detector parameter was varied each time by 1𝜎. We then examined
+the impact of each parameter variation on the extracted cross sections, by obtain-
+ing the differences with respect to the central value on a bin-by-bin basis. We note
+that the detection efficiency used for the cross section extraction is re-evaluated for
+each variation separately, including bin migration corrections. This procedure there-
+fore accounted for the systematic uncertainty in these corrections due to both the
+61
+
+cross-section and detector response modeling. We then defined the total detector
+1𝜎 systematic uncertainty by summing in quadrature the effect of each individual
+variation.
+A dedicated MC simulation was used to estimate possible background from events
+in which a neutrino interacts outside the MicroBooNE cryostat but produce parti-
+cles that enter the TPC and pass the event selection cuts [85]. No such events were
+found in that study, which is also supported by our observation that the 𝑧-vertex
+distributions for the measured events follows a uniform distribution, as can be seen
+in figure 3-3. The measured 𝑧-vertex distribution, after the beam related MC back-
+ground has been subtracted, does not show an excess at low-𝑧, which indicates that
+background events from interactions upstream of the detector are not accidentally
+entering our selection, which would show up as a small-𝑧 enhancement in our vertex
+distribution. The deficit at 𝑧 = 700 cm is due to dead wires in our detector and its
+effect has been incorporated in our simulation.
+0
+200
+400
+600
+800
+1000
+vertex z [cm]
+0
+10
+20
+30
+40
+Beam On Events
+0
+200
+400
+600
+800
+1000
+vertex z [cm]
+0
+25
+50
+75
+100
+125
+150
+175
+Efficiency Corrected Beam On Events
+Figure 3-3: Vertex 𝑧 distribution for the measured events, after the beam related MC
+background has been subtracted, before (left) and after (right) detection efficiency
+corrections. No small-𝑧 enhancement is observed and, with efficiency corrections, the
+measured distribution is consistent with that of a uniform neutrino interaction vertex.
+The MC simulation used to estimate the backgrounds and effective efficiency con-
+tains real cosmic data overlayed onto a neutrino interaction simulation that uses
+GENIE [103,104] to simulate both the signal events and the beam backgrounds [55].
+For the simulated portion, the particle propagation is based on GEANT4 [58], while
+62
+
+the simulation of the MicroBooNE detector is performed in the LArSoft frame-
+work [106,107]. The beam-related background subtracted from the candidate CC1p0𝜋
+events in the data sample is simulated.
+3.1.2
+Quasielastic-like Cross-Section Results
+Figure 3-4 shows the flux integrated single differential CC1p0𝜋 cross section as a func-
+tion of the cosine of the measured muon scattering angle. The data were compared to
+several theoretical calculations and to our GENIE-based MC prediction. This predic-
+tion is the result of analyzing a sample of MC events produced using our “nominal”
+GENIE model and propagated through the full detector simulation in the same way as
+data. This model (GENIE v2.12.2) [103,104] treats the nucleus as a the Bodek-Ritchie
+Fermi Gas (RFG), used the Llewellyn-Smith CCQE scattering prescription [108], the
+empirical MEC model [109], the Rein-Sehgal resonance (RES) model, the coherent
+(COH) scattering model [110], and a data driven FSI model denoted as “hA” [111].
+In addition, theoretical predictions by several other event generators are shown
+at the cross-section level without any detector effects [112]. These include GENIE
+v2.12.2 and v3.0.6 [103, 104], NuWro 19.02.1 [113], and NEUT v5.4.0 [114].
+The
+agreement between the “nominal” GENIE calculation (v2.12.2) and the MC predic-
+tion constitutes a closure test for our analysis. The other generators all improve on
+GENIE v2.12.2 by using updated nuclear interaction models, among which is the use
+of a Local Fermi Gas model (LFG) [37] and Random Phase Approximation (RPA)
+correction [115]. GENIE v3.0.6 and NEUT also include Coulomb corrections for the
+outgoing muon [116]. The theoretical models implemented in these event generators
+include free parameters that are typically fit to data, with different generators using
+different data sets. We also consider the GiBUU 2019 [117] event generator which
+fundamentally differs from the others due to its use of a transport equation approach.
+A brief discussion of the underlying model configuration used in the different event
+generator predictions included in this analysis is shown below.
+• GENIE Nominal: Uses the aforementioned GENIE v2.12.2 modeling.
+63
+
+0.7
+0.4
+0.0
+0.4
+0.8
+1.0
+cos(
+)
+0
+3
+6
+9
+12
+15
+18
+d
+dcos(
+) [10 38cm2
+Ar ]
+Simulation
+MicroBooNE Data   4.59×1019 POT
+MC
+ 
+GENIE Nominal
+GENIE v3.0.6
+NuWro 19.02.1
+NEUT  v5.4.0
+GiBUU 2019
+Figure 3-4: The flux integrated single differential CC1p0𝜋 cross sections as a function
+of the cosine of the measured muon scattering angle. Inner and outer error bars show
+the statistical and total (statistical and systematic) uncertainty at the 1𝜎, or 68%,
+confidence level. Colored lines show the results of theoretical absolute cross section
+calculations using different event generators (without passing through a detector sim-
+ulation). The blue band shows the extracted cross section obtained from analyzing
+MC events propagated through our full detector simulation. The width of the band
+denotes the simulation statistical uncertainty.
+• NuWro 19.02.1 [113]: Using the LFG ground state model [37], the Llewellyn-
+Smith CCQE scattering prescription [108], the Transverse Enhancement model
+for two–body currents [118], the Adler-Rarita-Schwinger formalism to calculate
+the ∆ resonance explicitly [119], the BS COH [120] scattering model and an
+intranuclear cascade model for FSI.
+• NEUT v5.4.0 [114]: Using the LFG ground state model [37], the Nieves CCQE
+scattering prescription [121], the Nieves MEC model [122], the BS RES [123–
+126] and Rein-Sehgal COH [110] scattering models, and FSI with Oset medium
+correction for pions [103,104].
+• GiBUU 2019: Using somewhat similar models, but unlike other generators,
+those are implemented in a coherent way, by solving the Boltzmann-Uehling-
+64
+
+Uhlenbeck transport equation [117].
+The models include: Local Fermi Gas
+model [37], standard CCQE expression [127], empirical MEC model and a dedi-
+cated spin dependent resonance amplitude calculation following the MAID anal-
+ysis [128]. The DIS model is as in PYTHIA [129] and the FSI treatment is
+different as the hadrons propagate through the residual nucleus in a nuclear
+potential which is consistent with the initial state.
+As can be seen in figure 3-4, all models are in overall good agreement with our
+data, except for the highest cos(𝜃𝜇) bin with cos(𝜃𝜇)> 0.8, where the measured cross
+section is significantly lower than the theoretical predictions. This discrepancy cannot
+be explained by the systematic uncertainties and is therefore indicative of an issue
+with the theoretical models. Specifically, high cos(𝜃𝜇) correspond to low momentum
+transfer events which were previously observed to not be well reproduced by theory
+in inclusive reactions [84,85] and is now also seen in exclusive reactions.
+As the differential cross sections in proton kinematics and muon momentum in-
+clude contributions from all muon scattering angles, their agreement with the the-
+oretical calculation is affected by this disagreement in the forward muon scattering
+angle. Therefore, for the results presented below, we repeated the cross-section ex-
+traction exercise twice, where the first time we included all the events that satisfy
+our selection criteria, and the second time we excluded those events with cos(𝜃𝜇)>
+0.8. The corresponding integrated measured CC1p0𝜋 cross sections are summarized
+in table 3.2. The same table also lists the 𝜒2 for the agreement of the different models
+with the data for differential cross sections for the full available phase-space and for
+cos(𝜃𝜇) < 0.8. The values reported in the table are the simple sum of those 𝜒2 values
+obtained for each distribution separately. Systematic uncertainties and correlations
+were accounted for using covariance matrices.
+As can be seen in table 3.2, GENIE v3.0.6 is the only model that reaches a 𝜒2/d.o.f.
+close to unity for the full phase-space. It is also the closest model to the data at the
+highest cos(𝜃𝜇) bin. For all other models, the 𝜒2/d.o.f. in the cos(𝜃𝜇) < 0.8 sample
+is reduced by a factor of ∼ 2 as compared to the full phase-space sample. GENIE
+v3.0.6 shows a smaller reduction in this case, and GiBUU 2019 obtains a consistently
+65
+
+Table 3.2: Integrated cross section values and 𝜒2 values for the agreement between
+the measured cross sections and various event generators. Results are listed for the
+full measured phase space and for a limited one of cos(𝜃𝜇) < 0.8.
+Integrated Cross Section [10−38cm2]
+(Differential Cross Section 𝜒2/d.o.f)
+−0.75 < cos(𝜃𝜇) < 0.95
+−0.75 < cos(𝜃𝜇) < 0.8
+Data CC1𝑝0𝜋
+4.93 ± 1.55
+4.05 ± 1.40
+Generators
+GENIE Nominal
+6.18 (63.2/28)
+4.04 (30.1/27)
+GENIE v3.0.6
+5.45 (34.6/28)
+3.66 (21.4/27)
+NuWro 19.02.1
+6.67 (76.7/28)
+4.39 (29.9/27)
+NEUT v5.4.0
+6.64 (78.5/28)
+4.39 (32.2/27)
+GiBUU 2019
+7.00 (82.2./28)
+4.78 (40.0/27)
+higher 𝜒2/d.o.f. for both the full and limited phase-space samples.
+0.4
+0.8
+1.2
+p  [GeV/c]
+0
+5
+10
+-0.65 < cos(
+) < 0.8
+0
+5
+10
+-0.65 < cos(
+) < 0.95
+d
+dp  [10 38
+cm2
+GeV/c Ar ]
+d
+dp  [10 38
+cm2
+GeV/c Ar ]
+0.2
+0.4
+0.6
+0.8
+cos( p)
+0
+5
+10
+-0.65 < cos(
+) < 0.8
+5
+10
+15
+-0.65 < cos(
+) < 0.95
+Simulation
+GENIE Nominal
+GENIE v3.0.6
+NuWro 19.02.1
+NEUT  v5.4.0
+GiBUU 2019
+MicroBooNE Data
+MC
+d
+dcos( p) [10 38cm2
+Ar ]
+d
+dcos( p) [10 38cm2
+Ar ]
+0.4
+0.6
+0.8
+1.0
+pp [GeV/c]
+5
+10
+-0.65 < cos(
+) < 0.8
+0
+5
+10
+15
+-0.65 < cos(
+) < 0.95
+d
+dpp  [10 38
+cm2
+GeV/c Ar ]
+d
+dpp  [10 38
+cm2
+GeV/c Ar ]
+Figure 3-5: As figure 3-4, but for the differential cross sections as a function of mea-
+sured muon momentum (left) and measured proton scattering angle (middle) and
+momentum (right). Cross sections are shown for the full measured phase-space (top)
+and for events with cos(𝜃𝜇) < 0.8 (bottom). Inner and outer error bars show the
+statistical and total (statistical and systematic) uncertainty at the 1𝜎, or 68%, con-
+fidence level.
+Colored lines show the results of theoretical absolute cross section
+calculations using different event generators (without passing through a detector sim-
+ulation). The blue band shows the extracted cross section obtained from analyzing
+MC events passed through our full detector simulation.
+Figure 3-5 shows this comparison between the relevant cross sections in the full
+available phase-space (top) and in the case where events with cos(𝜃𝜇) > 0.8 are
+excluded (bottom). Removing this part of the phase-space significantly improves the
+agreement between data and theory. The improved agreement with the data observed
+66
+
+for GENIE v3.0.6, especially for the full phase-space sample, is intriguing. Specifically,
+GENIE v3.0.6 and NEUT v5.4.0 are quite similar, using the same nuclear, QE, and
+MEC models, which are the most significant processes in our energy range. They
+do differ in the Coulomb corrections that only GENIE v3.0.6 and NEUT have, their
+free parameter tuning process, and the implementation of RPA correction, that are
+known to be important at low momentum transfer [115]. Our data indicates that
+these seemingly small differences can have a highly significant impact, as seen in
+table 3.2.
+0.0
+0.4
+0.8
+1.2
+Q2
+CCQE[GeV2/c2]
+0
+5
+10
+-0.65 < cos(
+) < 0.8
+0
+5
+10
+15
+Simulation
+-0.65 < cos(
+) < 0.95
+  
+  GENIE Nominal
+  GENIE v3.0.6
+  NuWro 19.02.1
+  NEUT  v5.4.0
+  GiBUU 2019
+ MicroBooNE Data
+ MC
+d
+dQ2
+CCQE  [10 38
+cm2
+GeV2/c2 Ar ]
+d
+dQ2
+CCQE  [10 38
+cm2
+GeV2/c2 Ar ]
+0.4
+0.8
+1.2
+Ecal [GeV]
+0
+5
+10
+-0.65 < cos(
+) < 0.8
+5
+10
+-0.65 < cos(
+) < 0.95
+d
+dEcal  [10 38
+cm2
+GeV Ar ]
+d
+dEcal  [10 38
+cm2
+GeV Ar ]
+Figure 3-6: The flux integrated single differential CC1p0𝜋 cross sections as a function
+of 𝑄2
+𝐶𝐶𝑄𝐸 = (𝐸𝑐𝑎𝑙
+𝜈
+− 𝐸𝜇)2 − (⃗𝑝𝜈 − ⃗𝑝𝜇)2 and 𝐸𝑐𝑎𝑙
+𝜈
+= 𝐸𝜇 + 𝑇𝑝 + 𝐵𝐸, where 𝐵𝐸 = 40
+MeV and ⃗𝑝𝜈 = (0, 0, 𝐸𝑐𝑎𝑙
+𝜈 ).
+Colored lines show the results of theoretical absolute
+cross section calculations using different event generators (without passing through a
+detector simulation). The blue band shows the extracted cross section obtained from
+analyzing MC events passed through our full detector simulation.
+Lastly, figure 3-6 shows the flux-integrated single differential cross sections as a
+function of calorimetric measured energy and reconstructed momentum transfer, with
+and without events with cos(𝜃𝜇) > 0.8. The former is defined as 𝐸𝑐𝑎𝑙
+𝜈
+= 𝐸𝜇+𝑇𝑝+𝐵𝐸,
+and the latter as 𝑄2
+𝐶𝐶𝑄𝐸 = (⃗𝑝𝜈 − ⃗𝑝𝜇)2 − (𝐸𝑐𝑎𝑙
+𝜈
+− 𝐸𝜇)2, where E𝜇 is the muon energy,
+T𝑝 is the proton kinetic energy, BE = 40 MeV is the effective nucleon binding energy
+for argon, and ⃗𝑝𝜈 = (0, 0, 𝐸𝑐𝑎𝑙
+𝜈 ) is the reconstructed interacting neutrino momentum.
+67
+
+𝐸𝑐𝑎𝑙
+𝜈
+is often used as a proxy for the true neutrino energy. Overall, good agreement is
+observed between data and calculations for these complex variables, even for the full
+event sample without the cos(𝜃𝜇) < 0.8 requirement.
+The systematic uncertainty of our measurement summed up to 26.2% and included
+contributions from the neutrino flux prediction and POT estimation (18.7%), detector
+response modeling (18.4%), imperfect proton and muon efficiency decoupling (5.7%),
+and neutrino interaction cross section modeling (7.1%).
+3.1.3
+Quasielastic-like Cross-Section Analysis Conclusions
+In summary, the first measurement of 𝜈𝜇 CCQE-like differential cross sections on ar-
+gon was reported for event topologies with a single muon and a single proton detected
+in the final state using data sets from the MicroBooNE LArTPC. The data are in
+good agreement with simulation predictions, except at small muon scattering angles
+that correspond to low-momentum-transfer reactions. This measurement confirmed
+and constrained calculations essential for the extraction of oscillation parameters and
+highlights kinematic regimes where improvement of theoretical models is required.
+The benchmarking of exclusive CC1p0𝜋 cross sections on argon presented here sug-
+gests that measurements of CC1p0𝜋 interactions are a suitable choice for use in pre-
+cision neutrino oscillation analyses, especially after theoretical models are reconciled
+with the small scattering angle data.
+3.2
+First Multidimensional Measurement Of Kine-
+matic Imbalance Cross Sections On Argon
+3.2.1
+Kinematic Imbalance Neutrino Data Analysis
+Over the course of two years (2019-2021), the MicroBooNE collaboration made sig-
+nificant improvements to the pre-existing analysis framework. These improvements
+provided high statistics neutrino-argon data sets, improved signal processing [74],
+reduced detector systematics [130], a theory-driven interaction modeling [79], and
+68
+
+the creation of the first MicroBooNE tune [131]. Figure 3-7 illustrates the improved
+data-MC agreement after the implementation of these changes as a function of cos𝜃𝜇,
+where the disagreement in the forward direction is longer observed. The improved
+picture at cos𝜃𝜇 ≈ 1 is primarily driven by the improved modeling of the MC beam
+related backgrounds.
+1
+−
+0.5
+−
+0
+0.5
+1
+µ
+θ
+cos
+0
+5
+10
+15
+20
+Ar
+2
+cm
+ 
+-38
+10
+ 
+µ
+θ
+dcos
+σ
+d
+ (21.86/18)
+ (9.51/18)
+ (14.38/18)
+ (21.36/18)
+NuWro 19.02.1
+GiBUU 2021
+NEUT v5.4.0
+GENIE v3 G18 Tune
+MicroBooNE Data
+ Shape Unc)
+⊕
+(Stat 
+6.79e+20 POT
+Norm Unc
+Figure 3-7: Muon angular distribution after the implementation of the analysis frame-
+work improvements. No data-MC disagreement is observed in the forward direction.
+Motivated by these improvements, in this analysis the first study of kinematic
+imbalance variables on argon, which are sensitive to nuclear effects, is reported. These
+variables are studied using CC1p0𝜋 events within a neutrino slice, as defined by the
+Pandora reconstruction framework and detailed in section 2.8. Exactly one muon
+with 0.1 < 𝑝𝜇 < 1.2 GeV/c, exactly one proton with 0.3 < 𝑝𝑝 < 1 GeV/c, no charged
+pions above the 0.07 GeV/c threshold, and no other mesons of any momenta are
+required in the final state originating from charged-current 𝜈𝜇-Ar scattering events.
+The existence of any number of neutrons, electrons or photons is allowed.
+Such kinematic imbalance variables of interest include the transverse variables
+(TVs), namely 𝛿𝑝𝑇, 𝛿𝛼𝑇 and 𝛿𝜑𝑇 [132, 133]. These are built specifically to charac-
+terize and minimize the degeneracy between the nuclear effects most pertinent to
+long-baseline oscillation experiments. In particular, the TVs facilitate the possible
+69
+
+identification of the Fermi motion of the initial state nucleon, the final state re-
+interactions of the nucleons in the nucleus and the multi-nucleon interactions (2p2h).
+As shown in figure 3-8, the TVs are defined by projecting the lepton and proton
+momentum on the plane perpendicular to the neutrino direction.
+Figure 3-8: Schematic illustration of the single transverse variables 𝛿𝑝𝑇, 𝛿𝛼𝑇 and 𝛿𝜑𝑇.
+Figure adapted from [134].
+In the absence of any nuclear effects, the proton and muon momenta are equal and
+opposite in this plane and therefore the measured difference between their projections
+is a direct probe of nuclear effects in quasi-elastic (QE) events. 𝛿⃗𝑝𝑇 can be fully
+characterized in terms of the vector magnitude (𝛿𝑝𝑇) and the two angles (𝛿𝛼𝑇 and
+𝛿𝜑𝑇):
+𝛿𝑝𝑇 = |⃗𝑝𝑇 ℓ + ⃗𝑝𝑇 𝑝|
+(3.2)
+𝛿𝛼𝑇 = 𝑎𝑟𝑐𝑐𝑜𝑠( −⃗𝑝𝑇 ℓ·𝛿⃗𝑝𝑇
+𝑝𝑇 ℓ·𝛿𝑝𝑇 )
+(3.3)
+𝛿𝜑𝑇 = 𝑎𝑟𝑐𝑐𝑜𝑠( −⃗𝑝𝑇 ℓ·⃗𝑝𝑇 𝑝
+𝑝𝑇 ℓ·𝑝𝑇 𝑝 )
+(3.4)
+where ⃗𝑝𝑇 ℓ and ⃗𝑝𝑇 𝑝 are, respectively, the projections of the momentum of the
+outgoing lepton and proton on the transverse plane. Different nuclear effects alter
+the distributions of the TVs in different and predictable ways. Measurements of the
+70
+
+pi
+b!
+b
+pP
+bp
+pl
+p
+p!
+SpT
+saTTVs therefore have a unique sensitivity to identify nuclear effects. This allows cross
+sections extracted using these observables to act as a powerful tool to tune and dis-
+tinguish nuclear models. Furthermore, in case of disagreement, the TV distributions
+provide useful hints on the possible causes of the discrepancies.
+Another kinematic variable of interest corresponds to the total momentum of
+the struck nucleon. The formalism introduced in [132] is adopted. This formalism
+provides an approximation for the longitudinal component of the struck nucleon mo-
+mentum shown in equation 3.5, which is derived in appendix 8.1,
+𝛿𝑝𝐿 = 1
+2𝑅 − 𝑚2
+𝐴−1 + 𝛿𝑝2
+𝑇
+2𝑅
+.
+(3.5)
+For simplicity, we defined
+𝑅 ≡ 𝑚𝐴 + 𝑝𝜇
+𝐿 + 𝑝𝑝
+𝐿 − 𝐸𝜇 − 𝐸𝑝
+(3.6)
+Combining information from both the longitudinal and the transverse components
+gave us access to an approximation for the total struck nucleon momentum
+𝑝𝑛,𝑝𝑟𝑜𝑥𝑦 =
+√︁
+𝛿𝑝2
+𝐿 + 𝛿𝑝2
+𝑇
+(3.7)
+The muon-proton momentum imbalances introduced in [135] parallel and trans-
+verse to 𝛿⃗𝑝𝑇, as shown in figure 3-9, are explored,
+𝛿𝑝𝑇𝑥 = (ˆ𝑝𝜈 × ˆ𝑝𝜇
+𝑇) · 𝛿⃗𝑝𝑇
+𝛿𝑝𝑇𝑦 = −ˆ𝑝𝜇
+𝑇 · 𝛿⃗𝑝𝑇,
+(3.8)
+71
+
+and, in terms of the magnitudes,
+𝛿𝑝𝑇𝑥 = 𝛿𝑝𝑇 · 𝑠𝑖𝑛𝛿𝛼𝑇
+𝛿𝑝𝑇𝑦 = 𝛿𝑝𝑇 · 𝑐𝑜𝑠𝛿𝛼𝑇.
+(3.9)
+Figure 3-9: Schematic illustration of 𝛿𝑝𝑇𝑥 and 𝛿𝑝𝑇𝑦. Figure adapted from [135].
+The measured 𝛿𝑝𝑇𝑥 event distribution shown in figure 3-10 (left) using the “com-
+bined” MicroBooNE runs 1-3 exhibit a QE peak near 0. If the interaction had occurred
+on a free nucleon, then a delta function would be expected at 0 because the muon
+and proton final states must balance. The width of the QE peak mostly results from
+the Fermi motion. If no significant deviation is assumed in the non-QE distributions
+originating from MEC and RES/DIS events, then data-MC discrepancies could im-
+ply an overestimation of the argon Fermi momentum, and/or a difference in the total
+fraction of the FSI contribution.
+Unlike the 𝛿𝑝𝑇𝑥 distribution, a non-QE tail is observed towards the negative 𝛿𝑝𝑇𝑦
+values shown in figure 3-10 (right). Inelastic events such as 2p2h, resonance, and DIS
+are inefficient at transferring the lepton momentum to the final state nucleons, since
+multiple initial states particles are often involved. Therefore, the protons tagged in
+72
+
+OpT
+qT = 
+pp
+pTy
+SPTc
+py
+N
+P
+Puthe non-QE events will in general have less momenta then the muons and the 𝛿𝑝𝑇𝑦
+distribution is shifted to the left.
+BeamOn (9051)
+CCQE (4744)
+CCMEC (1547)
+ExtBNB (642)
+CCRES (1035)
+CCDIS (181)
+ [GeV/c]
+T,x
+p
+δ
+0
+500
+1000
+1500
+2000
+2500
+# Events / 6.79e+20
+Combined
+CCQE = 57.7 %
+Cosmics = 7.8 %
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+ [GeV/c]
+T,x
+p
+δ
+0.9
+1
+1.1
+1.2
+1.3
+MC+ExtBNB
+BeamOn
+BeamOn (9051)
+CCQE (4744)
+CCMEC (1547)
+ExtBNB (642)
+CCRES (1035)
+CCDIS (181)
+ [GeV/c]
+T,y
+p
+δ
+0
+500
+1000
+1500
+2000
+2500
+# Events / 6.79e+20
+Combined
+CCQE = 57.7 %
+Cosmics = 7.8 %
+0.6
+−
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+ [GeV/c]
+T,y
+p
+δ
+0.9
+1
+1.1
+1.2
+1.3
+MC+ExtBNB
+BeamOn
+Figure 3-10: Interaction breakdown of the CC1p0𝜋 events as a function of 𝛿𝑝𝑇𝑥 (left)
+and 𝛿𝑝𝑇𝑦 (right). The data correspond to the “combined” MicroBooNE runs 1-3.
+To avoid multiple cosmic contributions and tracks from trajectories that exit the
+detector but their end-points are incorrectly reconstructed around its edges, a fiducial
+volume of
+10 < 𝑥 < 246, −105 < 𝑦 < 105, 10 < 𝑧 < 1026 cm
+(3.10)
+is defined. Candidate muon and proton tracks that were fully contained in this
+region were considered and their momenta were obtained based on their range [136,
+137].
+The log-likelihood ratio particle identification (LLR PID) score method [138] is
+used to obtain our muon and proton candidates. The candidate track with the greater
+LLR PID score was assigned the label of the candidate muon, while the one with the
+smaller 3-plane loglikelihood was our candidate proton.
+To minimize the contribution of misreconstructed tracks, we took advantage of
+the fact that we had two muon momentum reconstruction methods available for con-
+tained tracks, namely the momentum from range [137] and the one from Multiple
+Coulomb Scattering (MCS) [100]. A quality cut was applied on the contained muons
+by requiring the range and MCS momenta to be in agreement within 25%.
+In order to avoid flipped tracks, it was further required that the distance between
+73
+
+the track start points and the vertex is smaller than the corresponding distance be-
+tween the track end points and the vertex. It was also required that the distance
+between the start points of the two candidate tracks is smaller than the one between
+the two end points.
+The maximal possible signal contribution was ensured, while the majority of the
+cosmic contamination and the beam related MC backgrounds were rejected, by re-
+quiring that the proton LLP PID score is less than 0.05.
+The application of our event selection resulted in 9051 candidate events in our data
+sample. Using the MC, it was estimated that our event selection yielded a purity of
+≈ 70% and an efficiency of ≈ 10%.
+There was also some contribution from the
+remaining cosmic contamination (≈ 8%). After the application of the event selection,
+topological and interaction breakdowns for the kinematic variables of interest, such
+as the ones shown in figure 3-11 for 𝛿𝑝𝑇, were obtained.
+BeamOn (9051)
+Overlay CC1p (5673)
+Overlay NonCC1p (1837)
+ExtBNB (642)
+Dirt CC1p (0)
+Dirt NonCC1p (62)
+ [GeV/c]
+T
+p
+δ
+0
+500
+1000
+1500
+# Events / 6.79e+20
+Combined
+CC1p = 69 %
+Cosmics = 7.8 %
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+0.9
+1
+1.1
+1.2
+MC+ExtBNB
+BeamOn
+BeamOn (9051)
+CCQE (4744)
+CCMEC (1547)
+ExtBNB (642)
+CCRES (1035)
+CCDIS (181)
+ [GeV/c]
+T
+p
+δ
+0
+500
+1000
+1500
+# Events / 6.79e+20
+Combined
+CCQE = 57.7 %
+Cosmics = 7.8 %
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+0.9
+1
+1.1
+1.2
+MC+ExtBNB
+BeamOn
+Figure 3-11: Topological (left) and interaction (right) breakdown after the application
+of the event selection for 𝛿𝑝𝑇.
+3.2.2
+Cross-Section Extraction Technique
+The unfolded cross-section results reported in this analysis took advantage of the
+Wiener-SVD unfolding [139]. This method combines the use of the singular value
+decomposition (SVD) unfolding and a Wiener filter. SVD unfolding [140], such as
+the Tikhonov regularisation [141], unfolds a distribution by minimising a 𝜒2 function
+comparing a prediction to data. To avoid the large variance introduced, a penalty
+74
+
+term is added to regularise the curvature (second derivative) of the results.
+The
+strength of such a term is determined by finding an appropriate trade-off between the
+bias and the variance between the data and the MC. More details on the technique
+are included in appendix 8.2. In order to report the cross-section results, two key
+ingredients are required, namely the response and covariance matrices.
+The construction of the response matrices uses the selected MC CC1p0𝜋 events
+to construct a two-dimensional (2D) object, where each entry in true bin i and re-
+constructed bin j (𝑁 𝑡𝑟𝑢𝑒 𝑖,𝑟𝑒𝑐𝑜 𝑗) is divided by the true number of events generated in
+bin i (𝑆𝑡𝑟𝑢𝑒 𝑖). These response matrices serve as “2D local efficiencies”, as defined in
+equation 3.11 and can be seen in figure 3-12 for 𝛿𝑝𝑇.
+𝑀𝑖𝑗 = 𝑁 𝑡𝑟𝑢𝑒 𝑖,𝑟𝑒𝑐𝑜 𝑗
+𝑆𝑡𝑟𝑢𝑒 𝑖
+(3.11)
+0
+0.02
+0.04
+0.06
+ 0.00
+±
+0.05
+ 0.00
+±
+0.01
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.03
+ 0.00
+±
+0.06
+ 0.00
+±
+0.02
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.01
+ 0.00
+±
+0.02
+ 0.00
+±
+0.06
+ 0.00
+±
+0.02
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.02
+ 0.00
+±
+0.06
+ 0.00
+±
+0.02
+ 0.00
+±
+0.01
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.02
+ 0.00
+±
+0.05
+ 0.00
+±
+0.03
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.01
+ 0.00
+±
+0.06
+ 0.00
+±
+0.02
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.02
+ 0.00
+±
+0.05
+ 0.00
+±
+0.02
+ 0.00
+±
+0.01
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.01
+ 0.00
+±
+0.05
+ 0.00
+±
+0.02
+ 0.00
+±
+0.01
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.01
+ 0.00
+±
+0.06
+ 0.00
+±
+0.02
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.01
+ 0.00
+±
+0.05
+ 0.00
+±
+0.02
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.01
+ 0.00
+±
+0.05
+ 0.00
+±
+0.02
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.01
+ 0.01
+±
+0.05
+ 0.00
+±
+0.01
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.00
+ 0.00
+±
+0.01
+ 0.00
+±
+0.03
+Combined Response Matrix
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+True 
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+Reco 
+Combined Response Matrix
+Figure 3-12: Response matrices of 𝛿𝑝𝑇 using the selected CC1p0𝜋 MC events.
+The method also uses a covariance matrix constructed from the MC flux nor-
+malised event rate as an input. The total covariance matrix incorporates information
+related to the systematic and statistical uncertainties. The flux-normalized MC event
+75
+
+rates in reconstructed space were obtained as
+˜𝜎𝑟𝑒𝑐𝑜 𝑖 =
+𝑁 𝑟𝑒𝑐𝑜 𝑖
+Φ𝐶𝑉
+𝜈
+× 𝑁𝑡𝑎𝑟𝑔𝑒𝑡𝑠
+,
+(3.12)
+where 𝑁 𝑟𝑒𝑐𝑜 𝑖 = 𝑀𝑖𝑗 ×𝑆𝑡𝑟𝑢𝑒 𝑗 +𝐵𝑟𝑒𝑐𝑜 𝑖 is the total number of reconstructed events in
+bin i, 𝑀𝑖𝑗 is the response matrix corresponding to reco bin i and true bin j as defined in
+equation 3.11, 𝑆𝑡𝑟𝑢𝑒 𝑗 is the true signal without any detector or reconstruction effects
+in bin j, and 𝐵𝑟𝑒𝑐𝑜 𝑖 is the total number of reconstructed beam-related MC background
+events in bin i. Substituting 𝑁 𝑟𝑒𝑐𝑜 𝑖 into equation 3.12 yields
+˜𝜎𝑟𝑒𝑐𝑜 𝑖 = 𝑀 𝑢𝑛𝑖𝑣
+𝑖𝑗
+× 𝑆𝑡𝑟𝑢𝑒 𝑗 𝐶𝑉 + 𝐵𝑟𝑒𝑐𝑜 𝑖 𝑢𝑛𝑖𝑣
+Φ𝐶𝑉
+𝜈
+× 𝑁𝑡𝑎𝑟𝑔𝑒𝑡𝑠
+.
+(3.13)
+For each systematic variation, each term in equation 3.13 labelled with “univ” is
+reweighted/modified and each term labelled with “CV” is fixed to the central value.
+The integrated flux, Φ𝐶𝑉
+𝜈
+, remains fixed for each variation.
+In the case of the cross section variations, the calculation of the response matrix
+in each universe is slightly modified via the normalization to the true signal in a given
+universe 𝑆𝑡𝑟𝑢𝑒 𝑗 𝑢𝑛𝑖𝑣,
+𝑀 𝑢𝑛𝑖𝑣
+𝑖𝑗
+= 𝑁 𝑡𝑟𝑢𝑒 𝑗,𝑟𝑒𝑐𝑜 𝑖 𝑢𝑛𝑖𝑣
+𝑆𝑡𝑟𝑢𝑒 𝑗 𝑢𝑛𝑖𝑣
+(3.14)
+This treatment of the systematic uncertainties addresses both the signal and the
+beam related background uncertainties.
+Using these flux-normalized event rates ˜𝜎, a covariance matrix, 𝐸𝑖𝑗, can be calcu-
+lated using the central value and 𝑁𝑢𝑛𝑖𝑣 multisims with the covariance formalism,
+𝐸𝑖𝑗 =
+1
+𝑁𝑢𝑛𝑖𝑣
+𝑁𝑢𝑛𝑖𝑣
+∑︁
+𝑠=0
+(˜𝜎𝑢𝑛𝑖𝑣
+𝑖
+− ˜𝜎𝐶𝑉
+𝑖
+)(˜𝜎𝑢𝑛𝑖𝑣
+𝑗
+− ˜𝜎𝐶𝑉
+𝑗
+)
+(3.15)
+where 𝑁𝑢𝑛𝑖𝑣 is number of alternative universes, ˜𝜎𝑢𝑛𝑖𝑣
+𝑖
+corresponds to the variation
+and ˜𝜎𝐶𝑉
+𝑖
+to the central-value prediction.
+The unfolding model uncertainty is accessed by comparing the data spectra un-
+folded with G18 to the data spectra unfolded with two alternative G18 configurations.
+76
+
+The former one did not include the effect of the MicroBooNE tune and the latter in-
+cluded an additional weight of 2 on the MEC events. The spread between the three
+configurations on a bin-by-bin basis normalized to
+√︀
+(2) is assigned as an additional
+uncertainty [142].
+The statistical uncertainty of our measurement is 1.5%. The total uncertainty
+sums to 13% and includes contributions from the neutrino flux prediction (7.3%),
+unfolding model uncertainty (7.3%), neutrino interaction cross section modeling (5%),
+detector response modeling (4.9%), POT estimation (2.3%), number-of-scattering-
+targets (1.15%), reinteractions (1%), and out-of-cryostat interaction modeling (0.2%).
+Figure 3-13 shows the total covariance matrix due to the aforementioned sources
+of uncertainty for 𝛿𝑝𝑇.
+0
+0.2
+0.4
+3
+−
+10
+×
+Combined Total Covariance Matrix
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+i bin 
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+j bin 
+Combined Total Covariance Matrix
+Figure 3-13: Total covariance matrix for 𝛿𝑝𝑇.
+The Wiener SVD unfolding machinery returns an unfolded data cross section
+along with an unfolded covariance matrix and an additional smearing matrix, 𝐴𝑐.
+The corresponding 𝐴𝑐 matrix for 𝛿𝑝𝑇 is shown in figure 3-14. The smearing matrix
+𝐴𝑐 contains information about the regularisation of the measurement and is applied
+to the true model cross section predictions when compared to the data. Therefore, the
+result of our measurement lives in a “regularized” phase-space, which is not identical
+to the true phase-space.
+77
+
+0.1
+−
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+Combined
+Figure 3-14: Additional smearing matrix 𝐴𝑐 for 𝛿𝑝𝑇.
+3.2.3
+Event Generator Modeling And Configurations
+The extracted cross sections were compared to GENIE v3.0.6 G18_10a_02_11a
+(G18) and the theory-driven GiBUU 2021 (GiBUU) event generator. Additional com-
+parisons to the corresponding events generators when FSI are turned off were also
+included (G18 No FSI and GiBUU No FSI). G18 uses the Local Fermi Gas model [37],
+the Nieves CCQE scattering prescription [121] which includes Coulomb corrections
+for the outgoing muon [116] and Random Phase Approximation correction [115], the
+Nieves MEC model [122], the KLN-BS RES [123–126] and Berger-Sehgal COH [120]
+scattering models, the hA2018 FSI model [143], and the T2K tune weights [131].
+GiBUU uses somewhat similar models, but, unlike GENIE, those are imple-
+mented in a coherent way, by solving the Boltzmann-Uehling-Uhlenbeck transport
+equation [117]. The modeling includes the Local Fermi Gas model [37], a standard
+CCQE expression [127], an empirical MEC model and a dedicated spin dependent
+resonances amplitude calculation following the MAID analysis [128]. The DIS model
+is as in PYTHIA [129] and the FSI treatment is different as the hadrons propagate
+through the residual nucleus in a nuclear potential which is consistent with the initial
+state.
+78
+
+Apart from the nominal G18 prediction, we further included a comparison to the
+recently added theory driven GENIE v3.0.6 G21_11b_00_000 configuration (G21
+hN). The latter uses the SuSAv2 model for QE and MEC interactions [28], and the
+hN2018 FSI model [144]. The modeling options for RES, DIS, and COH interactions
+are the same as for G18. We investigated the effect of the FSI modeling choice by
+comparing the G21hN results to the ones obtained with G21 hA, where the hA2018
+FSI model was used instead, and to G21 G4 with the recently coupled Geant4 FSI
+framework [145].
+Lastly, our results present the comparison between the nominal G18 LFG model
+and predictions using the same G18 modeling configuration but different nuclear
+model options available in the GENIE event generator, namely the Bodek-Ritchie
+Fermi Gas (G18 RFG) [36] and an Effective Spectral Function (G18 EffSF) [38].
+Furthermore, the prediction without Random Phase Approximation (RPA) effects
+was used for comparison (G18 No RPA) [115].
+3.2.4
+Kinematic Imbalance Differential Cross-Section Results
+The single- and double- in 𝛿𝛼𝑇 bins differential unfolded cross sections as a func-
+tion of 𝛿𝑝𝑇 are presented in figure 3-15. The single-differential results as a function
+of 𝛿𝑝𝑇 using all the events that satisfy our selection are shown in the top panel.
+The peak height of both generator predictions is ≈ 30% higher when FSI effects are
+turned off. Yet, all distributions illustrate a transverse missing momentum tail that
+extends beyond the Fermi momentum whether FSI effects are activated or not. The
+ratio between the generator predictions with and without FSI is shown in the insert
+and illustrates significant shape variations across the range of interest. The double-
+differential result using events with 𝛿𝛼𝑇 < 45𝑜 shown in the bottom left panel of
+figure 3-15 is dominated by events that primarily occupy the region up to the Fermi
+momentum and do not exhibit a high momentum tail. The corresponding ratio in-
+sert illustrates a fairly uniform behavior indicative of transparency effects ranging
+between 50-70%. The double-differential results using events with 135𝑜 < 𝛿𝛼𝑇 < 180𝑜
+is shown in the bottom right panel of figure 3-15 and illustrate the high transverse
+79
+
+missing momentum up to 1 GeV/c. The case without FSI effects is strongly disfa-
+vored and the ratio insert illustrates strong shape variations. Therefore, the high 𝛿𝛼𝑇
+region is an appealing candidate for neutrino experiments to benchmark and tune the
+FSI modeling in event generators.
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+0
+10
+20
+30
+40
+50
+GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+σ
+d
+GiB No FSI (102.7/13)
+GiB FSI (21.6/13)
+G18 No FSI (53.6/13)
+G18 FSI (5.8/13)
+ Shape)
+⊕
+MicroBooNE Data (Stat 
+6.79e+20 POT
+Norm
+(a) All events
+0
+0.2
+0.4
+0.6
+2
+4
+6
+FSI/No FSI
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+deg GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+T
+α
+δ
+d
+σ
+2
+d
+GiB No FSI (58.8/11)
+GiB FSI (3.2/11)
+G18 No FSI (27.1/11)
+G18 FSI (9.9/11)
+o
+ < 45
+T
+α
+δ
+(b) 
+0
+0.2
+0.4
+0.6
+0.8
+0.5
+1
+FSI/No FSI
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+deg GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+T
+α
+δ
+d
+σ
+2
+d
+GiB No FSI (83.7/13)
+GiB FSI (9.3/13)
+G18 No FSI (91.9/13)
+G18 FSI (12.5/13)
+o
+ < 180
+T
+α
+δ
+ < 
+o
+(c) 135
+0
+0.2
+0.4
+0.6
+2
+4
+6
+FSI/No FSI
+Figure 3-15: The flux-integrated single- (top) and double- in 𝛿𝛼𝑇 bins (bottom)
+differential CC1p0𝜋 cross sections as a function of the transverse missing momentum
+𝛿𝑝𝑇. Inner and outer error bars show the statistical and total (statistical and shape
+systematic) uncertainty at the 1𝜎, or 68%, confidence level. The gray band shows the
+normalization systematic uncertainty. Colored lines show the results of theoretical
+absolute cross section calculations with and without FSI based on the GENIE and
+GiBUU event generators.
+The single-differential results as a function of 𝛿𝛼𝑇 using all the events that satisfy
+our selection are shown in top panel of figure 3-16. The result without FSI illustrates
+a uniform behavior across the whole distribution and is disfavored. The addition of
+FSI effects leads to a ≈ 30% asymmetry around 𝛿𝛼𝑇 = 90𝑜 due to the fact that the
+proton in our selection undergoes FSI. The three FSI models used here for comparison
+result in a comparable performance, also shown in terms of the ratio plot of the
+80
+
+different FSI options to the prediction without FSI. The double-differential result
+using events with 𝛿𝑝𝑇 < 0.2 GeV/c shown in the bottom left panel of figure 3-16
+illustrates a uniform distribution indicative of the suppressed FSI impact in that part
+of the phase-space. The double-differential result using events with 𝛿𝑝𝑇 > 0.4 GeV/c
+is shown in the bottom right panel of figure 3-16 and illustrates the presence of strong
+FSI effects. The case without FSI effects is disfavored and the asymmetry around 90𝑜
+is significantly enhanced. Therefore, the high 𝛿𝛼𝑇 region is an appealing candidate for
+neutrino experiments to benchmark and tune the FSI modeling in event generators.
+0
+20
+40
+60
+80
+100 120 140 160 180
+ [deg]
+T
+α
+δ
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+deg Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+α
+δ
+d
+σ
+d
+G21 No FSI (84.4/7)
+G21 hA (5.9/7)
+G21 hN (7.9/7)
+G21 G4 (5.8/7)
+ Shape)
+⊕
+MicroBooNE Data (Stat 
+6.79e+20 POT
+Norm
+(a) All events, MicroBooNE Preliminary
+0
+50
+100
+150
+0.6
+0.8
+1
+1.2
+FSI/No FSI
+0
+20
+40
+60
+80
+100 120 140 160 180
+ [deg]
+T
+α
+δ
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+deg GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+T
+α
+δ
+d
+σ
+2
+d
+G21 No FSI (40.6/7)
+G21 hA (1.4/7)
+G21 hN (2.1/7)
+G21 G4 (5.1/7)
+ < 0.2 GeV/c, MicroBooNE Preliminary
+T
+p
+δ
+(b) 
+0
+50
+100
+150
+0.6
+0.8
+FSI/No FSI
+0
+20
+40
+60
+80
+100 120 140 160 180
+ [deg]
+T
+α
+δ
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+deg GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+T
+α
+δ
+d
+σ
+2
+d
+G21 No FSI (7.2/7)
+G21 hA (1.7/7)
+G21 hN (2.0/7)
+G21 G4 (1.2/7)
+ > 0.4 GeV/c, MicroBooNE Preliminary
+T
+p
+δ
+(c) 
+0
+50
+100
+150
+1
+1.5
+2
+FSI/No FSI
+Figure 3-16: The flux-integrated single- (top) and double- in 𝛿𝑝𝑇 bins (bottom) dif-
+ferential CC1p0𝜋 cross sections as a function of the angle 𝛿𝛼𝑇. Inner and outer error
+bars show the statistical and total (statistical and shape systematic) uncertainty at
+the 1𝜎, or 68%, confidence level. The gray band shows the normalization systematic
+uncertainty. Colored lines show the results of theoretical absolute cross section calcu-
+lations with a number of FSI modeling options based on the GENIE event generator.
+Lastly, figure 3-17 shows the single- (top) and double- in 𝛿𝑝𝑇,𝑦 bins (bottom)
+differential unfolded cross sections as a function of 𝛿𝑝𝑇,𝑥. The event distributions of
+81
+
+𝛿𝑝𝑇,𝑥 and 𝛿𝑝𝑇,𝑦 have already been presented in figure 3-11. The single differential
+result (top panel) illustrates a fairly broad symmetric distribution centered around 0.
+The double-differential result for events where 𝛿𝑝𝑇,𝑦 < -0.15 GeV/c (bottom left panel)
+illustrates an even broader distribution where all predictions yield comparable results.
+Unlike the asymmetric part of the 𝛿𝑝𝑇,𝑦 tail, the double-differential result for events
+with -0.15 < 𝛿𝑝𝑇,𝑦 < 0.15 GeV/c (bottom right panel) shows a much narrower peak
+which strongly depends on the choice of the underlying model and the addition or
+not of nuclear effects, such as RPA ones. The G18 LFG and G18 No RPA predictions
+are favored in that part of the phase-space.
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+ [GeV/c]
+T,x
+p
+δ
+0
+10
+20
+30
+40
+50
+GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T,x
+p
+δ
+d
+σ
+d
+G18 LFG (7.8/11)
+G18 No RPA (3.5/11)
+G18 RFG (37.5/11)
+G18 EffSF (57.3/11)
+ Shape)
+⊕
+MicroBooNE Data (Stat 
+6.79e+20 POT
+Norm
+ = 0.17
+Data
+σ
+(a) All events
+MicroBooNE Preliminary
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+ [GeV/c]
+T,x
+p
+δ
+0
+2
+4
+6
+8
+10
+12
+14
+ Ar
+2
+/c
+2
+GeV
+2
+cm
+ 
+-38
+10
+ 
+T,y
+p
+δ
+ d
+T,x
+p
+δ
+d
+σ
+2
+d
+G18 LFG (3.9/11)
+G18 No RPA (3.2/11)
+G18 RFG (3.8/11)
+G18 EffSF (16.3/11)
+ = 0.25
+Data
+σ
+ < -0.15 GeV/c
+T,y
+p
+δ
+(b) 
+MicroBooNE Preliminary
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+ [GeV/c]
+T,x
+p
+δ
+0
+20
+40
+60
+80
+ Ar
+2
+/c
+2
+GeV
+2
+cm
+ 
+-38
+10
+ 
+T,y
+p
+δ
+ d
+T,x
+p
+δ
+d
+σ
+2
+d
+G18 LFG (4.9/11)
+G18 No RPA (4.1/11)
+G18 RFG (39.3/11)
+G18 EffSF (52.3/11)
+ = 0.14
+Data
+σ
+ < 0.15 GeV/c
+T,y
+p
+δ
+(c) -0.15 < 
+MicroBooNE Preliminary
+Figure 3-17: The flux-integrated single- (top) and double- in 𝛿𝑝𝑇,𝑦 bins (bottom) dif-
+ferential CC1p0𝜋 cross sections as a function of the angle 𝛿𝑝𝑇,𝑥. Inner and outer error
+bars show the statistical and total (statistical and shape systematic) uncertainty at
+the 1𝜎, or 68%, confidence level. The gray band shows the normalization system-
+atic uncertainty. Colored lines show the results of theoretical absolute cross section
+calculations with a number of event generators.
+The 𝜒2 per degree of freedom (d.o.f.) data comparison for each prediction shown
+82
+
+on the results in figures 3-15, 3-16, and 3-17 takes into account the total covariance
+matrix including the off-diagonal elements. Figures 3-18 - 3-26 show in the interaction
+breakdown of the aforementioned results.
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+0
+10
+20
+30
+40
+50
+GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+σ
+d
+QE
+MEC
+RES
+DIS
+MicroBooNE Data
+ Shape Unc)
+⊕
+(Stat 
+Norm Unc
+G18    All events
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+0
+10
+20
+30
+40
+50
+GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+σ
+d
+G18 No FSI    All events
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+0
+10
+20
+30
+40
+50
+GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+σ
+d
+GiBUU    All events
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+0
+10
+20
+30
+40
+50
+GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+σ
+d
+GiBUU No FSI    All events
+Figure 3-18: Cross-section interaction breakdown for all the selected events. The
+breakdown is shown for (top left) the G18 configuration with FSI effects, (top right)
+the G18 configuration without FSI effects, (bottom left) GiB with FSI effects, and
+(bottom right) GiB without FSI effects.
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+deg GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+T
+α
+δ
+d
+σ
+2
+d
+QE
+MEC
+RES
+DIS
+MicroBooNE Data
+ Shape Unc)
+⊕
+(Stat 
+Norm Unc
+o
+ < 45
+T
+α
+δ
+G18    
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+deg GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+T
+α
+δ
+d
+σ
+2
+d
+o
+ < 45
+T
+α
+δ
+G18 No FSI    
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+deg GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+T
+α
+δ
+d
+σ
+2
+d
+o
+ < 45
+T
+α
+δ
+GiBUU    
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+deg GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+T
+α
+δ
+d
+σ
+2
+d
+o
+ < 45
+T
+α
+δ
+GiBUU No FSI    
+Figure 3-19: Cross-section interaction breakdown for events with 𝛿𝛼𝑇 < 45𝑜. The
+breakdown is shown for (top left) the G18 configuration with FSI effects, (top right)
+the G18 configuration without FSI effects, (bottom left) GiB with FSI effects, and
+(bottom right) GiB without FSI effects.
+83
+
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+deg GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+T
+α
+δ
+d
+σ
+2
+d
+QE
+MEC
+RES
+DIS
+MicroBooNE Data
+ Shape Unc)
+⊕
+(Stat 
+Norm Unc
+o
+ < 180
+T
+α
+δ
+ < 
+o
+G18    135
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+deg GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+T
+α
+δ
+d
+σ
+2
+d
+o
+ < 180
+T
+α
+δ
+ < 
+o
+G18 No FSI    135
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+deg GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+T
+α
+δ
+d
+σ
+2
+d
+o
+ < 180
+T
+α
+δ
+ < 
+o
+GiBUU    135
+0
+0.2
+0.4
+0.6
+0.8
+ [GeV/c]
+T
+p
+δ
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+deg GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+T
+α
+δ
+d
+σ
+2
+d
+o
+ < 180
+T
+α
+δ
+ < 
+o
+GiBUU No FSI    135
+Figure 3-20: Cross-section interaction breakdown for events with 135𝑜 < 𝛿𝛼𝑇 < 180𝑜.
+The breakdown is shown for (top left) the G18 configuration with FSI effects, (top
+right) the G18 configuration without FSI effects, (bottom left) GiB with FSI effects,
+and (bottom right) GiB without FSI effects.
+0
+20
+40
+60
+80
+100 120 140 160 180
+ [deg]
+T
+α
+δ
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+deg Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+α
+δ
+d
+σ
+d
+QE
+MEC
+RES
+DIS
+MicroBooNE Data
+ Shape Unc)
+⊕
+(Stat 
+Norm Unc
+G21 hA    All events
+0
+20
+40
+60
+80
+100 120 140 160 180
+ [deg]
+T
+α
+δ
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+deg Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+α
+δ
+d
+σ
+d
+G21 hN    All events
+0
+20
+40
+60
+80
+100 120 140 160 180
+ [deg]
+T
+α
+δ
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+deg Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+α
+δ
+d
+σ
+d
+G21 G4    All events
+0
+20
+40
+60
+80
+100 120 140 160 180
+ [deg]
+T
+α
+δ
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+deg Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+α
+δ
+d
+σ
+d
+G21 No FSI    All events
+Figure 3-21: Cross-section interaction breakdown for all the selected events. The
+breakdown is shown for (top left) the G21 hA configuration with the hA2018 FSI
+model, (top right) the G21 hN configuration with the hN FSI model, (bottom left)
+the G21 G4 configuration with the G4 FSI model, and (bottom right) the G21 No
+FSI configuration without FSI effects.
+84
+
+0
+20
+40
+60
+80
+100 120 140 160 180
+ [deg]
+T
+α
+δ
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+deg GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+T
+α
+δ
+d
+σ
+2
+d
+QE
+MEC
+RES
+DIS
+MicroBooNE Data
+ Shape Unc)
+⊕
+(Stat 
+Norm Unc
+ < 0.2 GeV/c
+T
+p
+δ
+G21 hA    
+0
+20
+40
+60
+80
+100 120 140 160 180
+ [deg]
+T
+α
+δ
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+deg GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+T
+α
+δ
+d
+σ
+2
+d
+ < 0.2 GeV/c
+T
+p
+δ
+G21 hN    
+0
+20
+40
+60
+80
+100 120 140 160 180
+ [deg]
+T
+α
+δ
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+deg GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+T
+α
+δ
+d
+σ
+2
+d
+ < 0.2 GeV/c
+T
+p
+δ
+G21 G4    
+0
+20
+40
+60
+80
+100 120 140 160 180
+ [deg]
+T
+α
+δ
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+deg GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+T
+α
+δ
+d
+σ
+2
+d
+ < 0.2 GeV/c
+T
+p
+δ
+G21 No FSI    
+Figure 3-22: Cross-section interaction breakdown for events with 𝛿𝑝𝑇 < 0.2 GeV/c.
+The breakdown is shown for (top left) the G21 hA configuration with the hA2018
+FSI model, (top right) the G21 hN configuration with the hN FSI model, (bottom
+left) the G21 G4 configuration with the G4 FSI model, and (bottom right) the G21
+No FSI configuration without FSI effects.
+0
+20
+40
+60
+80
+100 120 140 160 180
+ [deg]
+T
+α
+δ
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+deg GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+T
+α
+δ
+d
+σ
+2
+d
+QE
+MEC
+RES
+DIS
+MicroBooNE Data
+ Shape Unc)
+⊕
+(Stat 
+Norm Unc
+ > 0.4 GeV/c
+T
+p
+δ
+G21 hA    
+0
+20
+40
+60
+80
+100 120 140 160 180
+ [deg]
+T
+α
+δ
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+deg GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+T
+α
+δ
+d
+σ
+2
+d
+ > 0.4 GeV/c
+T
+p
+δ
+G21 hN    
+0
+20
+40
+60
+80
+100 120 140 160 180
+ [deg]
+T
+α
+δ
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+deg GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+T
+α
+δ
+d
+σ
+2
+d
+ > 0.4 GeV/c
+T
+p
+δ
+G21 G4    
+0
+20
+40
+60
+80
+100 120 140 160 180
+ [deg]
+T
+α
+δ
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+deg GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T
+p
+δ
+d
+T
+α
+δ
+d
+σ
+2
+d
+ > 0.4 GeV/c
+T
+p
+δ
+G21 No FSI    
+Figure 3-23: Cross-section interaction breakdown for events with 𝛿𝑝𝑇 > 0.4 GeV/c.
+The breakdown is shown for (top left) the G21 hA configuration with the hA2018
+FSI model, (top right) the G21 hN configuration with the hN FSI model, (bottom
+left) the G21 G4 configuration with the G4 FSI model, and (bottom right) the G21
+No FSI configuration without FSI effects.
+85
+
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+ [GeV/c]
+T,x
+p
+δ
+0
+10
+20
+30
+40
+50
+GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T,x
+p
+δ
+d
+σ
+d
+QE
+MEC
+RES
+DIS
+MicroBooNE Data
+ Shape Unc)
+⊕
+(Stat 
+Norm Unc
+G18 LFG    All events
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+ [GeV/c]
+T,x
+p
+δ
+0
+10
+20
+30
+40
+50
+GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T,x
+p
+δ
+d
+σ
+d
+G18 RFG    All events
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+ [GeV/c]
+T,x
+p
+δ
+0
+10
+20
+30
+40
+50
+GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T,x
+p
+δ
+d
+σ
+d
+G18 EffSF    All events
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+ [GeV/c]
+T,x
+p
+δ
+0
+10
+20
+30
+40
+50
+GeV/c Ar
+2
+cm
+ 
+-38
+10
+ 
+T,x
+p
+δ
+d
+σ
+d
+G18 No RPA    All events
+Figure 3-24: Cross-section interaction breakdown for all the selected events. The
+breakdown is shown for (top left) the G18 LFG configuration, (top right) the G18
+RFG configuration, (bottom left) the G18 EffSF configuration, and (bottom right)
+the G18 No RPA configuration.
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+ [GeV/c]
+T,x
+p
+δ
+0
+2
+4
+6
+8
+10
+12
+14
+ Ar
+2
+/c
+2
+GeV
+2
+cm
+ 
+-38
+10
+ 
+T,y
+p
+δ
+ d
+T,x
+p
+δ
+d
+σ
+2
+d
+QE
+MEC
+RES
+DIS
+MicroBooNE Data
+ Shape Unc)
+⊕
+(Stat 
+Norm Unc
+ < -0.15 GeV/c
+T,y
+p
+δ
+G18 LFG    
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+ [GeV/c]
+T,x
+p
+δ
+0
+2
+4
+6
+8
+10
+12
+14
+ Ar
+2
+/c
+2
+GeV
+2
+cm
+ 
+-38
+10
+ 
+T,y
+p
+δ
+ d
+T,x
+p
+δ
+d
+σ
+2
+d
+ < -0.15 GeV/c
+T,y
+p
+δ
+G18 RFG    
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+ [GeV/c]
+T,x
+p
+δ
+0
+2
+4
+6
+8
+10
+12
+14
+ Ar
+2
+/c
+2
+GeV
+2
+cm
+ 
+-38
+10
+ 
+T,y
+p
+δ
+ d
+T,x
+p
+δ
+d
+σ
+2
+d
+ < -0.15 GeV/c
+T,y
+p
+δ
+G18 EffSF    
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+ [GeV/c]
+T,x
+p
+δ
+0
+2
+4
+6
+8
+10
+12
+14
+ Ar
+2
+/c
+2
+GeV
+2
+cm
+ 
+-38
+10
+ 
+T,y
+p
+δ
+ d
+T,x
+p
+δ
+d
+σ
+2
+d
+ < -0.15 GeV/c
+T,y
+p
+δ
+G18 No RPA    
+Figure 3-25: Cross-section interaction breakdown for events with 𝛿𝑝𝑇,𝑦 < -0.15 GeV/c.
+The breakdown is shown for (top left) the G18 LFG configuration, (top right) the G18
+RFG configuration, (bottom left) the G18 EffSF configuration, and (bottom right)
+the G18 No RPA configuration.
+86
+
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+ [GeV/c]
+T,x
+p
+δ
+0
+20
+40
+60
+80
+ Ar
+2
+/c
+2
+GeV
+2
+cm
+ 
+-38
+10
+ 
+T,y
+p
+δ
+ d
+T,x
+p
+δ
+d
+σ
+2
+d
+QE
+MEC
+RES
+DIS
+MicroBooNE Data
+ Shape Unc)
+⊕
+(Stat 
+Norm Unc
+ < 0.15 GeV/c
+T,y
+p
+δ
+G18 LFG    -0.15 < 
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+ [GeV/c]
+T,x
+p
+δ
+0
+20
+40
+60
+80
+ Ar
+2
+/c
+2
+GeV
+2
+cm
+ 
+-38
+10
+ 
+T,y
+p
+δ
+ d
+T,x
+p
+δ
+d
+σ
+2
+d
+ < 0.15 GeV/c
+T,y
+p
+δ
+G18 RFG    -0.15 < 
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+ [GeV/c]
+T,x
+p
+δ
+0
+20
+40
+60
+80
+ Ar
+2
+/c
+2
+GeV
+2
+cm
+ 
+-38
+10
+ 
+T,y
+p
+δ
+ d
+T,x
+p
+δ
+d
+σ
+2
+d
+ < 0.15 GeV/c
+T,y
+p
+δ
+G18 EffSF    -0.15 < 
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+ [GeV/c]
+T,x
+p
+δ
+0
+20
+40
+60
+80
+ Ar
+2
+/c
+2
+GeV
+2
+cm
+ 
+-38
+10
+ 
+T,y
+p
+δ
+ d
+T,x
+p
+δ
+d
+σ
+2
+d
+ < 0.15 GeV/c
+T,y
+p
+δ
+G18 No RPA    -0.15 < 
+Figure 3-26: Cross-section interaction breakdown for events with -0.15 < 𝛿𝑝𝑇,𝑦 <
+0.15 GeV/c. The breakdown is shown for (top left) the G18 LFG configuration, (top
+right) the G18 RFG configuration, (bottom left) the G18 EffSF configuration, and
+(bottom right) the G18 No RPA configuration.
+3.2.5
+Kinematic Imbalance Cross-Section Analysis Conclusions
+The first measurement of 𝜈𝜇 CC1p0𝜋 single and double differential cross sections on
+argon as a function of kinematic imbalance variables for event topologies with a single
+muon and a single proton detected in the final state was reported. The unfolded data
+results were compared to a number of event generators, available model configurations
+and FSI modeling options. This measurement identified regions of the phase-space
+which are ideal to provide constraints for nuclear and final state interaction effects in
+generator predictions essential for the extraction of oscillation parameters.
+3.3
+Prospects With Future Neutrino Experiments
+The MicroBooNE experiment is a crucial step in the understanding of the underlying
+neutrino-argon interactions that will be used to drastically reduce the uncertainties of
+forthcoming high precision neutrino oscillation measurements. Furthermore, Micro-
+BooNE is the first LArTPC in a neutrino beam with automated event reconstruction
+and selection.
+87
+
+This work paved the path towards precision 𝜈𝜇 CC cross section measurements
+with a single proton and no pions in the final state.
+However, only three out of
+the five available run periods are used in the results presented in this thesis, as
+shown in figure 3-27. Within the next year, runs 4 and 5 will also become available.
+Therefore, the statistical uncertainties will be further reduced, and the path to further
+multidimensional analyses will be further explored.
+Figure 3-27: MicroBooNE total Protons on Target (POT) collected with the Booster
+Neutrino Beam (BNB) during the five run periods. In this thesis, the first three run
+periods were used.
+Beyond MicroBooNE, experiments of the Short-Baseline Neutrino (SBN) Pro-
+gram at Fermilab, namely ICARUS and SBND, have already turned (or will very
+soon turn) on. The progress of SBN, which shares the same technology and beam
+as MicroBooNE, will largely benefit from the LArTPC expertise developed by the
+MicroBooNE collaboration. With these experiments, the largest neutrino-argon scat-
+tering data sets will be collected and will be used to test the performance of our
+theory predictions in multiple variables simultaneously.
+Despite the abundance of neutrino-argon cross sections that will be reported at the
+SBN program, the relevant energy range is significantly lower than the DUNE energy
+spectrum. However, there is a wealth of cross-section opportunities that will become
+available once the ArgonCube demonstrator, a 2×2 grid engineering prototype for
+the LArTPC DUNE Near Detector (ND) module design, is commissioned in Fall
+2022. ArgonCube will provide sufficiently high statistics to measure cross sections
+88
+
+2.0E21
+DeliveredPOT
+POTontape
+Run 1
+Run 2
+Run 3l
+Run 4
+Run5
+1.5E21
+Cumulative POT
+1.0E21
+5.0E20
+0.0E00
+20151
+2016
+2017
+2018
+2019
+2020and to evaluate the corresponding systematic uncertainties using both multi-particle
+channels and multidimensional analyses and will pave the path towards the final
+design of the DUNE ND.
+All these neutrino experimental efforts will be complemented by the continuous
+benchmarking of the neutrino event generators predictions against external data sets,
+such as against electron scattering data sets. A major step in this direction is made
+in chapter 4 with the unification of the event generation process and of the modeling
+across the two particle species. Furthermore, chapters 5 and 6 detail the analysis of
+electron scattering data sets from the CLAS detector at Thomas Jefferson Laboratory
+following neutrino data analysis methods and the testing against the performance of
+commonly used GENIE event generator.
+89
+
+90
+
+Chapter 4
+Inclusive Electron Scattering And
+The GENIE Event Generator
+[Phys. Rev. D 103, 113003 (2021)]
+4.1
+Electron-Nucleus Modeling Development
+As already discussed in section 1.2, the extraction of neutrino mixing parameters
+from neutrino oscillation experiments [23, 146, 147] relies on comparing the energy-
+dependent neutrino event distribution for a particular neutrino flavor near the neu-
+trino production point with the corresponding event distribution at a significant dis-
+tance away.
+In practice, the yield at each neutrino energy is extracted from the
+measured neutrino-nucleus interactions in a detector, as reconstructed from the mea-
+sured particles ejected in the neutrino-nucleus interaction.
+This requires detailed
+knowledge of the neutrino-nucleus interactions.
+Unfortunately, measuring the neutrino-nucleus interaction is difficult due to the
+wide energy spread of accelerator-produced neutrino beams [148], as can be seen in
+figure 4-1, and the tiny neutrino-nucleus cross section. A relatively small body of data
+has been published [24], which suffers from poor statistics and is flux-averaged over
+a wide range of neutrino energies. This data is then supplemented with theoretical
+models and implemented into event generator codes such as GENIE [33], which is
+extensively used across US-based neutrino experiments, to simulate the neutrino-
+nucleus interactions across a wide range of energies and targets. GENIE simulations
+are then used to aid in extraction of the incident neutrino flux as a function of
+91
+
+energy from the neutrino-nucleus scattering events measured in neutrino detectors.
+However, the theoretical models need to describe many different interaction processes
+for medium to heavy nuclei (typically C, O, or Ar) where nuclear effects complicate the
+interactions. As a result, the uncertainties in the extraction of oscillation parameters
+are often dominated by the lack of knowledge of the neutrino-nucleus interactions [23,
+147].
+1
+2
+3
+4
+5
+0
+10
+20
+30
+DIS
+RES
+MEC
+QE
+(32%)
+(37%)
+(7%)
+(22%)
+1
+2
+3
+4
+5
+0
+20
+40
+60
+DIS
+RES
+MEC
+QE
+(28%)
+(36%)
+(8%)
+(26%)
+ [Arb Units]
+ν
+dE
+σ
+d
+ [GeV]
+ν
+E
+ [Arb Units]
+ν
+dE
+σ
+d
+ [GeV]
+ν
+E
+Figure 4-1: Charged-current cross sections as a function of neutrino energy obtained
+using GENIE for muon neutrino scattering using the DUNE near detector (left) and
+far detector (right) oscillated fluxes. The shaded bands show the fractional contri-
+bution for each interaction mechanism, quasielastic scattering (QE), meson-exchange
+currents (MEC), resonance excitation (RES), and deep inelastic scattering (DIS). See
+text for details of the interaction mechanisms. The numbers in parentheses indicate
+the percentage of the cross section due to each interaction mechanism.
+Figure 4-1 shows such a wide energy spectrum for the DUNE near detector flux-
+averaged cross sections (left) and the far detector oscillated flux-averaged cross sec-
+tions (right) using one model configuration in GENIE. All four neutrino-nucleus re-
+action mechanisms contribute significantly and all four need to be well understood.
+This is especially true because different reaction mechanisms contribute differently in
+the different oscillation peaks. Understanding one reaction mechanism better than
+the others could have significant implications for oscillation analyses.
+To improve our understanding of neutrino-nucleus interactions, we can take ad-
+vantage of the fact that neutrinos and electrons are both leptons. Thus, they interact
+92
+
+with atomic nuclei in similar ways via the same reaction mechanisms, as illustrated
+in figure 4-2 and detailed in sections 1.4 and 1.5.
+Figure 4-2: (Left) electron-nucleus inclusive scattering via one-photon exchange and
+(right) charged current neutrino-nucleus inclusive scattering via 𝑊 exchange with a
+final state charged lepton.
+The most common lepton-nucleus interaction mechanisms are shown in figure 4-
+3 include: (a) quasielastic (QE) scattering from individual moving nucleons in the
+nucleus; (b) two-nucleon knockout, due to interactions with a meson being exchanged
+between two nucleons referred to two-particle two-hole excitations, 2p2h or its major
+component, meson exchange currents (MEC); (c) interactions which leave the struck
+nucleon in an excited resonance state (RES); and (d) nonresonant interactions with
+a quark within the nucleon (DIS).
+Figure 4-3: Reaction mechanisms for lepton-nucleus scattering (a) quasielastic scat-
+tering (QE) where one nucleon is knocked out of the nucleus, (b) 2p2h where two
+nucleons are knocked out of the nucleus, (c) RES resonance production where a nu-
+cleon is excited to a resonance which decays to a nucleon plus meson(s), and (d) DIS
+where the lepton interacts with a quark in the nucleon.
+All these interaction processes have to be described in detailed in event generators
+like GENIE. GENIE started as an event generator that could exclusively handle
+93
+
+V1
+SSneutrino interactions. In recognition of the importance of electron scattering, the
+latter was added as a new option in close conjunction with the neutrino scattering
+section. As much as possible, the neutrino cross section references vector and axial
+contributions separately and uses the same modeling for vector interactions as the
+electron section. Some models were developed separately for electrons and others
+were developed for both applications in tandem.
+An earlier electron version of GENIE (v2.12.10) had already been tested by com-
+paring with inclusive (𝑒, 𝑒′) data [149]. Although the QE peak was well-described for
+a variety of energies and nuclei, the RES region was poorly described. However, the
+establishment of full compatibility between the electron and neutrino versions was
+then still in its early stages.
+With this work, we significantly improved both neutrino and electron versions of
+GENIE to address these and other issues. We fixed significant errors in the previous
+version, including an error in the Mott cross section in the electron QE Rosenbluth
+interaction, a missing Lorentz boost in the MEC interaction affecting both electron
+and neutrino interactions, and incorrect electron couplings used in the RES interac-
+tions. We worked to better integrate the electron and neutrino codes for QE and
+MEC models. We also added more up-to-date models such as SuSAv2 [150]. These
+changes have been incorporated in the latest GENIE version (v3.2.0). We refer to
+the electron-scattering component of the widely-used GENIE [33] event generator as
+eGENIE.
+The GENIE improvements can be seen in figure 4-4 [151].
+The QE peak (at
+𝜔 ≈ 0.15 GeV/c) predicted by the older GENIE v2 is too large and is slightly shifted
+to higher energy transfer than the data, while the first simulated resonance peak
+is at a larger energy transfer than the one observed in the data.
+The QE peak
+predicted by the updated GENIE v3 has about the correct integral and is at the
+correct energy transfer (but is slightly too narrow) and the first resonance peak is
+located at 𝑚Δ − 𝑚 ≈ 300 MeV beyond the QE peak, as expected. Details of the
+calculations and of the discrepancies between GENIE v3 and the data are discussed
+in detail below.
+94
+
+0.1
+0.2
+0.3
+0.4
+0
+2
+4
+6
+3
+10
+×
+o
+ = 60
+θ
+0.56 GeV, 
+v2
+v3
+b/sr/GeV]
+µ
+ [
+ dE
+Ω
+d
+σ
+2
+d
+Energy Transfer [GeV]
+Figure 4-4: Comparison between GENIE v2 and v3 descriptions of inclusive C(𝑒, 𝑒′)
+scattering cross sections at 𝐸0 = 0.56 GeV, 𝜃𝑒 = 60∘ and 𝑄2
+𝑄𝐸 ≈ 0.24 GeV2. Black
+points show the data, solid black line shows the GENIE v3 results and dashed black
+line shows the GENIE v2 results.
+In this analysis, we specifically focused on testing our knowledge of the electron-
+nucleus cross section by benchmarking eGENIE against existing inclusive electron
+scattering data for different target nuclei, beam energies and scattering angles. The
+goals are very similar to the ones in reference [149], but we test a much more modern
+version of eGENIE and we also compare different models within eGENIE. In addition,
+if eGENIE describes electron-nucleus scattering well, then it would be an improvement
+on the former empirical fit [152] and would be valuable for helping simulate a variety
+of electron experiments.
+For fixed incident beam energy and scattered electron angle, the dominant process
+changes from QE at low energy transfer (𝜔 ≈ 𝑄2/2𝑚) through MEC to RES and to
+DIS at high energy transfer. Therefore, examining the agreement of eGENIE with
+data as a function of energy transfer can provide valuable insight into the specific
+shortcomings of the eGENIE models and their implementations.
+This separation
+according to the underlying physics interactions gives valuable insights which are not
+presently possible with neutrino cross sections, because only broad-energy beams are
+available.
+The GENIE simulation framework offers several models of the nuclear ground
+state, multiple models for each of the electron- or neutrino-nucleus scattering mecha-
+95
+
+nisms accompanied by various tunable model parameters, and a number of models for
+hadronic final state interactions (FSI), i.e., intranuclear rescattering of the outgoing
+hadrons [33, 104, 153]. We describe the different models relevant for this work and
+the electron-specific effects that we accounted for during the eGENIE development
+below [42].
+Since our goal is to use electron scattering data to validate neutrino interaction
+modeling in GENIE, the GENIE code for electron and neutrino interactions are unified
+in many places. The neutrino interacts with a nucleus via the weak interaction and
+massive 𝑊 or 𝑍 exchange, whereas the electron interacts mostly electromagnetically
+via massless photon exchange, as shown in figure 4-2. This causes the cross sections
+to differ by an overall factor of
+8𝜋2𝛼2
+𝐺2
+𝐹
+1
+𝑄4
+(4.1)
+when equations 1.18 and 1.19 are compared. In the code, both interactions use
+the same nuclear ground state and many of the nuclear reaction effects, such as FSI,
+are very similar or identical. Except for mass effects and form factors, the electron
+nucleus cross section can be obtained by setting the axial part of the interaction to
+zero. We also accounted for isoscalar and isovector terms appropriately.
+Many of the models reported in this work, except for SuSAv2, use the GENIE
+implementation of the Local Fermi gas (LFG) model to describe the nuclear ground
+state. In the simplest Fermi gas model, nucleons occupy all momentum states up to
+the global Fermi momentum 𝑘𝐹 with equal probability. In the LFG model, the Fermi
+momentum at a given radial position depends on the local nuclear density obtained
+from measurements of nuclear charge densities. To account for this radial depen-
+dence, GENIE selects an initial momentum for the struck nucleon by first sampling
+an interaction location 𝑟 inside the nucleus according to the nuclear density. The
+nucleon momentum is then drawn from a Fermi distribution using the local Fermi
+momentum 𝑘𝐹(𝑟).
+Another commonly used nuclear model is the Relativistic Fermi Gas (RFG). Here
+96
+
+a global momentum distribution is used for the entire nucleus, independent of the
+interaction location in the nucleus. However, a high-momentum tail of nucleons with
+momenta above the Fermi-momentum is included. This tail is meant to approximately
+account for the effects of two-nucleon short-range correlations [154,155] and follows
+a 1/𝑘4 distribution, where 𝑘 is the nucleon momentum.
+We consider two distinct sets of eGENIE configurations:
+• G2018, which uses the LFG nuclear model, the Rosenbluth cross section for
+QE scattering, and the empirical MEC model [109]. This model set is formally
+marked as the G18_10a_02_11a configuration of GENIE v3.
+• GSuSAv2 [28], which follows the universal SuSAv2 super-scaling approach
+to lepton scattering.
+This new model set is included in the latest GENIE
+v3.2.0 release as the GEM21_11b_00_000 configuration for electron scattering
+and G21_11b_00_000 for neutrino interactions.
+In both model sets, RES is modeled using the Berger-Sehgal model [125] and DIS
+reactions are modeled using Bodek and Yang [156]. The models are described in more
+detail below.
+In QE interactions, a lepton scatters on a single nucleon, removing it from the
+spectator 𝐴 − 1 nucleus unless final-state interactions lead to reabsorption.
+The
+electron QE interaction in the G2018 configuration of GENIE uses the Rosenbluth
+cross section with the vector structure function parametrization of reference [157].
+We corrected the implementation of this model for eGENIE and modified the cross
+section to account for the identified issues. This electron QE cross section differs in
+important ways from the Valencia CCQE model [158] used in the G2018 configuration
+for neutrinos. Most notably, the Rosenbluth treatment lacks medium polarization
+corrections.
+A new QE model in GENIE, based on the SuSAv2 approach [28, 159, 160], uses
+superscaling to write the inclusive cross section in terms of a universal function inde-
+pendent of momentum transfer and nucleus. For EM scattering, the scaling function
+may be expressed in the form
+97
+
+𝑓(𝜓′) = 𝑘𝐹
+𝑑2𝜎
+𝑑Ω𝑒𝑑𝜈
+𝜎𝑀𝑜𝑡𝑡(𝑣𝐿𝐺𝑒𝑒′
+𝐿 + 𝑉𝑇𝐺𝑒𝑒′
+𝑇 ) ,
+(4.2)
+where 𝜓′ is a dimensionless scaling variable, 𝑘𝐹 is the nuclear Fermi momentum,
+the denominator is the single-nucleon elastic cross section, 𝑣𝐿 and 𝑣𝑇 are known
+functions of kinematic variables, and 𝐺𝑒𝑒′
+𝐿 (𝑞, 𝜔) and 𝐺𝑒𝑒′
+𝑇 (𝑞, 𝜔) are the longitudinal
+and transverse nucleon structure functions linearly related to 𝐹 𝑒
+1 and 𝐹 𝑒
+2 [161]. For
+eGENIE, we extended the original neutrino implementation [28] to the electron case.
+The original SuSAv2 QE cross section calculations used a Relativistic Mean Field
+(RMF) model of the nuclear ground state [162, 163].
+This approach includes the
+effects of the real part of the nucleon-nucleus potential on the outgoing nucleons
+which creates a “distorted” nucleon momentum distribution.
+Although GENIE lacks the option to use an RMF nuclear model directly, we
+achieve approximate consistency with the RMF-based results by using a two-step
+strategy for QE event generation. First, an energy and scattering angle for the out-
+going lepton are sampled according to the inclusive double-differential cross section.
+This cross section is computed by interpolating precomputed values of the nuclear
+responses 𝐺𝑒𝑒′
+𝐿 (𝑞, 𝜔) and 𝐺𝑒𝑒′
+𝐿 (𝑞, 𝜔) which are tabulated on a two-dimensional grid in
+(𝑞, 𝜔) space.
+The responses were obtained using the original RMF-based SuSAv2
+calculation. Second, the outgoing nucleon kinematics are determined by choosing its
+initial momentum from an LFG distribution. The default nucleon binding energy used
+in GENIE for the LFG model is replaced for SuSAv2 with an effective value tuned to
+most closely duplicate the RMF distribution. The outgoing nucleon kinematics are
+not needed for the comparisons to inclusive (𝑒, 𝑒′) data shown in this work.
+MEC describes an interaction that results in the ejection of two nucleons from
+the nucleus, thus is often referred to as 2p2h. It typically proceeds via lepton inter-
+action with a pion being exchanged between two nucleons or by interaction with a
+nucleon in an Short Range Correlated (SRC) pair. MEC is far less understood than
+other reaction mechanisms because, unlike the others, it involves scattering from two
+98
+
+nucleons simultaneously. GENIE has several models for MEC.
+The G2018 configuration of eGENIE uses the empirical MEC model [109] that
+is useable for both electron- and neutrino-nucleus scattering. It assumes that the
+MEC peak for inclusive scattering has a Gaussian distribution in 𝑊 and is located
+between the QE and first RES peaks. Although both versions of the model use the
+same effective form factors, the amplitude of the MEC peak was tuned separately to
+electron and neutrino scattering data. This model was developed in the context of
+empirically fitting GENIE to MiniBooNE inclusive neutrino scattering data and is still
+used for neutral-current interactions [109]. For charged-current neutrino interactions,
+𝜈GENIE G2018 uses the very different Valencia 2p2h model [158,164] instead of the
+empirical model.
+For the description of the 2p2h MEC contributions, the SuSAv2 model uses the
+fully relativistic calculations from [165]. This treatment allows for a proper separation
+of neutron-proton and proton-proton pairs in the final state via the analysis of the
+direct-exchange interference terms [166]. This approach is capable of reproducing the
+nuclear dynamics and superscaling properties observed in inclusive electron-nucleus
+scattering reactions [167–169]. The latter serves as a robust test for nuclear models.
+It further provides an accurate description of existing neutrino data [169–173]. As in
+the case for the SuSAv2 QE model, we extended the original GENIE implementation
+of SuSAv2 MEC for neutrinos to the electron case for eGENIE [28,159,174,175].
+The SuSAv2 MEC approach is the only fully relativistic model that can be ex-
+tended without approximations to the full-energy range of interest for neutrino scat-
+tering events. Therefore, it is a very promising modeling choice for present and future
+neutrino experiments for one of the least understood interaction channels.
+RES production in GENIE is simulated using the Berger-Sehgal model [125], in
+which the lepton interacts with a single moving nucleon and excites it to one of
+16 resonances. The cross sections are calculated based on the Feynman-Kislinger-
+Ravndal (FKR) model [176], without any interferences between them. Form factors
+are derived separately for vector and axial probes [110] but have not been updated
+to include recent electron scattering results.
+99
+
+The GENIE treatment of DIS used in this work is based on that of Bodek and
+Yang [156]. Hadronization is modeled using an approach which transitions gradually
+as a function of the hadronic invariant mass 𝑊 between the AGKY model [177] and
+the PYTHIA 6 model [178].
+At low 𝑊 values, the Bodek-Yang differential cross
+section is scaled by tunable parameters that depend on the multiplicity of hadrons in
+the final-state [104].
+Integration of the RES and DIS contributions is complicated by the need for a
+model of nonresonant meson production. There is no definite separation of RES and
+DIS contributions. GENIE makes a sharp cutoff at 𝑊 = 1.93 GeV in the latest tune
+and uses a suppression factor to enable usage of the Bodek-Yang cross section at low
+𝑊 in place of a true nonresonant model. These features were recently retuned by the
+GENIE collaboration using measurements of charged-current 𝜈𝜇 and ¯𝜈𝜇 scattering on
+deuterium [179]. The W cutoff and suppression factors apply to both electron- and
+neutrino-nucleus models.
+Final state interactions of outgoing hadrons with the residual nuclei are calcu-
+lated in eGENIE using the INTRANUKE [153,180] package and one of two options.
+The first, hA, an empirical data-driven method, uses the cross-section of pions and
+nucleons with nuclei as a function of energy up to 1.2 GeV and the CEM03 [181]
+calculation for higher energies. The second, hN, is a full intra-nuclear cascade cal-
+culation of the interactions of pions, kaons, photons, and nucleons with nuclei. In
+the hN model, each outgoing particle can interact successively with any or all the
+nucleons it encounters on its path leaving the nucleus, and any particles created in
+those interactions can also subsequently reinteract. The ability of the two models to
+describe hadron-nucleus data is very similar. The eGENIE G2018 configuration uses
+the hA FSI model, while GSuSAv2 uses hN. However, the choice of FSI model has no
+effect on the inclusive cross sections considered in the present work.
+100
+
+4.2
+Inclusive Electron Scattering Data Comparisons
+To test eGENIE, we compared inclusive electron scattering data to theoretical pre-
+dictions made using two different program configurations which differ in their choice
+of QE and MEC models. Namely, G2018 adopts the Rosenbluth model for QE in-
+teractions and the empirical Dytman model for MEC events, while GSuSAv2 uses
+SuSAv2 for both QE and MEC interactions.
+Figures 4-5, 4-6 and 4-7 show the inclusive C(𝑒, 𝑒′) cross sections for a wide range
+of beam energies and scattering angles compared to the G2018 and GSuSAv2 mod-
+els [151,182–189].
+The QE peak is the one at lowest energy transfer (𝜈 ≈ 𝑄2/2𝑚) in each plot.
+The next peak at about 300 MeV larger energy transfer corresponds to the ∆(1232)
+excitation and the “dip-region” is between the two peaks. The ∆ peak in the data
+is separated from the QE peak by less than the 300 MeV ∆-nucleon mass difference,
+indicating that it is shifted in the nuclear medium. This shift is more visible at lower
+momentum transfer where the ∆ peak is more prominent.
+GSuSAv2 clearly describes the QE and dip-regions much better than G2018, es-
+pecially at the three lowest momentum transfers, as shown in figure 4-5. G2018 has
+particular difficulty describing the data for 𝐸0 = 0.24 GeV and 𝜃𝑒 = 60∘, where
+𝑄2 = 0.05 GeV2 at the QE peak. G2018 also predicts too small a width for the QE
+peak and too small a MEC contribution for 𝐸0 = 0.56 GeV and 𝜃𝑒 = 60∘. GSuSAv2
+describes both features far better.
+At intermediate momentum transfers shown in figure 4-6, GSuSAv2 describes
+the data somewhat better than G2018, although it overpredicts the dip-region cross
+section at 𝐸0 = 1.299 GeV and 𝜃𝑒 = 37.5∘. The MEC contribution for G2018 appears
+to be much too small for 𝐸0 = 2.222 GeV and 𝜃𝑒 = 15.54∘ (𝑄2
+𝑄𝐸 = 0.33 GeV2).
+Both model sets significantly disagree with the data in the resonance region, where
+they use the same RES and DIS models. The 0.961 GeV, 37.5∘ and the 2.222 GeV,
+15.54∘ data are taken at almost identical 𝑄2
+𝑄𝐸. The lower beam-energy data is more
+transverse, since it is at larger scattering angle. The GSuSAv2 MEC contribution is
+101
+
+0.02 0.04 0.06 0.08 0.1
+0
+50
+3
+10
+×
+o
+ = 60
+θ
+C, 0.24 GeV, 
+12
+0.02 0.04 0.06 0.08 0.1
+0
+50
+3
+10
+×
+0
+0.1
+0.2
+0.3
+0.4
+0
+50
+3
+10
+×
+o
+ = 36
+θ
+C, 0.56 GeV, 
+12
+0
+0.1
+0.2
+0.3
+0.4
+0
+50
+3
+10
+×
+0
+0.1
+0.2
+0.3
+0.4
+0
+2
+4
+6
+3
+10
+×
+o
+ = 60
+θ
+C, 0.56 GeV, 
+12
+0
+0.1
+0.2
+0.3
+0.4
+0
+2
+4
+6
+3
+10
+×
+GSuSav2
+G2018
+b/sr/GeV]
+µ
+ [
+ dE
+Ω
+d
+σ
+2
+d
+Energy Transfer [GeV]
+Energy Transfer [GeV]
+Figure 4-5: Comparison of inclusive C(𝑒, 𝑒′) scattering cross sections for data and for
+GENIE. (left) data vs GSuSAv2 and (right) data vs G2018. (top) 𝐸0 = 0.24 GeV,
+𝜃𝑒 = 60∘ and 𝑄2
+𝑄𝐸 ≈ 0.05 GeV2 (middle) 𝐸0 = 0.56 GeV, 𝜃𝑒 = 36∘ and 𝑄2
+𝑄𝐸 ≈ 0.11
+GeV2, and (bottom) 𝐸0 = 0.56 GeV, 𝜃𝑒 = 60∘ and 𝑄2
+𝑄𝐸 ≈ 0.24 GeV2. Black points
+show the data, solid black lines show the total GENIE prediction, colored lines show
+the contribution of the different reaction mechanisms: (blue) QE, (red) MEC, (green)
+RES and (orange) DIS.
+similar for both data sets, but the G2018 MEC contribution is far smaller for the
+higher beam-energy data. The GSuSAv2 MEC contribution describes the dip-region
+better in the higher beam-energy data set. The RES model appears to agree with the
+102
+
+0.2
+0.4
+0.6
+0
+5
+3
+10
+×
+o
+ = 37.5
+θ
+C, 0.961 GeV, 
+12
+0.2
+0.4
+0.6
+0
+5
+3
+10
+×
+0.2
+0.4
+0.6
+0.8
+0
+1
+2
+3
+10
+×
+o
+ = 37.5
+θ
+C, 1.299 GeV, 
+12
+0.2
+0.4
+0.6
+0.8
+0
+1
+2
+3
+10
+×
+0
+0.2
+0.4
+0.6
+0.8
+0
+20
+40
+3
+10
+×
+o
+ = 15.54
+θ
+C, 2.222 GeV, 
+12
+0
+0.2
+0.4
+0.6
+0.8
+0
+20
+40
+3
+10
+×
+GSuSav2
+G2018
+b/sr/GeV]
+µ
+ [
+ dE
+Ω
+d
+σ
+2
+d
+Energy Transfer [GeV]
+Energy Transfer [GeV]
+Figure 4-6: Comparison of inclusive C(𝑒, 𝑒′) scattering cross sections for data and
+for GENIE. (left) data vs GSuSAv2 and (right) data vs G2018. (top) 𝐸0 = 0.96
+GeV, 𝜃𝑒 = 37.5∘ and 𝑄2
+𝑄𝐸 ≈ 0.32 GeV2, (middle) 𝐸0 = 1.30 GeV, 𝜃𝑒 = 37.5∘ and
+𝑄2
+𝑄𝐸 ≈ 0.54 GeV2, and (bottom) 𝐸0 = 2.22 GeV, 𝜃𝑒 = 15.5∘ and 𝑄2
+𝑄𝐸 ≈ 0.33 GeV2.
+Black points show the data, solid black lines show the total GENIE prediction, colored
+lines show the contribution of the different reaction mechanisms: (blue) QE, (red)
+MEC, (green) RES and (orange) DIS.
+data slightly better for the lower beam-energy, more transverse, data set.
+At the highest momentum transfers (𝑄2 ≈ 1 GeV2) shown in figure 4-7, the dis-
+agreement at the larger energy transfers is far greater. The empirical G2018 MEC
+103
+
+0.4
+0.6
+0.8
+0
+0.5
+1
+1.5
+3
+10
+×
+o
+ = 37.5
+θ
+C, 1.501 GeV, 
+12
+0.2
+0.4
+0.6
+0.8
+0
+0.5
+1
+1.5
+3
+10
+×
+0.5
+1
+1.5
+0
+2
+4
+3
+10
+×
+o
+ = 16
+θ
+C, 3.595 GeV, 
+12
+0.5
+1
+1.5
+0
+2
+4
+3
+10
+×
+0.4
+0.5
+0.6
+0.7
+0.8
+0
+0.2
+0.4
+0.6
+o
+ = 20
+θ
+C, 3.595 GeV, 
+12
+0.4
+0.5
+0.6
+0.7
+0.8
+0
+0.2
+0.4
+0.6
+GSuSav2
+G2018
+b/sr/GeV]
+µ
+ [
+ dE
+Ω
+d
+σ
+2
+d
+Energy Transfer [GeV]
+Energy Transfer [GeV]
+Figure 4-7: Comparison of inclusive C(𝑒, 𝑒′) scattering cross sections for data and for
+GENIE. (left) data vs GSuSAv2 and (right) data vs G2018. (top) 𝐸0 = 1.501 GeV,
+𝜃𝑒 = 37.5∘ and 𝑄2
+𝑄𝐸 ≈ 0.92 GeV2, (middle) 𝐸0 = 3.595 GeV, 𝜃𝑒 = 16∘ and 𝑄2
+𝑄𝐸 ≈ 1.04
+GeV2, and (bottom) 𝐸0 = 3.595 GeV, 𝜃𝑒 = 20∘ and 𝑄2
+𝑄𝐸 ≈ 1.3 GeV2. Black points
+show the data, solid black lines show the total GENIE prediction, colored lines show
+the contribution of the different reaction mechanisms: (blue) QE, (red) MEC, (green)
+RES and (orange) DIS.
+model contributions are negligible, in marked contrast to the GSuSAv2 MEC con-
+tributions. The RES and DIS contributions are very significant at high 𝑄2 and in
+general the GENIE model is larger than the data in the region dominated by RES
+104
+
+0.1
+0.2
+0.3
+0.4
+0
+10
+20
+3
+10
+×
+o
+ = 60
+θ
+Fe, 0.56 GeV, 
+56
+0.1
+0.2
+0.3
+0.4
+0
+10
+20
+3
+10
+×
+0.2
+0.4
+0.6
+0
+20
+40
+3
+10
+×
+o
+ = 37.5
+θ
+Fe, 0.961 GeV, 
+56
+0.2
+0.4
+0.6
+0
+20
+40
+3
+10
+×
+0.2
+0.4
+0.6
+0.8
+0
+5
+10
+3
+10
+×
+o
+ = 37.5
+θ
+Fe, 1.299 GeV, 
+56
+0.2
+0.4
+0.6
+0.8
+0
+5
+10
+3
+10
+×
+GSuSav2
+G2018
+b/sr/GeV]
+µ
+ [
+ dE
+Ω
+d
+σ
+2
+d
+Energy Transfer [GeV]
+Energy Transfer [GeV]
+Figure 4-8: Comparison of inclusive Fe(𝑒, 𝑒′) scattering cross sections for data and for
+GENIE. (left) data vs GSuSAv2 and (right) data vs G2018. (top) Fe(𝑒, 𝑒′), 𝐸0 = 0.56
+GeV, 𝜃𝑒 = 60∘ and 𝑄2
+𝑄𝐸 ≈ 0.24 GeV2, (middle) Fe(𝑒, 𝑒′), 𝐸0 = 0.96 GeV, 𝜃𝑒 = 37.5∘
+and 𝑄2
+𝑄𝐸 ≈ 0.32 GeV2, (bottom) Fe(𝑒, 𝑒′), 𝐸0 = 1.30 GeV, 𝜃𝑒 = 37.5∘ and 𝑄2
+𝑄𝐸 ≈ 0.54
+GeV2. Black points show the data, solid black lines show the total GENIE prediction,
+colored lines show the contribution of the different reaction mechanisms: (blue) QE,
+(red) MEC, (green) RES and (orange) DIS.
+interactions [149]. In addition, GENIE does not include the nuclear medium depen-
+dent ∆-peak shift, so that the predicted location of the ∆-peak is at larger energy
+transfer than that of the data.
+105
+
+0
+0.2
+0.4
+0.6
+0.8
+0
+50
+100
+3
+10
+×
+o
+ = 15.54
+θ
+Ar, 2.222 GeV, 
+40
+0
+0.2
+0.4
+0.6
+0.8
+0
+50
+100
+3
+10
+×
+GSuSav2
+G2018
+b/sr/GeV]
+µ
+ [
+ dE
+Ω
+d
+σ
+2
+d
+Energy Transfer [GeV]
+Energy Transfer [GeV]
+Figure 4-9: Comparison of inclusive Ar(𝑒, 𝑒′) scattering cross sections for data and for
+GENIE at 𝐸0 = 2.22 GeV, 𝜃𝑒 = 15.5∘ and 𝑄2
+𝑄𝐸 ≈ 0.33 GeV2. (left) data vs GSuSAv2
+and (right) data vs G2018. Black points show the data, solid black lines show the
+total GENIE prediction, colored lines show the contribution of the different reaction
+mechanisms: (blue) QE, (red) MEC, (green) RES and (orange) DIS.
+Figure 4-8 shows the inclusive Fe(𝑒, 𝑒′) cross sections for several beam energies and
+scattering angles compared to the G2018 and GSuSAv2 models. The GSuSAv2 model
+describes the QE region better for all three data sets. As described previously, the
+GSuSAv2 MEC model is independently calculated. The empirical G2018 MEC model
+was fit using GENIE v2 QE and RES models. The fit will have to be redone once the
+QE and RES models stabilize. The GSuSAv2 MEC contributions are significantly
+larger than the empirical G2018 MEC contributions and match the dip-region data
+far better at 𝑄2
+𝑄𝐸 = 0.24 and 0.32 GeV2. However, it overpredicts the dip-region
+cross section at 𝑄2
+𝑄𝐸 = 0.54 GeV2. The RES and DIS models describe the Fe data
+better than the C data at large energy transfers.
+Figure 4-9 shows the inclusive Ar(𝑒, 𝑒′) cross sections for 𝐸0 = 2.222 GeV and
+𝜃𝑒 = 15.54∘ [185] compared to the G2018 and GSuSAv2 models. The GSuSAv2 model
+reproduces the data very well in the QE-peak region and the G2018 reproduces the
+data moderately well.
+The GSuSAv2 MEC model describes the dip-region much
+better than the G2018 model. Again, there is significant disagreement with the RES
+and DIS models at larger energy transfers.
+The quality of the agreement between data and GENIE depends more on the
+beam energy and angle than on the target mass from C to Fe. There is a possible
+106
+
+0.5
+1
+1.5
+0
+500
+1000
+o
+ = 20
+θ
+2.445 GeV, 
+0.5
+1
+1.5
+0
+500
+1000
+1
+1.5
+2
+0
+50
+o
+ = 26.98
+θ
+3.245 GeV, 
+1
+1.5
+2
+0
+50
+3.4
+3.6
+3.8
+4
+0
+0.5
+1
+o
+ = 41
+θ
+5.5 GeV, 
+3.4
+3.6
+3.8
+4
+0
+0.5
+1
+Proton
+Deuterium
+ [nb/sr/GeV]
+ dE
+Ω
+d
+σ
+2
+d
+Energy Transfer [GeV]
+Energy Transfer [GeV]
+Figure 4-10: Comparison of inclusive proton (left) and deuterium (right) (𝑒, 𝑒′) scat-
+tering cross sections for data and for GENIE using G2018. (top) 𝐸0 = 2.445 GeV
+and 𝜃𝑒 = 20∘, (middle) 𝐸0 = 3.245 GeV and 𝜃𝑒 = 26.98∘, and (bottom) 𝐸0 = 5.5 GeV
+and 𝜃𝑒 = 41∘. Black points show the data, solid black lines show the total GENIE
+prediction, colored lines show the contribution of the different reaction mechanisms:
+(green) RES and (orange) DIS. The first peak at lowest energy transfer is the ∆(1232)
+resonance.
+momentum-transfer dependent shift in the location of the SuSAv2 QE peak in Fe due
+to the extrapolation via scaling from C to Fe. The GSuSAv2 QE model generally
+describes the data as well as or better than the G2018 model.
+107
+
+0.5
+1
+1.5
+0
+100
+200
+3
+10
+×
+o
+ = 4
+θ
+4.499 GeV, 
+0.5
+1
+1.5
+0
+100
+200
+3
+10
+×
+0.5
+1
+1.5
+0
+50
+100
+3
+10
+×
+o
+ = 4
+θ
+6.999 GeV, 
+0.5
+1
+1.5
+0
+50
+100
+3
+10
+×
+0.5
+1
+1.5
+0
+20
+40
+3
+10
+×
+o
+ = 4
+θ
+9.993 GeV, 
+0.5
+1
+1.5
+0
+20
+40
+3
+10
+×
+Proton
+Deuterium
+b/sr/GeV]
+µ
+ [
+ dE
+Ω
+d
+σ
+2
+d
+Energy Transfer [GeV]
+Energy Transfer [GeV]
+Figure 4-11: Comparison of inclusive proton (left) and deuterium (right) (𝑒, 𝑒′) scat-
+tering cross sections for data and for GENIE using G2018. (top) 𝐸0 = 4.499 GeV
+and 𝜃𝑒 = 4∘, (middle) 𝐸0 = 6.699 GeV and 𝜃𝑒 = 4∘, and (bottom) 𝐸0 = 9.993 GeV
+and 𝜃𝑒 = 4∘. Black points show the data, solid black lines show the total GENIE
+prediction, colored lines show the contribution of the different reaction mechanisms:
+(green) RES and (orange) DIS. The first peak at lowest energy transfer is the ∆(1232)
+resonance.
+The GSuSAv2 MEC model appears to be significantly superior to the empirical
+MEC model, especially at 𝑄2 < 0.5 GeV2 or at smaller scattering angles. The empir-
+ical MEC contribution is often much smaller than needed to explain the dip-region
+108
+
+cross section. However, as an empirical model, it can be tuned to better describe the
+data.
+eGENIE dramatically overpredicts the large-energy transfer data at higher mo-
+mentum transfers (𝑄2 > 0.5 GeV2), indicating issues with the RES (Berger-Sehgal)
+and DIS (Bodek and Yang) models used.
+This discrepancy at larger momentum and energy transfers is due to the elemen-
+tary electron-nucleon cross section in the RES and DIS regions, rather than to the
+nuclear models, since eGENIE also significantly overpredicts the proton and deuteron
+cross sections. That is the case especially above the ∆ peak, as shown in figures 4-10
+and 4-11. The discrepancy becomes even more pronounced due to the double counting
+of processes common across the two interaction channels. This shows that tuning the
+RES and DIS models to neutrino data [179] is not sufficient to constrain the vector
+part of the cross section.
+4.3
+Implications For Neutrinos
+Electron-scattering data can be a very effective tool for testing neutrino event gener-
+ators due to the similarity between the interactions. Figure 4-12 shows the remark-
+ably similar cross-section shapes for electron-nucleus and neutrino-nucleus scattering
+for semi-exclusive 1.16 GeV lepton-carbon scattering with exactly one proton with
+𝑄2 ≥ 0.1 GeV2 and 𝑃𝑝 ≥ 300 MeV/c, no charged pions with 𝑃𝜋 ≥ 70 MeV/c and
+no neutral pions or photons of any momenta. This corresponds approximately to
+the JLab CLAS detector thresholds. When comparing electron and neutrino distri-
+butions, the electron events are each weighted by 𝑄4 to reflect the difference in the
+electron and neutrino elementary interactions.
+Exploiting these similarities within the same code is invaluable for minimizing the
+systematic uncertainties of future high-precision neutrino-oscillation experiments. Os-
+cillation analysis uncertainties exceeding 1% for signal and 5% for backgrounds may
+substantially degrade the experimental sensitivity to CP violation and mass hierar-
+chy [146]. Such uncertainties already include the relevant neutrino-nucleus interaction
+109
+
+0
+0.2
+0.4
+0.6
+0.8
+1
+Energy Transfer [GeV]
+0
+0.05
+0.1
+0.15
+0.2
+Area Normalized
+ν
+e
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+]
+2
+/c
+2
+ [GeV
+2
+Q
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+Area Normalized
+ν
+e
+Figure 4-12:
+Comparison of semi-exclusive 1.16 GeV lepton-carbon scattering for
+𝑄2 ≥ 0.1 GeV2. The number of generated events is plotted versus energy transfer
+(left) and 4-momuntm transfer squared (right) for events with exactly one proton
+with 𝑃𝑝 ≥ 300 MeV/c, no charged pions with 𝑃𝜋 ≥ 70 MeV/c and no neutral pions or
+photons of any momentum for eGENIE electrons (orange) and GENIE CC 𝜈𝜇 (blue).
+The electron events have been weighted by 𝑄4. Both curves are area normalized.
+uncertainties. These uncertainties are driven by the choices of the nuclear models and
+cross-section configurations available in event generators like GENIE.
+Figure 4-13 shows that there is a larger difference among QE scattering models
+than there is between QE electron and neutrino scattering using the same nuclear
+model. All six panels show a “ridge”, a maximum in the cross section as a function of
+energy transfer and momentum transfer. The length of the ridge, namely the decrease
+in intensity as the energy and momentum transfers increase, reflects the momentum
+transfer dependence of the nucleon form factors used in the cross section model. The
+width of the distribution perpendicular to the ridge reflects the width of the nuclear
+momentum distribution.
+The momentum distribution of the LFG model cuts off at about 260 MeV/c for
+C, whereas the RMF and the RFG models have “tails” that extend to much larger
+momenta, as shown in figure 4-14. The Nieves cross section decreases more slowly
+with momentum transfer than the others. For GSuSAv2, the electron cross section
+appears to decrease slightly faster with momentum transfer than the neutrino cross
+section, possibly reflecting differences in the axial and vector nucleon form factors.
+Our ability to use the GENIE code to transfer knowledge gained from electron
+110
+
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+ [GeV/c]
+3
+q
+0.2
+0.4
+0.6
+0.8
+1
+ [GeV]
+ω
+Relativistic MF
+ SuSav2
+ν
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+ [GeV/c]
+3
+q
+0.2
+0.4
+0.6
+0.8
+1
+ [GeV]
+ω
+Local FG
+ Nieves
+ν
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+ [GeV/c]
+3
+q
+0.2
+0.4
+0.6
+0.8
+1
+ [GeV]
+ω
+Relativistic FG
+ Llewellyn-Smith
+ν
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+ [GeV/c]
+3
+q
+0.2
+0.4
+0.6
+0.8
+1
+ [GeV]
+ω
+Relativistic MF
+e SuSav2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+ [GeV/c]
+3
+q
+0.2
+0.4
+0.6
+0.8
+1
+ [GeV]
+ω
+Local FG
+e Rosenbluth
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+ [GeV/c]
+3
+q
+0.2
+0.4
+0.6
+0.8
+1
+ [GeV]
+ω
+Relativistic FG
+e Rosenbluth
+Figure 4-13:
+Number of simulated events for QE scattering on 12C at 1.161 GeV
+with 𝑄2 ≥ 0.1 shown as a function of the energy transfer 𝜔 and the momentum
+transfer 𝑞3 = |⃗𝑞 | for all the available nuclear models in GENIE for neutrinos (top)
+and for electrons (bottom). (left) the GSuSAv2 model which uses a Relativistic Mean
+Field momentum distribution, (middle) the Nieves or Rosenbluth cross section with
+the Local Fermi Gas momentum distribution, and (right) the Llewellyn-Smith or
+Rosenbluth cross section with the Relativistic Fermi Gas momentum distribution.
+The electron events have been weighted by 𝑄4.
+scattering depends critically on the implementation of its components. Because of
+its modular design, all reaction models in GENIE use the same nuclear model, for
+instance RFG or LFG. Although the electron scattering capability was added after
+the initial code release, many of the reaction models used electron scattering data
+to construct the vector components of neutrino interactions. This was true for the
+RES [110,125] and the DIS [156] interactions. The difference between vector neutrino-
+and electron-scattering is an overall factor illustrated in equation 4.1 and an appro-
+priate change in the form factors.
+Similarly, figure 4-15 shows that the distribution of MEC events is very similar for
+electrons and for neutrinos within the same model. Thus, measurements of electron
+scattering will be able to significantly constrain models of neutrino scattering.
+Both QE and MEC models use the same vector form factors for neutrino and for
+111
+
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+ [GeV/c]
+p
+p
+0
+0.02
+0.04
+0.06
+0.08
+Probability Density
+LFG
+RFG
+Figure 4-14:
+Initial momentum distribution of protons in simulated QE C(𝑒, 𝑒′𝑝)
+events at 𝐸 = 1.161 GeV for the Local Fermi Gas (solid histogram) and Relativistic
+Fermi Gas (dotted histogram) models. The two curves are normalized to have the
+same area.
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+ [GeV/c]
+3
+q
+0
+0.2
+0.4
+0.6
+0.8
+1
+ [GeV]
+ω
+SuSav2
+MEC
+ν
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+ [GeV/c]
+3
+q
+0
+0.2
+0.4
+0.6
+0.8
+1
+ [GeV]
+ω
+e
+Figure 4-15: Number of simulated events as a function of the energy transfer 𝜔 and
+of the momentum transfer 𝑞3 = |⃗𝑞 | for neutrinos (left) and for electrons (right) using
+GSuSav2 for MEC interactions. The electron events have been scaled by 𝑄4 and all
+the samples have been generated with 𝑄2 ≥ 0.1.
+electron scattering. QE models can use nucleon form factors from electron scatter-
+ing [157], but MEC models must calculate the form factors.
+The GENIE DIS cross section comes from the Bodek-Yang model [156] for the
+full cross section which extends to 𝜋N threshold. The cross section is scaled in the
+RES region so that it agrees with neutrino-deuterium data [179]. Since a single factor
+112
+
+is used to fit the model to the neutrino data, the high-quality electron-proton and
+electron-deuterium data will be poorly described. While the total neutrino cross sec-
+tion and some of the hadronic content of the final state are loosely constrained by the
+neutrino-deuterium data, the vector component of the models is poorly constrained.
+The QE models describe the data reasonably well in the low energy-transfer region.
+Similarly, the largest energy-transfer portions of figures 4-7 and 4-10 show a reasonable
+agreement between GENIE and data. However, at intermediate energy transfer, the
+RES modeling disagrees with the data for both nuclear and nucleon targets, also
+observed in [149]. This is due to the use of RES form factors that are not up-to-date
+and the way the nonresonant contribution was modeled.
+Improvements are in progress but are not simple and therefore not available at this
+time. A possible short-term fix would be to include the electron-proton and electron-
+deuterium inclusive electron-scattering models of Bosted and Christy [190,191]. Alter-
+natively, the vector resonant form factors could be updated using electroproduction
+data from JLab and elsewhere. A fit to that data is available [192] and partially
+implemented in GENIE, but it does not include nonresonant scattering.
+A more
+comprehensive solution would be to use the recent DCC model [193, 194] to simul-
+taneously describe both resonant and nonresonant scattering of both electrons and
+neutrinos.
+The comparisons presented in this work are focused on inclusive electron cross-
+section measurements. Yet, forthcoming neutrino oscillation experiments like DUNE
+will use 4𝜋-coverage liquid argon time projection chamber tracking detectors in order
+to investigate exclusive interaction channels. Until 2021, no prior tests of the event
+generator performance against such exclusive channels existed with electron scattering
+events. Chapters 5 and 6 present the work that alleviated this shortage using wide
+phase-space electron scattering data sets from Hall B and the e2a experiment at JLab.
+113
+
+114
+
+Chapter 5
+The CLAS Electrons-For-Neutrinos
+Experiment At Thomas Jefferson
+National Laboratory
+5.1
+Continuous Electron Beam Accelerator Facility
+To test the event generator performance against wide phase-space exclusive interac-
+tion channels with electron-nucleus scattering events, data sets from Hall B and the
+e2a experiment at Thomas Jefferson Laboratory (JLab) in Newport News VA were
+used.
+The Continuous Electron Beam Accelerator Facility (CEBAF) electron accelera-
+tor [195] provided electron beams up to 6 GeV to three experimental halls, referred
+to as halls A, B, and C, until 2015. CEBAF has since been upgraded to 12 GeV and
+a new hall (Hall D) was constructed in the context of the 12 GeV upgrade. A laser
+shone on a gallium arsenide cathode to emit photo-electrons. The electrons were then
+accelerated by cavities operating at 1.497 GHz. Each of the halls received electron
+bunches every 2 ns. Those bunches can have different electron densities so that each
+hall could be supplied with a different beam current. A schematic of the facility is
+shown in figure 5-1.
+115
+
+Figure 5-1: Schematic view of the accelerator facility and the experimental halls at
+Jefferson Lab. Figure adapted from [196].
+The electrons are accelerated through the beamline. The latter consists of two
+parallel linacs that are connected with two arcs with curvature radii of 80 m. Each
+one of the linacs could increase the electron beam energy by 550 MeV. The beam was
+circulated up to 5 times, for a maximum energy of 6 GeV. The beam energy spread
+was ≈ 10−4.
+5.2
+The CEBAF Large Acceptance Spectrometer
+The e2a data sets were collected at Hall B using the CEBAF Large Acceptance
+Spectrometer (CLAS) spectrometer shown in figure 5-2. The spectrometer had ≈
+50% angular coverage and ran at luminosities up to 1034 cm−2 sec−1.
+CLAS used a toroidal magnet for momentum reconstruction.
+The magnet di-
+vided CLAS into six almost identical sectors. Each sector had three regions of drift
+chambers (DC) for charged particle trajectory measurements, threshold Cherenkov
+counters (CC) for electron identification, scintillator counters (SC) for timing and
+for charged hadron identification, and electromagnetic calorimeters (EC) for electron
+identification and for photon and neutron detection. The 𝜃-angle coverage for the DC
+is 8𝑜 − 140𝑜, for the SC 9𝑜 − 140𝑜, and for the EC 8𝑜 − 45𝑜.
+116
+
+Recirculation
+Arcs
+FELFacility
+0.4-GeV Linac
+(20 Cryomodules)
+0.4-GeV Linac
+45-MeVInjector
+(20 Cryomodules)
+Helium
+(2 1/4 Cryomodules)
+Refrigerator
+Extraction
+Elements
+End
+Stationse– beam
+Scintillators (timing)
+Drift chambers
+(tracking)
+Calorimeters
+(energy)
+Cherenkov (e– ID)
+Target
+≈8m
+Figure 5-2:
+Drawing of the CLAS detector showing the sector structure and the
+different detectors. The beam enters from the upper left side. The target is located
+at the center of the detector.
+5.3
+Toroidal Magnet
+The torus magnet consisted of six sectors and was made of iron-free super conducting
+coils oriented around the beam axis [197]. The torus had a 2 T maximum magnetic
+field. The coils had a length of 5 m, and a width of 2.5 m. The magnetic field was 5
+times smaller at large angles. The central region of the spectrometer had no magnetic
+field allowing for polarized target operation.
+5.4
+Drift Chambers
+The Drift Chambers (DC) are used for tracking and momentum measurements of the
+charged particles produced out of the interactions. The DC have the structure shown
+117
+
+Figure 5-3: The CLAS superconducting toroidal magnet. Figure adapted from [197].
+in figure 5-4 (left) and more details can be found in [198].
+Figure 5-4: (Left) Illustration of the region 3 drift chamber structure for one of the
+CLAS sectors. (Right) Schematic representation of the thee drift chamber regions.
+Figures adapted from [198].
+Each one of the modules is filled with a gas mixture of 90% Argon and 10% CO2.
+The charged particles that transverse the DC ionize the gas mixture. The produced
+electrons drift towards the anode wires and the corresponding ions drift towards the
+cathode wires. The relevant electrical signal is used to determine the distance from
+the charged particle trajectory to the wires.
+As the ionization electrons approach the wires, the increased electric field results in
+the multiplication of the electron-atom collisions. The latter results in the production
+of more electrons by a factor of ≈ 104 and results into an “avalanche”. The CO2
+molecule serves as a quenching gas and prevents the creation of secondary avalanches.
+118
+
+Coil
+Beam
+DirectionBeam
+Sector-plate
+Back-plate
+Gas window
+Endplate
+Wire direction
+Endplate
+Circuit boardRegion 3
+Region 1
+beam
+Region 2
+mThe DC consist of three radial regions as shown in figure 5-4 (right). Regions 1 and
+3 are located in a low magnetic field region. Each one of these regions is made of two
+superlayers, one axial and one stereo (at a stereo angle of 6𝑜), to allow for complete
+coordinate reconstruction. The layers consist of six layers of 192 hexagonal drift cells
+with the 20-micron sense wire at the center and six shared field wires creating the
+electric field, with the exception of the innermost layer in region 1, which is made of
+four layers. The DC structure provides a position resolution ≈ 400 𝜇m, ≈ 5 mrad for
+the angular variables and ≈ 1% (0.5%) for the hadron (electron) momenta.
+5.5
+Electromagnetic Calorimeter
+The electromagnetic calorimeter (EC) [199] was used to trigger on and to identify
+the electons. It further had a high momentum neutron detection efficiency of ≈ 50%
+and a high photon detection efficiency for energies above 300 MeV. It was also used
+to measure neutral particles with an efficiency of ≈ 60%. It consists of 6 modules
+corresponding to each one of the detector sectors. Each one of the modules has a
+triangular shape with a projected vertex at the CLAS target location ≈ 5 m. Each
+element has a thickness corresponding to 16 radiation lengths and is made of 39 layers
+of alternating 10 mm thick scintillator strips and 2.2 mm of lead sheets. All the layers
+have triangular shapes. In each layer, the scintillator strips are parallel to a given
+side of the triangle. Each subsequent layer is rotated by 120𝑜 with respect to the
+previous layer. That results in the formation of three views, namely u,v, and w, as
+can be seen in figure 5-5.
+Each view consists of 13 layers. The scintillator material thickness for each EC
+module is 39 cm and the lead one is 8.4 cm. This proportionality between the scin-
+tillator and the lead layers leads to one third of the shower energy being deposited
+in the scintillator part. The time resolution for electrons and neutrons is 200 and
+600 ps, respectively. The 13 layers are combined into an inner (5 layers) and an outer
+(8 layer) stack in order to provide longitudinal sampling of the showers and a hadron
+identification.
+119
+
+Figure 5-5: Schematic view of a CLAS electromagnetic calorimeter module. Figures
+adapted from [199].
+The EC energy resolution is given by the expression
+∆𝐸
+𝐸
+= 0.003 +
+0.093
+√︀
+𝐸 [𝐺𝑒𝑉 ]
+(5.1)
+5.6
+Cherenkov Counters
+There are six Cherenkov Counters (CC) [200], which are used to trigger on the elec-
+trons and to separate between pions and electrons. A schematic of the CC is shown
+in figure 5-6.
+Each CC covers the polar angular range of 𝜃 = 8 - 45𝑜 and consists of eighteen
+regions. Each region is made of two modules called segments. The CC is filled with
+perfluorobutane (𝐶4𝐹10) gas with an index of refraction of n = 1.00153. That yields
+a 2.5 GeV energy threshold for pions and a 10 MeV threshold for electrons. When
+the particle velocity (v) is greater than the speed of light (c) in the detector medium
+(c/n), Cherenkov light is emitted. The produced light is then directed to the light
+collections cone using elliptical and hyperbolic mirrors in order to get focused to the
+PMT.
+120
+
+Scintillatorbars
+U-plane
+-Leadsheets
+V - plane
+W-plane
+Fiber Light Guides
+(front)
+Fiber Light Guides
+(rear)
+PMT'sFigure 5-6: Optical arrangement of one of the optical modules of the CLAS Cherenkov
+detector, showing the optical and light collection components.
+Figures adapted
+from [200].
+5.7
+Time-Of-Flight Detector
+The time-of-flight detectors (TOF) [201] were made of scintillator paddles and are
+shown in figure 5-7. Each scintillator was 5 cm thick and 10-15 cm wide. They ranged
+in length between 25-450 cm. Using the time-of-flight (t) from the TOF subsystem
+and the distance (d) based on tracking information from the DC, the velocity of the
+charged particles would be obtained (v = d/t). The particle mass would be calculated
+by combining the information of the particle momentum from the DC, as shown in
+equation 5.2.
+𝑚 = 𝑝
+√︀
+1 − 𝛽2
+𝛽
+(5.2)
+There exist 57 TOF paddles in each sector of the detectororiented perpendicular
+to the beamline. Each one covers ≈ 2𝑜. The overall coverage in 𝜃 starts at 8𝑜 and
+goes up to 142𝑜, organized in four panels. There is a PMT attached at the end of
+each paddle. The time resolution for electrons in the TOF is 163 ps.
+121
+
+elliptical mirror
+sectorcenterline
+PMT
+magnetic
+shield
+lightcollection
+cone
+cylindrical
+mirror
+electron
+hyperbolicmirror
+trackFigure 5-7: Schematic view of the TOF counters in one sector illustrating the grouping
+into four panels. Figure adapted from [201].
+5.8
+E2a Targets
+The Hall B e2a experiment ran between April 15-May 27 1999. During the data taking
+period, data were collected using the energies and targets presented in table 5.1.
+Table 5.1: Target areal densities and integrated charges for the 𝑒4𝜈 data sets.
+Target
+Energy
+Areal Density [g/cm2]
+Integrated Charge [mC]
+3He
+1.159
+0.2770
+0.08
+3He
+2.257
+0.2770
+0.08
+3He
+4.453
+0.2770
+0.11
+4He
+2.257
+0.5375
+1.17
+4He
+4.453
+0.5375
+0.98
+12C
+1.159
+0.1786
+0.08
+12C
+2.257
+0.1786
+1.79
+12C
+4.453
+0.1786
+2.84
+56Fe
+2.257
+0.1181
+0.22
+56Fe
+4.453
+0.1181
+0.31
+The 2.257 and 4.453 GeV data sets used a torus current of 2250 A, while the
+1.159 GeV ones were obtained with a 750 A torus current. The beam current ranged
+122
+
+between 3-18 nA. The solid targets (12C and 56Fe) were 0.9 × 0.9 cm2 square plates
+with a thickness of 1 mm. The liquid targets (3He and 4He) were stored in cylindrical-
+shaped vessel with a diameter of 2.8 cm. More details on the used data sets can be
+found in the already published analyses [202–207].
+123
+
+124
+
+Chapter 6
+Electrons-For-Neutrinos Results
+[Nature 599, 565–570 (2021)]
+6.1
+Electron Data Mining Analysis
+As discussed in section 1.5, the incomplete lepton-nucleus knowledge can be lever-
+aged using the fact that electrons and neutrinos interact similarly with nuclei. While
+previous work has compared these interaction models with inclusive electron scatter-
+ing [42,149], this analysis is the first comparison of electron scattering data with these
+interaction models where events with one or more detected hadrons are used [208].
+The data presented here can therefore test and constrain neutrino-nucleus interaction
+models to be used in analysis of neutrino oscillation measurements.
+For the purposes of the “Electrons-For-Neutrinos” (e4𝜈) analysis, electron scatter-
+ing data sets from the CEBAF Large Acceptance Spectrometer (CLAS) [209] at the
+Thomas Jefferson National Accelerator Facility (JLab) were used. The components
+of the detector are described in detail in chapter 5.
+The e2a data sets detailed in section 5.8 were used, which were measured in 1999
+and reported in many published analyses [202–207]. These include electron scattering
+events on 4He, 12C, and 56Fe nuclei at beam energies of 1.159, 2.257 and 4.453 GeV.
+The beam energy equaled the injector energy plus the pass number times the linac
+energy. The three-pass beam energy was measured using the Hall A arc measurement
+and the four pass energy was measured using the Hall C arc measurement. These
+gave a central linac energy of 1.0979 GeV and the Hall B one-, two-, and four-pass
+125
+
+beam energies of 1.159, 2.257, and 4.453 GeV, respectively. An uncertainty of 2×10−3
+was assigned to these energies, based on the difference between the Hall A and Hall C
+measurements. The incident energies used in this analysis span the range of typical
+accelerator-based neutrino beam energies [20,210], as can be seen in figure 6-1. The
+carbon data are relevant for scintillator-based experiments such as MINER𝜈A and
+NO𝜈A [211] and similar to the oxygen in water-based Cherenkov detectors such as
+Super-Kamiokande (SK) [22, 23] and HK [212]. The iron is similar to the argon in
+the liquid argon time projection chambers of MicroBooNE [93], the Fermilab short-
+baseline oscillation program [94] and DUNE [213]. Many nuclear interaction processes
+are mass dependent, so it is important to measure a range of target nuclei.
+0
+1
+2
+3
+4
+5
+ [GeV]
+ν
+E
+ Flux [arb.]
+ν
+(a)
+1
+2
+3
+4
+5
+ [GeV]
+ν
+E
+ Flux [arb.]
+ν
+DUNE
+T2K
+NOvA
+(b)
+Figure 6-1: The expected energy distribution of different 𝜈𝜇 beams, (left) before
+oscillation at the near detector and (right) after oscillation at the far detector. The
+vertical lines show the three electron beam energies of this measurement. The NO𝜈A
+far-detector beam flux is calculated using the near detector flux and the neutrino
+oscillation parameters from [1].
+Electrons with energies 𝐸𝑒 ≥ 0.4, 0.55 and 1.1 GeV for 𝐸𝑏𝑒𝑎𝑚 = 1.159, 2.257,
+and 4.453 GeV respectively, and angles 15∘ ≤ 𝜃𝑒 ≤ 45∘ were detected.
+Protons
+with momenta 𝑝𝑝 ≥ 300 MeV/𝑐 and angles 𝜃𝑝 ≥ 10∘, charged pions with momenta
+𝑝𝜋 ≥ 150 MeV/𝑐 and angles 𝜃𝜋+ ≥ 10∘ and 𝜃𝜋− ≥ 22∘, and photons with energy
+𝐸𝛾 ≥ 300 MeV and 8 ≤ 𝜃𝛾 ≤ 45∘ were detected. Separate fiducial cuts were applied
+for electrons, negatively-charged pions, positively-charged particles, and photons, to
+select momentum-dependent regions of CLAS where the detection efficiency was con-
+126
+
+stant and close to one. These hadron detection thresholds are similar to those of
+neutrino detectors [87], however neutrino detectors have full angular coverage and
+lower lepton energy thresholds.
+Apart from the aforementioned momentum and angular cuts, additional angular
+outlines are applied to account for the detector acceptance. The minimum electron
+angle as a function of electron momentum 𝑝 for each beam energy was determined as
+𝜃1.1
+𝑒
+≥ 17∘ +
+7∘
+𝑝 [GeV]
+(6.1)
+𝜃2.2
+𝑒
+≥ 16∘ + 10.5∘
+𝑝 [GeV]
+(6.2)
+𝜃4.4
+𝑒
+≥ 13.5∘ +
+15∘
+𝑝 [GeV]
+(6.3)
+and the minimum 𝜋− angle as
+𝜃1.1
+𝜋− ≥ 17∘ +
+4∘
+𝑝 [GeV]
+(6.4)
+and
+𝜃2.2,4.4
+𝜋−
+≥ 25∘ +
+7∘
+𝑝 [GeV]
+(6.5)
+for 𝑝𝜋− < 0.35 GeV/c and
+𝜃2.2,4.4
+𝜋−
+≥ 16∘ +
+10∘
+𝑝 [GeV]
+(6.6)
+for 𝑝𝜋− ≥ 0.35 GeV/c. The minimum 𝜋+ and proton angle was 𝜃 > 12∘ for all
+data sets and momenta.
+The momentum and charge of the outgoing charged particles were obtained from
+their measured positions in the drift chambers and the curvature of their trajectories
+in the magnetic field. Electrons were identified by requiring that the track originated
+in the target, produced a time-correlated signal in the Cherenkov counter, and de-
+127
+
+posited enough energy in the electromagnetic calorimeter. The charged pions and
+protons were separated by requiring that the track originated in the target and that
+the measured time of flight agreed with that calculated from the particle’s momentum
+and assumed mass. Photons were identified by requiring a signal in the electromag-
+netic calorimeter which implied a velocity greater than ≈ 0.96 c [214].
+Elastic electron scattering from hydrogen was used to correct the electron momen-
+tum as a function of angle for uncertainties in the CLAS magnetic field and in the
+tracking chamber locations. These corrections also significantly narrowed the elastic
+peak width. Typical correction factors were less than 1%. The momentum correction
+factors at lower scattered electron energies were tested using the H(𝑒, 𝑒′𝜋+)𝑋 and
+3He(𝑒, 𝑒′𝑝𝑝)𝑋 reactions and found that they gave the correct missing mass for the
+undetected neutron [208].
+Low momentum protons were corrected for energy losses traversing the target
+and detector material. The CLAS GEANT Monte Carlo (MC) was used to simulate
+the proton energy loss in CLAS as a function of proton momentum. The maximum
+correction was about 20 MeV/c for a proton momentum of 300 MeV/c. The correction
+was negligible for protons with momenta greater than 600 MeV/c.
+The results from the e2a data sets were compared to predictions from the GE-
+NIE [33] simulation, which is used by most neutrino experiments in the USA and has
+an electron-scattering version (eGENIE) that was recently overhauled to be consistent
+with the neutrino counterpart (𝜈GENIE), as detailed in [42]. GENIE includes quasi-
+elastic lepton scattering (QE), meson exchange currents (MEC), resonance production
+(RES) and deep inelastic scattering (DIS), as well as rescattering via final state in-
+teractions (FSI) of the outgoing hadrons. The two GENIE configurations already
+presented in section 4.1 were compared. These include the significantly improved
+G2018 setup which reproduces measured neutrino [215] and electron inclusive cross
+sections, and newly implemented SuSAv2 that uses modern, theoretically-inspired,
+recently-implemented QE and MEC models [28].
+The resulting simulated events
+were then analyzed using the same code as the data and the two were compared.
+Electrons, unlike neutrinos, radiate bremsstrahlung photons in the electric field
+128
+
+of the nucleus.
+Events where the photons from scattered-electron radiation were
+detected in CLAS were vetoed. It was assumed that the photons came from either
+radiation by the outgoing electron approximately parallel to its motion or from 𝜋0
+decay. The radiated photons were identified by requiring that they be detected within
+∆𝜑𝛾,𝑒′ ≤ 30∘ and ∆𝜃𝛾,𝑒′ ≤ 40∘ of the scattered electron and removed them from the
+data set. The events that were removed are indicated by the red box in figure 6-2.
+1
+10
+2
+10
+3
+10
+Fe @ 2.2 GeV
+56
+Data 
+150
+−
+100
+−
+50
+−
+0
+50
+100 150
+ [deg]
+,e'
+γφ
+∆
+0
+20
+40
+60
+80
+100
+ [deg]
+,e'
+γθ
+∆
+ from radiation
+γ
+0
+π
+ from 
+γ
+0
+π
+ from 
+γ
+Fe @ 2.2 GeV
+56
+Data 
+Figure 6-2: ∆𝜃𝛾,𝑒′ as a function of ∆𝜑𝛾,𝑒′. The red box indicated the region with
+radiated photons which was removed in our analysis.
+The incoming and outgoing electrons can each radiate a real photon, which changes
+the kinematics of the interaction or the detected particles, and there can be vertex
+or propagator corrections that change the cross section. When comparing electron
+scattering data to models, either the data or the model needs to be corrected for ra-
+diative effects. Published electron scattering cross sections are typically corrected for
+radiative effects, but this correction is complicated and somewhat model-dependent.
+A framework for electron radiative corrections in GENIE was implemented for the
+first time to allow comparisons to nonradiatively corrected data. The framework al-
+lows electron radiation, which can change the kinematics of the event by changing
+either the incident or scattered electron energy through radiation of a real photon.
+129
+
+We modeled external radiation in the same way as the JLab SIMC event genera-
+tor [216, 217]. The implementation takes advantage of the peaking approximation
+that greatly simplifies the calculation of the angular distribution of the emitted pho-
+ton radiation by making the assumption that radiation along the direction of a given
+particle can be interpreted as radiation due to that particle [218]. Future versions
+of eGENIE will incorporate cross section changes due to vertex and propagator cor-
+rections. The radiative correction procedure was validated by comparing a simulated
+sample to electron scattering from protons at JLab. Figure 6-3 shows the data com-
+pared to the GENIE simulation with and without radiative corrections [219]. The
+radiatively corrected calculation is clearly much closer to the data. This correction
+can be used for comparisons with nonradiatively corrected data.
+ [GeV]
+cal
+H(e,e'p) E
+1
+4.3
+4.305
+4.31
+4.315
+4.32
+4.325
+4.33
+# events
+0
+2
+4
+6
+8
+10
+12
+3
+10
+×
+Data
+GENIE + radiative correction
+GENIE default
+Figure 6-3: Number of events vs 𝐸𝑐𝑎𝑙 = 𝐸𝑒′ + 𝑇𝑝 the scattered electron energy plus
+proton kinetic energy for 4.32 GeV H(𝑒, 𝑒′𝑝). Black points are data, the blue histogram
+shows the unradiated GENIE prediction and the black histogram shows the GENIE
+prediction with electron radiation. The GENIE calculations have been scaled to have
+the same integral as the data.
+The primary focus of this analysis was events with one electron and zero pions
+or photons from 𝜋0 decay above threshold, which are referred to as (e,e’)0𝜋. That
+choice was made to maximize the contribution of QE events where the incident lepton
+scattered from a single nucleon in the nucleus, as is done in many neutrino oscillation
+130
+
+analyses [1, 220]. Furthermore, events with one detected electron, one proton, and
+zero pions, denoted here as (e,e’p)1𝑝0𝜋 were examined. These events were expected to
+be dominated by well-understood QE events.
+Because the CLAS geometrical coverage is incomplete (≈ 50%), undetected pions
+and photons needed to be subtracted to obtain true (e,e’)0𝜋 and (e,e’p)1𝑝0𝜋 event sam-
+ples. The undetected pion, photon, and proton (if applicable) contribution was quan-
+tified from the events with detected pions, photons or protons. The pion-production
+cross section was assumed to be independent of 𝜑𝑞𝜋. The latter is the angle between
+the electron-scattering plane (the plane containing the incident and scattered elec-
+trons and the virtual photon) and the hadron plane (the plane containing the virtual
+photon and pion). The data-driven correction for these undetected hadrons is out-
+lined below for the (e,e’p)1𝑝0𝜋 case. For each detected (𝑒, 𝑒′𝑝𝜋) event, the proton-pion
+pair was rotated around the momentum transfer direction ⃗𝑞 randomly multiple times,
+as can be seen in figure 6-4. For each rotation, it was determined whether the particle
+was within the fiducial region of the detector. The particle acceptance would then
+be 𝐴𝜋 = 𝑁𝑑𝑒𝑡/𝑁𝑟𝑜𝑡, where 𝑁𝑟𝑜𝑡 is the number of rotations and 𝑁𝑑𝑒𝑡 is the number of
+times the pion would have been detected. The corresponding number of undetected
+(𝑒, 𝑒′𝑝𝜋) events for that detected (𝑒, 𝑒′𝑝𝜋) event is (𝑁𝑟𝑜𝑡 − 𝑁𝑑𝑒𝑡)/𝑁𝑑𝑒𝑡. That was used
+as a weight to subtract for the undetected pion events. For example, if one specific
+(𝑒, 𝑒′𝑝𝜋) event would have been detected 250 times out of 1000 rotations, then it was
+inferred that for each detected event, there were three more that were not detected.
+The appropriate variables were calculated for that event and subtracted it from the
+corresponding distributions with a weight of three. That was done separately for
+𝜋+, 𝜋− and photons in order to obtain a true (e,e’p)1𝑝0𝜋 sample. The same process is
+repeated for the (e,e’)0𝜋 channel, where the corrections included undetected protons.
+Higher multiplicity events were also accounted for, such as for events with two
+detected 𝜋± or photons. When these events were rotated, each rotated event could
+have been detected as a 2𝜋 event, a 1𝜋 event, or a 0𝜋 event. If it appeared as a
+0𝜋 event, its contribution was subtracted from the various 0𝜋 spectra as described
+above. If it appeared as a 1𝜋 event, it was included in the set of 1𝜋 events with the
+131
+
+Figure 6-4: Schematic illustration of the data driven background correction using
+detected (e,e’p𝜋) events.
+appropriate negative weight. It was then treated as a regular 1𝜋 event, which was
+then rotated and added to the 0𝜋 data set. Some of the detected 1𝜋 events were
+actually 2𝜋 events with an undetected pion. When the effect of these events was
+accounted for, there were fewer true 1𝜋 events left. This reduced the contamination
+of the 1𝜋 events in the 0𝜋 channel.
+In practice, the process was initiated with the highest multiplicity events. Then,
+their contributions to each of the detected lower multiplicity channels were subtracted.
+The process was repeated recursively by rotating the higher multiplicity events. In
+this way, their contributions to the lower multiplicity channels were determined and
+subtracted, and then each of the lower multiplicity channels in turn were considered.
+Event multiplicities up to three pions and photons (total) for the (𝑒, 𝑒′)0𝜋 channel and
+up to three protons, pions and photons (total) for the (𝑒, 𝑒′𝑝)1𝑝0𝜋 channel, where the
+subtraction converged, were considered. The effects of the subtraction and its con-
+vergence can be seen in figure 6-5 for 𝐸𝑄𝐸. The number of events with an undetected
+𝜋± or photon is about equal to the number of events with a detected 𝜋± or photon,
+consistent with the ≈ 50% CLAS geometrical acceptance. The effect of including two
+𝜋± or photon events is much less than that of the one 𝜋± or photon events and the
+132
+
+effect of including three 𝜋± or photon events is negligible.
+2
+3
+4
+5
+[GeV]
+QE
+E
+0
+0.1
+0.2
+Weighted Events / GeV
+γ/
+±
+π
+Detected 1 
+ (-)
+γ/
+±
+π
+Undetected 1
+ (+)
+γ/
+±
+π
+Undetected 2
+Fe
+56
+(a)
+2
+3
+4
+5
+ [GeV]
+QE
+E
+0
+0.2
+0.4
+0.6
+Weighted Events / GeV
+No cuts
+γ/
+±
+π
+No detected 
+γ/
+±
+π
+Subtract undetected 
+Fe
+56
+(b)
+Figure 6-5:
+The effect of undetected pion subtraction.
+The number of weighted
+events as a function of reconstructed energy 𝐸𝑄𝐸 for 4.453 GeV Fe(𝑒, 𝑒′) events for
+(left) events with a detected 𝜋± or photon (blue), events with one (red) or two (light
+brown) undetected 𝜋± or photons and (right) all (𝑒, 𝑒′𝑋) events with detected or
+undetected 𝜋± or photon (blue), (𝑒, 𝑒′) events with no detected 𝜋± or photon (red),
+and (𝑒, 𝑒′) events after subtraction for undetected 𝜋± or photon (light brown).
+The subtraction method was tested by applying it to eGENIE events. The result-
+ing subtracted spectra agreed reasonably with the true 1𝑝0𝜋 spectra as can be seen
+in figure 6-6. The method diverges for total hadron multiplicities greater than four
+due to the proton and pion multiplicity differences shown in figure 6-7. It is clear
+that eGENIE dramatically overpredicts the number of events with large proton and
+pion multiplicities.
+The CLAS acceptance maps were used to determine the probability that each
+particle produced by eGENIE was detected as a function of the momentum, the
+angular orientation, and the particle species. Figure 6-9 shows the electron acceptance
+map for 12C at 𝐸𝑏𝑒𝑎𝑚 = 1.159 GeV as a function of (left) cos𝜃𝑒 vs 𝜑𝑒 and (right) cos𝜃𝑒
+vs momentum 𝑝𝑒 illustrating an acceptance greater than 90% across the majority of
+the detector fiducial volume.
+The particle momenta were smeared with an effective CLAS resolution. Namely,
+electrons and proton momentum resolutions of 0.5% and 1%, respectively, for the
+2.257 and 4.453 GeV data and 1.5% and 3% for the 1.159 GeV data, which was taken
+with a lower torus magnetic field, were used.
+133
+
+0.8
+1
+1.2 1.4
+1.6 1.8
+2
+2.2 2.4
+ [GeV]
+cal
+ E
+π
+1p0
+(e,e'p)
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+Normalized Yield
+π
+1p0
+True (e,e'p)
+π
+1p0
+Subtracted (e,e'p)
+π
+1p0
+Unsubtracted (e,e'p)
+C @ 2.257 GeV
+12
+Figure 6-6: Illustration of the successful closure test of the data driven correction
+for undetected particles as a function of 𝐸𝐶𝑎𝑙 using the (e,e’p)1𝑝0𝜋 channel on 12C at
+𝐸𝑏𝑒𝑎𝑚 = 2.257 GeV. The contribution of the unsubtracted (e,e’p)1𝑝0𝜋 spectrum (black)
+is reduced to the subtracted (e,e’p)1𝑝0𝜋 spectrum (magenta), which is in reasonable
+agreement with the true (e,e’p)1𝑝0𝜋 spectrum (green).
+0
+1
+2
+3
+4
+Multiplicities
+4
+10
+5
+10
+6
+10
+7
+10
+# Events
+SuSav2
+G2018
+Protons
+±
+π
+Figure 6-7:
+The proton (black) and charged pion (blue) multiplicities for data
+(points), SuSav2 (solid histogram) and G2018 (dashed histogram) for 2.257 GeV
+carbon. Error bars show the 68% (1𝜎) confidence limits for the statistical and point-
+to-point systematic uncertainties added in quadrature.
+Error bars are not shown
+when they are smaller than the size of the data point. Normalization uncertainties of
+3% not shown.
+The cross section as a function of variables of interest for particles above the
+minimum angles shown in equations 6.1-6.6 was determined in several steps.
+All
+134
+
+0.6
+0.8
+1
+1.2
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+Radiation Correction
+(g)
+1
+1.5
+2
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+Radiation Correction
+(h)
+2
+3
+4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+Radiation Correction
+(i)
+0.6
+0.8
+1
+1.2
+0
+5
+10
+15
+Acceptance Uncertainty [%]
+(d)
+1
+1.5
+2
+0
+5
+10
+15
+Acceptance Uncertainty [%]
+(e)
+2
+3
+4
+0
+5
+10
+15
+Acceptance Uncertainty [%]
+(f)
+0.6
+0.8
+1
+1.2
+2
+4
+6
+8
+10
+12
+Acceptance Correction
+(a)
+C
+12
+1.159 GeV
+1
+1.5
+2
+2
+4
+6
+8
+10
+12
+Acceptance Correction
+(b)
+Fe
+56
+He
+4
+2.257 GeV
+2
+3
+4
+2
+4
+6
+8
+10
+12
+Acceptance Correction
+(c)
+4.453 GeV
+ [GeV]
+cal
+ E
+π
+1p0
+(e,e'p)
+Figure 6-8:
+(Top row) Acceptance correction factors, (middle row) acceptance cor-
+rection factor uncertainties, and (bottom row) electron radiation correction factors
+plotted vs E𝑐𝑎𝑙 for the three incident beam energies. Results for carbon are shown
+in black, helium in green and iron in magenta. The left column (a,d,g) shows the
+1.1 GeV results, the middle column (b,e,h) shows the 2.2 GeV results and the right
+column (c,f,i) shows the 4.4 GeV results.
+90
+92
+94
+96
+98
+100
+Acceptance (%)
+0
+50 100 150 200 250 300
+ [deg]
+e
+φ
+0.6
+0.7
+0.8
+0.9
+eθ
+cos
+90
+92
+94
+96
+98
+100
+Acceptance (%)
+0.6
+0.7
+0.8
+0.9
+eθ
+cos
+0.5
+1
+1.5
+2
+2.5
+ [GeV/c]
+ep
+Figure 6-9: Electron acceptance maps for 12C at 𝐸𝑏𝑒𝑎𝑚 = 1.159 GeV as a function of
+(left) cos𝜃𝑒 vs 𝜑𝑒 and (right) cos𝜃𝑒 vs momentum 𝑝𝑒.
+events were first weighted by a factor of 𝑄4 to account for the major difference in
+electron- and neutrino-nucleus scattering. The number of weighted events was then
+135
+
+determined and corrected (if appropriate) for events with undetected pions, photons,
+and extra protons. The background-subtracted event distribution was divided by the
+number of target nuclei per area and the number of incident beam electrons to get
+the normalized yield. The delivered integrated beam charge was measured using the
+CLAS Faraday Cup [209]. Electron radiation effects were corrected for by multiplying
+the resulting spectra by the ratio of eGENIE without electron radiation divided by
+eGENIE with electron radiation, as shown in panels g-i of figure 6-8. This includes
+a multiplicative factor to account for the effects of internal radiation. Electron and
+proton acceptance and other detector effects were corrected for using eGENIE. The
+acceptance correction factor is the ratio of the number of true signal events without
+detector effects to the number of true signal events with detector effects. The detector
+effects included momentum resolution, fiducial cuts, and acceptance map effects. The
+fiducial cuts determine the useful areas of the detector as a function of particle mo-
+menta and angles, and the acceptance maps describe the efficiency of the detector as
+a function of particle momenta and angles. This factor corrects the effective electron
+and proton solid angles to almost 4𝜋. It excludes all electrons, pions and protons
+below their minimum angles defined in equation 6.1-6.6. The acceptance correction
+factor was obtained using both G2018 and SuSav2 shown in panels a-c of figure 6-8
+as the bin-by-bin average of the two configurations. The G2018 results were shifted
+so that the energy reconstruction peaks lined up at the correct beam energy. Finally,
+the bin width division was taken into account.
+In order to perform a sanity check of our cross-section extraction procedure, the
+inclusive 1.159 37.5∘ GeV C(𝑒, 𝑒′) cross section was determined as follows:
+𝑑𝜎
+𝑑Ω𝑑𝜔 =
+𝑁𝑒
+∆Ω𝑁𝑖𝑁𝑡
+(6.7)
+where 𝑁𝑒 is the number of detected electrons in Sector 1 within 36∘ ≤ 𝜃𝑒 ≤ 39∘
+and a 12∘ range in 𝜑𝑒, ∆Ω = sin 𝜃𝑒𝑑𝜃𝑒𝑑𝜑𝑒 = 6.68 msr, 𝑁𝑖 is the number of incident
+electrons, and 𝑁𝑡 = 0.179 g/cm2 = 8.95×10−9 nuclei/𝜇b. The extracted cross section
+136
+
+was further compared to measurements from SLAC [182], as can be seen in figure 6-
+10. The measured JLab cross section is in reasonable agreement with the GENIE
+predictions and also consistent with the SLAC measurements at lower and higher
+energies [182].
+Figure 6-10:
+Comparison between the inclusive C(𝑒, 𝑒′) cross sections measured at
+37.5∘ for data (points) and SuSav2 (lines) for the 0.961 and 1.299 GeV SLAC data
+and our 1.159 GeV CLAS data.
+Several major sources of systematic uncertainties were considered, including the
+angular dependence of the pion-production cross section for the undetected-pion sub-
+traction, the effects of fiducial cuts on undetected particle subtraction, photon iden-
+tification cuts, the sector-to-sector variation of the data to eGENIE ratio, the model-
+dependence of the acceptance correction, and uncertainties in the normalization mea-
+surement. Table 6.1 shows the summary of the total systematic uncertainties used in
+the e4𝜈 analysis.
+When events containing pions were rotated around the momentum transfer vec-
+tor, the cross section was assumed to not depend on 𝜑𝑞𝜋. The 𝜑𝑞𝜋 independence of
+the pion-production cross section was tested by weighting the subtraction using the
+measured 𝜑𝑞𝜋-dependent 𝐻(𝑒, 𝑒′𝑝𝜋) cross sections of reference [221]. This changed
+the subtracted spectra by about 1% and was included as a systematic uncertainty.
+137
+
+0 = 37.5°
+SLAC 0.961 GeV
+GeV
+JLab
+A
+1.159 GeV
+S
+dQdE
+2
+d
+SLAC 1.299 GeV
+0.1
+0.2
+0.3
+0.40.5
+0.6
+0.7
+0.8
+Energy Transfer [GeV]Table 6.1: Summary of the total systematic uncertainties used in the e4𝜈 analysis.
+Source
+Uncertainty (%)
+Detector acceptance
+Identification cuts
+Number of rotations
+𝜑𝑞𝜋 cross-section dependence
+2, 2.1, 4.7
+(@1.1,2.2,4.4 GeV)
+Sector dependence
+6
+Acceptance correction
+2-15
+Overall normalization
+3
+Electron inefficiency
+2
+The subtraction of events with undetected pions depends on the CLAS acceptance
+for such particles.
+The final spectrum should be independent of the CLAS pion
+acceptance. The effect of varying the CLAS acceptance on the undetected particle
+subtraction was estimated by comparing the results using the nominal fiducial cuts
+and using fiducial cuts with the 𝜑 acceptance in each CLAS sector reduced by 6∘
+or about 10-20%. This changed the resulting subtracted spectra by about 1% at
+1.159 and 2.257 GeV and by 4% at 4.453 GeV. This difference was included as a
+point-to-point systematic uncertainty.
+The photon identification cuts were also varied.
+Photons were also identified
+as neutral particle hits in the calorimeter with a velocity greater than 2𝜎 (3𝜎 at
+1.159 GeV) below the mean of the photon velocity peak at 𝑣 = 𝑐. This limit was
+varied by ±0.25𝜎. This gave an uncertainty in the resulting subtracted spectra of
+0.1%, 0.5% and 2% at 1.159, 2.257 and 4.453 GeV, respectively.
+CLAS had six almost identical sectors. The primary difference among the sectors
+is the distribution of dead detector channels. These dead channels were accounted for
+in our fiducial cuts and in our acceptance maps, where the effect of the dead detectors
+on the particle detection efficiency was measured and applied that efficiency to the
+particles generated in the eGENIE simulation. If our fiducial cuts and acceptance
+maps completely accounted for the effect of the dead and inefficient detector channels,
+then the ratio of data to eGENIE should be the same for all six sectors. Sectors with
+anomalous data to eGENIE ratios were discarded. More precisely, sectors 3/5 were
+138
+
+discarded at 𝐸𝑏𝑒𝑎𝑚 = 1.159 GeV, as well as sectors 3/4/5 at 𝐸𝑏𝑒𝑎𝑚 = 2.257 GeV. All the
+sectors were used at 𝐸𝑏𝑒𝑎𝑚 = 4.453 GeV. The variance of the ratios for the remaining
+sectors was used as a measure of the uncertainty in the measured normalized yields.
+This gave a point-to-point systematic uncertainty of 6%.
+The acceptance correction factor uncertainty was obtained using both G2018 and
+SuSav2 and their bin-by-bin difference divided by
+√
+12. The uncertainty was averaged
+over the entire peak to avoid large uncertainties due to small misalignments, as shown
+in panels d-f of figure 6-8.
+The overall normalization was determined using inclusive 4.4 GeV H(𝑒, 𝑒′) mea-
+surements. The measured and simulated H(𝑒, 𝑒′) cross sections agreed to within an
+uncertainty of 3%, which is used as a normalization uncertainty [222].
+The statistical uncertainty and the point-to-point systematic uncertainties were
+added in quadrature and displayed on the data points.
+The total point-to-point
+systematic uncertainties ranged between 7-25%, with the largest uncertainties for the
+smallest cross sections.
+6.2
+Incident Energy Reconstruction Results
+There are two general approaches for reconstructing the incident neutrino energy
+based on the particle detection capabilities of the neutrino detector.
+Water Cherenkov detectors only measure charged leptons and pions. If the neu-
+trino scattered quasi-elastically from a stationary nucleon in the nucleus, its energy
+can be reconstructed from the measured lepton as:
+𝐸𝑄𝐸 = 2𝑀𝑁𝜖 + 2𝑀𝑁𝐸𝑙 − 𝑚2
+𝑙
+2(𝑀𝑁 − 𝐸𝑙 + 𝑘𝑙 cos 𝜃𝑙),
+(6.8)
+where 𝜖 ≈ 20 MeV is the average nucleon separation energy, 𝑀𝑁 is the nucleon
+mass, and (𝑚𝑙, 𝐸𝑙, 𝑘𝑙, 𝜃𝑙) are the scattered lepton mass, energy, momentum, and angle.
+Figure 6-11 shows the 𝐸𝑄𝐸 distribution for 1.159 GeV C(𝑒, 𝑒′)0𝜋 events, which are
+most relevant for T2K and HK. A broad peak is observed centered at the real beam
+energy with a large tail extending to lower energies. The peak is doppler-broadened
+139
+
+by the motion of the nucleons in the nucleus. The tail is caused by non-QE reactions
+that pass the (𝑒, 𝑒′)0𝜋 selection. The tail is cut off at the lowest energies by the CLAS
+minimum detected electron energy of 0.4 GeV. The SuSAv2 eGENIE peak has the
+correct width, but is somewhat larger than the data. It overestimates the tail by
+about 25%. The G2018 eGENIE peak also exceeds the data, but is too narrow, with
+a Gaussian width of 𝜎 = 76 MeV, compared to 89 MeV for the data. This is due to
+inexact modeling of the nuclear ground state momentum distribution. The tail dips
+below the data at around 0.9 GeV, and is larger than the data at lower reconstructed
+energies. Neither model describes the data quantitatively well.
+0.6
+0.8
+1
+1.2
+1.4
+ [GeV]
+QE
+ E
+π
+0
+C(e,e')
+0
+0.5
+1
+GeV
+b
+µ
+ 
+QE
+dE
+σ
+d
+Data
+SuSav2 (Total)
+QE
+MEC
+RES
+DIS
+G2018
+beam
+E
+⇓
+Figure 6-11: The 1.159 GeV C(𝑒, 𝑒′)0𝜋 cross section plotted as a function of the
+reconstructed energy 𝐸𝑄𝐸 for data (black points), GENIE SuSAv2 (solid black curve)
+and GENIE G2018 (dotted black curve). The colored lines show the contributions
+of different processes to the GENIE SuSAv2 cross section: QE (blue), MEC (red),
+RES (green) and DIS (orange).
+Error bars show the 68% (1𝜎) confidence limits
+for the statistical and point-to-point systematic uncertainties added in quadrature.
+Error bars are not shown when they are smaller than the size of the data point.
+Normalization uncertainty of 3% not shown.
+Figure 6-12 shows the cross section as a function of 𝐸𝑄𝐸 for 1.159, 2.257 and
+4.453 GeV C(𝑒, 𝑒′)0𝜋 events and 2.257 and 4.453 GeV Fe(𝑒, 𝑒′)0𝜋 events. It is clear
+that the mismodeling already observed in the 1.159 GeV C(𝑒, 𝑒′)0𝜋 sample becomes
+140
+
+even more pronounced for higher energies and heavier nuclei.
+0.6
+0.8
+1
+1.2
+1.4
+0
+0.2
+0.4
+1
+1.5
+2
+2.5
+0
+0.2
+0.4
+1
+1.5
+2
+2.5
+0
+0.5
+1
+1.5
+2
+2
+3
+4
+5
+0
+0.2
+0.4
+2
+3
+4
+5
+0
+0.5
+1
+1.5
+2
+C
+12
+Fe
+56
+1.159 GeV (x1/2)
+2.257 GeV
+4.453 GeV (x5)
+ [GeV]
+QE
+ E
+π
+0
+(e,e')
+Data
+SuSav2 (Total)
+QE
+MEC
+RES DIS
+G2018
+GeV
+b
+µ
+ 
+QE
+dE
+σ
+d
+GeV
+b
+µ
+ 
+QE
+dE
+σ
+d
+Figure 6-12: The 𝐴(𝑒, 𝑒′𝑝)0𝜋 cross section plotted as a function of the reconstructed
+quasielastic energy 𝐸𝑄𝐸 for data (black points), SuSAv2 (black solid curve) and G2018
+(black dotted curve). Different panels show results for different beam energy and
+target nucleus combinations: (top row) Carbon target at (left to right) 1.159, 2.257
+and 4.453 GeV, and (bottom) Iron target at (left) 2.257 and (right) 4.453 GeV incident
+beam. The 1.159 GeV yields have been scaled by 1/2 and the 4.453 GeV yields have
+been scaled by 5 to have the same vertical scale. Colored lines show the contributions
+of different processes to the SuSAv2 GENIE simulation: QE (blue), MEC (red),
+RES (green) and DIS (orange).
+Error bars show the 68% (1𝜎) confidence limits
+for the statistical and point-to-point systematic uncertainties added in quadrature.
+Error bars are not shown when they are smaller than the size of the data point.
+Normalization uncertainties of 3% not shown.
+Tracking detectors measure all charged particles above their detection thresholds.
+The “calorimetric” incident neutrino energy is then the sum of all the detected particle
+energies:
+𝐸𝑐𝑎𝑙 =
+∑︁
+𝐸𝑖 + 𝜖,
+(6.9)
+where 𝐸𝑖 are the detected nucleon kinetic energies and the lepton and meson total
+energies and 𝜖 is the average total removal energy for the detected particles. This
+quantity 𝜖, used in reconstructing the incident energies in equations 6.8 and 6.9, was
+141
+
+determined from the data. It is defined using the difference in the binding energies
+for knocking a proton out of nucleus 𝐴 as 𝜖 = |𝑀𝐴 − 𝑀𝐴−1 − 𝑚𝑝| + ∆𝜖. The removal
+energy correction ∆𝜖 was adjusted so that the peaks in the 𝐸𝑐𝑎𝑙 spectrum for low
+transverse missing momentum events reconstructed to the correct beam energy. It
+was found that ∆𝜖 = 5 and 11 MeV for 12C and 56Fe, respectively, which are consistent
+with average excitation energies from single-nucleon knockout from nuclei.
+Figure 6-13 shows the cross section as a function of 𝐸𝑐𝑎𝑙 for 1.159, 2.257 and
+4.453 GeV C(𝑒, 𝑒′𝑝)1𝑝0𝜋 events and 2.257 and 4.453 GeV Fe(𝑒, 𝑒′𝑝)1𝑝0𝜋 events.
+All
+spectra show a sharp peak at the real beam energy, followed by a large tail at lower
+energies. For carbon, only 30-40% of the events reconstruct to within 5% of the real
+beam energy, as illustrated in table 6.2.
+Table 6.2: (𝑒, 𝑒′𝑝)1𝑝0𝜋 events reconstructed to the correct beam energy. Peak Fraction
+refers to the fraction of events reconstructed to the correct beam energy and Peak Sum
+refers to the integrated weighted cross section (as shown in Fig. 6-13) reconstructed
+to the correct beam energy. The peak integration windows are 1.1 ≤ 𝐸𝑐𝑎𝑙 ≤ 1.22
+GeV, 2.19 ≤ 𝐸𝑐𝑎𝑙 ≤ 2.34 GeV, and 4.35 ≤ 𝐸𝑐𝑎𝑙 ≤ 4.60 GeV, respectively, for the three
+incident beam energies. SuSAv2 is not intended to model nuclei lighter than 12C.
+1.159 GeV
+2.257 GeV
+4.453 GeV
+Peak
+Peak
+Peak
+Peak
+Peak
+Peak
+Fraction
+Sum [𝜇b]
+Fraction
+Sum [𝜇b]
+Fraction
+Sum [𝜇b]
+12C
+Data
+39
+4.13
+31
+1.26
+32
+0.34
+SuSAv2
+44
+5.33
+27
+1.76
+12
+0.20
+G2018
+51
+6.53
+37
+2.44
+23
+0.43
+56Fe
+Data
+-
+-
+20
+3.73
+23
+1.01
+SuSAv2
+-
+-
+21
+5.28
+10
+0.58
+G2018
+-
+-
+30
+8.22
+19
+1.48
+For iron this fraction is only 20-25%, highlighting the crucial need to well model
+the low-energy tail of these distributions. eGENIE overpredicts the fraction of events
+in the peak at 1.159 GeV and significantly underpredicts it at 4.453 GeV. eGENIE
+using SuSAv2 dramatically overpredicts the peak cross section at 1.159 and 2.257
+GeV, and significantly underestimates the peak cross section at 4.453 GeV, as shown
+142
+
+0.6
+0.8
+1
+1.2
+0
+0.5
+1
+1.5
+2
+Cross Section
+(a)
+0.1
+0.2
+1
+1.5
+2
+0
+0.5
+1
+1.5
+2
+Cross Section
+(b)
+0.1
+0.2
+1
+1.5
+2
+0
+2
+4
+6
+Cross Section
+(d)
+0.2
+0.4
+0.6
+2
+3
+4
+0
+0.5
+1
+1.5
+2
+Cross Section
+(c)
+0.1
+0.2
+2
+3
+4
+0
+2
+4
+6
+Cross Section
+(e)
+0.2
+0.4
+0.6
+C
+12
+Fe
+56
+1.159 GeV (x1/2)
+2.257 GeV
+4.453 GeV (x5)
+ [GeV]
+cal
+ E
+π
+1p0
+(e,e'p)
+Data
+SuSav2 (Total)
+QE
+MEC
+RES DIS
+G2018
+GeV
+b
+µ
+ 
+cal
+dE
+σ
+d
+GeV
+b
+µ
+ 
+cal
+dE
+σ
+d
+Figure 6-13: The 𝐴(𝑒, 𝑒′𝑝)1𝑝0𝜋 cross section plotted as a function of the reconstructed
+calorimetric energy 𝐸𝑐𝑎𝑙 for data (black points), SuSAv2 (black solid curve) and G2018
+(black dotted curve). Different panels show results for different beam energy and
+target nucleus combinations: (top row) Carbon target at (left to right) 1.159, 2.257
+and 4.453 GeV, and (bottom) Iron target at (left) 2.257 and (right) 4.453 GeV incident
+beam.
+The 1.159 GeV yields have been scaled by 1/2 and the 4.453 GeV yields
+have been scaled by 5 to have the same vertical scale. The insets show the cross
+sections with the same horizontal scale and an expanded vertical scale. Colored lines
+show the contributions of different processes to the SuSAv2 GENIE simulation: QE
+(blue), MEC (red), RES (green) and DIS (orange). Error bars show the 68% (1𝜎)
+confidence limits for the statistical and point-to-point systematic uncertainties added
+in quadrature. Error bars are not shown when they are smaller than the size of the
+data point. Normalization uncertainties of 3% not shown.
+in table 6.2. eGENIE using the older G2018 models overestimates the peak cross
+section at all three incident energies. It also reconstructs the peak position (i.e. the
+incident energy) to be 10, 25 and 36 MeV too low for 4He, C and Fe, respectively,
+at all three beam energies.
+This is due to an error in the G2018 QE modeling.
+This beam-energy dependence of the data-GENIE discrepancy could have significant
+implications for the neutrino flux reconstruction.
+At 1.159 GeV, eGENIE using SuSAv2 slightly overpredicts the low energy tail
+and eGENIE using G2018 is reasonably close. Both models dramatically overpredict
+143
+
+the low energy tail at the higher beam energies shown in the insets of figure 6-
+13. The tail seems to be dominated by RES and DIS at 4.453 GeV that did not
+result in the production of other charged particles above detection threshold. This
+overprediction has already been observed in inclusive electron scattering from the
+proton and deuteron, and thus appears to be due to the electron-nucleon interaction,
+rather than to the nuclear modeling [42].
+SuSAv2 describes the peak cross section - the part of the cross section that re-
+constructs to the correct beam energy - equally well for C and for Fe, while G2018
+over estimates the peak cross section more for Fe than for C. Both models predict a
+greater peak fraction relative to the data for Fe than for C, particularly at 2.2 GeV,
+as shown in table 6.2. While the (𝑒, 𝑒′)0𝜋 QE reconstruction of equation 6.8 gives a
+much broader peak at the true beam energy than the calorimetric energy 𝐸𝑐𝑎𝑙 due to
+the effects of nucleon motion, as shown in figures 6-14 and 6-15, it has the same tail
+of lower energy events for the same (𝑒, 𝑒′𝑝)1𝑝0𝜋 data set.
+0.6
+−
+0.4
+−
+0.2
+−
+0
+ Feeddown
+cal
+ E
+π
+1p0
+(e,e'p)
+2
+−
+10
+1
+−
+10
+1
+b
+µ
+ 
+Feed
+cal
+dE
+σ
+d
+(a)
+/  1.159 GeV
+/  2.257 GeV
+/  4.453 GeV(x4)
+C
+12
+Data/SuSav2
+0.6
+−
+0.4
+−
+0.2
+−
+0
+0.2
+ Feeddown
+QE
+ E
+π
+0
+(e,e')
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+b
+µ
+ 
+Feed
+QE
+dE
+σ
+d
+(b)
+/  1.159 GeV
+/  2.257 GeV
+/  4.453 GeV(x4)
+C
+12
+Data/SuSav2
+Figure 6-14:
+Energy feed-down cross-sections (𝐸𝑟𝑒𝑐 − 𝐸𝑡𝑟𝑢𝑒)/𝐸𝑡𝑟𝑢𝑒 for data (points)
+and SuSav2 (lines) for 1.159 GeV (red triangles and dotted lines), 2.257 GeV (green
+squares and dashed lines) and 4.453 GeV (blue dots and solid lines) on carbon for (a)
+𝐸𝑐𝑎𝑙, and (b) 𝐸𝑄𝐸.
+6.3
+Kinematic Imbalance Results
+Neutrino experiments use the “transverse variables” (TVs) outlined in section 3.2.1
+to enhance their sensitivity to different aspects of the reaction mechanism. These
+144
+
+0.6
+−
+0.4
+−
+0.2
+−
+0
+ Feeddown
+cal
+ E
+π
+1p0
+(e,e'p)
+2
+−
+10
+1
+−
+10
+1
+10
+b
+µ
+ 
+Feed
+cal
+dE
+σ
+d
+(c)
+/  2.257 GeV
+/  4.453 GeV(x4)
+Fe
+56
+Data/SuSav2
+0.6
+−
+0.4
+−
+0.2
+−
+0
+0.2
+ Feeddown
+QE
+ E
+π
+0
+(e,e')
+2
+4
+6
+b
+µ
+ 
+Feed
+QE
+dE
+σ
+d
+(d)
+/  2.257 GeV
+/  4.453 GeV(x4)
+Fe
+56
+Data/SuSav2
+Figure 6-15:
+Energy feed-down cross-sections (𝐸𝑟𝑒𝑐 − 𝐸𝑡𝑟𝑢𝑒)/𝐸𝑡𝑟𝑢𝑒 for data (points)
+and SuSav2 (lines) for 1.159 GeV (red triangles and dotted lines), 2.257 GeV (green
+squares and dashed lines) and 4.453 GeV (blue dots and solid lines) on iron for (c)
+𝐸𝑐𝑎𝑙, and (d) Fe 𝐸𝑄𝐸.
+TVs are independent of the neutrino energy and use the momentum of the detected
+particles transverse to the incident lepton [132, 223, 224] as shown in equation 6.12,
+where ⃗𝑃 𝑒′
+𝑇 and ⃗𝑃 𝑝
+𝑇 are the three-momenta of the detected lepton and proton perpen-
+dicular to the direction of the incident lepton, respectively. The ⃗𝑃𝑇 vector is intended
+to characterize the nuclear ground state, 𝛿𝛼𝑇 the FSI and ∆𝜑𝑇 is intended to probe
+regions where MEC events dominate [132,223,224].
+⃗𝑃𝑇
+=
+⃗𝑃 𝑒′
+𝑇 + ⃗𝑃 𝑝
+𝑇
+(6.10)
+𝛿𝛼𝑇
+=
+arccos(−
+⃗𝑃 𝑒′
+𝑇 · ⃗𝑃𝑇
+𝑃 𝑒′
+𝑇 𝑃𝑇
+)
+(6.11)
+𝛿𝜑𝑇
+=
+arccos(−
+⃗𝑃 𝑒′
+𝑇 · ⃗𝑃 𝑝
+𝑇
+𝑃 𝑒′
+𝑇 𝑃 𝑝
+𝑇
+)
+(6.12)
+Purely QE events without final state interactions, where the lepton scattered
+from a bound moving proton, will have small 𝑃𝑇, consistent with the motion of the
+struck nucleon. Events with small 𝑃𝑇 should thus reconstruct to the correct incident
+energy. Non-QE events, where neutral or sub-detection-threshold charged particles
+were produced, will have larger 𝑃𝑇 and will not reconstruct to the correct incident
+energy.
+𝑃𝑇 is thus an ideal observable for tuning reaction models to ensure they
+145
+
+correctly account for non-QE processes.
+The 𝑃𝑇 distribution for 2.257 GeV C(𝑒, 𝑒′𝑝)1𝑝0𝜋 is shown in figure 6-16 and the
+other targets and energies are shown in figure 6-17. Both data and eGENIE peak
+at relatively low momenta, as expected, and both have a large tail extending out to
+1 GeV/𝑐 and containing about half of the measured events. The high-𝑃𝑇 tail is pre-
+dominantly due to resonance production that did not result in an additional pion or
+nucleon above the detection threshold. eGENIE using SuSAv2 reproduces the shape
+of the data moderately well, suggesting adequate reaction modeling, including the
+contribution of non-QE processes such as resonance production. As expected, both
+data and eGENIE/SuSAv2 events with 𝑃𝑇
+< 200 MeV/𝑐 almost all reconstruct to
+the correct incident energy. However, events with 𝑃𝑇≥ 400 MeV/𝑐 do not reconstruct
+to the correct energy and are poorly reproduced by eGENIE. This disagreement be-
+comes even more pronounced at higher energies and heavier nuclei, and indicates that
+including high-𝑃𝑇 data in oscillation analyses could bias the extracted parameters.
+As high-𝑃𝑇 data accounts for 25 − 50% of the measured events, care must be taken
+to improve the models implemented in GENIE, so that they can reproduce the high-
+𝑃𝑇 data. This will be especially true at the higher incident neutrino energies expected
+for DUNE.
+The opening angle 𝛿𝛼𝑇 measures the angle between 𝑃𝑇 and the transverse mo-
+mentum transfer (⃗𝑞𝑇 = −⃗𝑃 𝑒′
+𝑇 ) in the transverse plane and is isotropic in the absence
+of final state interactions. 𝛿𝜑𝑇 measures the opening angle between the detected pro-
+ton momentum and the transverse momentum transfer and is forward peaked. The
+𝛿𝛼𝑇 distributions become progressively less isotropic at higher energies and heavier
+targets, indicating the increasing importance of FSI and of non-QE reaction mech-
+anisms. GENIE agrees best with data at the lowest beam energy. At the higher
+beam energies GENIE describes the relatively flat smaller angles much better than
+the back-angle peak. GENIE also describes the lowest energy 𝛿𝜑𝑇 distribution. At
+higher energies, GENIE overestimates the height of the forward peak, as shown in
+figure 6-18.
+In line with the neutrino-based MicrobooNE analysis presented in section 3.2.4,
+146
+
+0
+0.2
+0.4
+0.6
+0.8
+1
+ [GeV/c]
+T
+ P
+π
+1p0
+C(e,e'p)
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+GeV/c
+b
+µ
+ 
+T
+dP
+σ
+d
+Data
+SuSav2 (Total)
+QE
+MEC
+RES
+DIS
+G2018
+2.257 GeV
+1
+1.5
+2
+0
+0.05
+ > 400 [MeV/c]
+T
+P
+(c)
+ > 400 [MeV/c]
+T
+P
+(c)
+ > 400 [MeV/c]
+T
+P
+(c)
+0
+0.1
+0.2
+ < 400 [MeV/c]
+T
+200 < P
+(b)
+ < 400 [MeV/c]
+T
+200 < P
+(b)
+ < 400 [MeV/c]
+T
+200 < P
+(b)
+0
+1
+2
+ < 200 [MeV/c]
+T
+0 < P
+(a)
+ < 200 [MeV/c]
+T
+0 < P
+(a)
+ < 200 [MeV/c]
+T
+0 < P
+(a)
+0.02
+0.04
+0.06
+0.08
+ [GeV]
+cal
+ E
+π
+1p0
+(e,e'p)
+GeV
+b
+µ
+ 
+cal
+dE
+σ
+d
+Figure 6-16: (Left) the 2.257 GeV C(𝑒, 𝑒′𝑝)1𝑝0𝜋 cross section plotted versus missing
+transverse momentum, 𝑃𝑇, for data (black points), SuSav2 (black solid line) and
+G2018 (black dashed line). The vertical lines at 200 MeV/𝑐 and at 400 MeV/𝑐 separate
+the three bins in 𝑃𝑇. Colored lines show the contributions of different processes to the
+SuSAv2 GENIE simulation: QE (blue), MEC (red), RES (green) and DIS (orange).
+(Right) The cross section plotted versus the calorimetric energy 𝐸𝑐𝑎𝑙 for different bins
+in 𝑃𝑇: (top) 𝑃𝑇< 200 MeV/𝑐, (middle) 200 MeV/𝑐 ≤ 𝑃𝑇≤ 400 MeV/𝑐, and (bottom)
+𝑃𝑇> 400 MeV/𝑐.
+these kinematic variables were further investigated in the form of multidimensional
+cross sections. Figure 6-19 shows the data-simulation cross-section comparisons for
+12C at 2.261 GeV as a function of 𝑃𝑇 for (top) all the events, (bottom left) events with
+𝛿𝛼𝑇 < 45𝑜 dominated by QE interactions and no reinteractions, and (bottom right)
+events with 135𝑜 < 𝛿𝛼𝑇 < 180𝑜 maximally affected by FSI and multi-hadron channels.
+Using all the events (top) that satisfy our selection yielded a QE-rich region up to
+≈ 300 MeV/c and a RES-dominated tail that extended to ≈ 1 GeV/c. Slicing the
+available 𝑃𝑇 phase-space in 𝛿𝛼𝑇 regions revealed regions with specific features. More
+precisely, the region with 𝛿𝛼𝑇 < 45𝑜 can be used to isolate primarily QE events and
+to test nuclear models in event generators. Even in this region though, differences
+of 20-30% in the QE strength are observed by both model configurations used for
+147
+
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+(a)
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+(b)
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.5
+1
+1.5
+2
+(d)
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+(c)
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.5
+1
+1.5
+2
+(e)
+C
+12
+Fe
+56
+1.159 GeV
+2.257 GeV
+4.453 GeV (x4)
+ [GeV/c]
+T
+ P
+π
+1p0
+(e,e'p)
+Data
+SuSav2 (Total)
+QE
+MEC
+RES DIS
+G2018
+GeV/c
+b
+µ
+ 
+T
+dP
+σ
+d
+GeV/c
+b
+µ
+ 
+T
+dP
+σ
+d
+Figure 6-17: The cross section plotted vs transverse missing momentum 𝑃𝑇 for data
+(black points), SuSAv2 (black solid curve) and G2018 (black dotted curve). Different
+panels show results for different beam energy and target nucleus combinations: (top
+row) Carbon target at (left to right) 1.159, 2.257 and 4.453 GeV, and (bottom) Iron
+target at (left) 2.257 and (right) 4.453 GeV. The 4.453 GeV yields have been scaled by
+four to have the same vertical scale. Colored lines show the contributions of different
+processes to the SuSAv2 GENIE simulation: QE (blue), MEC (red), RES (green) and
+DIS (orange).
+comparison in this analysis. Meanwhile, the 135𝑜 < 𝛿𝛼𝑇 < 180𝑜 region is dominated
+by RES interactions and could be used to tune FSI parameters and to improve the
+RES modeling.
+Figure 6-20 shows the data-simulation comparisons for 12C at 2.261 GeV as a
+function of 𝛿𝛼𝑇 for (left) all the events, (middle) events with 𝑃𝑇 < 0.2 GeV/c, and
+(right) events with 𝑃𝑇 > 0.4 GeV/c. The 𝑃𝑇 < 0.2 GeV/c slice is dominated by QE
+events with minimal FSI effects that result in a fairly uniform distribution with a
+slight enhancement in the forward direction due to the RES contamination.
+The
+𝑃𝑇 > 0.4 GeV/c slice is dominated by multi-hadron and enhanced-FSI events that
+result in a sharp peak in the region close to 150𝑜. Therefore, this investigation of the
+𝛿𝛼𝑇 phase-space in slices of 𝑃𝑇 is complimentary to the one illustrated in figure 6-19.
+Using the exact same 𝑒4𝜈 data sets and cross-section extraction technique, the
+148
+
+0
+50
+100
+150
+0
+1
+2
+3
+(a)
+0
+50
+100
+150
+0
+1
+2
+3
+(b)
+0
+50
+100
+150
+0
+5
+10
+(d)
+0
+50
+100
+150
+0
+1
+2
+3
+(c)
+0
+50
+100
+150
+0
+5
+10
+(e)
+C
+12
+Fe
+56
+1.159 GeV
+2.257 GeV
+4.453 GeV (x2)
+ [deg]
+T
+α
+δ
+ 
+π
+1p0
+(e,e'p)
+Data
+SuSav2 (Total)
+QE
+MEC
+RES DIS
+G2018
+deg
+nb
+ 
+T
+α
+d
+σ
+d
+deg
+nb
+ 
+T
+α
+d
+σ
+d
+0
+50
+100
+150
+0
+2
+4
+6
+8
+(f)
+0
+50
+100
+150
+0
+2
+4
+6
+8
+(g)
+0
+50
+100
+150
+0
+5
+10
+15
+20
+(i)
+0
+50
+100
+150
+0
+2
+4
+6
+8
+(h)
+0
+50
+100
+150
+0
+5
+10
+15
+20
+(j)
+C
+12
+Fe
+56
+(x2)
+ [deg]
+T
+φ
+δ
+ 
+π
+1p0
+(e,e'p)
+deg
+nb
+ 
+T
+φ
+d
+σ
+d
+deg
+nb
+ 
+T
+φ
+d
+σ
+d
+Figure 6-18: The cross section plotted vs 𝛿𝛼𝑇 (a-e) and vs 𝛿𝜑𝑇 (f-j) for data (black
+points), SuSAv2 (black solid curve) and G2018 (black dotted curve). Different panels
+show results for different beam energy and target nucleus combinations: (top row)
+Carbon target at (left to right) 1.159, 2.257 and 4.453 GeV, and (bottom) Iron target
+at (left) 2.257 and (right) 4.453 GeV. The 4.453 GeV yields have been scaled by two
+to have the same vertical scale.
+Colored lines show the contributions of different
+processes to the SuSAv2 GENIE simulation: QE (blue), MEC (red), RES (green)
+and DIS (orange).
+cross section as a function of the total struck nucleon momentum approximation,
+derived following the approach in section 3.2.1, was reported. The only difference
+compared to the already outlined formalism on 40Ar is that B = 0.09216 GeV is the
+149
+
+0
+0.2
+0.4
+0.6
+0.8
+1
+ [GeV/c]
+T
+ P
+π
+1p0
+(e,e'p)
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+GeV/c
+b
+µ
+ 
+T
+dP
+σ
+d
+12C @ 2.257 GeV, All events
+Data
+SuSav2 (Total)
+QE
+MEC
+RES
+DIS
+G2018
+12C @ 2.257 GeV, All events
+0
+0.2
+0.4
+0.6
+0.8
+1
+ [GeV/c]
+T
+ P
+π
+1p0
+(e,e'p)
+0
+0.5
+1
+1.5
+2
+3
+−
+10
+×
+deg GeV/c
+b
+µ
+ 
+T
+dP
+T
+α
+δ
+d
+σ
+2
+d
+o
+ < 45
+T
+α
+δ
+                       12C @2.257 GeV, 
+o
+ < 45
+T
+α
+δ
+                       12C @2.257 GeV, 
+0
+0.2
+0.4
+0.6
+0.8
+1
+ [GeV/c]
+T
+ P
+π
+1p0
+(e,e'p)
+0
+1
+2
+3
+4
+3
+−
+10
+×
+deg GeV/c
+b
+µ
+ 
+T
+dP
+T
+α
+δ
+d
+σ
+2
+d
+o
+ < 180
+T
+α
+δ
+ < 
+o
+                       12C @2.257 GeV, 135
+o
+ < 180
+T
+α
+δ
+ < 
+o
+                       12C @2.257 GeV, 135
+Figure 6-19: Data-simulation comparisons for 12C at 2.261 GeV showing the cross
+section results as a function of 𝑃𝑇 for (top) all the events, (bottom left) events with
+𝛿𝛼𝑇 < 45𝑜 dominated by QE interactions and no reinteractions, and (bottom right)
+events with 135𝑜 < 𝛿𝛼𝑇 < 180𝑜 maximally affected by FSI and multi-hadron channels.
+Colored lines show the contributions of different processes to the SuSAv2 GENIE
+simulation: QE (blue), MEC (red), RES (green) and DIS (orange).
+binding energy for 12C, and 𝜖𝑁 = 0.024 GeV is the corresponding removal energy [225].
+This approximation, refered to as 𝑃𝑛,𝑝𝑟𝑜𝑥𝑦, is compared to the true missing momentum
+𝑃𝑀𝑖𝑠𝑠 calculated as shown in equation 6.13,
+𝑃𝑀𝑖𝑠𝑠 = |⃗𝑞 − ⃗𝑝|,
+(6.13)
+where ⃗𝑞 is the 3-vector for the momentum transfer based on the difference between
+the kinematics of the incoming and the outgoing lepton, and ⃗𝑝 is the 3-vector of the
+outgoing proton.
+Figure 6-21 shows the data-simulation comparisons for 12C at 2.261 GeV as a
+function of (top left) the true missing momentum 𝑃𝑀𝑖𝑠𝑠 and (top right) the missing
+150
+
+0
+50
+100
+150
+ [deg]
+T
+α
+δ
+ 
+π
+1p0
+(e,e'p)
+0
+0.5
+1
+1.5
+2
+2.5
+3
+−
+10
+×
+deg
+b
+µ
+ 
+T
+α
+δ
+d
+σ
+d
+12C @ 2.257 GeV, All events
+Data
+SuSav2 (Total)
+QE
+MEC
+RES
+DIS
+G2018
+12C @ 2.257 GeV, All events
+0
+50
+100
+150
+ [deg]
+T
+α
+δ
+ 
+π
+1p0
+(e,e'p)
+0
+1
+2
+3
+3
+−
+10
+×
+deg GeV/c
+b
+µ
+ 
+T
+dP
+T
+α
+δ
+d
+σ
+2
+d
+ < 0.2 GeV/c
+T
+                       12C @2.257 GeV, P  < 0.2 GeV/c
+T
+                       12C @2.257 GeV, P
+0
+50
+100
+150
+ [deg]
+T
+α
+δ
+ 
+π
+1p0
+(e,e'p)
+0
+0.5
+1
+1.5
+2
+3
+−
+10
+×
+deg GeV/c
+b
+µ
+ 
+T
+dP
+T
+α
+δ
+d
+σ
+2
+d
+ > 0.4 GeV/c
+T
+                       12C @2.257 GeV, P  > 0.4 GeV/c
+T
+                       12C @2.257 GeV, P
+Figure 6-20: Data-simulation comparisons for 12C at 2.261 GeV showing the cross
+section results as a function of 𝛿𝛼𝑇 for (top) all the events, (bottom left) events
+with 𝑃𝑇 < 0.2 GeV/c dominated by QE interactions and no reinteractions, and (bot-
+tom right) events with 𝑃𝑇 > 0.4 GeV/c maximally affected by FSI and multi-hadron
+channels. Colored lines show the contributions of different processes to the SuSAv2
+GENIE simulation: QE (blue), MEC (red), RES (green) and DIS (orange).
+momentum approximation 𝑃𝑛,𝑝𝑟𝑜𝑥𝑦 commonly used by neutrino experiments. The bot-
+tom panel shows the overlay of the two extracted data cross sections illustrating that
+𝑃𝑛,𝑝𝑟𝑜𝑥𝑦 fails to reproduce 𝑃𝑀𝑖𝑠𝑠. As can be seen in the interaction breakdown plots,
+the differences are primarily driven by the RES interactions which are reconstructed
+at lower values when the QE-like assumption deployed for 𝑃𝑛,𝑝𝑟𝑜𝑥𝑦 is used.
+Figures 6-22 and 6-23 also illustrate the equivalent of the multidimensional Mi-
+croBooNE analysis using the 𝑃𝑇,𝑥 cross sections in slices of 𝑃𝑇,𝑦 (figure 6-22) and vice
+versa (figure 6-23). As can be seen in the interaction breakdown plots, the regions
+close to 0 for both kinematic variables (bottom left panels) are the ideal place to iso-
+late QE events with small contributions from more complex events. Yet, even in this
+151
+
+0
+0.5
+1
+1.5
+ [GeV/c]
+Miss
+ P
+π
+1p0
+(e,e'p)
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+]
+GeV/c
+b
+µ
+ [
+Miss
+dP
+σ
+d
+12C @2.257 GeV, All events
+Data
+SuSav2 (Total)
+QE
+MEC
+RES
+DIS
+G2018
+12C @2.257 GeV, All events
+0
+0.5
+1
+1.5
+ [GeV/c]
+n,proxy
+ P
+π
+1p0
+(e,e'p)
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+]
+GeV/c
+b
+µ
+ [
+n,proxy
+dP
+σ
+d
+12C @2.257 GeV, All events
+Data
+SuSav2 (Total)
+QE
+MEC
+RES
+DIS
+G2018
+12C @2.257 GeV, All events
+0
+0.5
+1
+1.5
+ [GeV/c]
+Miss
+ P
+π
+1p0
+(e,e'p)
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+]
+GeV/c
+b
+µ
+ [
+Miss
+dP
+σ
+d
+12C @2.257 GeV, All events
+Miss
+P
+n,proxy
+P
+12C @2.257 GeV, All events
+Figure 6-21: (Top) data-simulation comparisons on 12C at 2.261 GeV showing the
+cross section results as a function of (top left) the true missing momentum 𝑃𝑀𝑖𝑠𝑠 and
+(top right) the missing momentum approximation 𝑃𝑛,𝑝𝑟𝑜𝑥𝑦 commonly used by neutrino
+experiments. (Bottom) overlay of the two extracted data cross sections illustrating
+the differences between 𝑃𝑀𝑖𝑠𝑠 and 𝑃𝑛,𝑝𝑟𝑜𝑥𝑦.
+very QE-dominated region, significant data-simulation differences are observed, most
+likely due to the already-observed RES overestimation. On the other hand, events
+occupying the phase-space outside that QE-rich band (bottom right) are dominated
+primarily by most complicated interactions (RES and DIS) and QE events with strong
+FSI effects, as indicated by the relevant pointy shapes. These populations result in
+much more broader and smeared distributions.
+Of particular interest is the asymmetric behavior observed in figure 6-23.
+As
+discussed in section 3.2.1, this clearly-observed asymmetry is caused by multi-nucleon
+and FSI effects. More precisely, the asymmetry in the QE-enhanced region (bottom
+left) is entirely driven by the overpredicted RES events. In the non-QE-dominated
+slice (bottom right), data-simulation disagreements by a factor of ∼ 2 are further
+152
+
+0.5
+−
+0
+0.5
+ [GeV/c]
+T,x
+ P
+π
+1p0
+(e,e'p)
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+]
+GeV/c
+b
+µ
+ [
+T,x
+dP
+σ
+d
+12C @2.257 GeV, All events
+Data
+SuSav2 (Total)
+QE
+MEC
+RES
+DIS
+G2018
+12C @2.257 GeV, All events
+0.5
+−
+0
+0.5
+ [GeV/c]
+T,x
+ P
+π
+1p0
+(e,e'p)
+0
+0.2
+0.4
+0.6
+0.8
+1
+2
+/c
+2
+GeV
+b
+µ
+ 
+T,y
+dP
+T,x
+dP
+σ
+2
+d
+ < 0.15 GeV/c
+T,y
+                       12C @2.257 GeV, -0.15 < P
+ < 0.15 GeV/c
+T,y
+                       12C @2.257 GeV, -0.15 < P
+0.5
+−
+0
+0.5
+ [GeV/c]
+T,x
+ P
+π
+1p0
+(e,e'p)
+0
+5
+10
+15
+20
+25
+3
+−
+10
+×
+2
+/c
+2
+GeV
+b
+µ
+ 
+T,y
+dP
+T,x
+dP
+σ
+2
+d
+ < -0.15 GeV/c
+T,y
+                       12C @2.257 GeV, P
+ < -0.15 GeV/c
+T,y
+                       12C @2.257 GeV, P
+Figure 6-22: Data-simulation comparisons for 12C at 2.261 GeV showing the cross
+section results as a function of 𝑃𝑇,𝑥 for (top) all the events, (bottom left) events
+with -0.15 < 𝑃𝑇,𝑦 < 0.15 GeV/c dominated by QE interactions and no reinteractions,
+and (bottom right) events with 𝑃𝑇,𝑦 < -0.15 GeV/c maximally affected by FSI and
+multi-hadron channels.
+observed and are driven by RES/DIS events and FSI-enhanced QE interactions.
+6.4
+Electrons-For-Neutrinos Conclusions
+In this electron-based analysis, the similarities between electron- and neutrino-nucleus
+interactions were exploited, and electron scattering data with known beam energies
+to test energy reconstruction methods and interaction models were used [208]. Even
+in simple interactions where no pions are detected, only a small fraction of events was
+found to reconstruct to the correct incident energy. More importantly, widely-used
+interaction models reproduced the reconstructed energy distribution only qualita-
+tively and the quality of the reproduction varied strongly with beam energy. This
+153
+
+1.5
+−
+1
+−
+0.5
+−
+0
+ [GeV/c]
+T,y
+ P
+π
+1p0
+(e,e'p)
+0
+0.1
+0.2
+0.3
+0.4
+]
+GeV/c
+b
+µ
+ [
+T,y
+dP
+σ
+d
+12C @2.257 GeV, All events
+Data
+SuSav2 (Total)
+QE
+MEC
+RES
+DIS
+G2018
+12C @2.257 GeV, All events
+0.5
+−
+0
+0.5
+ [GeV/c]
+T,y
+ P
+π
+1p0
+(e,e'p)
+0
+0.5
+1
+2
+/c
+2
+GeV
+b
+µ
+ 
+T,y
+dP
+T,x
+dP
+σ
+2
+d
+ < 0.15 GeV/c
+T,x
+                       12C @2.257 GeV, -0.15 < P
+ < 0.15 GeV/c
+T,x
+                       12C @2.257 GeV, -0.15 < P
+0.5
+−
+0
+0.5
+ [GeV/c]
+T,y
+ P
+π
+1p0
+(e,e'p)
+0
+2
+4
+6
+8
+10
+3
+−
+10
+×
+2
+/c
+2
+GeV
+b
+µ
+ 
+T,y
+dP
+T,x
+dP
+σ
+2
+d
+ < -0.15 GeV/c
+T,x
+                       12C @2.257 GeV, P
+ < -0.15 GeV/c
+T,x
+                       12C @2.257 GeV, P
+Figure 6-23: Data-simulation comparisons for 12C at 2.261 GeV showing the cross
+section results as a function of 𝑃𝑇,𝑦 for (top) all the events, (bottom left) events
+with -0.15 < 𝑃𝑇,𝑥 < 0.15 GeV/c dominated by QE interactions and no reinteractions,
+and (bottom right) events with 𝑃𝑇,𝑥 < -0.15 GeV/c maximally affected by FSI and
+multi-hadron channels.
+shows both the need and the pathway to improve current models to meet the re-
+quirements of next-generation, high-precision experiments such as DUNE [146] and
+Hyper-Kamiokande (HK) [226].
+6.5
+Prospects With CLAS12
+Jefferson Lab Experiment E12-17-006, “Electrons for Neutrinos: Addressing Critical
+Neutrino–Nucleus Issues” (scientific rating: A) [227] has already taken further data on
+more targets with a greater kinematical range using the upgraded CLAS12 detector
+shown in figure 6-24. The approved experiment includes measurements on 4He, C, Ar
+and Sn with 1-, 2-, 4-, and 6-GeV electron beams, as well as measurements on O with
+154
+
+1- and 2-GeV electron beams. The 1- and 2-GeV measurements will be performed
+with a minimum electron scattering angle of 5𝑜, compared to a minimum CLAS angle
+of about 15𝑜. This will extend the measurements down to the much lower momentum
+transfers, typical of some neutrino experiments, and to multi-hadron topologies. It
+will therefore allow comparisons with the lower beam-energy data of T2K and HK.
+The first part of the experiment ran in the second half of 2021 and the second one in
+the beginning of 2022. The majority of the 2022 data sets have already been collected,
+with the exception of the 1 GeV and 16O data sets.
+Cherenkov
+Time-of-Flight
+BAND
+Calorimeter
+e– beam
+Target
+Tracker
+Figure 6-24: Schematic view of the upgraded CLAS12 detector components.
+155
+
+LTCC
+FTOF
+EC & PCAL
+HTCC
+CTOF + CND
+BAND156
+
+Chapter 7
+Summary
+The findings presented in this thesis report on both neutrino and electron cross-
+section modeling and analysis results in order to critically improve the understanding
+of lepton-nucleus interactions. This knowledge will be used to significantly reduce the
+cross-section related uncertainties of forthcoming experiments that aim to extract the
+neutrino oscillation parameters with high-accuracy measurements.
+The outlined work took a significant step towards this high-precision era with the
+use of neutrino data sets from the MicroBooNE liquid argon time projection chamber
+detector at Fermi National Laboratory. The reported neutrino analyses isolated parts
+of the phase space where significant model improvements are required. Furthermore,
+valuable kinematic variables were identified and established as tools to probe specific
+nuclear effects with multi-differential measurements.
+This thesis further realized the connections between electron and neutrino inter-
+actions. This realization resulted in significant improvements of the modeling used
+in neutrino oscillation analyses. Yet, these improved predictions failed to reproduce
+exclusive electron scattering results from the CLAS detector at Thomas Jefferson
+Laboratory using high-statistics data sets and monoenergetic beams. However, such
+comparisons against electron scattering data sets can definitively constrain the vector
+part and the nuclear effects in lepton-nucleus interactions in event generators that
+will be used for the forthcoming oscillation analyses.
+157
+
+158
+
+Chapter 8
+Appendices
+8.1
+Total Struck Nucleon Momentum Derivation
+The total momentum of the struck nucleon can be obtained following the derivation
+outlined below for two-body interactions. We followed the approach introduced by
+the Minerva collaboration to reconstruct the longitudinal and total nucleon momenta
+in [132]. We focused on quasielastic-like (QE-like) CC1p0𝜋 processes,
+𝜈𝜇 + 𝐴 → 𝜇− + 𝑝 + (𝐴 − 1),
+(8.1)
+while using the formalism detailed in [228] for the nuclear masses,
+𝑚𝐴 = 22 × 𝑀𝑛 + 18 × 𝑀𝑝 − 𝐵 [𝐺𝑒𝑉 ]
+𝑚𝐴−1 = 𝑚𝐴 − 𝑀𝑛 + 𝜖𝑁 [𝐺𝑒𝑉 ].
+(8.2)
+Here 𝑀𝑝 and 𝑀𝑛 denote the proton and neutron masses, respectively, B = 0.34381
+GeV is the argon binding energy (obtained from page 3 in [228] for 40 nucleons with
+an average binding energy of 9 MeV) and 𝜖𝑁 = 0.0309 GeV is the removal energy,
+159
+
+with
+𝜖𝑁 = 𝑆𝑁 + 𝐸𝑁
+𝑥 + ⟨𝑇𝐴−1⟩
+(8.3)
+where 𝑆𝑁 = 9.9 MeV is the neutron separation energy obtained for argon from
+table 7 in [225], 𝐸𝑁
+𝑥 is the excitation energy, and ⟨𝑇𝐴−1⟩ is the average kinetic energy
+of the remnant system, which is negligible.
+In equation 8.1, the incident neutrino energy 𝐸𝜈 is unknown, but the dependence
+of 𝛿⃗𝑝 (struck nucleon momentum before the interaction) on 𝐸𝜈 can be removed under
+a QE-like approximation. This was achieved with the process detailed below.
+First, we decomposed 𝛿⃗𝑝 into longitudinal and transverse components with respect
+to the neutrino direction, used energy/momentum conservation equations and that
+𝑝𝜈 = 𝐸𝜈,
+𝛿⃗𝑝 ≡ (𝛿⃗𝑝𝑇, 𝛿𝑝𝐿)
+(8.4)
+⃗𝑝𝜈 = ⃗𝑝𝜇 + ⃗𝑝𝑝 + ⃗𝑝𝐴−1
+(8.5)
+𝐸𝜈 + 𝛿𝑝𝐿 = 𝑝𝜇
+𝐿 + 𝑝𝑝
+𝐿
+(8.6)
+𝛿⃗𝑝𝑇 = ⃗𝑝𝜇
+𝑇 + ⃗𝑝𝑝
+𝐿
+(8.7)
+𝐸𝜈 + 𝑚𝐴 = 𝐸𝜇 + 𝐸𝑝 + 𝐸𝐴−1
+(8.8)
+where ⃗𝑝𝜇 and ⃗𝑝𝑝 are the muon and proton momenta, respectively, 𝐸𝜇 = √︀𝑝2
+𝜇 + 𝑚2
+𝜇
+for the muon candidate and 𝐸𝑝 = √︀𝑝2
+𝑝 + 𝑚2
+𝑝 for the proton candidate, where the
+corresponding momenta are obtained based on the particles’ ranges [137], and 𝐸𝐴−1
+is the energy of remnant nuclear system.
+Under the assumption that no final state interactions (FSI) take place,
+⃗𝑝𝜈 + 𝛿⃗𝑝 = ⃗𝑝𝜇 + ⃗𝑝𝑝.
+(8.9)
+160
+
+If we combine equations 8.5 and 8.9, 𝛿⃗𝑝 gives the magnitude of its recoil momen-
+tum, 𝛿⃗𝑝 = −⃗𝑝𝐴−1 and 𝛿𝑝 = 𝑝𝐴−1. Combining equations 8.6 and 8.8 to eliminate 𝐸𝜈
+yields
+𝛿𝑝𝐿 = 𝑚𝐴 + 𝑝𝜇
+𝐿 + 𝑝𝑝
+𝐿 − 𝐸𝜇 − 𝐸𝑝 − 𝐸𝐴−1.
+(8.10)
+Using the fact that 𝐸𝐴−1 =
+√︁
+𝑚2
+𝐴−1 + 𝑝2
+𝐴−1, 𝛿𝑝 = 𝑝𝐴−1 and the decomposition of
+𝛿⃗𝑝 into longitudinal and transverse components in equation 8.4,
+𝛿𝑝𝐿 = 𝑚𝐴 + 𝑝𝜇
+𝐿 + 𝑝𝑝
+𝐿 − 𝐸𝜇 − 𝐸𝑝 −
+√︁
+𝑚2
+𝐴−1 + 𝛿𝑝2
+𝑇 + 𝛿𝑝2
+𝐿
+(8.11)
+For simplicity, we defined
+𝑅 ≡ 𝑚𝐴 + 𝑝𝜇
+𝐿 + 𝑝𝑝
+𝐿 − 𝐸𝜇 − 𝐸𝑝
+(8.12)
+Using R to simplify equation 8.11,
+𝛿𝑝𝐿 = 𝑅 −
+√︁
+𝑚2
+𝐴−1 + 𝛿𝑝2
+𝑇 + 𝛿𝑝2
+𝐿
+(8.13)
+Rearranging the terms, squaring each side and solving for 𝛿𝑝𝐿 yields
+𝛿𝑝𝐿 = 1
+2𝑅 − 𝑚2
+𝐴−1 + 𝛿𝑝2
+𝑇
+2𝑅
+.
+(8.14)
+Finally, combining the longitudinal and the transverse components, we obtain the
+total struck nucleon momentum. Given that this momentum is an approximation
+following the procedure and the assumptions mentioned above (QE-like scattering
+161
+
+and no FSI), we will be referring to it as 𝑝𝑛,𝑝𝑟𝑜𝑥𝑦 as opposed to 𝛿𝑝, where
+𝑝𝑛,𝑝𝑟𝑜𝑥𝑦 =
+√︁
+𝛿𝑝2
+𝐿 + 𝛿𝑝2
+𝑇
+(8.15)
+162
+
+8.2
+Wiener SVD Regularization Technique
+The procedure detailed below relies on [139]. We choose to work with a 𝜒2(𝑠) metric.
+𝜒2(𝑠) = (m − 𝑟 · 𝑠)𝑇𝐶𝑜𝑣−1(m − 𝑟 · 𝑠)
+(8.16)
+Notation
+Explanation
+Dimension
+And Format
+m
+measured spectrum in data (signal and background events)
+m × 1 vector
+s
+free variable in 𝜒2(𝑠) function
+n × 1 vector
+ˆ𝑠
+estimator of true signal, obtained after minimizing 𝜒2
+n × 1 vector
+𝑠𝑡𝑟𝑢𝑒
+true signal
+n × 1 vector
+𝑠
+nominal MC signal prediction
+n × 1 vector
+r
+response matrix
+m × n matrix
+Cov
+symmetric covariance matrix
+with statistical and systematic uncertainties
+m × m matrix
+Note that we use 𝑠𝑡𝑟𝑢𝑒 to represent the true signal in order to differentiate from s
+which is a variable in the function 𝜒2(𝑠). We use ˆ𝑠 to represent the estimator of the
+true signal 𝑠𝑡𝑟𝑢𝑒, which is obtained after minimizing the 𝜒2(𝑠) function. We further
+restrict ourselves to the m ≥ n case.
+Our objective is to minimize 𝜒2(𝑠). Since the covariance matrix Cov is symmetric,
+the inverse of it 𝐶𝑜𝑣−1 is also symmetric. Hence, 𝐶𝑜𝑣−1 can be decomposed with
+Cholesky decomposition [229] into
+𝐶𝑜𝑣−1 = 𝑄𝑇 · 𝑄
+(8.17)
+where Q is a uniquely defined lower triangular matrix and 𝑄𝑇 is its transpose.
+163
+
+We further define
+𝑀 ≡ 𝑄 · m
+𝑅 ≡ 𝑄 · 𝑟
+(𝑝𝑟𝑒 − 𝑠𝑐𝑎𝑙𝑖𝑛𝑔)
+(8.18)
+Notation
+Explanation
+Dimension
+And Format
+M
+measured spectrum after pre-scaling
+m × 1 vector
+R
+response matrix after pre-scaling
+m × n matrix
+Q
+Lower triangular matrix from Cholesky decomposition of 𝐶𝑜𝑣−1
+m × m matrix
+𝑀
+expectation spectrum after pre-scaling (R · 𝑠)
+m × 1 vector
+Replacing the definitions in equation 8.18 into equation 8.16 yields
+𝜒2(𝑠) = (𝑀 − 𝑅 · 𝑠)𝑇(𝑀 − 𝑅 · 𝑠) =
+∑︁
+𝑖
+(𝑀𝑖 −
+∑︁
+𝑗
+𝑅𝑖𝑗 · 𝑠𝑗)2
+(8.19)
+Ideally, we want to achieve 𝜒2(𝑠) = 0. That is obtained when
+𝑀 − 𝑅 · 𝑠 = 0 ⇒ 𝑀 = 𝑅 · 𝑠 ⇒ 𝑅𝑇 · 𝑀 = 𝑅𝑇 · 𝑅 · 𝑠
+(8.20)
+We use the fact that 𝑅𝑇 · 𝑅 is a square n×n invertable matrix. Thus, the exact
+solution ˆ𝑠 is uniquely defined as
+ˆ𝑠 = (𝑅𝑇 · 𝑅)−1 · 𝑅𝑇 · 𝑀
+(8.21)
+We decompose the measured spectrum into the signal (R · s) part and the noise
+/ background (N).
+𝑀 = 𝑅 · 𝑠𝑡𝑟𝑢𝑒 + 𝑁
+(8.22)
+164
+
+Substituting equation 8.22 into equation 8.21 yields
+ˆ𝑠 = (𝑅𝑇 · 𝑅)−1 · 𝑅𝑇 · (𝑅 · 𝑠𝑡𝑟𝑢𝑒 + 𝑁)
+(8.23)
+with N representing the “noise” coming from uncertainties (statistical and system-
+atic uncertainties associated with both m and r). Each term in the noise vector after
+pre-scaling follows a normal distribution with 𝜇 = 0 and 𝜎 = 1, since the denominator
+of the 𝜒2 function in equation 8.19 (i.e. square of error) is unity. Given the fact that
+each term in the noise vector is independent (i.e. uncorrelated), we refer to the basis
+in this domain as orthogonal.
+The response matrix after pre-scaling (R) can be decomposed, using the singular
+value decomposition (SVD) approach [230], as
+𝑅 = 𝑈 · 𝐷 · 𝑉 𝑇
+(8.24)
+with both 𝑈𝑚×𝑚 and 𝑉𝑛×𝑛 being orthogonal matrices that satisfy 𝑈 𝑇 · 𝑈 = 𝑈 · 𝑈 𝑇
+= 𝐼𝑚×𝑚 and 𝑉 𝑇 ·𝑉 = 𝑉 ·𝑉 𝑇 = 𝐼𝑛×𝑛, with U,V and D being uniquely defined. I is the
+identity matrix and the subscript represents the dimension. D is an m×n diagonal
+matrix with positive definite diagonal elements (known as singular values) 𝐷𝑖𝑖 = 𝑑𝑖
+arranged in descending order as i increases.
+Notation
+Explanation
+Dimension
+And Format
+𝑉 𝑇
+right orthogonal matrix from decomposition of R
+n × n vector
+D
+diagonal matrix from decomposition of R
+m × n matrix
+U
+left orthogonal matrix from decomposition of R
+m × m matrix
+165
+
+Inserting equation 8.24 into 8.23, we have
+ˆ𝑠 = 𝑉 · 𝐷−1 · 𝑈 𝑇 · (𝑅 · 𝑠𝑡𝑟𝑢𝑒 + 𝑁)
+= 𝑉 · 𝐷−1 · (𝑅𝑈 · 𝑠𝑡𝑟𝑢𝑒 + 𝑁𝑈)
+= 𝑉 · 𝐷−1 · 𝑀𝑈
+(8.25)
+where 𝑅𝑈 ≡ 𝑈 𝑇 · 𝑅, 𝑁𝑈 ≡ 𝑈 𝑇 · 𝑁, and 𝑀𝑈 ≡ 𝑈 𝑇 · 𝑀 are transformations of the
+smearing matrix R, the noise N, and the measured signal M, respectively. Note that,
+since U is an orthogonal matrix and the elements of the original noise vector N are
+uncorrelated, the elements of the new noise vector 𝑁𝑈 are still uncorrelated. Each
+element follows a normal distribution with 𝜇 = 0 and 𝜎 = 1. Thus, the basis in this
+new domain is still orthogonal.
+However, there are cases where the unbiased solution to an unfolding problem
+via a direct inversion can have catastrophic oscillations via the introduction of huge
+variances.
+A proposed solution is the introduction of a trade-off term between the bias and
+the variance to suppress the oscillations. We refer to that term as “regularization” and
+is introduced in the form of an n×n matrix 𝐹 that is applied on the exact solution ˆ𝑠.
+ˆ𝑠 = 𝐹 · 𝑉 · 𝐷−1 · (𝑅𝑈 · 𝑠𝑡𝑟𝑢𝑒 + 𝑁𝑈)
+= 𝐹 · 𝑉 · 𝐷−1 · 𝑈 𝑇 · 𝑀
+= 𝐹 · 𝑉 · 𝐷−1 · 𝑀𝑈
+(8.26)
+Using equation 8.26 (focusing on the diagonal elements ii, using the fact that V
+is orthogonal 𝑉 · 𝑉 𝑇 = 𝐼, and that D is diagonal 𝐷−1
+𝑖𝑖 = 1/𝑑𝑖),
+(𝑉 𝑇 ˆ𝑠)𝑖 = 𝐹𝑖𝑖 · 𝑀𝑈,𝑖
+𝑑𝑖
+ˆ𝑆(𝜔) = 𝐹(𝜔) · 𝑀(𝜔)
+𝑅(𝜔)
+(8.27)
+166
+
+It is easy to see the similarities between the two lines in equation 8.27. Therefore,
+following the same terminology as that for the signal processing, we refer to 𝑀𝑈 after
+the SVD transformation as the measurement in the effective frequency domain in
+analogy to the frequency domain in the signal processing.
+The additional matrix F can be decomposed as
+𝐹 = 𝑉 · 𝑊 · 𝑉 𝑇.
+(8.28)
+At this point, the F and W matrices are still unknown.
+Plugging equation 8.28 into equation 8.26 yields
+ˆ𝑠 = 𝑉 · 𝑊 · 𝐷−1 · (𝑅𝑈 · 𝑠𝑡𝑟𝑢𝑒 + 𝑁𝑈).
+(8.29)
+We consider the expectation value of the signal in the effective frequency domain:
+𝑀𝑈 = 𝑈 𝑇 · 𝑀 = 𝑈 𝑇 · 𝑅 · 𝑠
+(8.30)
+In general, the 𝑠𝑡𝑟𝑢𝑒 is unknown, so the expectation signal 𝑠 using the nominal
+simulation prediction is used.
+The construction of W is based on the Wiener filter
+𝑅2(𝜔)·𝑆2(𝜔)
+𝑅2(𝜔)·𝑆2(𝜔)+𝑁2(𝜔).
+Taking
+equation 8.30 at bin i, we have
+⟨𝑅2 · 𝑆2⟩ = 𝑀 2
+𝑈,𝑖 =𝐸𝑞. 8.27 𝑑2
+𝑖 · (
+∑︁
+𝑗
+𝑉 𝑇
+𝑖𝑗 · 𝑠𝑗)2
+(8.31)
+⟨𝑁 2⟩ = 1
+(8.32)
+resulting in a Wiener filter of
+𝑊𝑖𝑘 =
+𝑑2
+𝑖 · (∑︀
+𝑗 𝑉 𝑇
+𝑖𝑗 · 𝑠𝑗)2
+𝑑2
+𝑖 · (∑︀
+𝑗 𝑉 𝑇
+𝑖𝑗 · 𝑠𝑗)2 + 1 · 𝛿𝑖𝑘
+(8.33)
+At this point, W is uniquely defined. Here, equation 8.32 is obtained, since each
+167
+
+element of noise 𝑁𝑈 follows a normal distribution with 𝜇 = 0 and 𝜎 = 1. We have
+(𝑊 · 𝐷−1)𝑖𝑗 =
+𝑑𝑖 · (∑︀
+𝑗 𝑉 𝑇
+𝑖𝑗 · 𝑠𝑗)2
+𝑑2
+𝑖 · (∑︀
+𝑗 𝑉 𝑇
+𝑖𝑗 · 𝑠𝑗)2 + 1 · 𝛿𝑖𝑘
+(8.34)
+The small value of 𝑑𝑖 is balanced by the finite value of the expectation value of
+𝑁 2 ≡ 1 and, thus, equation 8.29 doesn’t suffer from catastrophic oscillations. From
+equation 8.33, the construction of the Wiener filter takes into account the strengths
+of both the signal and noise expectations and is independent of the regularization
+strength 𝜏 used in traditional regularization techniques [231].
+As shown in [231], the regularization can be applied on the curvature of the
+spectrum instead of the strength of the spectrum, which involves an additional matrix
+C. This is also be achieved in the Wiener-SVD approach:
+𝑀 = 𝑅 · 𝐶−1 · 𝐶 · 𝑠
+(8.35)
+by including an additional matrix C that has the commonly used regularization
+forms, such as the first and second order derivatives. Since the effective frequency
+domain is determined by the smearing matrix R, the inclusion of C would alter the
+basis of the effective frequency domain. In this case, the SVD decomposition becomes
+(just like shown in equation 8.24)
+𝑅 · 𝐶−1 = 𝑈𝐶 · 𝐷𝐶 · 𝑉 𝑇
+𝐶 .
+(8.36)
+Notation
+Explanation
+Dimension
+And Format
+𝑉 𝑇
+𝐶
+right orthogonal matrix from decomposition of R ·𝐶−1
+n × n vector
+𝐷𝐶
+diagonal matrix from decomposition of R ·𝐶−1
+m × n matrix
+𝑈𝐶
+left orthogonal matrix from decomposition of R ·𝐶−1
+m × m matrix
+The final solution of the regularisation becomes (just like shown in equation 8.25
+168
+
+by inserting it into equation 8.23)
+ˆ𝑠 = 𝐶−1 · 𝑉𝐶 · 𝑊𝐶 · 𝑉 𝑇
+𝐶 · 𝐶 · (𝑅𝑇𝑅)−1 · 𝑅𝑇 · 𝑀
+(8.37)
+or, equivalently,
+ˆ𝑠 = 𝐴𝐶 · (𝑅𝑇𝑅)−1 · 𝑅𝑇 · 𝑀
+(8.38)
+where
+𝐴𝐶 = 𝐶−1 · 𝑉𝐶 · 𝑊𝐶 · 𝑉 𝑇
+𝐶 · 𝐶
+(8.39)
+The corresponding Wiener filter would be (once again, by realizing the similarities
+between the two expressions in equation 8.27 and by using equation 8.39)
+𝑊𝐶,𝑖𝑖 =
+𝑑2
+𝐶𝑖 · (∑︀
+𝑗 𝑉 𝑇
+𝐶𝑖𝑗 · (∑︀
+𝑙 𝐶𝑗𝑙𝑠𝑙))2
+𝑑2
+𝐶𝑖 · (∑︀
+𝑗 𝑉 𝑇
+𝐶𝑖𝑗 · (∑︀
+𝑙 𝐶𝑗𝑙𝑠𝑙))2 + 1
+(8.40)
+where 𝐶𝑗𝑙, 𝑉 𝑇
+𝐶𝑖𝑗, and 𝑑𝐶𝑖 are matrix elements of matrices C, 𝑉𝐶, and 𝐷𝐶, respec-
+tively.
+Since the unfolded results are a linear transformation of the measurement, we
+can easily evaluate the uncertainties associated with them. Equation 8.38 can be
+rewritten as
+ˆ𝑠 = 𝑅𝑡𝑜𝑡 · 𝑚
+(8.41)
+with
+𝑅𝑡𝑜𝑡 = 𝐴𝐶 · (𝑅𝑇𝑅)−1 · 𝑅𝑇 · 𝑄.
+(8.42)
+Then, the covariance matrix of ˆ𝑠 can be deduced from the covariance matrix of
+m as
+𝐶𝑜𝑣^𝑠,m = 𝑅𝑡𝑜𝑡 · 𝐶𝑜𝑣m · 𝑅𝑇
+𝑡𝑜𝑡.
+(8.43)
+The variances of the unfolded data can also be easily calculated given that their
+169
+
+origin N in equation 8.37 is uniquely defined. Defining N(i) as a vector with the i-th
+element being 1 and the rest of elements being 0, we can calculate the variance in s
+due to i-th element in N as:
+𝑇𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛(𝑖) = 𝐴𝐶 · (𝑅𝑇𝑅)−1 · 𝑅𝑇 · 𝑁(𝑖),
+(8.44)
+with 𝑇𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛(i) being a vector. The variance of the j-th element of 𝑇𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛,𝑗 can
+thus be written as:
+𝑇𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛,𝑗 =
+√︃∑︁
+𝑖
+𝑇 2
+𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛,𝑗(𝑖),
+(8.45)
+after summing the contribution from each independent noise source. The square
+of 𝑇𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛,𝑗 corresponds to the j-th diagonal element of the covariance matrix 𝐶𝑜𝑣𝑠
+in equation 8.43.
+Given equation 8.37, we can understand the entire process of unfolding as to
+“remove” the effect of R through multiplying (𝑅𝑇𝑅)−1 · 𝑅𝑇 and then replace it with
+a new smearing matrix 𝐴𝐶. Therefore, it is straightforward to estimate the bias on
+the unfolded results:
+𝑇𝑏𝑖𝑎𝑠 = (𝐴𝐶 − 𝐼) · (𝑅𝑇𝑅)−1 · 𝑅𝑇 · 𝑀
+= (𝐴𝐶 − 𝐼) · 𝑠
+(8.46)
+with I being identity matrix and 𝑠 being the expectation of the nominal MC signal
+prediction.
+170
+
+8.3
+Electrons-For-Neutrinos Fiducials
+Fiducial cuts for e2a have been defined and used by several analyses [232–234] and
+follow the same general procedure for all charged particle species. A series of event
+selection cuts are applied to data to produce event samples suitable for defining the
+fiducial regions of CLAS. From these event samples, regions of good acceptance are
+found, defined as flat regions of 𝜑 space in bins of momentum and 𝜃, which are then
+parameterised into functions. The specific procedures for each particle species for
+which fiducial cuts have been defined are summarised in the following sections.
+(a)
+(b)
+Figure 8-1: The 𝜃 vs 𝜑 distributions for 𝜋−, before (a) and after (b) the application
+of fiducial cuts as defined in [235] for 2.2 GeV analysis of 4He.
+Electron fiducial cuts were defined in [236], using electron candidates identified by
+the CLAS triggering requirements for e2a. These electron samples are then subjected
+to geometric cuts in the 𝑢, 𝑣, and 𝑤 “views” corresponding to the orientation of the
+scintillator layers of the electromagnetic calorimeter. This cut accounts for poorly
+understood electron acceptance at the edges of the calorimeter, and the difference in
+acceptance between the electromagnetic calorimeter and the Cerenkov Counter, which
+can be seen in data as a characteristic “smile” feature in the uncut 𝜃 vs 𝜑 distributions
+for electrons. Additionally, a cut is applied on the ratio of energy deposition by an
+electron in the calorimeter to electron momentum as measured by the drift chambers,
+𝐸𝑡𝑜𝑡/𝑝𝑒′. This ratio is known as the ‘sampling fraction’ and is fixed by the design of
+the detector to 0.3𝑐. At the detector edges, this value can decrease due to shower
+171
+
+, (degrees)
+180
+160
+450
+140
+400
+350
+120
+300
+100
+.
+..
+250
+80
+200
+60
+150
+40
+100
+20
+50
+0.
+50
+100
+150
+200
+250
+300
+350
+0
+(degrees) (degrees)
+180
+400
+160
+350
+140
+300
+120
+250
+100
+200
+80
+150
+60
+100
+40
+20
+50
+0
+50
+100
+150
+200
+250
+300
+350
+0
+(degrees)leakage, where the electron energy is not fully deposited in the calorimeter.
+The electrons distributions for each beam energy and torus field setting are then
+split into sectors and momentum bins 50 𝑀𝑒𝑉/𝑐 wide, spanning the electron momen-
+tum range of the data. A plot of 𝜃𝑒 against 𝜑𝑒 is produced for each momentum bin,
+and for each degree in 𝜃𝑒, the plot was projected onto the 𝜑𝑒 axis and fitted with a
+trapezoid function.
+(a)
+(b)
+Figure 8-2: The 𝜃 vs 𝜑 distributions for electrons, before (left) and after (right) the
+application of fiducial cuts for 2.2 GeV analysis of 4He.
+Positively charged hadrons, i.e. protons and 𝜋+, have their fiducial cuts defined as
+a single species, under the assumption that their identical charge means their fiducial
+regions will be the same. The positive hadron fiducial cuts were defined in [237].
+Event samples are identified as tracks with good drift chamber status, a good hit
+in the time-of-flight system, and particle identification via a 𝜒2 cut on the DCPB
+bank
+In order to eliminate protons in quasi-free reactions, which are not uniformly
+distributed in 𝜑, an energy transfer cut is applied
+The positively charged hadron distributions for each beam energy and torus field
+setting are then split into sectors and momentum bins 50 𝑀𝑒𝑉/𝑐 wide, spanning the
+momentum range of the data. A plot of 𝜃 against 𝜑 is produced for each momentum
+bin, and for each degree in 𝜃, the plot was projected onto the 𝜑 axis and fitted with
+a trapezoid function.
+172
+
+120
+5000
+100
+4000
+80
+60
+3000
+40
+2000
+20
+1000
+0
+0
+50
+100
+150
+200
+250
+300
+350(degrees)
+120
+3500
+100
+3000
+80
+2500
+60
+2000
+1500
+40
+1000
+20
+5000
+0
+50
+100
+150
+200
+250
+300
+350
+-
+(degrees)(a)
+(b)
+Figure 8-3: The 𝜃 vs 𝜑 distributions for protons, before (left) and after (right) the
+application of fiducial cuts for 2.2 GeV analysis of 4He.
+(a)
+(b)
+Figure 8-4: The 𝜃 vs 𝜑 distributions for 𝜋+, before (left) and after (right) the appli-
+cation of fiducial cuts for 2.2 GeV analysis of 4He.
+As neutral particles, the fiducial cuts for photons take the from of a cut on the
+detection area of the electromagnetic calorimeter, with no momentum dependence.
+Photon events are selected after applying the same 𝑢, 𝑣 and 𝑤 calorimeter cuts as
+used for electrons, and the fiducial regions for each energy and target defined on
+distributions of 𝑐𝑜𝑠𝜃 vs 𝜑.
+Two first order polynomials are used to describe the
+outline of the sides of the sector and two second order polynomials to describe the
+top and bottom edges, as shown in figure 8-5. Additionally, two ‘hot spots’ in the
+corners of sector four were removed.
+The originally defined fiducial cut functions for e2a were implemented in C++ and
+FORTRAN versions for the ROOT and PAW data analysis frameworks respectively.
+173
+
+180
+160
+4000
+140
+3500
+120
+3000
+100
+2500
+80
+2000
+60
+1500
+40
+1000
+20
+500
+50
+100
+150
+200
+250
+350
+0.
+300180
+160
+4000
+140
+3500
+120
+3000
+100
+2500
+80
+2000
+60
+1500
+40
+1000
+20
+500
+50
+100
+150
+200
+250
+300
+350
+0
+0
+(degrees)
+proton.- (degrees)
+180
+160
+006
+800
+140
+700
+120
+600
+100
+500
+80
+400
+60
+300
+40
+200
+20
+100
+0
+0.
+50
+100
+150
+200
+250
+300
+350
+0
+(degrees)
+T* (degrees)
+180
+900
+160
+800
+140
+700
+120
+600
+100
+500
+80
+400
+60
+人
+300
+40
+200
+20
+100
+50
+100
+150
+200
+250
+300
+350
+0
+0
+(degrees)
+T(a) 4.4 GeV.
+Figure 8-5: The cos 𝜃 vs 𝜑 distributions for photons, with fiducial cut outline indicated
+by red for 4.4 GeV analysis for 3He.
+They were checked into the CLAS CVS repository for e2 analysis software, and have
+been applied as part of the various analyses performed on this data since it was
+collected in 1999.
+The e4nu analyses of e2a data have used the C++/ROOT implementation of the
+fiducial cuts, incorporating them into their analysis software, with minor modifica-
+tions to ensure compatibility with contemporary software environments.
+In the analysis of [235], 𝜋− fiducial cuts were defined, initially by direct reuse of
+the electron fiducial cut parameters to cut fiducial regions for 𝜋−. This was considered
+to offer a reasonable approximation for 𝜋−, having the same charge and thus similar
+behaviour in the torus field of CLAS. However, because the electron fiducial cuts are
+only defined to a minimum momentum of 350 MeV/c, and the minimum momentum
+threshold used for pion detection in CLAS is 150 MeV/c, a new set of cut parameters
+had to be defined for the 𝜋− in the momentum range 150 − 350 MeV/c. These used
+50 MeV/c momentum bins on 2.2 GeV 12C data, and follow the same procedure to
+obtain the 𝜃 vs 𝜑 outline cuts and the 𝜃 gaps corresponding to malfunctioning TOF
+paddles at this low momentum as described in [234] for the 𝑝 > 350 MeV/c region.
+At 1.1 GeV beam energy, this procedure was used to define 𝜋− fiducial cuts for
+the whole momentum range, as the electron cuts offer minimal coverage for 𝜋−.
+The theta gap functions for dead channels in 𝜋− were updated to account for
+previously missed gaps, and to skip electron gaps from the CC, a detector not used
+174
+
+160
+140
+120
+0.9
+100
+0.8
+80
+60
+0.7
+40
+0.6
+20
+0.5
+0
+50
+100
+150
+200
+250
+300
+350
+o[Deg.]in determining the 𝜋−. Several functions were refit to extend to the lower momentum
+range of the pions, removing the need to apply them only in specific momentum
+ranges, and allowing the clumsy box cuts previously used to be eliminated.
+Figure 8-6: The 𝜃 vs momentum distributions for 𝜋− in sector 3, after the application
+of the original fiducial and theta gap cuts for 2.2 GeV analysis of 4He.
+At low
+momentum, the theta gap cuts fail, and have been replaced box cuts that do not
+appropriately describe the gaps at low momenta.
+Figure 8-7 shows the polar angle versus momentum distributions for 𝜋− in each
+sector of CLAS, after the application of fiducial cuts. Of the theta gap functions used
+for the electrons, several are carried over unchanged, some are updated in order to
+appropriately apply to the full momentum range of 𝜋−, while gaps defined for the CC
+are omitted, as this subsystem is not used to identify 𝜋−.
+Figure 8-8 shows the polar angle versus momentum distributions for 𝜋− in each
+sector of CLAS, after the application of fiducial cuts. As with the 1 GeV case, several
+theta gap functions are carried over unchanged from electrons, some are updated in
+order to appropriately apply to the full momentum range seen for 𝜋−, and CC gaps
+are omitted. Additionally at 2.2 and 4.4 GeV, the maximum polar angle condition
+for 𝑝 > 350 MeV/c, imposed by the electron fiducial cut parameters, is removed.
+At 1 GeV beam energy, dedicated fiducial cut parameters for 𝜋− at 750 A torus
+current were defined as part of the analysis of [235]. The 1500 A data was not used in
+175
+
+rees)
+180
+180
+(degre
+160
+160
+140
+140
+120
+120
+100
+100
+80
+80
+60
+60
+40
+40
+20
+20
+U
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+元momentum(GeV)Figure 8-7: The 𝜃 vs momentum distributions for 𝜋−, showing the 𝜋− updated theta
+gap cuts (green), retained (red) and not used (yellow) electron theta gap cuts for
+1.1 GeV analysis of 3He.
+the analysis of [235], therefore no bespoke 𝜋− fiducial cut parameters were produced
+for this field setting. Figure 8-9 shows the 𝜃 vs momentum distributions for the 𝜋−
+at 1.1 GeV, after the application of the updated fiducial cuts.
+Figure 8-10 shows the 𝜃 vs momentum distributions for the 𝜋− at 2.2 GeV, after
+the application of the updated fiducial cuts. The same cuts are used in 4.4 GeV
+analyses.
+As seen in the reuse of electron theta gap cuts on the 𝜋−, where the gap functions
+were not defined to the lower momentum of the pions, a similar effect is seen in 𝜋+,
+with the gap functions defined for protons directly reused. This has resulted in similar
+box cuts being applied at low momentum, as can be seen in figure 8-11, removing
+good events rather than low and poorly understood acceptance regions.
+Unlike the 𝜋−, the positive hadron fiducial cut functions, defined for protons, are
+for the most part valid at the lower pion momentum range. We have extended the
+application of several of these functions to define cuts at lower 𝜋+ momenta, where
+the functions continue to appropriately describe the gap. This is shown for sector 3
+176
+
+180
+35
+180
+180
+160
+160
+160F
+35
+30
+140
+30
+140E
+140
+30
+120E
+25
+120 E
+25
+120E
+25
+100E
+20
+100
+20
+100
+20
+80E
+415
+80
+80
+15
+15
+60
+60
+60E
+10
+10
+40F
+10
+40
+40
+20F
+5
+20
+5
+20
+5
+0.2
+0.4
+0.6
+0.8
+1.2
+0.2
+0.4
+0.6
+0.8
+1.2
+0.2
+0.4
+0.6
+0.8
+1.2
+元momentum(GeV)
+180
+180
+180
+35
+35
+160
+160
+160
+30
+140
+30
+140
+30
+140
+25
+120E
+25
+120
+120
+25
+20
+100
+100
+100F
+20
+20
+80
+80
+80F
+15
+15
+15
+60
+60
+60E
+10
+40F
+10
+10
+40
+40F
+20F
+5
+20
+5
+20F
+0.2
+0.4
+0.6
+0.8
+1.2
+0.2
+0.4
+0.6
+0.8
+1.2
+0.2
+0.4
+0.6
+0.8
+1.2Figure 8-8: The 𝜃 vs momentum distributions for 𝜋−, showing the 𝜋− updated theta
+gap cuts (green), retained (red) and not used (yellow) electron theta gap cuts for
+2.2 GeV analysis of 3He. A new parameterisation of the maximum polar angle cut is
+also shown (black line). The same gap functions are applied at 4.4 GeV
+(a)
+Figure 8-9: The 𝜃 vs momentum distributions for 𝜋−, after the application of the
+updated theta gap cuts for 1.1 GeV analysis of 3He.
+177
+
+180
+180
+180
+18C
+16C
+160
+14(
+160
+160E
+16C
+14C
+140
+12(
+140
+140
+14C
+12C
+120E
+120 E
+120E
+10C
+12C
+10C
+100F
+100F
+100E
+10C
+80
+80E
+80F
+80
+80
+80
+60
+60
+60F
+60
+60
+60
+40F
+40
+40
+40
+40
+40
+20F
+20
+20
+20
+20
+20
+0.2
+0.40.60.8
+1.2
+1.4
+1.6
+2
+0.20.40.60.8
+1.2
+1.4
+1.61.8
+2
+0.20.40.60.8
+1.2
+1.4
+1.61.8
+2
+元 momentum (GeV)
+180
+18(
+180
+180
+16C
+16C
+160
+16(
+160 
+160
+140
+140
+14C
+140
+14C
+14
+120E
+12(
+120
+12C
+120
+12C
+100
+100
+10C
+100
+10C
+10
+80F
+80
+80E
+80
+80F
+80
+60F
+60
+60
+60
+60E
+60
+40
+40
+40F
+40
+40
+40
+20F
+20
+20
+20
+20F
+20
+1.2
+1.4
+0.2
+0.40.6
+0.8
+1.2
+0.60.8
+1.2
+1.8180
+180
+180
+35
+35
+160
+160
+160
+35
+30
+30
+140
+140
+140
+30
+120E
+25
+120 E
+25
+120E
+25
+100E
+20
+100
+20
+100E
+20
+80F
+80E
+80F
+15
+15
+60E
+60
+60E
+10
+10
+40F
+10
+40
+40F
+ 20E
+5
+20
+5
+20E
+5
+0.2
+0.4
+0.6
+0.8
+1.2
+0.2
+0.4
+0.6
+0.8
+1.2
+0.2
+0.4
+0.6
+0.8
+1.2
+元momentum (GeV)
+180
+180
+180
+35
+35
+160
+160
+30
+140
+30
+140
+30
+140
+120E
+120E
+25
+25
+120
+20
+100
+100
+100
+20
+20
+80F
+80
+80
+15
+15
+15
+60F
+60
+60E
+10
+ 40F
+10
+10
+40F
+40F
+5
+5
+20F
+5
+20
+20
+0.2
+0.4
+0.6
+0.8
+1.2
+0.2
+0.4
+0.6
+0.8
+1.2
+0.2
+0.4
+0.6
+0.8
+1.2(a)
+Figure 8-10: The 𝜃 vs momentum distributions for 𝜋−, after the application of the
+updated theta gap cuts for 2.2 GeV analysis of 3He. The same cuts are applied in
+the 4.4 GeV analysis.
+Figure 8-11: The 𝜃 vs momentum distributions for 𝜋+, showing the original theta
+gap cuts for analysis of 2.2 GeV 4He data. The same gap functions are applied at
+4.4 GeV.
+178
+
+180
+180
+180
+18C
+16C
+140
+160
+160
+16C
+14C
+140
+12(
+140
+140
+14C
+12C
+120
+120
+120E
+100
+12C
+10C
+100E
+100
+100E
+10C
+80
+80E
+80F
+80
+80
+80
+60
+60
+60
+60
+60
+60
+40F
+40
+40
+40
+40
+40
+20
+20
+20
+20
+20
+20
+0.20.40.60.8
+1.21.4
+1.61.8
+2
+0.20.40.60.8
+1.2
+1.41.61.8
+2
+0.20.40.60.8
+1.41.61.8
+2
+元momentum (GeV)
+180
+18(
+180
+180
+16C
+16C
+160
+16(
+160
+160
+140
+140
+14C
+140
+14C
+14(
+120E
+120
+12C
+120E
+12C
+12(
+100
+100
+10C
+100
+10C
+10
+80F
+80
+80
+80
+80
+80
+60元
+60
+60
+60
+60
+60
+40
+40
+40F
+40
+40F
+40
+20F
+20
+20
+20
+20
+20
+1.2
+0.20.40.60.8
+1.41.61.8
+0.20.40.60.8
+1.2
+2180
+180
+20C
+18
+160
+18C
+160
+18C
+16(
+140F
+16C
+140
+140
+16C
+14
+14C
+120E
+120
+120E
+14C
+12(
+12C
+12C
+100
+100
+100F
+10
+10C
+10C
+80F
+80
+80
+80
+80
+80
+60E
+60
+60
+60
+60
+60
+40F
+40
+40
+40
+40F
+40
+20F
+20
+20
+20
+20
+20
+%
+0
+0.5
+1.5
+2.5
+0.5
+1.5
+2
+2.5
+0.5
+2
+2.5
+元*momentum (GeV)
+180
+180
+180
+200
+18C
+160
+160
+18C
+160
+18
+16C
+16C
+140
+16(
+140
+140
+14C
+14C
+120E
+14
+120
+120
+12C
+12C
+100
+12(
+100
+100
+10C
+10C
+80
+100
+80
+80F
+80
+80
+80
+60F
+60
+60E
+60
+60
+60
+40F
+40
+40
+40F
+40
+40
+20
+20
+20
+20F
+20
+%
+0.5
+0.
+0.5
+2.5
+0.5
+2.5in figure 8-12.
+(a)
+(b)
+Figure 8-12: The 𝜃 vs momentum distributions for 𝜋+ in sector 3, showing the original
+(a) and new (b) theta gap cuts for 2.2 GeV analysis of 3He. The same gap functions
+are applied at 4.4 GeV beam energy.
+179
+
+u (degrees
+180
+20C
+160
+18C
+140
+16C
+14C
+120
+12C
+100
+10C
+80
+80
+60
+60
+40
+40
+20
+20
+0
+0
+0.5
+1
+1.5
+2
+2.5
+*momentum (GeV)u(degrees
+180
+20C
+160
+18C
+140
+16C
+14C
+120
+12C
+100
+10C
+80
+80
+60
+60
+40
+40
+20
+20
+0
+0.5
+1
+1.5
+2
+2.5
+元*momentum(GeV)180
+
+List of Figures
+1-1
+The Standard Model of particle physics illustrating the three genera-
+tions of fermions, the gauge bosons, and the scalar Higgs boson. Figure
+adapted from ScienceAlert [4]. . . . . . . . . . . . . . . . . . . . . . .
+12
+1-2
+Graphic illustrating the left-handed (right-handed) nature of neutri-
+nos (anti-neutrinos) via the orientation of the momentum-spin vectors.
+Figure adapted from [6]. . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+1-3
+The experimental setup of an accelerator-based long-baseline neutrino
+oscillation experiment. . . . . . . . . . . . . . . . . . . . . . . . . . .
+20
+1-4
+Neutrino energy spectra reconstruction depends on our ability to model
+the interaction of neutrinos with atomic nuclei and the propagation
+of particles through the atomic nucleus.
+This flow chart shows the
+process, starting with an oscillated far-detector incident-energy spec-
+trum (green), differentiating the physical neutrino interactions (green
+arrows) from the experimental analysis (blue arrows), and ending up
+with an inferred incident-energy spectrum that hopefully matches the
+actual one.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+22
+1-5
+The four main interaction processes for neutrino-nucleus scattering
+events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+25
+1-6
+Neutrino cross-section evolution as a function of the neutrino energy
+illustrating the energy range where each one of the four main processes
+dominates. Figure adapted from [34]. . . . . . . . . . . . . . . . . . .
+26
+1-7
+Feyman diagrams illustrating charged current (CC) and neutral current
+(NC) processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+26
+181
+
+1-8
+Nucleon momentum distribution options available in commonly used
+neutrino event generators. Figure adapted from [35].
+. . . . . . . . .
+27
+1-9
+Schematic illustration the wealth of possible final state interactions
+that the final state hadrons can undergo. Figure adapted from [39]. .
+27
+1-10 Quasielastic lepton-nucleus scattering where the outgoing nucleon does
+not reinteract as it leaves the nucleus. . . . . . . . . . . . . . . . . . .
+30
+1-11 Graphic illustration of the three complementary approaches used in
+this thesis to improve our understanding of lepton-nucleus interactions.
+31
+2-1
+The Fermilab accelerator complex showing the accelerator components
+and the different beams. Figure adapted from [45].
+. . . . . . . . . .
+34
+2-2
+The ingredients for a neutrino beam include the accelerated protons,
+the target, the magnetic horn, the decay pipe, and the absorbers. Fig-
+ure adapted from [45].
+. . . . . . . . . . . . . . . . . . . . . . . . . .
+35
+2-3
+The BNB neutrino flux prediction through the MicroBooNE detector
+for 𝜈𝜇, ¯𝜈𝜇, 𝜈𝑒, and ¯𝜈𝑒. A TPC volume with dimensions 2.56 m × 2.33 m
+× 10.37 m is assumed. Figure adapted from [46]. . . . . . . . . . . . .
+35
+2-4
+Schematic illustration of the MicroBooNE detector and its dimensions.
+Figure adapted from [55]. . . . . . . . . . . . . . . . . . . . . . . . . .
+37
+2-5
+The MicroBooNE coordinate system. . . . . . . . . . . . . . . . . . .
+37
+2-6
+The design of CRT planes as part of the MicroBooNE detector. Simula-
+tion of cosmic rays crossing the CRT, the brown lines represent possible
+cosmic ray trajectories. There are four CRT planes: top plane, bottom
+plane, pipe side plane and feedthrough side plane. The beam direction
+is along the z axis. Figure adapted from [56]. . . . . . . . . . . . . . .
+38
+2-7
+Working principle of a LArTPC detector. Figure adapted from [61]. .
+39
+182
+
+2-8
+(Left) bipolar (U and V induction planes) and unipolar (Y collec-
+tion plane) signal induction on the three MicroBooNE planes. Figure
+adapted from [63]. (Right) schematic view of the wire planes. The
+vertical collection Y wires are shown in pink, the induction U wires,
+angled at +60𝑜 are shown in blue and the induction V wires, angled at
+-60𝑜, are shown in green. Figure adapted from [64].
+. . . . . . . . . .
+40
+2-9
+Entry/exit points of cosmic muon tracks with a signal from a muon
+counter located outside of the cryostat. In the absence of space charge
+effects, the points should be located along the TPC boundaries indi-
+cated by the dashed lines. Figure adapted from [66]. . . . . . . . . . .
+41
+2-10 The production of scintillation light in liquid argon. A charged particle
+can either excite or ionise the argon. Figure adapted from [6].
+. . . .
+42
+2-11 The MicroBooNE light collection system with the 32 PMTs. Figure
+adapted from [69]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+44
+2-12 (Left) cosmic-induced event that was stored because of the coincidence
+of a 1.6 𝜇s accelerator BNB signal and light detected by the PMTs.
+(Right) neutrino-induced event where the light was coming from a neu-
+trino interaction. Figure adapted from [70]. . . . . . . . . . . . . . . .
+45
+2-13 Optically reconstructed flash object recorded by the MicroBooNE PMT
+light collection system. The dark orange regions represent a higher PE
+yield. Figure adapted from [64]. . . . . . . . . . . . . . . . . . . . . .
+46
+2-14 Neutrino-induced tracks (black) are matched to the corresponding light
+signals collected by PMTs (red circles) and are clearly separated from
+the cosmic-induced ones (dimmed color tracks). Figure adapted from [72]. 47
+2-15 Candidate neutrino event display from MicroBooNE data on one of the
+induction planes. (a) The raw waveform image. (b) The image after
+noise-filtering. (c) The image after 2D deconvolution. The image qual-
+ity near the neutrino interaction vertex significantly improves after the
+2D deconvolution and the latter leads to improvements in the pattern
+recognition. Figure adapted from [74].
+. . . . . . . . . . . . . . . . .
+49
+183
+
+2-16 Illustration of the hierarchical structure of particles reconstructed for
+a simulated charged current 𝜈𝜇 event in MicroBooNE. The interaction
+includes a muon, proton and charged pion in the visible final state.
+Figure adapted from [75]. . . . . . . . . . . . . . . . . . . . . . . . . .
+50
+2-17 Illustration of the measured dQ/dx vs dE/dx distribution with the
+modified recombination model in the MC simulation with the Ar-
+goNeuT parametrization. Figure adapted from [76]. . . . . . . . . . .
+51
+2-18 MicroBooNE event display of an event in a cosmic overlay sample.
+A simulated neutrino event (orange box) is overlaid on top of cosmic
+events (blue box). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+52
+3-1
+Interaction breakdown of the cos𝜃𝜇 plot illustrating the dominance of
+CCQE interactions after the application of our selection cuts. . . . . .
+59
+3-2
+Interaction breakdown of the cos𝜃𝜇 plot illustrating the dominance of
+RES interactions after the application of our selection cuts for the non-
+CC1p0𝜋 background part of the MC sample. . . . . . . . . . . . . . .
+60
+3-3
+Vertex 𝑧 distribution for the measured events, after the beam related
+MC background has been subtracted, before (left) and after (right)
+detection efficiency corrections. No small-𝑧 enhancement is observed
+and, with efficiency corrections, the measured distribution is consistent
+with that of a uniform neutrino interaction vertex. . . . . . . . . . . .
+62
+3-4
+The flux integrated single differential CC1p0𝜋 cross sections as a func-
+tion of the cosine of the measured muon scattering angle. Inner and
+outer error bars show the statistical and total (statistical and system-
+atic) uncertainty at the 1𝜎, or 68%, confidence level. Colored lines
+show the results of theoretical absolute cross section calculations using
+different event generators (without passing through a detector simula-
+tion). The blue band shows the extracted cross section obtained from
+analyzing MC events propagated through our full detector simulation.
+The width of the band denotes the simulation statistical uncertainty.
+64
+184
+
+3-5
+As figure 3-4, but for the differential cross sections as a function of
+measured muon momentum (left) and measured proton scattering an-
+gle (middle) and momentum (right). Cross sections are shown for the
+full measured phase-space (top) and for events with cos(𝜃𝜇) < 0.8 (bot-
+tom). Inner and outer error bars show the statistical and total (statis-
+tical and systematic) uncertainty at the 1𝜎, or 68%, confidence level.
+Colored lines show the results of theoretical absolute cross section cal-
+culations using different event generators (without passing through a
+detector simulation). The blue band shows the extracted cross section
+obtained from analyzing MC events passed through our full detector
+simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+66
+3-6
+The flux integrated single differential CC1p0𝜋 cross sections as a func-
+tion of 𝑄2
+𝐶𝐶𝑄𝐸 = (𝐸𝑐𝑎𝑙
+𝜈
+− 𝐸𝜇)2 − (⃗𝑝𝜈 − ⃗𝑝𝜇)2 and 𝐸𝑐𝑎𝑙
+𝜈
+= 𝐸𝜇 + 𝑇𝑝 + 𝐵𝐸,
+where 𝐵𝐸 = 40 MeV and ⃗𝑝𝜈 = (0, 0, 𝐸𝑐𝑎𝑙
+𝜈 ). Colored lines show the
+results of theoretical absolute cross section calculations using different
+event generators (without passing through a detector simulation). The
+blue band shows the extracted cross section obtained from analyzing
+MC events passed through our full detector simulation. . . . . . . . .
+67
+3-7
+Muon angular distribution after the implementation of the analysis
+framework improvements. No data-MC disagreement is observed in
+the forward direction. . . . . . . . . . . . . . . . . . . . . . . . . . . .
+69
+3-8
+Schematic illustration of the single transverse variables 𝛿𝑝𝑇, 𝛿𝛼𝑇 and
+𝛿𝜑𝑇. Figure adapted from [134]. . . . . . . . . . . . . . . . . . . . . .
+70
+3-9
+Schematic illustration of 𝛿𝑝𝑇𝑥 and 𝛿𝑝𝑇𝑦. Figure adapted from [135]. .
+72
+3-10 Interaction breakdown of the CC1p0𝜋 events as a function of 𝛿𝑝𝑇𝑥 (left)
+and 𝛿𝑝𝑇𝑦 (right). The data correspond to the “combined” MicroBooNE
+runs 1-3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+73
+3-11 Topological (left) and interaction (right) breakdown after the applica-
+tion of the event selection for 𝛿𝑝𝑇. . . . . . . . . . . . . . . . . . . . .
+74
+3-12 Response matrices of 𝛿𝑝𝑇 using the selected CC1p0𝜋 MC events. . . .
+75
+185
+
+3-13 Total covariance matrix for 𝛿𝑝𝑇. . . . . . . . . . . . . . . . . . . . . .
+77
+3-14 Additional smearing matrix 𝐴𝑐 for 𝛿𝑝𝑇. . . . . . . . . . . . . . . . . .
+78
+3-15 The flux-integrated single- (top) and double- in 𝛿𝛼𝑇 bins (bottom) dif-
+ferential CC1p0𝜋 cross sections as a function of the transverse missing
+momentum 𝛿𝑝𝑇. Inner and outer error bars show the statistical and
+total (statistical and shape systematic) uncertainty at the 1𝜎, or 68%,
+confidence level. The gray band shows the normalization systematic
+uncertainty. Colored lines show the results of theoretical absolute cross
+section calculations with and without FSI based on the GENIE and
+GiBUU event generators. . . . . . . . . . . . . . . . . . . . . . . . . .
+80
+3-16 The flux-integrated single- (top) and double- in 𝛿𝑝𝑇 bins (bottom)
+differential CC1p0𝜋 cross sections as a function of the angle 𝛿𝛼𝑇. Inner
+and outer error bars show the statistical and total (statistical and shape
+systematic) uncertainty at the 1𝜎, or 68%, confidence level. The gray
+band shows the normalization systematic uncertainty. Colored lines
+show the results of theoretical absolute cross section calculations with
+a number of FSI modeling options based on the GENIE event generator. 81
+3-17 The flux-integrated single- (top) and double- in 𝛿𝑝𝑇,𝑦 bins (bottom)
+differential CC1p0𝜋 cross sections as a function of the angle 𝛿𝑝𝑇,𝑥.
+Inner and outer error bars show the statistical and total (statistical and
+shape systematic) uncertainty at the 1𝜎, or 68%, confidence level. The
+gray band shows the normalization systematic uncertainty. Colored
+lines show the results of theoretical absolute cross section calculations
+with a number of event generators.
+. . . . . . . . . . . . . . . . . . .
+82
+3-18 Cross-section interaction breakdown for all the selected events. The
+breakdown is shown for (top left) the G18 configuration with FSI ef-
+fects, (top right) the G18 configuration without FSI effects, (bottom
+left) GiB with FSI effects, and (bottom right) GiB without FSI effects.
+83
+186
+
+3-19 Cross-section interaction breakdown for events with 𝛿𝛼𝑇 < 45𝑜. The
+breakdown is shown for (top left) the G18 configuration with FSI ef-
+fects, (top right) the G18 configuration without FSI effects, (bottom
+left) GiB with FSI effects, and (bottom right) GiB without FSI effects.
+83
+3-20 Cross-section interaction breakdown for events with 135𝑜 < 𝛿𝛼𝑇 <
+180𝑜. The breakdown is shown for (top left) the G18 configuration
+with FSI effects, (top right) the G18 configuration without FSI effects,
+(bottom left) GiB with FSI effects, and (bottom right) GiB without
+FSI effects.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+84
+3-21 Cross-section interaction breakdown for all the selected events. The
+breakdown is shown for (top left) the G21 hA configuration with the
+hA2018 FSI model, (top right) the G21 hN configuration with the hN
+FSI model, (bottom left) the G21 G4 configuration with the G4 FSI
+model, and (bottom right) the G21 No FSI configuration without FSI
+effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+84
+3-22 Cross-section interaction breakdown for events with 𝛿𝑝𝑇 < 0.2 GeV/c.
+The breakdown is shown for (top left) the G21 hA configuration with
+the hA2018 FSI model, (top right) the G21 hN configuration with the
+hN FSI model, (bottom left) the G21 G4 configuration with the G4
+FSI model, and (bottom right) the G21 No FSI configuration without
+FSI effects.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+85
+3-23 Cross-section interaction breakdown for events with 𝛿𝑝𝑇 > 0.4 GeV/c.
+The breakdown is shown for (top left) the G21 hA configuration with
+the hA2018 FSI model, (top right) the G21 hN configuration with the
+hN FSI model, (bottom left) the G21 G4 configuration with the G4
+FSI model, and (bottom right) the G21 No FSI configuration without
+FSI effects.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+85
+187
+
+3-24 Cross-section interaction breakdown for all the selected events. The
+breakdown is shown for (top left) the G18 LFG configuration, (top
+right) the G18 RFG configuration, (bottom left) the G18 EffSF con-
+figuration, and (bottom right) the G18 No RPA configuration.
+. . .
+86
+3-25 Cross-section interaction breakdown for events with 𝛿𝑝𝑇,𝑦 < -0.15 GeV/c.
+The breakdown is shown for (top left) the G18 LFG configuration, (top
+right) the G18 RFG configuration, (bottom left) the G18 EffSF con-
+figuration, and (bottom right) the G18 No RPA configuration.
+. . .
+86
+3-26 Cross-section interaction breakdown for events with -0.15 < 𝛿𝑝𝑇,𝑦 <
+0.15 GeV/c. The breakdown is shown for (top left) the G18 LFG con-
+figuration, (top right) the G18 RFG configuration, (bottom left) the
+G18 EffSF configuration, and (bottom right) the G18 No RPA config-
+uration.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+87
+3-27 MicroBooNE total Protons on Target (POT) collected with the Booster
+Neutrino Beam (BNB) during the five run periods. In this thesis, the
+first three run periods were used.
+. . . . . . . . . . . . . . . . . . . .
+88
+4-1
+Charged-current cross sections as a function of neutrino energy ob-
+tained using GENIE for muon neutrino scattering using the DUNE near
+detector (left) and far detector (right) oscillated fluxes. The shaded
+bands show the fractional contribution for each interaction mechanism,
+quasielastic scattering (QE), meson-exchange currents (MEC), reso-
+nance excitation (RES), and deep inelastic scattering (DIS). See text
+for details of the interaction mechanisms. The numbers in parenthe-
+ses indicate the percentage of the cross section due to each interaction
+mechanism.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+92
+4-2
+(Left) electron-nucleus inclusive scattering via one-photon exchange
+and (right) charged current neutrino-nucleus inclusive scattering via
+𝑊 exchange with a final state charged lepton. . . . . . . . . . . . . .
+93
+188
+
+4-3
+Reaction mechanisms for lepton-nucleus scattering (a) quasielastic scat-
+tering (QE) where one nucleon is knocked out of the nucleus, (b) 2p2h
+where two nucleons are knocked out of the nucleus, (c) RES resonance
+production where a nucleon is excited to a resonance which decays to
+a nucleon plus meson(s), and (d) DIS where the lepton interacts with
+a quark in the nucleon. . . . . . . . . . . . . . . . . . . . . . . . . . .
+93
+4-4
+Comparison between GENIE v2 and v3 descriptions of inclusive C(𝑒, 𝑒′)
+scattering cross sections at 𝐸0 = 0.56 GeV, 𝜃𝑒 = 60∘ and 𝑄2
+𝑄𝐸 ≈ 0.24
+GeV2. Black points show the data, solid black line shows the GENIE
+v3 results and dashed black line shows the GENIE v2 results.
+. . . .
+95
+4-5
+Comparison of inclusive C(𝑒, 𝑒′) scattering cross sections for data and
+for GENIE. (left) data vs GSuSAv2 and (right) data vs G2018. (top)
+𝐸0 = 0.24 GeV, 𝜃𝑒 = 60∘ and 𝑄2
+𝑄𝐸 ≈ 0.05 GeV2 (middle) 𝐸0 = 0.56
+GeV, 𝜃𝑒 = 36∘ and 𝑄2
+𝑄𝐸 ≈ 0.11 GeV2, and (bottom) 𝐸0 = 0.56 GeV,
+𝜃𝑒 = 60∘ and 𝑄2
+𝑄𝐸 ≈ 0.24 GeV2. Black points show the data, solid
+black lines show the total GENIE prediction, colored lines show the
+contribution of the different reaction mechanisms: (blue) QE, (red)
+MEC, (green) RES and (orange) DIS. . . . . . . . . . . . . . . . . . .
+102
+4-6
+Comparison of inclusive C(𝑒, 𝑒′) scattering cross sections for data and
+for GENIE. (left) data vs GSuSAv2 and (right) data vs G2018. (top)
+𝐸0 = 0.96 GeV, 𝜃𝑒 = 37.5∘ and 𝑄2
+𝑄𝐸 ≈ 0.32 GeV2, (middle) 𝐸0 = 1.30
+GeV, 𝜃𝑒 = 37.5∘ and 𝑄2
+𝑄𝐸 ≈ 0.54 GeV2, and (bottom) 𝐸0 = 2.22 GeV,
+𝜃𝑒 = 15.5∘ and 𝑄2
+𝑄𝐸 ≈ 0.33 GeV2. Black points show the data, solid
+black lines show the total GENIE prediction, colored lines show the
+contribution of the different reaction mechanisms: (blue) QE, (red)
+MEC, (green) RES and (orange) DIS. . . . . . . . . . . . . . . . . . .
+103
+189
+
+4-7
+Comparison of inclusive C(𝑒, 𝑒′) scattering cross sections for data and
+for GENIE. (left) data vs GSuSAv2 and (right) data vs G2018. (top)
+𝐸0 = 1.501 GeV, 𝜃𝑒 = 37.5∘ and 𝑄2
+𝑄𝐸 ≈ 0.92 GeV2, (middle) 𝐸0 =
+3.595 GeV, 𝜃𝑒 = 16∘ and 𝑄2
+𝑄𝐸 ≈ 1.04 GeV2, and (bottom) 𝐸0 = 3.595
+GeV, 𝜃𝑒 = 20∘ and 𝑄2
+𝑄𝐸 ≈ 1.3 GeV2. Black points show the data,
+solid black lines show the total GENIE prediction, colored lines show
+the contribution of the different reaction mechanisms: (blue) QE, (red)
+MEC, (green) RES and (orange) DIS. . . . . . . . . . . . . . . . . . .
+104
+4-8
+Comparison of inclusive Fe(𝑒, 𝑒′) scattering cross sections for data and
+for GENIE. (left) data vs GSuSAv2 and (right) data vs G2018. (top)
+Fe(𝑒, 𝑒′), 𝐸0 = 0.56 GeV, 𝜃𝑒 = 60∘ and 𝑄2
+𝑄𝐸 ≈ 0.24 GeV2, (middle)
+Fe(𝑒, 𝑒′), 𝐸0 = 0.96 GeV, 𝜃𝑒 = 37.5∘ and 𝑄2
+𝑄𝐸 ≈ 0.32 GeV2, (bot-
+tom) Fe(𝑒, 𝑒′), 𝐸0 = 1.30 GeV, 𝜃𝑒 = 37.5∘ and 𝑄2
+𝑄𝐸 ≈ 0.54 GeV2.
+Black points show the data, solid black lines show the total GENIE
+prediction, colored lines show the contribution of the different reaction
+mechanisms: (blue) QE, (red) MEC, (green) RES and (orange) DIS. .
+105
+4-9
+Comparison of inclusive Ar(𝑒, 𝑒′) scattering cross sections for data and
+for GENIE at 𝐸0 = 2.22 GeV, 𝜃𝑒 = 15.5∘ and 𝑄2
+𝑄𝐸 ≈ 0.33 GeV2. (left)
+data vs GSuSAv2 and (right) data vs G2018. Black points show the
+data, solid black lines show the total GENIE prediction, colored lines
+show the contribution of the different reaction mechanisms: (blue) QE,
+(red) MEC, (green) RES and (orange) DIS.
+. . . . . . . . . . . . . .
+106
+4-10 Comparison of inclusive proton (left) and deuterium (right) (𝑒, 𝑒′) scat-
+tering cross sections for data and for GENIE using G2018. (top) 𝐸0 =
+2.445 GeV and 𝜃𝑒 = 20∘, (middle) 𝐸0 = 3.245 GeV and 𝜃𝑒 = 26.98∘,
+and (bottom) 𝐸0 = 5.5 GeV and 𝜃𝑒 = 41∘. Black points show the
+data, solid black lines show the total GENIE prediction, colored lines
+show the contribution of the different reaction mechanisms: (green)
+RES and (orange) DIS. The first peak at lowest energy transfer is the
+∆(1232) resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+107
+190
+
+4-11 Comparison of inclusive proton (left) and deuterium (right) (𝑒, 𝑒′) scat-
+tering cross sections for data and for GENIE using G2018.
+(top)
+𝐸0 = 4.499 GeV and 𝜃𝑒 = 4∘, (middle) 𝐸0 = 6.699 GeV and 𝜃𝑒 = 4∘,
+and (bottom) 𝐸0 = 9.993 GeV and 𝜃𝑒 = 4∘. Black points show the
+data, solid black lines show the total GENIE prediction, colored lines
+show the contribution of the different reaction mechanisms: (green)
+RES and (orange) DIS. The first peak at lowest energy transfer is the
+∆(1232) resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+108
+4-12 Comparison of semi-exclusive 1.16 GeV lepton-carbon scattering for
+𝑄2 ≥ 0.1 GeV2. The number of generated events is plotted versus en-
+ergy transfer (left) and 4-momuntm transfer squared (right) for events
+with exactly one proton with 𝑃𝑝 ≥ 300 MeV/c, no charged pions with
+𝑃𝜋 ≥ 70 MeV/c and no neutral pions or photons of any momentum for
+eGENIE electrons (orange) and GENIE CC 𝜈𝜇 (blue). The electron
+events have been weighted by 𝑄4. Both curves are area normalized. .
+110
+4-13 Number of simulated events for QE scattering on 12C at 1.161 GeV with
+𝑄2 ≥ 0.1 shown as a function of the energy transfer 𝜔 and the momen-
+tum transfer 𝑞3 = |⃗𝑞 | for all the available nuclear models in GENIE for
+neutrinos (top) and for electrons (bottom). (left) the GSuSAv2 model
+which uses a Relativistic Mean Field momentum distribution, (middle)
+the Nieves or Rosenbluth cross section with the Local Fermi Gas mo-
+mentum distribution, and (right) the Llewellyn-Smith or Rosenbluth
+cross section with the Relativistic Fermi Gas momentum distribution.
+The electron events have been weighted by 𝑄4. . . . . . . . . . . . . .
+111
+4-14 Initial momentum distribution of protons in simulated QE C(𝑒, 𝑒′𝑝)
+events at 𝐸 = 1.161 GeV for the Local Fermi Gas (solid histogram)
+and Relativistic Fermi Gas (dotted histogram) models. The two curves
+are normalized to have the same area. . . . . . . . . . . . . . . . . . .
+112
+191
+
+4-15 Number of simulated events as a function of the energy transfer 𝜔 and
+of the momentum transfer 𝑞3 = |⃗𝑞 | for neutrinos (left) and for electrons
+(right) using GSuSav2 for MEC interactions. The electron events have
+been scaled by 𝑄4 and all the samples have been generated with 𝑄2 ≥
+0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+112
+5-1
+Schematic view of the accelerator facility and the experimental halls
+at Jefferson Lab. Figure adapted from [196]. . . . . . . . . . . . . . .
+116
+5-2
+Drawing of the CLAS detector showing the sector structure and the
+different detectors. The beam enters from the upper left side. The
+target is located at the center of the detector.
+. . . . . . . . . . . . .
+117
+5-3
+The CLAS superconducting toroidal magnet. Figure adapted from [197].118
+5-4
+(Left) Illustration of the region 3 drift chamber structure for one of
+the CLAS sectors. (Right) Schematic representation of the thee drift
+chamber regions. Figures adapted from [198].
+. . . . . . . . . . . . .
+118
+5-5
+Schematic view of a CLAS electromagnetic calorimeter module. Fig-
+ures adapted from [199]. . . . . . . . . . . . . . . . . . . . . . . . . .
+120
+5-6
+Optical arrangement of one of the optical modules of the CLAS Cherenkov
+detector, showing the optical and light collection components. Figures
+adapted from [200]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+121
+5-7
+Schematic view of the TOF counters in one sector illustrating the
+grouping into four panels. Figure adapted from [201]. . . . . . . . . .
+122
+6-1
+The expected energy distribution of different 𝜈𝜇 beams, (left) before
+oscillation at the near detector and (right) after oscillation at the far
+detector. The vertical lines show the three electron beam energies of
+this measurement.
+The NO𝜈A far-detector beam flux is calculated
+using the near detector flux and the neutrino oscillation parameters
+from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+126
+6-2
+∆𝜃𝛾,𝑒′ as a function of ∆𝜑𝛾,𝑒′. The red box indicated the region with
+radiated photons which was removed in our analysis.
+. . . . . . . . .
+129
+192
+
+6-3
+Number of events vs 𝐸𝑐𝑎𝑙 = 𝐸𝑒′ + 𝑇𝑝 the scattered electron energy plus
+proton kinetic energy for 4.32 GeV H(𝑒, 𝑒′𝑝). Black points are data,
+the blue histogram shows the unradiated GENIE prediction and the
+black histogram shows the GENIE prediction with electron radiation.
+The GENIE calculations have been scaled to have the same integral as
+the data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+130
+6-4
+Schematic illustration of the data driven background correction using
+detected (e,e’p𝜋) events. . . . . . . . . . . . . . . . . . . . . . . . . .
+132
+6-5
+The effect of undetected pion subtraction. The number of weighted
+events as a function of reconstructed energy 𝐸𝑄𝐸 for 4.453 GeV Fe(𝑒, 𝑒′)
+events for (left) events with a detected 𝜋± or photon (blue), events with
+one (red) or two (light brown) undetected 𝜋± or photons and (right)
+all (𝑒, 𝑒′𝑋) events with detected or undetected 𝜋± or photon (blue),
+(𝑒, 𝑒′) events with no detected 𝜋± or photon (red), and (𝑒, 𝑒′) events
+after subtraction for undetected 𝜋± or photon (light brown). . . . . .
+133
+6-6
+Illustration of the successful closure test of the data driven correc-
+tion for undetected particles as a function of 𝐸𝐶𝑎𝑙 using the (e,e’p)1𝑝0𝜋
+channel on 12C at 𝐸𝑏𝑒𝑎𝑚 = 2.257 GeV. The contribution of the un-
+subtracted (e,e’p)1𝑝0𝜋 spectrum (black) is reduced to the subtracted
+(e,e’p)1𝑝0𝜋 spectrum (magenta), which is in reasonable agreement with
+the true (e,e’p)1𝑝0𝜋 spectrum (green). . . . . . . . . . . . . . . . . . .
+134
+6-7
+The proton (black) and charged pion (blue) multiplicities for data
+(points), SuSav2 (solid histogram) and G2018 (dashed histogram) for
+2.257 GeV carbon. Error bars show the 68% (1𝜎) confidence limits
+for the statistical and point-to-point systematic uncertainties added in
+quadrature. Error bars are not shown when they are smaller than the
+size of the data point. Normalization uncertainties of 3% not shown. .
+134
+193
+
+6-8
+(Top row) Acceptance correction factors, (middle row) acceptance cor-
+rection factor uncertainties, and (bottom row) electron radiation cor-
+rection factors plotted vs E𝑐𝑎𝑙 for the three incident beam energies.
+Results for carbon are shown in black, helium in green and iron in ma-
+genta. The left column (a,d,g) shows the 1.1 GeV results, the middle
+column (b,e,h) shows the 2.2 GeV results and the right column (c,f,i)
+shows the 4.4 GeV results. . . . . . . . . . . . . . . . . . . . . . . . .
+135
+6-9
+Electron acceptance maps for 12C at 𝐸𝑏𝑒𝑎𝑚 = 1.159 GeV as a function
+of (left) cos𝜃𝑒 vs 𝜑𝑒 and (right) cos𝜃𝑒 vs momentum 𝑝𝑒. . . . . . . . .
+135
+6-10 Comparison between the inclusive C(𝑒, 𝑒′) cross sections measured at
+37.5∘ for data (points) and SuSav2 (lines) for the 0.961 and 1.299 GeV
+SLAC data and our 1.159 GeV CLAS data. . . . . . . . . . . . . . . .
+137
+6-11 The 1.159 GeV C(𝑒, 𝑒′)0𝜋 cross section plotted as a function of the re-
+constructed energy 𝐸𝑄𝐸 for data (black points), GENIE SuSAv2 (solid
+black curve) and GENIE G2018 (dotted black curve). The colored lines
+show the contributions of different processes to the GENIE SuSAv2
+cross section: QE (blue), MEC (red), RES (green) and DIS (orange).
+Error bars show the 68% (1𝜎) confidence limits for the statistical and
+point-to-point systematic uncertainties added in quadrature.
+Error
+bars are not shown when they are smaller than the size of the data
+point. Normalization uncertainty of 3% not shown.
+. . . . . . . . . .
+140
+194
+
+6-12 The 𝐴(𝑒, 𝑒′𝑝)0𝜋 cross section plotted as a function of the reconstructed
+quasielastic energy 𝐸𝑄𝐸 for data (black points), SuSAv2 (black solid
+curve) and G2018 (black dotted curve). Different panels show results
+for different beam energy and target nucleus combinations: (top row)
+Carbon target at (left to right) 1.159, 2.257 and 4.453 GeV, and (bot-
+tom) Iron target at (left) 2.257 and (right) 4.453 GeV incident beam.
+The 1.159 GeV yields have been scaled by 1/2 and the 4.453 GeV yields
+have been scaled by 5 to have the same vertical scale. Colored lines
+show the contributions of different processes to the SuSAv2 GENIE
+simulation: QE (blue), MEC (red), RES (green) and DIS (orange).
+Error bars show the 68% (1𝜎) confidence limits for the statistical and
+point-to-point systematic uncertainties added in quadrature.
+Error
+bars are not shown when they are smaller than the size of the data
+point. Normalization uncertainties of 3% not shown.
+. . . . . . . . .
+141
+6-13 The 𝐴(𝑒, 𝑒′𝑝)1𝑝0𝜋 cross section plotted as a function of the reconstructed
+calorimetric energy 𝐸𝑐𝑎𝑙 for data (black points), SuSAv2 (black solid
+curve) and G2018 (black dotted curve). Different panels show results
+for different beam energy and target nucleus combinations: (top row)
+Carbon target at (left to right) 1.159, 2.257 and 4.453 GeV, and (bot-
+tom) Iron target at (left) 2.257 and (right) 4.453 GeV incident beam.
+The 1.159 GeV yields have been scaled by 1/2 and the 4.453 GeV yields
+have been scaled by 5 to have the same vertical scale. The insets show
+the cross sections with the same horizontal scale and an expanded verti-
+cal scale. Colored lines show the contributions of different processes to
+the SuSAv2 GENIE simulation: QE (blue), MEC (red), RES (green)
+and DIS (orange).
+Error bars show the 68% (1𝜎) confidence limits
+for the statistical and point-to-point systematic uncertainties added in
+quadrature. Error bars are not shown when they are smaller than the
+size of the data point. Normalization uncertainties of 3% not shown.
+143
+195
+
+6-14 Energy feed-down cross-sections (𝐸𝑟𝑒𝑐 − 𝐸𝑡𝑟𝑢𝑒)/𝐸𝑡𝑟𝑢𝑒 for data (points)
+and SuSav2 (lines) for 1.159 GeV (red triangles and dotted lines),
+2.257 GeV (green squares and dashed lines) and 4.453 GeV (blue dots
+and solid lines) on carbon for (a) 𝐸𝑐𝑎𝑙, and (b) 𝐸𝑄𝐸.
+. . . . . . . . .
+144
+6-15 Energy feed-down cross-sections (𝐸𝑟𝑒𝑐 − 𝐸𝑡𝑟𝑢𝑒)/𝐸𝑡𝑟𝑢𝑒 for data (points)
+and SuSav2 (lines) for 1.159 GeV (red triangles and dotted lines),
+2.257 GeV (green squares and dashed lines) and 4.453 GeV (blue dots
+and solid lines) on iron for (c) 𝐸𝑐𝑎𝑙, and (d) Fe 𝐸𝑄𝐸.
+. . . . . . . . .
+145
+6-16 (Left) the 2.257 GeV C(𝑒, 𝑒′𝑝)1𝑝0𝜋 cross section plotted versus missing
+transverse momentum, 𝑃𝑇, for data (black points), SuSav2 (black solid
+line) and G2018 (black dashed line). The vertical lines at 200 MeV/𝑐
+and at 400 MeV/𝑐 separate the three bins in 𝑃𝑇. Colored lines show
+the contributions of different processes to the SuSAv2 GENIE simula-
+tion: QE (blue), MEC (red), RES (green) and DIS (orange). (Right)
+The cross section plotted versus the calorimetric energy 𝐸𝑐𝑎𝑙 for dif-
+ferent bins in 𝑃𝑇: (top) 𝑃𝑇< 200 MeV/𝑐, (middle) 200 MeV/𝑐 ≤ 𝑃𝑇≤
+400 MeV/𝑐, and (bottom) 𝑃𝑇> 400 MeV/𝑐.
+. . . . . . . . . . . . . .
+147
+6-17 The cross section plotted vs transverse missing momentum 𝑃𝑇 for data
+(black points), SuSAv2 (black solid curve) and G2018 (black dotted
+curve). Different panels show results for different beam energy and
+target nucleus combinations: (top row) Carbon target at (left to right)
+1.159, 2.257 and 4.453 GeV, and (bottom) Iron target at (left) 2.257
+and (right) 4.453 GeV. The 4.453 GeV yields have been scaled by four
+to have the same vertical scale. Colored lines show the contributions
+of different processes to the SuSAv2 GENIE simulation: QE (blue),
+MEC (red), RES (green) and DIS (orange).
+. . . . . . . . . . . . . .
+148
+196
+
+6-18 The cross section plotted vs 𝛿𝛼𝑇 (a-e) and vs 𝛿𝜑𝑇 (f-j) for data (black
+points), SuSAv2 (black solid curve) and G2018 (black dotted curve).
+Different panels show results for different beam energy and target nu-
+cleus combinations: (top row) Carbon target at (left to right) 1.159,
+2.257 and 4.453 GeV, and (bottom) Iron target at (left) 2.257 and
+(right) 4.453 GeV. The 4.453 GeV yields have been scaled by two to
+have the same vertical scale. Colored lines show the contributions of
+different processes to the SuSAv2 GENIE simulation: QE (blue), MEC
+(red), RES (green) and DIS (orange). . . . . . . . . . . . . . . . . . .
+149
+6-19 Data-simulation comparisons for 12C at 2.261 GeV showing the cross
+section results as a function of 𝑃𝑇 for (top) all the events, (bottom
+left) events with 𝛿𝛼𝑇 < 45𝑜 dominated by QE interactions and no
+reinteractions, and (bottom right) events with 135𝑜 < 𝛿𝛼𝑇 < 180𝑜
+maximally affected by FSI and multi-hadron channels. Colored lines
+show the contributions of different processes to the SuSAv2 GENIE
+simulation: QE (blue), MEC (red), RES (green) and DIS (orange).
+.
+150
+6-20 Data-simulation comparisons for 12C at 2.261 GeV showing the cross
+section results as a function of 𝛿𝛼𝑇 for (top) all the events, (bottom
+left) events with 𝑃𝑇 < 0.2 GeV/c dominated by QE interactions and no
+reinteractions, and (bottom right) events with 𝑃𝑇 > 0.4 GeV/c maxi-
+mally affected by FSI and multi-hadron channels. Colored lines show
+the contributions of different processes to the SuSAv2 GENIE simula-
+tion: QE (blue), MEC (red), RES (green) and DIS (orange).
+. . . .
+151
+6-21 (Top) data-simulation comparisons on 12C at 2.261 GeV showing the
+cross section results as a function of (top left) the true missing mo-
+mentum 𝑃𝑀𝑖𝑠𝑠 and (top right) the missing momentum approximation
+𝑃𝑛,𝑝𝑟𝑜𝑥𝑦 commonly used by neutrino experiments. (Bottom) overlay of
+the two extracted data cross sections illustrating the differences be-
+tween 𝑃𝑀𝑖𝑠𝑠 and 𝑃𝑛,𝑝𝑟𝑜𝑥𝑦.
+. . . . . . . . . . . . . . . . . . . . . . . .
+152
+197
+
+6-22 Data-simulation comparisons for 12C at 2.261 GeV showing the cross
+section results as a function of 𝑃𝑇,𝑥 for (top) all the events, (bottom
+left) events with -0.15 < 𝑃𝑇,𝑦 < 0.15 GeV/c dominated by QE inter-
+actions and no reinteractions, and (bottom right) events with 𝑃𝑇,𝑦 <
+-0.15 GeV/c maximally affected by FSI and multi-hadron channels.
+.
+153
+6-23 Data-simulation comparisons for 12C at 2.261 GeV showing the cross
+section results as a function of 𝑃𝑇,𝑦 for (top) all the events, (bottom
+left) events with -0.15 < 𝑃𝑇,𝑥 < 0.15 GeV/c dominated by QE inter-
+actions and no reinteractions, and (bottom right) events with 𝑃𝑇,𝑥 <
+-0.15 GeV/c maximally affected by FSI and multi-hadron channels.
+.
+154
+6-24 Schematic view of the upgraded CLAS12 detector components. . . . .
+155
+8-1
+The 𝜃 vs 𝜑 distributions for 𝜋− at 2.2 GeV. . . . . . . . . . . . . . . .
+171
+8-2
+The 𝜃 vs 𝜑 distributions for electrons at 2.2 GeV. . . . . . . . . . . .
+172
+8-3
+The 𝜃 vs 𝜑 distributions for protons.
+. . . . . . . . . . . . . . . . . .
+173
+8-4
+The 𝜃 vs 𝜑 distributions for 𝜋+. . . . . . . . . . . . . . . . . . . . . .
+173
+8-5
+The cos 𝜃 vs 𝜑 distributions for photons.
+. . . . . . . . . . . . . . . .
+174
+8-6
+The 𝜃 vs momentum distributions for 𝜋−. . . . . . . . . . . . . . . . .
+175
+8-7
+The 𝜃 vs momentum distributions for 𝜋−. . . . . . . . . . . . . . . . .
+176
+8-8
+The 𝜃 vs momentum distributions for 𝜋−. . . . . . . . . . . . . . . . .
+177
+8-9
+The 𝜃 vs momentum distributions for 𝜋−. . . . . . . . . . . . . . . . .
+177
+8-10 The 𝜃 vs momentum distributions for 𝜋−. . . . . . . . . . . . . . . . .
+178
+8-11 The 𝜃 vs momentum distributions for 𝜋−. . . . . . . . . . . . . . . . .
+178
+8-12 The 𝜃 vs momentum distributions for 𝜋+. . . . . . . . . . . . . . . . .
+179
+198
+
+List of Tables
+1.1
+Summary table of the 3𝜈 oscillation parameters. The 1𝜎 intervals for
+both the case normal are inverse ordering are shown. Table adapted
+from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+18
+3.1
+Interaction breakdown after the application of our selection cuts. . . .
+59
+3.2
+Integrated cross section values and 𝜒2 values for the agreement be-
+tween the measured cross sections and various event generators. Re-
+sults are listed for the full measured phase space and for a limited one
+of cos(𝜃𝜇) < 0.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+66
+5.1
+Target areal densities and integrated charges for the 𝑒4𝜈 data sets. . .
+122
+6.1
+Summary of the total systematic uncertainties used in the e4𝜈 analysis. 138
+6.2
+(𝑒, 𝑒′𝑝)1𝑝0𝜋 events reconstructed to the correct beam energy. Peak Frac-
+tion refers to the fraction of events reconstructed to the correct beam
+energy and Peak Sum refers to the integrated weighted cross section (as
+shown in Fig. 6-13) reconstructed to the correct beam energy. The peak
+integration windows are 1.1 ≤ 𝐸𝑐𝑎𝑙 ≤ 1.22 GeV, 2.19 ≤ 𝐸𝑐𝑎𝑙 ≤ 2.34
+GeV, and 4.35 ≤ 𝐸𝑐𝑎𝑙 ≤ 4.60 GeV, respectively, for the three incident
+beam energies. SuSAv2 is not intended to model nuclei lighter than 12C.142
+199
+
+200
+
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+page_content='  Deepto Chakrabarty  Associate Department Head of Physics, MIT  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=', National Kapodistrian University of Athens (2016)  Submitted to the Department of Physics  on May 12, 2022, in partial fulfillment of the  requirements for the degree of  Doctor of Philosophy  Abstract  Currently running and forthcoming precision neutrino oscillation experiments aim  to unambiguously determine the neutrino mass ordering, the charge-parity violating  phase in the lepton sector and the possible existence of physics Beyond the Standard  Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' To have an understanding of all the effects necessary for the success of these  experiments, lepton-nucleus interactions must be modeled in unprecedented detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  With this thesis, expertise in both neutrino and electron cross-section modeling and  analysis was leveraged in order to make fundamental and critical improvements to  our understanding of these interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The outlined work takes a significant step  towards this high-precision measurement era with three complementary approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Cross sections are reported using neutrino data sets from the MicroBooNE liquid  argon time projection chamber detector at Fermi National Laboratory, as well as  electron scattering data from the CLAS detector at Thomas Jefferson National Labo-  ratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Furthermore, the modeling development of the commonly used GENIE event  generator is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  3  Neutrino Cross Sections  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' 125, 201803 (2020)  Event Generator  Electron  Modeling  Cross Sections  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' D 103, 113003 (2021)  Nature 599, 565-570 (2021)Thesis Supervisor: Or Hen  Title: Associate Professor  4  Acknowledgments  First and foremost, my eternal gratitude goes to my advisor, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Or Hen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Or has  provided me with all the resources necessary to proceed in my academic career.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' I am  also grateful to Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Lawrence Weinstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Their guidance has been crucial in my  formation as a physicist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  I’m grateful to all the members of our group for their invaluable support and to  all the MIT graduate students in my year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Many thanks to all the members of the  MicroBooNE, “Electrons-For-Neutrinos”, CLAS, GENIE, and GlueX collaborations  for all their input and guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' I’m also grateful to my entire family and all my  friends for their continuous support and encouragement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  5  6  Contents  1  Introduction  11  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  Neutrinos In The Standard Model .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  11  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  Neutrino Oscillations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  Long-Baseline Accelerator-Based Neutrino  Oscillation Experiments  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='4  Neutrino-Nucleus Interactions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='5  Connections To Electron Scattering .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='6  Thesis Scope .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  31  2  The MicroBooNE Experiment At Fermi National Laboratory  33  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  Fermilab Neutrino Beamlines  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  33  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  The MicroBooNE Detector .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  36  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  Liquid Argon Time Projection Chambers .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  38  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  Scintillation Light .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  74  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  Event Generator Modeling And Configurations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='4  Kinematic Imbalance Differential Cross-Section Results .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  87  4  Inclusive Electron Scattering And The GENIE Event Generator  [Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' D 103, 113003 (2021)]  91  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  Electron-Nucleus Modeling Development .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  109  5  The CLAS Electrons-For-Neutrinos Experiment At Thomas Jeffer-  son National Laboratory  115  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='2  The CEBAF Large Acceptance Spectrometer .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  122  8  6  Electrons-For-Neutrinos Results  [Nature 599, 565–570 (2021)]  125  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  Electron Data Mining Analysis  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='2  Incident Energy Reconstruction Results .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='3  Kinematic Imbalance Results  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  180  List of Tables .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  199  References .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  200  9  10  Chapter 1  Introduction  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  Neutrinos In The Standard Model  The Standard Model (SM) of particle physics was developed to describe the parti-  cles that are considered to be fundamental and their interactions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Though an  extremely accurate theory, we are already aware of particles exhibiting behaviors not  predicted within the scope of the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These peculiar particles are called neutrinos,  which travel enormous distances before they interact with matter, exclusively via  weak interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The recent realization of neutrino oscillations was a historic dis-  covery [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The SM had been incredibly successful, yet it requires neutrinos to  be massless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The new observations clearly showed that neutrinos oscillate between  different identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This behavior is driven by their non-zero masses and is indicative  of new physics Beyond the Standard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Though very challenging to detect them, the study of neutrinos is a promising  venue towards the generalisation of the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The latter, initially developed more than  40 years ago, introduced the notion of fundamental particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Those particles are  either the building blocks of matter referred to as fermions with semi-integer spins, or  mediators of the interactions referred to as bosons with integer spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The collection  of the fundamental particles and the carriers is shown in figure 1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Prior to the SM, the electroweak theory incorporated neutrinos (𝜈𝑒 , 𝜈𝜇, 𝜈𝜏) as  left-handed partners of the three families of charged leptons (e, 𝜇, 𝜏) and the corre-  11  Figure 1-1: The Standard Model of particle physics illustrating the three generations  of fermions, the gauge bosons, and the scalar Higgs boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure adapted from  ScienceAlert [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  sponding anti-neutrinos as right-handed partners, which is illustrated in figure 1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The experimental verification that neutrinos (anti-neutrinos) are left-handed (right-  handed) was established with the Goldhaber experiment [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 1-2: Graphic illustrating the left-handed (right-handed) nature of neutrinos  (anti-neutrinos) via the orientation of the momentum-spin vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted  from [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  In the context of the SM, neutrinos are assumed to be massless and their individ-  ual lepton number is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Under the assumption that neutrinos are massless,  only the left-handed component of the Dirac spinor interacts via the weak force and  12  FERMIONS  (matter particles)  BOSONS (force carriers)  C  t  H  QUARKS  up  charm  top  gluon  Higgs boson  6  down  strange  bottom  photon  e  LEPTONS  electron  muon  tau  Z boson  Ve  VT  W  electron  muon  tau  w boson  neutrino  neutrino  neutrinoMomentum  Spin  Momentum  Spinright-handed components are completely absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Prior to the discovery of neutrino  oscillations, there was no experimental evidence of violation of the individual lepton  number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A typical example of such searches is the muon- and electron-number violat-  ing decay 𝜇± → 𝑒± + 𝛾, where only upper limits have been placed on the branching  ratio [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Yet, the experimental discovery of neutrino oscillations established that  the conservation of individual lepton number is not universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' That raises a wealth  of questions related to the possible right-handed nature and the non-zero mass of  neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  If neutrinos are massive, helicity is not exactly conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A massive neutrino  can have its spin and momentum anti-aligned, and that would correspond to a left-  handed neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' However, in the reference frame of an observer travelling faster than  the neutrino itself, the neutrino spin and momentum can be aligned, and that would  correspond to a right-handed particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' On the other hand, if neutrinos are massless,  they must travel at the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Since there exists no reference frame travelling  faster than that, the neutrino helicity cannot change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  At the same time, the neutrino masses are significantly smaller compared to other  particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The latest measurements by KATRIN indicate that electrons are 𝒪(106)  more massive than neutrinos [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Massless particles require no extra terms to be  added in the SM Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' On the other hand, the tiny neutrino masses constitute  a riddle that requires fine tuning of the additional Lagrangian terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Instead, many  theorists argue in favor of a fundamental reason why the neutrino masses are so small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  In order to accommodate a non-zero neutrino mass, there are two potential ap-  proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The first one includes new particles, namely Dirac neutrinos, and the second  introduces a new particle type, namely Majorana neutrinos [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  A Dirac neutrino can acquire mass via its coupling to the Higgs field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Those parti-  cles that interact with the Higgs field change helicity and, thus, left-handed particles  become right-handed and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Experimental results to-date indicate that in-  teracting neutrinos are left-handed [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A potential extension of the SM includes such  right-handed neutrinos that obtain mass via the Higgs field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' However, these neutri-  nos do not have electroweak charge, thus they interact only via the mixing with the  13  left-handed counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' If indeed neutrinos are Dirac particles and obtain their mass  due to the coupling with the Higgs field, their masses should be comparable to the  other fermions predicted by the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' However, the neutrino interaction coupling can  be tuned to retrieve masses that are comparable with the experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Another possibility would be the introduction of Majorana particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' With this  approach, there is no distinction between neutrinos and anti-neutrinos, with the latter  being plausible since neutrinos are electrically neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Massive neutrinos can be  accommodated in this extension of the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Earlier the argument that an observer  traveling at the speed of light might observe a flip of the neutrino helicity was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  However, if neutrinos are their own anti-particles, there is no helicity change within  this massive neutrino hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The simplest SM extension with Majorana-like  particles is the type 1 seesaw mechanism [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  In this framework, a left-handed  neutrino interacts with the Higgs boson and a really heavy right-handed neutrino is  briefly produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The latter further interacts with the Higgs field to produce a light  left-handed Majorana neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  Neutrino Oscillations  Neutrinos are produced in specific flavor eigenstates, namely 𝜈𝑒 , 𝜈𝜇 or 𝜈𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The  neutrino mass eigenstates can be expressed as a superposition of flavor eigenstates  and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The flavor eigenstates evolve as a function of the distance that neu-  trinos travel, the neutrino energy and the neutrino mass differences squared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Such  an evolution is referred to as “neutrino oscillation” and is a natural consequence of  the fact that a neutrino flavor state is composed of multiple mass eigestates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For  simplicity, the derivation of the formalism for the two-neutrino oscillation case with  masses 𝑚𝑖 for i ∈ {1,2} is outlined here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The neutrino wavefunction is treated as a  quantum-mechanical plane wave 𝜓 that evolves in time and space as:  𝜓(𝐿) = 𝜓(0) · 𝑒𝑖·𝑝·𝑥,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1)  with 𝑝·𝑥 = 𝐸𝑡−⃗𝑝·⃗𝑥 corresponding to the Lorentz-invariant phase and with (𝐸,⃗𝑝 )  14  being the energy and the three-momentum, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For a particle of mass 𝑚,  its momentum 𝑝 can be obtained in the highly relativistic limit where 𝐸 ≫ 𝑚 via a  Taylor expansion as:  𝑝 =  √  𝐸2 − 𝑚2 ≈ 𝐸 − 𝑚2/2𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2)  If a neutrino travels some distance 𝐿 in vacuum, a phase shift is introduced in its  wavefunction given by equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1, and equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 becomes:  𝜈𝑖(𝐿) = 𝜈𝑖(0) · 𝑒−𝑖·𝑚2  𝑖 ·𝐿/2𝐸  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3)  where all the constants have been absorbed in the global phase 𝜈𝑖(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' It is impor-  tant to stress the fact that these expressions are accurate for plane waves propagat-  ing in vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Additional phase shifts have to be introduced when neutrinos travel  through a high density material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  In the two-neutrino case, the flavor eigenstates 𝜈𝑒 and 𝜈𝜇 can be expressed as a  linear superposition of the mass eigenstates 𝜈1 and 𝜈2:  𝜈𝑒 = 𝑐𝑜𝑠𝜃 · 𝜈1 + 𝑠𝑖𝑛𝜃 · 𝜈2,  𝜈𝜇 = −𝑠𝑖𝑛𝜃 · 𝜈1 + 𝑐𝑜𝑠𝜃 · 𝜈2,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4)  where 𝜃 is the mixing angle between the two states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Consider now a pure beam  of electron neutrinos produced at the source, effectively at a distance of 𝐿 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  wavefunction evolves following the formalism in equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3,  𝜈𝑒(𝐿) = 𝑐𝑜𝑠𝜃 · 𝑒−𝑖·𝑚2  1·𝐿/2𝐸 · 𝜈1(0) + 𝑠𝑖𝑛𝜃 · 𝑒−𝑖·𝑚2  2·𝐿/2𝐸 · 𝜈2(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5)  Given that in our detectors the products of weak interactions are reconstructed  15  based on the neutrino flavor, equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5 needs to be rewritten using the flavor basis,  𝜈𝑒(𝐿) = [𝑐𝑜𝑠2𝜃 · 𝑒−𝑖·𝑚2  1·𝐿/2𝐸 + 𝑠𝑖𝑛2𝜃 · 𝑒−𝑖·𝑚2  2·𝐿/2𝐸] · 𝜈𝑒(0)  −𝑠𝑖𝑛𝜃 · 𝑐𝑜𝑠𝜃 · [𝑒𝑖·𝑚2  1·𝐿/2𝐸 − 𝑒𝑖·𝑚2  2·𝐿/2𝐸] · 𝜈𝜇(0)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6)  The probability of detecting a neutrino of a given flavor is obtained via the square  of the amplitude,  𝑃𝜈𝑒→𝜈𝑒 = ⟨𝜈𝑒(𝐿)|𝜈𝑒(𝐿)⟩ = 1 − 𝑠𝑖𝑛2(2𝜃)𝑠𝑖𝑛2(∆𝑚2𝐿  4𝐸  ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7)  𝑃𝜈𝑒→𝜈𝜇 = ⟨𝜈𝑒(𝐿)|𝜈𝜇(𝐿)⟩ = 𝑠𝑖𝑛2(2𝜃)𝑠𝑖𝑛2(∆𝑚2𝐿  4𝐸  ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8)  where ∆𝑚2 = 𝑚2  2 − 𝑚2  1 is the neutrino mass difference squared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Provided that  𝜃 ̸= 0 or 𝜋/2 and ∆𝑚2 ̸= 0 for oscillations to take place, the neutrino beam evolves  as a function of 𝐿/𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The amplitude of this oscillation is given by 𝑠𝑖𝑛2(2𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  wavelength, expressed in commonly used units, is obtained by  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='27∆𝑚2 [𝑒𝑉 2]𝐿 [𝑘𝑚]  𝐸 [𝐺𝑒𝑉 ]  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9)  The experimental observation of neutrino oscillations serves as proof of evidence  that the initial state indeed undergoes a phase shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The explanation for such a shift is  that at least some neutrinos have non-zero mass and that the transformation between  the mass and the flavor eigenstates involves a non-zero mixing angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  gives the neutrino oscillation probability for the simplified case of only two types of  neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The full three-flavor probability for 𝜈𝜇 → 𝜈𝑒 oscillation in vacuum is given  by  𝑃𝜈𝜇→𝜈𝑒(𝐸, 𝐿)  ≈  𝐴 sin2 ∆𝑚2  13𝐿  4𝐸  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='10)  −𝐵 cos  (︂∆𝑚2  13𝐿  4𝐸  + 𝛿𝐶𝑃  )︂  sin ∆𝑚2  13𝐿  4𝐸  ,  16  where ∆𝑚2  13 = 𝑚2  1 − 𝑚2  3 is the neutrino mass difference squared that determines  the oscillation wavelength as a function of 𝐿/𝐸 and 𝛿𝐶𝑃 is the charge-parity (CP)  symmetry violating phase [11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The coefficients 𝐴 and 𝐵 depend primarily on the  neutrino oscillation mixing angles,  𝐴 = sin2 𝜃23 sin2 2𝜃13, 𝐵 = −sin 2𝜃12 sin 2𝜃23  2 sin 𝜃13  sin ∆𝑚2  21𝐿  4𝐸  sin2 2𝜃13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='11)  The different flavor neutrinos (labelled 𝜈𝑒, 𝜈𝜇 and 𝜈𝜏) are linear combinations of  the different mass neutrinos labelled 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The outlined neutrino mixing can be generalized to N neutrino eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  probability of finding an 𝛼 flavor eigenstate neutrino in a flavor eigenstate 𝛽, after  traveling some distance 𝐿, is given by:  𝑃𝛼→𝛽 = 𝛿𝛼𝛽 − 4  ∑︁  𝑗>𝑖  𝑅𝑒[𝑈 *  𝛼𝑖𝑈𝛽𝑖𝑈𝛼𝑖𝑈 *  𝛽𝑗]𝑠𝑖𝑛2  (︂[︂1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='27 𝐺𝑒𝑉  𝑒𝑉 2 𝑘𝑚  ]︂ ∆𝑚2  𝑗𝑖𝐿  𝐸  )︂  +2  ∑︁  𝑗>𝑖  𝐼𝑚[𝑈 *  𝛼𝑖𝑈𝛽𝑖𝑈𝛼𝑖𝑈 *  𝛽𝑗]𝑠𝑖𝑛  (︂[︂2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='54 𝐺𝑒𝑉  𝑒𝑉 2 𝑘𝑚  ]︂ ∆𝑚2  𝑗𝑖𝐿  𝐸  )︂  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='12)  where U is the N×N unitary neutrino mixing matrix and ∆𝑚2  𝑗𝑖 = 𝑚2  𝑗 − 𝑚2  𝑖 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The corresponding antineutrino oscillation probability can be obtained by replacing  𝑈 → 𝑈 †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Within the widely accepted neutrino model, there exist three active neutrinos [15],  resulting into two squared mass splittings, ∆𝑚2  21 and ∆𝑚2  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The corresponding 3 × 3  mixing matrix, referred to as the Pontecorvo-Maki-Nakagawa-Sakata (PNMS) matrix,  relates the mass eigenstates (𝜈1, 𝜈2, 𝜈3) to the flavor eigenstates (𝜈𝑒, 𝜈𝜇, 𝜈𝜏):  ⎛  ⎜  ⎜  ⎜  ⎜  ⎜  ⎜  ⎝  𝜈𝑒  𝜈𝜇  𝜈𝜏  ⎞  ⎟  ⎟  ⎟  ⎟  ⎟  ⎟  ⎠  =  ⎛  ⎜  ⎜  ⎜  ⎜  ⎜  ⎜  ⎝  𝑈𝑒1  𝑈𝑒2  𝑈𝑒3  𝑈𝜇1  𝑈𝜇2  𝑈𝜇3  𝑈𝜏1  𝑈𝜏2  𝑈𝜏3  ⎞  ⎟  ⎟  ⎟  ⎟  ⎟  ⎟  ⎠ ⎛  ⎜  ⎜  ⎜  ⎜  ⎜  ⎜  ⎝  𝜈1  𝜈2  𝜈3  ⎞  ⎟  ⎟  ⎟  ⎟  ⎟  ⎟  ⎠  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='13)  17  The mixing of the mass-flavor eigenstates is parametrised with three mixing angles  (𝜃12, 𝜃23, 𝜃13) and the 𝛿𝐶𝑃 violating phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The best measured values of the these  angular parameters to date are obtained from global fits as documented in the latest  Review of Particle Physics edition [1] and listed in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1: Summary table of the 3𝜈 oscillation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The 1𝜎 intervals for both  the case normal are inverse ordering are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Table adapted from [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Normal Ordering  Inverted Ordering  𝑠𝑖𝑛2𝜃12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='320+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='020  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='016  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='320+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='020  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='016  𝑠𝑖𝑛2𝜃23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='547+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='020  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='030  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='551+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='018  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='030  𝑠𝑖𝑛2𝜃13  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='160+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='083  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='069) · 10−2  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='220+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='074  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='076) · 10−2  ∆𝑚2  21  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='55+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='16) · 10−5 𝑒𝑉 2  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='55+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='16) · 10−5 𝑒𝑉 2  ∆𝑚2  32  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='42+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='03  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='03) · 10−3 𝑒𝑉 2  (−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='50+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='04  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='03) · 10−3 𝑒𝑉 2  The 𝛿𝐶𝑃 phase and the neutrino mass hierarchy (𝑚2 < 𝑚3 or the other way  around) are still not determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A potential way to resolve these open questions is  via the Mikheyev-Smirnov-Wolfenstein (MSW) effect [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In the presence of matter,  the neutrino wavefunction propagation is modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' When a neutrino travels through  a dense material, the energy-momentum relationship is affected via coherent interac-  tions with matter particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Given that regular matter on Earth contains electrons,  but not muons or taus, charged current interactions with the medium only affect the  electron neutrino propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This interaction flavor dependence results in mea-  surable changes in the contribution of the different flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Thus, the MSW effect  is exploited to perform long-baseline neutrino experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The Deep Underground  Neutrino Experiment (DUNE) is a forthcoming multi-billion-dollar international ex-  periment aiming to resolve the aforementioned open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' To successfully achieve  that goal, DUNE will utilize an intense muon beam, with a near detector located at  Fermi National Laboratory IL and a far detector ≈ 1300 km away from the neutrino  source at Sanford Underground Research Facility SD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  18  Before DUNE starts taking data, a number of other neutrino experiments will  have already provided constraints for 𝛿𝐶𝑃 and the mass hierarchy [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For these  results to be obtained, different neutrino oscillation experiments utilize a number of  neutrino sources covering a wide range of energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Reactor neutrino experiments exploit the large electron anti-neutrino fluxes pro-  duced in nuclear reactors by 𝛽 decays of heavy nuclei (nuclear fissions of 235𝑈, 238𝑈,  239𝑃𝑢, 241𝑃𝑢).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The typical energy scale of reactor 𝜈𝑒’s is a few MeVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Atmospheric neutrino experiments take advantage of neutrinos produced by cos-  mic rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The latter interact with the upper layers of the atmosphere producing  a large flux of pions and kaons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  These decay in the atmosphere into muons and  muon neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These muons might decay into electrons, electron anti-neutrinos,  and muon neutrinos before they reach the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Atmospheric neutrino experiments  aim to detect these muon neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The energy of these neutrinos spans a really  wide range up to 𝒪(100) GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In some cases, like in IceCube [18], even neutrinos  with PeV energies can be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Such neutrinos cover distances between 𝒪(10) km  for neutrinos produced in the upper layers of the atmosphere directly above the de-  tector - to 𝒪(104) km - for netrinos that are produced on the other side of the Earth  and travel through the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Atmospheric experiments provide the current best limit  of ∆𝑚2  32 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='56+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='13  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='11 × 10−3𝑒𝑉 2 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Solar neutrino experiments detect the electron neutrinos produced in the Sun core  due to nuclear fusion processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Solar neutrino experiments are designed to detect  the 𝜈𝑒’s produced via these processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The currently best limit corresponds to an  extremely small value of ∆𝑚2  21 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='37+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='59  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='44 × 10−5𝑒𝑉 2 [1], much smaller than the  ∆𝑚2  32 splitting mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Accelerator-based experiments use muon neutrino beams produced via the decay of  primarily pions produced when a proton beam hits a heavy target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Such experiments  are further classified into appearance and disappearance experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The former  search for electron neutrinos oscillated from the initial muon neutrino beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  latter look for the reduction of muon neutrino interactions due to oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Different experiments are designed to be sensitive to different values of ∆𝑚2 by  19  choosing the appropriate 𝐿/𝐸 ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Building on equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9, the value of ∆𝑚2 for  which  ∆𝑚2𝐿  2𝐸  ≃ 1  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='14)  corresponds to the ∆𝑚2 sensitivity of a given experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Neutrino oscillation experiments are further classified depending on the average  value of this 𝐿/𝐸 ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Using equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='14, short baseline experiments with 𝐿/𝐸 ≲ 1  km/GeV are sensitive to ∆𝑚2 ≳ 1 𝑒𝑉 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Long baseline experiments where 𝐿/𝐸 ≲ 103  km/GeV are sensitive to ∆𝑚2 ≳ 10−3 𝑒𝑉 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  Long-Baseline Accelerator-Based Neutrino  Oscillation Experiments  Figure 1-3: The experimental setup of an accelerator-based long-baseline neutrino  oscillation experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Long-baseline accelerator-based experiments consist of a near detector (ND) posi-  tioned close to the neutrino source, and a far detector (FD), which is positioned close  to the oscillation maximum„ as shown in figure 1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The number of interactions of  20  Laboratory 1  Laboratory 2  Neutrino  detector  Neutrino  sourceflavor 𝛼 in the near detector (𝑁 𝛼  𝑁𝐷) is obtained as  𝑁 𝑁𝐷  𝛼  (𝐸𝑟𝑒𝑐𝑜) =  ∫︁  Φ𝑁𝐷  𝛼  (𝐸𝑡𝑟𝑢𝑒)𝜎𝛼(𝐸𝑡𝑟𝑢𝑒)𝜖𝛼(𝐸𝑡𝑟𝑢𝑒)𝑓 𝑁𝐷(𝐸𝑡𝑟𝑢𝑒, 𝐸𝑟𝑒𝑐𝑜)𝑑𝐸𝑡𝑟𝑢𝑒  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15)  where Φ𝑁𝐷  𝛼  (𝐸𝑡𝑟𝑢𝑒) is the neutrino flux of flavor 𝛼 close to the source, 𝜎𝛼(𝐸𝑡𝑟𝑢𝑒)  is the cross section for a given flavor 𝛼, 𝜖𝛼(𝐸𝑡𝑟𝑢𝑒) is the reconstruction efficiency for  flavor 𝛼, and 𝑓 𝑁𝐷(𝐸𝑡𝑟𝑢𝑒, 𝐸𝑟𝑒𝑐𝑜) is the detector response function, describing how 𝐸𝑡𝑟𝑢𝑒  is mapped to 𝐸𝑟𝑒𝑐𝑜 in the near detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For 𝜈𝛼 → 𝜈𝛽 oscillations, the number of  events in the far detector can be obtained as  𝑁 𝐹𝐷  𝛼→𝛽(𝐸𝑟𝑒𝑐𝑜) =  ∫︁  Φ𝐹𝐷  𝛼 (𝐸𝑡𝑟𝑢𝑒)𝑃𝛼→𝛽(𝐸𝑡𝑟𝑢𝑒)𝜎𝛽(𝐸𝑡𝑟𝑢𝑒)𝜖𝛽(𝐸𝑡𝑟𝑢𝑒)𝑓 𝐹𝐷(𝐸𝑡𝑟𝑢𝑒, 𝐸𝑟𝑒𝑐𝑜)𝑑𝐸𝑡𝑟𝑢𝑒  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='16)  where 𝑁 𝐹𝐷  𝛼→𝛽(𝐸𝑟𝑒𝑐𝑜) is the number of 𝛽 flavor interactions, Φ𝐹𝐷  𝛼 (𝐸𝑡𝑟𝑢𝑒) is the neu-  trino flux of flavor 𝛼, 𝑃𝛼→𝛽(𝐸𝑡𝑟𝑢𝑒) is the oscillation probability for 𝛼 → 𝛽, 𝜎𝛽(𝐸𝑡𝑟𝑢𝑒)  is the cross section for flavor 𝛽, 𝜖𝛽(𝐸𝑡𝑟𝑢𝑒) is the reconstruction efficiency for flavor  𝛽, and 𝑓 𝐹𝐷(𝐸𝑡𝑟𝑢𝑒, 𝐸𝑟𝑒𝑐𝑜) is the detector response function, describing how 𝐸𝑡𝑟𝑢𝑒 is  reconstructed in 𝐸𝑟𝑒𝑐𝑜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Accelerator-produced neutrino beams predominantly contain muon neutrinos [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Therefore, long-baseline accelerator-based neutrino oscillation experiments focus on  muon neutrino disappearance and electron neutrino appearance studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The former  are sensitive to the oscillation parameters 𝜃23 and ∆𝑚2  23, while the latter have sensi-  tivity to 𝜃13 and 𝛿𝐶𝑃 [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' At a distance 𝐿 from the neutrino production point, some  muon neutrinos will oscillate to electron neutrinos, resulting in fluxes of approximately  Φ𝐹𝐷  𝑒  (𝐸𝑡𝑟𝑢𝑒)  ∝  𝑃𝜈𝜇→𝜈𝑒(𝐸𝑡𝑟𝑢𝑒, 𝐿) Φ𝑁𝐷  𝜇  (𝐸𝑡𝑟𝑢𝑒),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='17)  Φ𝐹𝐷  𝜇 (𝐸𝑡𝑟𝑢𝑒)  ∝  [︀  1 − 𝑃𝜈𝜇→𝜈𝑒(𝐸𝑡𝑟𝑢𝑒, 𝐿)  ]︀  Φ𝑁𝐷  𝜇  (𝐸𝑡𝑟𝑢𝑒),  where the proportionality constant depends on the experiment geometry, that can  21  be affected by the different experimental acceptances at the ND and FD locations,  and 𝑃𝜈𝜇→𝜈𝑒 is the electron neutrino appearance probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  𝜈𝜇 → 𝜈𝑒 oscillations  are thus observed by measuring the neutrino fluxes Φ𝐹𝐷  𝑒  (𝐸𝑡𝑟𝑢𝑒) and Φ𝐹𝐷  𝜇 (𝐸𝑡𝑟𝑢𝑒) as  a function of energy and distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The three-flavor oscillation equations are similar  but include additional terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Charge-parity (CP) symmetry violation in the leptonic  sector would add a phase 𝛿𝐶𝑃 to the three-flavor oscillation with an opposite sign for  neutrinos and anti-neutrinos [21,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Therefore, the precision to which oscillation parameters can be determined exper-  imentally depends on our ability to extract Φ𝐹𝐷  𝛼 (𝐸) from 𝑁 𝐹𝐷  𝛼  (𝐸𝑟𝑒𝑐𝑜), as can be seen  in equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='16 and is graphically illustrated in figure 1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 1-4: Neutrino energy spectra reconstruction depends on our ability to model  the interaction of neutrinos with atomic nuclei and the propagation of particles  through the atomic nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  This flow chart shows the process, starting with an  oscillated far-detector incident-energy spectrum (green), differentiating the physical  neutrino interactions (green arrows) from the experimental analysis (blue arrows),  and ending up with an inferred incident-energy spectrum that hopefully matches the  actual one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  While experimental effects are generally understood and can be minimized us-  22  PHYSICS PROCESS  n  10  Oscillated v Flux  8  元  6  4  元°  2  0  p  particles  E[GeV]  EXPERIMENTALANALYSISing improved detectors, nuclear effects are irreducible and must be accounted for  using theoretical models, typically implemented in neutrino event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Thus,  the experimental sensitivity is largely determined by the accuracy of the theoreti-  cal models used to calculate the interaction cross sections and the accuracy of the  energy reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The available models have many free parameters that are  poorly constrained and are “tuned” by each neutrino experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Current oscillation  experiments report significant systematic uncertainties due to these interaction mod-  els [22–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Simulations further show that energy reconstruction errors can lead to  significant biases in extracting 𝛿𝐶𝑃 at DUNE [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' There is a robust theoretical effort  to improve these models [27–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Since there are no mono-energetic high-energy neutrino beams, these models can-  not be tested for individual neutrino energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Instead, experiments tune models to  their near-detector data, where the unoscillated flux Φ𝑁𝐷  𝛼  (𝐸𝑡𝑟𝑢𝑒) is calculated from  hadronic reaction rates [30–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' While highly informative, such integrated constraints  are insufficient to ensure that the models are correct for each value of 𝐸𝑡𝑟𝑢𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Thus, for  precision measurements using a broad-energy neutrino beam, the degree to which the  near-detector data alone can constrain models is unclear for a number of reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In  some cases, the near and far detectors use different target nuclei, which demands the  cross-section extrapolation between the two targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' At the same time, the fluxes at  the two detectors are not identical due to their distance from the source, and the fact  that neutrinos oscillate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The near detector measures neutrino interactions originat-  ing from pion decays in a long pipe with a length of 𝒪(50 m), while the far detector  measures neutrinos from a much smaller solid angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Thus, the particle acceptance is  different between the two detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Furthermore, the selection efficiencies 𝜖𝛼(𝐸𝑡𝑟𝑢𝑒)  in equations 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='16, both for the signal and the background events at the two  detectors, are model dependent and different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Yet another challenge for neutrino oscillation experiments is the accuracy of the  true neutrino energy reconstruction that enters the oscillation probability formula  shown in equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The neutrino energy is reconstructed using the measured en-  ergy deposition of the final state particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' However, energy losses due to the energy  23  of neutral particles, particles below thresholds, and inactive detector regions have to  be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Inevitably, assumptions about these effects have to be made  based on the underlying modeling choices and on the detector capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This pro-  cedure introduces systematic uncertainties that might limit the precise reconstruction  of the true neutrino energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Thus, the success of forthcoming neutrino experiments  like DUNE relies on the accurate identification and reconstruction of all particles  produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Hence, tracking detectors with low detection thresholds are key elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Furthermore, neutrinos traveling through matter experience a potential due to  the coherent elastic scattering with electrons and nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Coherent scattering takes  place when a neutrino wavefunction interacts with matter as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The implication  of this behavior is that neutrinos and antineutrinos are affected in different ways, due  to the lack of positrons in regular matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This effect can mimic a CP violating picture  with 𝑃(𝜈𝛼 → 𝜈𝛽) ̸= 𝑃(¯𝜈𝛼 → ¯𝜈𝛽), though it contains no fundamental information  related to the matter-antimatter asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Therefore, accounting for matter effects  in oscillation experiments that aim to extract the oscillation parameters is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  All these open questions need to be addressed in order to ensure the success of  forthcoming high-precision neutrino oscillation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Hence, this thesis pro-  gresses in that direction by improving the understanding of lepton-nucleus interactions  described in sections 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  Neutrino-Nucleus Interactions  Neutrino-nucleus interactions are extremely complicated processes that become even  more complex on heavier nuclei, such as on argon, the target nucleus used in liquid  argon time projection chambers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These interactions are approximated by deploying  a two-step approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The first step involves the primary neutrino interaction with a single nucleon or  a pair of nucleons with the nucleons being treated as quasi-free objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' There are  four main types of neutrino scattering processes that play a major role in the energy  spectrum of neutrino experiments like MicroBooNE and DUNE, namely quasielastic  24  (QE), meson exchange currents (MEC), resonant production (RES), and deep inelas-  tic scattering (DIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These four interaction types and the relevant outgoing particles  are shown in figure 1-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 1-5: The four main interaction processes for neutrino-nucleus scattering events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Each one of those processes dominates in different energy ranges, as illustrated in  figure 1-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' At neutrino energies below ≈ 1 GeV, QE interactions are the ones that  dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' With this process, leptons scatter off, and liberate, a single nucleon from  the target nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In the region around 𝒪(1 GeV), a set of nuclear forces include  the exchange of virtual mesons between two (or more) nucleons, which is referred to  as 2 particle-2 hole effects (2p2h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Such an effect has major contributions from MEC  events that lead to the emission of two nucleons from the primary neutrino interaction  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For interaction energies greater than the ∆ baryon mass of 1232 MeV, RES  interactions become energetically allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Such processes are the dominant ones in  the energy range between 1-4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' With such an interaction type, the struck nucleon  is brought into an excited state that is called a baryon resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The resonance  deexcites and that leads to the emission of a single nucleon and a single pion in the  final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' At even higher energies starting at ≈ 4 GeV, the incoming neutrino scatters  off a quark in the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Such an energy transfer results in the production of multiple  hadrons in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Interference terms across the different interaction processes  need to be added coherently at the quantum-mechanical amplitude level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' However,  commonly used neutrino event generators, such as GENIE [33], do not include such  interference terms between the different reaction modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In other words, the total  cross section is obtained by adding the individual cross sections 𝜎𝑖(𝐸) incoherently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  For neutrino scattering events, interaction processes can take place in the form of  either charged current (CC) or neutral current (NC) interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The former includes  25  O  MEC  RES  DISFigure 1-6: Neutrino cross-section evolution as a function of the neutrino energy  illustrating the energy range where each one of the four main processes dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure adapted from [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  the exchange of charged W bosons, and the latter of neutral Z bosons respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The corresponding Feyman diagrams are shown in figure 1-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 1-7: Feyman diagrams illustrating charged current (CC) and neutral current  (NC) processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Beyond the possible interactions and processes mentioned above, neutrino-nucleon  interactions become even more sophisticated due to the fact that the interaction takes  place in a dense nuclear medium, such as an argon nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These complex nuclear  effects can be further classified into initial and final state interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Initial state interactions are associated with the nucleon-nucleon correlations in  the target nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The nucleons inside the target nucleus are not at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Their  26  1  TOTAL  DIS  cross  RES  >  10-1  1  10  102  E, (GeV)CC  NC  μ  Vu W  1  Z V  n  p  p  pFigure 1-8: Nucleon momentum distribution options available in commonly used  neutrino event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  momentum distribution can be approximated in a number of ways, typical examples  of which would be a Bodek-Richie Fermi Gas [36], a Local Fermi Gas [37], or an  Effective Spectral Function [38], as illustrated in figure 1-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The precise behavior of  the Fermi motion is unknown, and it results in the smearing of the reconstructed  value of the true neutrino energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 1-9: Schematic illustration the wealth of possible final state interactions that  the final state hadrons can undergo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  27  ntranuclearMedium  HadronAbsorption  n  n元  ChargeExchange  CCQEInteraction  Pion Absorption  b元  p  (In)elastic  Scattering  PionProduction  ElasticScattering  p元8  Probability / GeV  Bodek-Ritchie Fermi Gas  6  Local Fermi Gas  Effective Spectral Function  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  Momentum [GeV]In the second part of the factorization process, after the primary neutrino-nucleus  interaction, the outgoing nucleons can undergo a wealth of final state interactions  while transversing the nuclear medium, before they exit the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Such re-interactions  might result in processes such as the emission of further hadrons, the absorption of the  initially emitted hadrons, charge exchange processes, and/or acceleration/deceleration  of the primary hadrons, as illustrated in figure 1-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  All the effects described above have a direct impact on the ability to accurately  reconstruct the energy of the incoming neutrino using the properties of the final state  particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  Connections To Electron Scattering  Neutrinos and electrons interact with atomic nuclei by exchanging intermediate vector  bosons, a massive 𝑊 ± or 𝑍 for the neutrino and a massless photon for the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Electrons interact via a vector current 𝑗𝜇  𝐸𝑀 = ¯𝑢𝛾𝜇𝑢 and neutrinos interact via vector  and axial-vector 𝑗𝜇  𝐶𝐶 = ¯𝑢𝛾𝜇(1 − 𝛾5)𝑢 −𝑖𝑔𝑊  2  √  2 currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Fundamental considerations give an inclusive (𝑒, 𝑒′) electron-nucleon elastic scat-  tering cross section that depends on only two structure functions [40],  𝑑2𝜎𝑒  𝑑𝑥𝑑𝑄2 = 4𝜋𝛼2  𝑄4  [︂1 − 𝑦  𝑥  𝐹 𝑒  2 (𝑥, 𝑄2) + 𝑦2𝐹 𝑒  1 (𝑥, 𝑄2)  ]︂  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='18)  Here 𝐹 𝑒  1 and 𝐹 𝑒  2 are the standard electromagnetic vector structure functions, 𝑄2 =  q2 − 𝜈2 is the squared momentum transfer and q and 𝜈 are the three-momentum  and energy transfers, 𝑥 = 𝑄2/(2𝑚𝜈) is the Bjorken scaling variable, 𝑚 is the nucleon  mass, 𝑦 = 𝜈/𝐸𝑒 is the electron fractional energy loss, and 𝛼 is the fine structure  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This formula shows the simplest case where 𝑄2 ≫ 𝑚2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The corresponding inclusive charged-current (CC) (𝜈, 𝑙±) neutrino-nucleon elastic  cross section has a similar form, where 𝑙± is the outgoing charged lepton [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  vector part of the current is subject to the same fundamental considerations as above,  28  but the axial-vector part of the current does not conserve parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This leads to a third,  axial, structure function,  𝑑2𝜎𝜈  𝑑𝑥𝑑𝑄2 =𝐺2  𝐹  2𝜋  [︂1 − 𝑦  𝑥  𝐹 𝜈  2 (𝑥, 𝑄2) + 𝑦2𝐹 𝜈  1 (𝑥, 𝑄2)  −𝑦(1 − 𝑦/2)𝐹 𝜈  3 (𝑥, 𝑄2)  ]︀  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='19)  Here 𝐹 𝜈  1 and 𝐹 𝜈  2 are the parity-conserving neutrino-nucleus vector structure functions,  𝐹 𝜈  3 is the axial structure function, and 𝐺𝐹 is the Fermi constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The vector form  factors, 𝐹 𝜈  1 and 𝐹 𝜈  2 , have both vector-vector and axial-axial contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  These simple equations are very similar for electron-nucleus scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In the limit  of electron-nucleon elastic scattering (𝑥 = 1), the two structure functions reduce to the  Dirac and Pauli form factors, which are linear combinations of the electric 𝐺𝐸(𝑄2) and  magnetic 𝐺𝑀(𝑄2) form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Neutrino-nucleon elastic scattering has an additional  axial form factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In the simplest case where a lepton scatters quasielastically from  a nucleon in the nucleus and the nucleon does not reinteract as it leaves the nucleus  shown in figure 1-10, the lepton-nucleus cross section is the integral over all initial  state nucleons,  𝑑𝜎  𝑑𝐸𝑑Ω =  ∫︁  pi  ∫︁  𝐸𝑏  𝑑3pi𝑑𝐸𝑏𝐾𝑆(pi, 𝐸𝑏)𝑑𝜎𝑓𝑟𝑒𝑒  𝑑Ω  𝛿3(q − pf − pr)𝛿(𝜔 − 𝐸𝑏 − 𝑇𝑓 − 𝑇𝑟),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='20)  where pi and pf = q + pi are the initial and final momenta of the struck nucleon  in the absense of any reinteraction, pr = −pi is the momentum of the recoil 𝐴 − 1  nucleus, 𝐸𝑏 is the nucleon binding energy, 𝑆(pi, 𝐸𝑏) is the probability of finding a  nucleon in the nucleus with momentum pi and binding energy 𝐸𝑏, 𝑇𝑓 and 𝑇𝑟 are  the kinetic energies of the final state nucleon and 𝐴 − 1 system, 𝑑𝜎𝑓𝑟𝑒𝑒/𝑑Ω is the  lepton-bound nucleon elastic cross section, and 𝐾 is a known kinematic factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  This simple picture is complicated by nucleon reinteractions which change the  29  Figure 1-10: Quasielastic lepton-nucleus scattering where the outgoing nucleon does  not reinteract as it leaves the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  overlap integral between the initial and final states, and thus the cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  latter further changes the momentum and angle of the outgoing nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Thus,  to calculate even the simplest type of lepton-nucleus interaction, the momentum and  binding energy distribution of all nucleons in the nucleus need to be known, as well as  how the outgoing nucleon wave function is distorted by the nucleon-nucleus potential,  and how the outgoing nucleon kinematics is changed by final state interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Electron-nucleus scattering is much easier to understand than neutrino-nucleus  scattering for three reasons: (a) electron beams have a single, well-known, energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (b)  electron experiments typically have low statistical uncertainty because electron beams  have high flux, and electron-nucleus cross sections are far higher than their neutrino  counterparts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' and (c) electron cross sections are purely vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The strength of the  interaction is very different: 4𝜋𝛼2/𝑄4 for electrons versus 𝐺2  𝐹/(2𝜋) for neutrinos,  where the factor of 1/𝑄4 in the electron cross section is due to the exchanged boson  mass (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' massless photon) in the propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' When compensating for the factor  of 1/𝑄4, the shapes of the electron- and neutrino-nucleus cross sections are very  similar [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Nuclear medium effects such as nucleon motion, binding energy, two-  body currents, and final state interactions are similar or identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Therefore electron-  nucleus scattering can be used to constrain models of neutrino-nucleus scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Any  model which fails to accurately describe electron-nucleus (vector-vector) scattering  data cannot be used with confidence to simulate neutrino-nucleus (vector-vector +  axial-axial + vector-axial) interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Thus, models of the neutrino-nucleus cross  30  Final StateMuon  FinalStateProtonsection must be able to describe the more limited electron-nucleus cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  Thesis Scope  As detailed in section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3, the success of future precision neutrino oscillation experi-  ments depends on an unprecedented understanding of neutrino-nucleus interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Insufficient knowledge of either the energy reconstruction or the cross section will limit  the experimental precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In this thesis, the expertise in both neutrino and electron  cross-section modeling and analysis is leveraged in order to alleviate this insufficient  knowledge using three complementary approaches, illustrated in figure 1-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 1-11: Graphic illustration of the three complementary approaches used in this  thesis to improve our understanding of lepton-nucleus interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Namely, neutrino scattering data sets from the MicroBooNE detector at Fermi  National Laboratory were analysed (chapter 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The first measurements of exclusive  cross sections with a single proton and no pions detected in the final state were  reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These results were used to identify regions where modeling improvements  are required and specific nuclear effects can be studied in detail (chapter 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  exact same event topology was investigated using electron scattering data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For  the connection across neutrinos and electrons to be established, significant modeling  improvements took place to ensure a consistent modeling across the two particle  species in the commonly used GENIE event generator (chapter 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Building on those  improvements, the “Electrons-For-Neutrinos” analysis reported on the first use of wide  31  Neutrino Cross Sections  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' 125, 201803 (2020)  Event Generator  Electron  Modeling  Cross Sections  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' D 103, 113003 (2021)  Nature 599, 565-570 (2021)phase-space electron scattering data sets from the CLAS detector at Thomas Jefferson  National Laboratory (chapter 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The analysis identified significant shortcomings in  our lepton-nucleus interaction understanding by reporting cross sections as a function  of energy reconstruction methods and testing the validity of models commonly used  in neutrino oscillation analyses (chapter 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  32  Chapter 2  The MicroBooNE Experiment At  Fermi National Laboratory  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  Fermilab Neutrino Beamlines  Fermi National Laboratory (Fermilab) takes advantage of artificially produced neu-  trino beams in order to study neutrino oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The advantage of that choice is  the better control over the energy spectrum range and the neutrino flavor content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Thus, the spectrum can be tuned so that, given the distance of the detector from the  source, the experiment is located at an oscillation maximum or minimum, depending  on the objective of the intended measurement [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' On top of that, a narrower energy  spectrum can be obtained via the off-axis technique [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The neutrino beam production includes a well-defined procedure that is outlined  below for the Fermilab beamlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The Fermilab linac is used to accelerate a proton  beam up to a kinetic energy of 400 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Then the Booster synchrotron is used to  further accelerate the protons up to a kinetic energy of 8 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  For the Booster Neutrino Beam (BNB), a kicker extracts the protons in a single  turn and those protons are redirected to the relevant target hall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For all the other  experiments, the protons are redirected to the Main Injector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The latter is the high-  est energy US-based accelerator facility and the source of neutrinos from the Main  Injector (NuMI), the source of muons for the muon campus experiments, and the  33  forthcoming high intensity DUNE neutrino beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure 2-1 illustrates the series of  the relevant accelerator components and the different beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 2-1: The Fermilab accelerator complex showing the accelerator components  and the different beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The BNB takes advantage of the 8 GeV protons from the Booster synchrotron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  These protons collide with a beryllium target which is located within a pulsed elec-  tromagnet called horn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Such collisions lead to the production of mesons, mainly 𝜋± ,  𝐾±, and 𝐾0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The channel that dominates is p + Be → 𝜋+ + X, with X corresponding  to all the other hadrons produced out of the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' When operating in neutrino  mode, the horn focuses the positively charged particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The focused particle beam  then enters the decay pipe via a concrete-made collimator with a length of ≈ 2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In  the decay pipe, some of the particles decay, a process that results in the production  of neutrinos primarily via the channel 𝜋+ → 𝜇+ + 𝜈𝜇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The massive particles are  stopped using a beam absorber made of steel and concrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' On the other hand, neu-  trinos transverse the absorber and, at the end of this process, a neutrino-dominated  beam is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A graphic illustration of the process is shown in figure 2-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Apart from the desired 𝜈𝜇 beam, neutrinos can also be produced via the decay of  anti-muons coming out of the proton-target collision and the channel 𝜇+ → 𝑒+ +𝜈𝑒 +  ¯𝜈𝜇 [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Such interactions are the main sources of intrinsic 𝜈𝑒 contamination to the  main 𝜈𝜇 beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Similar hadron decays result in further contamination of the 𝜈𝜇 beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The main source of the ¯𝜈𝜇 contamination comes from 𝜋−’s which are not separated  34  Main Injector  Recycler Ring  Low-Energy  Neutrino  Experiments  High-Energy  Neutrino  Experiments  Muon  Delivery  Booster  Ring  Fixed-Target  Experiments,  Test Beam  Facility  Linac  Muon  lon Source  ExperimentsFigure 2-2: The ingredients for a neutrino beam include the accelerated protons,  the target, the magnetic horn, the decay pipe, and the absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted  from [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  from the main beam by the horn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure 2-3 illustrates the BNB energy spectrum  while operated in neutrino mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 2-3: The BNB neutrino flux prediction through the MicroBooNE detector for  𝜈𝜇, ¯𝜈𝜇, 𝜈𝑒, and ¯𝜈𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A TPC volume with dimensions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='56 m × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='33 m × 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='37 m is  assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The Fermilab-based NuMI neutrino beamline is used by the MINOS, Miner𝜈a,  ArgoNeuT, NO𝜈A, and MINOS+ experiments [47–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' On top of that, the Mini-  35  Neutrino BeamRecipe  Rock,  BeamAbsorber  DecayPipe  Neutrinos  MagneticFocusing  Horn  Muons,  Neutrinos  Target  PionmKaons  Protons2  10  9  v/POT/GeV/cm  MicroBooNE Simulation Preliminary  10-11  10-12  10  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0  Energy (GeV)BooNE and MicroBooNE experiments use the beamline for off-axis studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In this  case, neutrinos are produced via the collision of a 120 GeV beam on a graphite tar-  get.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Due to the large angle with respect to the NuMI beam dump, MicroBooNE  collects a significant number of low-energy neutrinos originating from kaons decay-  ing at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Using the off-axis NuMI beamline, MicroBooNE records events with a  narrower energy spectrum and with a higher electron-neutrino contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  The MicroBooNE Detector  The Micro Booster Neutrino Experiment (MicroBooNE) at Fermilab was proposed  to succeed MiniBooNE in order to resolve the Low Energy Excess (LEE) [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' It is  located ≈ 20 m away from the MiniBooNE detector, thus establishing almost the same  𝐿/𝐸 ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' MicroBooNE uses the same beamline as MiniBooNE, therefore the two  experiments share similar flux uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' At the core of the MicroBooNE detector,  there exists a Liquid Argon Time Projection Chamber (LArTPC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This detector  technology offers high spacial and momentum resolution, allowing an unprecedented  energy reconstruction accuracy and a precise particle identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Some neutrinos from the two beamlines enter the MicroBooNE LArTPC and  interact with the argon nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Out of these interactions, charged and neutral particles  are produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  When the charged particles transverse the liquid argon, the argon  atoms get excited and they further ionize the medium, a process that results in  the emission of ionization electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  In the presence of a strong electric field of  273 V/cm, the ionization electrons drift towards the anode plane, where three wire  planes are located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Apart from the ionization electrons, scintillation light is also  produced and is collected by 32 photomultiplier tubes (PMTs), which are located  behind the anode plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Both the TPC and the light collection system are embedded  withing a cylindrical 170 ton liquid argon cryostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure 2-4 shows the size of the  detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The readout electronics are embedded withing the liquid argon in order to sig-  nificantly reduce the electronic noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The analogue-to-digital conversion (ADC) and  36  Figure 2-4: Schematic illustration of the MicroBooNE detector and its dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure adapted from [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  PMT electronics are located outside the cryostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Given the baseline of ≈ 470 m, neutrinos in the two beamlines need to be charac-  terized by two angles in the MicroBooNE coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' As shown in figure 2-5,  𝜃 is the angle with respect to the z-direction along the beam direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Further-  more, 𝜑 defines the orientation with respect to the XY-plane, orthogonal to the beam  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 2-5: The MicroBooNE coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  A Cosmic Ray Tagger (CRT) system shown in figure 2-6 was installed in 2017 [56]  to improve on the rejection of cosmics, which constitute the greatest source of back-  grounds on MicroBooNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This detector sub-system consists of 73 scintillating modules  made of interleaved layers of scintillating plastic strips situated on the top, bottom,  37  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6 m  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3 m  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 mY  A  Z  Xand two sides parallel to the neutrino beam [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Based on simulation predictions  from CORSIKA [57] and GEANT [58], an estimated coverage of 85% is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The CRT installation aimed to improve on the identification and rejection of the  dominant cosmic background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Cosmic muons transversing the CRT result in the pro-  duction of scintillation light that can be reconstructed as hits on the CRT channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Such hits allow the identification of the cosmic-induced muon tracks with a time pre-  cision of ≈ 100 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The latter offers a complementary way to resolve the x-direction  ambiguity in the TPC reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 2-6: The design of CRT planes as part of the MicroBooNE detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Simula-  tion of cosmic rays crossing the CRT, the brown lines represent possible cosmic ray  trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' There are four CRT planes: top plane, bottom plane, pipe side plane  and feedthrough side plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The beam direction is along the z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted  from [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  Liquid Argon Time Projection Chambers  The TPC technology was introduced in the 1970s by David Nygren [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Carlo Rubbia  designed a LArTPC in 1977 using the same TPC principles as Nygren, but with liquid  argon instead of gas [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure 2-7 shows the working principle of a LArTPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  cuboid volume of the LArTPC is filled with ultra pure liquid argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The presence  38  Top  Top  Pipe Side  Top  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4m  Pipe Side  Top  PipeSide  Bottom  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4m  Extra  Feedthrough  Side  Bottom  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4ml  Botton  Pipe Side  Bottom  Cryostat(wrapping TPC)  LArTF Pit Ground  xof a high-voltage cathode on one side of the detector and a grounded anode on the  other side create a homogeneous electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 2-7: Working principle of a LArTPC detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  When a neutrino interaction takes place in the TPC, charged particles are pro-  duced in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Along their propagation path in LAr, such particles excite  and ionize the argon nuclei, a process that results in the emission of ionization elec-  trons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In the presence of a strong electric field, these ionization electrons drift towards  the anode plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' To ensure an electron drift time of 𝒪(ms) before recombination ef-  fects take place, a very small 𝑂2 contamination of 10 parts per trillion has to be  guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  On the anode plane, there exist three wire planes with a 3 mm spacing where the  clouds of the arriving ionization electrons create signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The first two planes corre-  spond to induction planes and are oriented ±60𝑜 with respect to the vertical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In  order to obtain 3D views of the particle trajectories, at least two planes with different  orientations are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The third collection plane removes the ambiguities due to  dead wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' On top of that, the calorimetric and tracking abilities are improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Bias  voltages are applied on the wire planes so that the two induction planes satisfy the  transparency condition outlined in [62,63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The condition requires that the drifting  39  Sense Wires  V  Vwire plane waveforms  Liquid Argon TPC  Charged Particles  Cathode  Plane  >0  Edrift  Y wire plane waveformselectrons pass the two induction planes and are fully captured on the third collection  plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The drifting charge induces a bipolar signal on the two induction planes and is  collected on the third one, where a unipolar signal is produced, as shown in figure 2-8  (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The signal area is proportional to the ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 2-8: (Left) bipolar (U and V induction planes) and unipolar (Y collection  plane) signal induction on the three MicroBooNE planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (Right) schematic view of the wire planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The vertical collection Y wires are shown  in pink, the induction U wires, angled at +60𝑜 are shown in blue and the induction  V wires, angled at -60𝑜, are shown in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The MicroBooNE field cage has a height of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3 m, a width of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6 m and a length  along the beam direction of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The liquid argon is kept at a pressure of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 atm,  a boiling temperature of 89 K, and a resulting density of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='38 g/cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The TPC is  merged into the liquid argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The active TPC volume is 86 tons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The cathode is kept  at -70 kV and the anode is grounded, which results into a homogeneous electric field of  273 V/cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' That translates into a drift velocity of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='14 m/ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Thus, the MicroBooNE  readout window is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The induction planes U and V are biased at -110 V and  the collection V plane is biased at 230 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The induction planes include 2400 wires and  the collection plane consists of 3456 wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The distance between both the different  planes and the plane wires is 3 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure 2-8 (right) shows the schematic view of  the wire planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Within the liquid argon, the wire signals are fed into a front end ASIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Inter-  mediate amplifiers further amplify the signal and pass it to the feed-through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Then,  outside the cryogenic environment, the signal is digitized by readout modules with a  frequency of 16 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The next step is to downsample the signal to lower frequencies  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15  [e-  sn  U plane  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  V plane  Y plane  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='05  50  40  30  20  10  0  10  Time [us]of 2 Hz [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  There exist three 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6 ms signal readout windows for each event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These frames  are further truncated to the range between -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 ms to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The hardware-defined  trigger 0 time is obtained from the accelerator division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Given the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3 ms MicroBooNE  drift time, the extra buffer of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 ms makes sure that there is enough time to isolate  the neutrino interaction from the cosmic rays that arrive close to the neutrino beam  trigger time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 2-9: Entry/exit points of cosmic muon tracks with a signal from a muon  counter located outside of the cryostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In the absence of space charge effects, the  points should be located along the TPC boundaries indicated by the dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure adapted from [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The ion drift velocity in liquid argon is ≈ 5 mm/s, orders of magnitude smaller  than the electron one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Thus, the argon ions result in the build-up of positive charge in  the LArTPC for minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' On top of that, the continuous interaction of cosmic rays in  the TPC at ≈ 5 kHz also results in a continuous build-up of positively charged argon  ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The existence of this positive charge leads to a distortion of the homogeneous  electric field within the TPC, as shown in figure 2-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This distortion is referred to as  41  MicroBooNE  y100  50  0  50  100  _150  0  50  100  150  200  250  x [cm]“space charge effect” (SCE) [66], which leads to a displacement of the reconstructed  signal ionization source by up to 𝒪(10 cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  Scintillation Light  The scintillation light production and propagation is almost instantaneous and of the  order of 𝒪(ns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Thus, collecting the emitted scintillation light is crucial to identify  the time that the event took place and to determine the x-position along the drift  direction in the TPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' When charged particles transverse the liquid argon, scintillation  light is produced that results in the deposition of ≈ 104 𝛾 / MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This light can be  produced via two mechanisms, shown in figure 2-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 2-10: The production of scintillation light in liquid argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A charged particle  can either excite or ionise the argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The first mechanism is the self-trapped exciton luminescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In that case, charged  particles transverse the liquid argon and they leave some of the argon atoms in an  excited state called excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These states are molecules with another argon atom  with a short lifetime and are called dimers or excimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Roughly 65% of them are in  a singlet state 1Σ𝑢 and the remaining ones are in a triplet state 3Σ𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  With the second mechanism, the charged particles ionize the argon atoms and  42  quenching by impurity  excited state  singlet  65%  scintillation light  35%  Ar  excimer states  50%  50%  absorption by impurity  ionisation  recombination  tripletthat results in the creation of free electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  These electrons recombine with the  positive argon ions, a process that also creates excited dimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' With this mechanism,  the probability of creating either singlets or triplets is equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The singlet states result in the emission of fast scintillation light with a decay  time of ≈ 6 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The triplet states result in a slow component and in a decay time of  ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5 𝜇s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Both the fast and the slow components have a peak wavelength at 128 nm  in the Vacuum Ultra-Violet (VUV) region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Both states have an energy minimum  which is equivalent to a distance between the atoms of ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8 Å [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The liquid argon  inter-atom separation is ≈ 4 Å, which is greater than the one that corresponds to  the excimer energy minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Thus, liquid argon is transparent to its own light and,  therefore, the light can be detected over long distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The second mechanism is related to the recombination luminescence and relies on  the free electron and argon ion local density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Thus, recombination effects are stronger  for particles with higher energy deposition per unit length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This implies a particle-  type dependence of the energy deposition in the detector to the light-equivalent pro-  duction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' That also means that the scintillation light yield depends on the choice for  the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A strong electric field will reduce the recombination effects and,  therefore, the recombination-induced luminescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The amount of scintillation light  can be further reduced due to impurities that result into quenching and absorption  effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' With that in mind, the liquid argon purity in MicroBooNE is monitored to  ensure an absorption length greater than the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5 m-long TPC width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The MicroBooNE light detection system consists of 32 8-inch Hamamatsu cryo-  genic PMTs located behind the anode plane, shown in figure 2-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A Tetra-Phenyl  Butadiene (TPB) coated acrylic plate is located in front of the PMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The TPB con-  verts the 128 nm light argon scintillation light to 425 nm visible light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The tube is  surrounded by mu-metal to shield the electronics against any magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  PMTs operate at a voltage of 1300 V and have a gain of 107 [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The PMT signal is  sent to a splitter board, where it gets pre-amplified and digitized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The signals have a  60 ns rise time, much smaller than the spill windows for the BNB (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6 𝜇s) and NuMI  (10 𝜇s) beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  43  Figure 2-11: The MicroBooNE light collection system with the 32 PMTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure  adapted from [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  Hardware And Software Triggers  Even when accelerator-induced neutrino beams transverse the MicroBooNE detec-  tor, it is highly unlikely that any neutrino will interact due to the extremely small  neutrino-nucleus cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Each BNB bunch includes ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 × 1012 protons and  the TPC surface area is ≈ 60 × 103 cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' That yields a neutrino interaction every ≈  500 bunches [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In order to significantly reduce the amount of data that is stored on  tape, a minimal amount of optical activity is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Regardless, background events  with light activity within the beam spill arrival time are still the ones that dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 2-12 shows the differences between two such events [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The first one (left)  corresponds to pure cosmic activity that arrives in coincidence with the beam spill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The second one (right) includes a neutrino interaction of interest, although there is  still a significant cosmic contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Cosmic-induced background events are accounted for by recording events when  the neutrino beam is off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' That leads to the existence of two trigger types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The first  one, referred to as “Beam On”, corresponds to data samples recorded when the beam  is on and the light system yielded a signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The second one, referred to as “Beam Off”  or “External BNB” (ExtBNB), uses the same configuration as in the Beam On case,  but those data sets are recorded when the beam is off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The MicroBooNE readout system is triggered with the arrival of a signal [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  signal might originate from the BNB/NuMI accelerator clocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Alternatively, there  44  Figure 2-12: (Left) cosmic-induced event that was stored because of the coincidence  of a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6 𝜇s accelerator BNB signal and light detected by the PMTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (Right) neutrino-  induced event where the light was coming from a neutrino interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted  from [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  is a function generator in the trigger rack producing pulses at a fixed frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  latter is used to record Beam Off events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The trigger board sends a signal to all the  readout crates to start recording data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' On top of that, the trigger keeps track of the  trigger type and the time that the signal was received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The accelerator-based signals  are produced in couples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The former (early signal) vetos Beam Off triggers just before  the beam triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' That process aims to avoid any trigger overlap that might result  in the reduction of the exposure to the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The latter signal is the one used to  trigger the readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The MicroBooNE TPC readout is completely unbiased, thus all  the time ticks are stored and the readout is not zero-suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The PMT readout  is biased though, with data being stored during specific time intervals determined by  the “discriminators”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  There exist two PMT data-taking configurations that differ by their duration and  by the suppression level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The duration is frequently expressed in tick units, with each  tick corresponding to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='625 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For the BNB triggering, the beam discriminator  starts the data recording simultaneously for all PMTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' That happens by replicating  the trigger signal and by redirecting it to all the PMT boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The duration of this  beam window is 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 𝜇s or 1500 optical ticks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' On average, the neutrino arrival time  is ≈ 4 𝜇s after the opening of the beam window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The cosmic discriminators span a  range outside the 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 𝜇s window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The ultimate goal of the cosmic discriminators is to  45  "Cosmic Rays Only"  "Neutrino+CosmicRays  A  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6μs beam  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3 ms drift time  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6 μs beam  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3 ms drift time  spill time  spill timesuppress the amount of data that is recorded over a long time interval, once the signal  for a triggered event has arrived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The duration of the cosmic window is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8 ms and  spans the range of [-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6,+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2] ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Only waveforms with more than 130 ADC counts  are stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' That number corresponds to ≈ 7 photo-electrons (PE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The accelerator  signals are meticulously timed so that neutrinos originating from the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6 𝜇s BNB spill  arrive in MicroBooNE during this time window of 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 𝜇s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' When the beamline is fully  operational, that takes place ≈ 5 times/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The majority of the triggered events based  on the hardware-driven signal do not include neutrino interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Thus, a software  trigger is further applied to determine whether an event is a neutrino candidate or  not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This trigger uses PMT optical waveforms and searches for light activity within  the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6 𝜇s BNB beam-spill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' That action takes place after the TPC data has been sent  to the DAQ crates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  Optical Event Reconstruction  In order to perform a MicroBooNE data analysis, the low-level PMT signals have  to be converted to high-level data products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  That is achieved by using the raw  PMT waveforms as an input quantity and, based on these, flashes are reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Such flashes are indicative of optical activity in the TPC recorded by multiple PMTs  simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This light activity is collected by the 32 8-inch PMTs oriented as  shown in figure 2-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The neutrino-induced light activity results in a higher PE yield  compared to the cosmic-induced one, as shown in figure 2-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 2-13: Optically reconstructed flash object recorded by the MicroBooNE PMT  light collection system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The dark orange regions represent a higher PE yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure  adapted from [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The PMTs have a low (≈ 20 ADC/PE) and a high (≈ 2 ADC/PE) gain read-  46  80  OHO  CH1  CH7  CHB  CH13  CH14  CH19  CH20  CH25  CH26  60  40  20  CH3  GHO  OLHO  CH15  CH16  CH21  CH22  CH28  0  CH4  20  CHG  LLHO  CH12  CH17  CH18  CH23  CH24  40  CH3  60  80  500  400  300  200  100  100  200  300  400  500out [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The optical reconstruction merges the two streams into a corrected waveform  which corrects saturated high-gain pulses based on the information obtained from the  low-gain pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Depending on the discriminator type, a different baseline estimation  is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The cosmic discriminator uses a constant baseline, called pedestal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The  beam discriminator uses a time dependent baseline estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This is a more accurate  estimation and addresses potential overestimations or underestimations in the signal  baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Once the pedestal has been identified, the waveform ADC counts are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Those pulses above threshold are identified and propagated to the next stage for the  “hit” reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These pulses result in the creation of data products based on this  optical hit reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Each optical hit refers to optical PMT activity, namely  the number of produced PEs and the event time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The optical hits are clustered and  the PMT PE production is summed in order to reconstruct flashes [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 2-14: Neutrino-induced tracks (black) are matched to the corresponding light  signals collected by PMTs (red circles) and are clearly separated from the cosmic-  induced ones (dimmed color tracks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7  TPC Signal Processing  While drifting towards the anode plane, the ionization electrons repel each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  That results in a diffused signal arriving on the anode plane and the level of the  diffusion depends on the position that the interaction took place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Those drifting  electrons result in the induction of current on the neighboring wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Therefore,  waveforms produced on a specific wire might have an effect on those produced on a  47  neighboring wire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The current induced on the wires is amplified and shaped by the  ASICs located within the liquid argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The objective of the noise filtering and signal processing on MicroBooNE is to  convert those raw digitized waveforms into the number of ionization electrons passing  through a specific wire plane at a given time [73,74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For that to be achieved, the first  step is the application of noise filters to remove the external noise and the electronics-  induced one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The major sources of external noise originate from the TPC drift high-  voltage power supply and the low-voltage regulators for the front-end ASICs [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Then, the application of a deconvolution of the digitized TPC wire signals follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  That takes place in two dimensions, with the first one being over time and the second  one being the effect across multiple wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A region of interest (ROI) is identified  based on the deconvoluted charge distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The ionization charge is obtained  with a linear baseline subtraction within the start/end bins of the ROI window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' At  the final step, the processed signals accounting for the number of electrons on a given  wire at a certain time are used as input to the MicroBooNE event reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The signals are calibrated before the conversion to deposited energy is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 2-15 illustrates the necessity of signal processing before any reconstruction is  enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  Pandora Reconstruction Framework  The processed signals from the previous stage are used as the input for higher level re-  construction of objects such as vertices, tracks, and showers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In order to reconstruct  objects in the TPC, the Pandora reconstruction framework is deployed [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Pan-  dora uses pattern-recognition algorithms, along with the use of multiple algorithms  completing specific tasks for a given topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The waveforms obtained with 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8 ms  window are used as an input to the reconstruction framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The first step includes fitting the processed waveforms with Gaussian distributions  to each peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This fitting process results into the creation of a 2D hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Then, Pandora-  Cosmic is a track-focused selection that aims to tag the cosmic muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The selection  48  Figure 2-15: Candidate neutrino event display from MicroBooNE data on one of  the induction planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (a) The raw waveform image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (b) The image after noise-  filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (c) The image after 2D deconvolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The image quality near the neutrino  interaction vertex significantly improves after the 2D deconvolution and the latter  leads to improvements in the pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  results in the creation of a cosmic-free hit collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' PandoraNu identifies neutrino  interaction vertices and uses them to reconstruct tracks and showers originating from  the vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A parent neutrino particle is defined and the reconstructed objects are  added as daughter particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Creating a “slice” is the first step that Pandora performs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  A slice is defined as a collection of reconstructed particles originating from the same  interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For the creation of the slices, the PandoraCosmic algorithm is first ran  over all the hits in order to identify the cosmic-induced muon tracks and the associated  𝛿-rays and Michel electrons under a cosmic hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The obvious cosmic activity  of through-going muon tracks is identified based on the geometric information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  remaining hits are used by the PandoraNu algorithm and objects are reconstructed  under the neutrino hypothesis, creating a slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Each slice is reconstructed using both  49  Raw  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  AfterNoiseFiltering  After2-DDeconvolution  750(a)  40000  (b)  (c)  MicroBooNE  30000  600  20000  10000  [sr  450  m  Time  0  300  10000  20000  150  30000  40000  0  0  20  40  60  80  0  20  40  60  80  0  20  40  60  80  Wire[3mmspacing]hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For the interactions to be reconstructed in a three-dimensional space,  Pandora requires information from at least two wire planes [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The 2D hits are  clustered on each wire plane and for each slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A collection of 3D candidate vertices  is produced by identifying locations that project to the same points of the 2D clus-  ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' All the candidate vertices are propagated into a Support Vector Machine (SVM)  selection and the candidate with the highest score is isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The cluster matching  algorithms are ran around this candidate vertex on each plane and are compared to  improve the matching of the reconstructed objects [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' With this process, a collec-  tion of reconstructed Particle Flow Particles (PFParticles) is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Such a  PFParticle is created by combining 2D cluster objects on the three wire planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Each  one of those PFParticles is associated with a vertex location and has a collection of 3D  points, which contain the charge details from the relevant 2D hits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These 3D points  are referred to as SpacePoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The reconstructed PFParticles in the neutrino slice  are organized with a hierarchical structure based on parent-daughter assocciations,  as shown in figure 2-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 2-16: Illustration of the hierarchical structure of particles reconstructed for a  simulated charged current 𝜈𝜇 event in MicroBooNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The interaction includes a muon,  proton and charged pion in the visible final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  A candidate neutrino PFParticle is assigned at the very top of the hierarchy  by the PandoraNu algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' That candidate neutrino will be having at least one  50  Track (p), daughter of primary p  Track (p),primary daughter ofvμ  Shower (e+),daughterofprimary  Track (π+),primary daughter of vμ  w,wireposition  Parent Vμ  interactionvertex  Track (μ), primary daughter of vμ  x, drift positiondaughter PFParticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These daughter particles are assigned a score that classifies  them as either-track-like or shower-like objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A Support Vector Machine uses the  collection of hits to determine the nature of reconstructed object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Track-like objects  have a score close to 1 and shower-like objects score closer to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Based on that score,  a shower- or a track-like data product is constructed for each particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  In the case of a track-like classification, Pandora uses a linear fit, described in detail  in [75], and returns the direction and the position of each point across the particle  trajectory in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For each one of the points, the charge deposition dQ/dx and the  residual range - the distance from the end of the track - are stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This approach  allows accurate measurements of dx that might include deflections and displacements  due to space charge effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For track-like objects, dQ/dx is converted to dE/dx using  the inverse Modified Box model [76], as shown in figure 2-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The advantage of this  model is that the non-linear dependence of the local density of ions is taken into  consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 2-17: Illustration of the measured dQ/dx vs dE/dx distribution with the mod-  ified recombination model in the MC simulation with the ArgoNeuT parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure adapted from [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  51  MicroBooNE Simulation  dQ/dx (e/cm)  200X103  180  100(  160  140  800  120  600  100  80  400  60  40  200  20  T  Modified box with ArgoNeuT parametrization  2  4  6  10  12  14161820  0  8  dE/dx (MeV/cm)For shower-like objects, Pandora creates a 3D cone along the hit collection with  a fixed 3D orientation, solid angle and length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The shower energy is obtained and  calibrated using the same techniques as the ones used on the 𝜋0 reconstruction [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Furthermore, showers are fitted with a Kalman filter [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' With this fit, hits that are  longitudinally or transversely displaced from the main shower cone are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This  fitting process returns a track-like object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Therefore, the calorimetric tools mentioned  in the previous paragraph become available for shower-like objects too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9  Cosmic Overlay Simulation  For the purposes of the oscillation and cross section analyses on MicroBooNE, neutrino-  induced interactions in our detector need to be induced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' However, MicroBooNE is  a surface detector dominated by the cosmic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In order to simulate this cos-  mic contamination as accurately as possible, real cosmic events collected with the  unbiased trigger are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This trigger stores events outside the beam-related trig-  ger windows and does not demand the existence of any optical activity in a specific  part of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Such events are overlaid on top of GENIE simulated neutrino  interactions [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These resulting samples are referred to as “cosmic overlays” and an  example event display is shown in figure 2-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 2-18: MicroBooNE event display of an event in a cosmic overlay sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A  simulated neutrino event (orange box) is overlaid on top of cosmic events (blue box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  52  μBooNE  BNBRun1Simulation  5cm  Run 7021Subrun 857Event 42856For the simulated part of the cosmic overlays, the reconstructed information is  “backtracked” to the underlying truth-level information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' That is achieved by associ-  ating the hits on each plane to the GEANT particles that resulted in the production  of these hits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Thus, it is feasible to relate reconstructed PFParrticles to the under-  lying simulated interaction products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' It is further possible to identify the amount of  charge originating from the cosmic part of the overlay samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Reconstructed track-  and shower-like objects are matched to a simulated object when they have more hits  in common than any of the other simulated particles or cosmic tracks in a given event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Motivated by the analyzer’s involvement in the development and validation of  these samples, the analyses presented in sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 were the first ones  to adopt the aforementioned overlay technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Due to the success in accurately  describing the cosmic background, this technique is currently used as the default  simulation option across the MicroBooNE collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  53  54  Chapter 3  MicroBooNE Quasielastic-like  Cross-Section Results  [Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' 125, 201803 (2020)]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  First Measurement of Differential Charged Cur-  rent Quasielastic-like Scattering Cross Sections  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  Quasielastic-like Neutrino Data Analysis  As outlined in section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4, understanding the interaction of neutrinos with argon nuclei  is of particular importance as a growing number of neutrino oscillation experiments  employ liquid argon time projector chamber (LArTPC) neutrino detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Exper-  imentally, the energy of interacting neutrinos is determined from the measured mo-  menta of particles that are emitted following the neutrino interaction in the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Many accelerator-based oscillation studies focus on measurements of charged-current  (CC) neutrino-nucleon quasielastic (QE) scattering interactions [80–89], where the  neutrino removes a single intact nucleon from the nucleus without producing any  additional particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  This choice is guided by the fact that CCQE reactions can  be reasonably well approximated as two-body interactions, and their experimental  signature of a correlated muon-proton pair is relatively straightforward to measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Therefore, precise measurements of CCQE processes are expected to allow precise  reconstruction of neutrino energies with discovery-level accuracy [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  55  A working definition for identifying CCQE interactions in experimental measure-  ments requires the identification of a neutrino interaction vertex with an outgoing  lepton, exactly one outgoing proton, and no additional particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These events are  referred to herein as “CCQE-like”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  This definition can include contributions from  non-CCQE interactions that lead to the production of additional particles that are  absent from the final state due to nuclear effects, such as pion absorption, or have  momenta that are below the experimental detection threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Pre-existing data on  neutrino CCQE-like interactions came from experiments using various energies and  target nuclei [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These primarily included measurements of CCQE-like muon neu-  trino (𝜈𝜇) cross sections for interactions where a muon and no pions were detected,  with [86–89] and without [80–85] requiring the additional detection of a proton in  the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' While most relevant for LArTPC based oscillation experiments, no  measurements of CCQE-like cross sections on argon with the detection of a proton in  the final state existed until 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  This analysis presents the first measurement of exclusive CCQE-like neutrino-  argon interaction cross sections, measured using the MicroBooNE LArTPC [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Our  data serve as the first study of exclusive CCQE-like differential cross sections on ar-  gon as well as a benchmark for theoretical models of 𝜈𝜇-Ar interactions, which are  key for performing a precise extraction of oscillation parameters by future LArTPC  oscillation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We focused on a specific subset of CCQE-like interactions, de-  noted here as CC1p0𝜋, where the contribution of CCQE interactions is enhanced [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  These include charged-current 𝜈𝜇-Ar scattering events with a detected muon and ex-  actly one proton, with momenta greater than 100 MeV/𝑐 and 300 MeV/𝑐, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The measured muon-proton pairs were required to be co-planar with small missing  transverse momentum and minimal residual activity near the interaction vertex that  was not associated with the measured muon or proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For these CC1p0𝜋 events,  the flux-integrated 𝜈𝜇-Ar total and differential cross sections in muon and proton mo-  mentum and angle were extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The relevant cross sections were further reported  as a function of the calorimetric measured energy and the reconstructed momentum  transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  56  The measurement used data from the MicroBooNE LArTPC detector [93], which  is the first of a series of LArTPCs to be used for precision oscillation measure-  ments [94–99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' As described in chapter 2, the MicroBooNE detector has an active  mass of 85 tons and is located along the Booster Neutrino Beam (BNB) at Fermilab,  463 m downstream from the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The BNB energy spectrum extends to 2 GeV and  peaks around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7 GeV [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A neutrino is detected by its interaction with an argon  nucleus in the LArTPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The secondary charged particles produced in the interac-  tion travel through the liquid argon, leaving a trail of ionization electrons that drift  horizontally and transverse to the neutrino beam direction in an electric field of 273  V/cm, to a system of three anode wire planes located 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5 m from the cathode plane  detailed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The Pandora tracking package [75] described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8 is  used to form individual particle tracks from the measured ionization signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Particle  momenta are determined from the measured track length for protons and multiple  Coulomb scattering pattern for muons [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The analysis presented here was performed on data collected from the BNB beam,  with an exposure of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='59 × 1019 protons on target (POT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' At nominal running con-  ditions, one neutrino interaction is expected in ≈ 500 BNB beam spills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The trigger,  based on the scintillation light detected by the 32 photomultiplier tubes (PMTs),  increases the fraction of recorded spills with a neutrino interaction to ≈ 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Appli-  cation of additional software selection further rejects background events, mostly from  cosmic muons, to provide a sample that contains a neutrino interaction in ≈ 15% of  the selected spills [74,101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A CCQE-like event selection, further cosmic rejection and  neutrino-induced background rejection cuts, described in detail in [55], are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Muon-proton pair candidates are identified by requiring two tracks with a common  vertex and an energy deposition profile consistent with a proton and a muon [102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Further cuts on the track pair opening angle (|∆𝜃𝜇,𝑝 − 90∘| < 55∘) and the muon and  proton track lengths (𝑙𝜇 > 𝑙𝑝) reduce the cosmic background rate to less than 1% [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The selected CC1p0𝜋 event definition includes events with any number of protons  with momenta below 300 MeV/𝑐, neutrons at any momenta, and charged pions with  momentum lower than 70 MeV/𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The minimal proton momentum requirement of  57  300 MeV/𝑐 is guided by its stopping range in LAr and corresponds to five wire pitches  in the TPC, to ensure an efficient particle identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  To avoid contributions from cosmic tracks, our CC1p0𝜋 selection considered only  pairs of tracks with a fully-contained proton candidate, and a fully or partially con-  tained muon candidate in the fiducial volume of the MicroBooNE detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The  fiducial volume is defined by 3 < 𝑥 < 253 cm, -110 < 𝑦 < 110 cm, and 5 < 𝑧 <  1031 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The 𝑥 axis points along the negative drift direction with 0 cm placed at  the anode plane, 𝑦 points vertically upward with 0 cm at the center of the detector,  and 𝑧 points along the direction of the beam, with 0 cm at the upstream edge of the  detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Tracks are fully contained if both the start point and end point are within  this volume, and partially contained if only the start point is within this volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  We limited our analysis to a phase space region where the detector response to  our signal is well understood and its effective detection efficiency is higher than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  This corresponds to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 < 𝑝𝜇 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5 GeV/𝑐, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3 < 𝑝𝑝 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0 GeV/𝑐, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='75 < cos 𝜃𝜇 <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='95, and cos 𝜃𝑝 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Additional kinematical selections were used to enhance the  contribution of CCQE interactions in our CC1p0𝜋 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These include requiring  that the measured muon-proton pairs be coplanar (|∆𝜑𝜇,𝑝−180∘| < 35∘) relative to the  beam axis, have small missing transverse momentum relative to the beam direction  (𝑝𝑇 = |⃗𝑝 𝜇  𝑇 + ⃗𝑝 𝑝  𝑇| < 350 MeV/𝑐), and have a small energy deposition around the  interaction vertex that is not associated with the muon or proton tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This event  selection results in a CCQE dominated sample, where table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 shows the fractional  contribution for each interaction channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure 3-1 shows the relevant interaction  breakdown for the entire sample of selected events as a function of cos𝜃𝜇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The same  nominal MC sample was also used to compute the purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  After the application of the event selection requirement on the data sample, we  retained 410 CC1p0𝜋 candidate events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' It is estimated that our CC1p0𝜋 CCQE-  like event selection purity equals ≈ 84% [55], with ≈ 81% of the measured events  originating from an underlying CCQE interaction as defined by the GENIE event  generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The efficiency for detecting CC1p0𝜋 events, out of all generated CC1p0𝜋  with an interaction vertex within our fiducial volume, was estimated using our Monte  58  Carlo (MC) simulation and equals ≈ 20% [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We note that this efficiency includes  acceptance effects, as the typical LArTPC efficiency for reconstructing a contained  high-momentum proton or muon track is grater than ≈ 90% [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1: Interaction breakdown after the application of our selection cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Interaction  Mode  Fractional  Contribution (%)  QE  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  MEC  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9  RES  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  DIS  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0  cos(  )  0  500  1000  1500  2000  2500  3000  Event Count  QE  MEC  RES  DIS  Figure 3-1: Interaction breakdown of the cos𝜃𝜇 plot illustrating the dominance of  CCQE interactions after the application of our selection cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Single differential cross sections are reported in measured proton and muon kine-  matics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The differential cross section is given by:  d𝜎  d𝑋𝑛  =  𝑁 on  𝑛 − 𝑁 off  𝑛 − 𝐵𝑛  𝜖𝑛 · Φ𝜈 · 𝑁target · ∆𝑝  𝑛,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1)  59  where 𝑋 stands for the kinematical variable that the cross section is differential  in and 𝑛 marks the cross-section bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In each bin 𝑛, 𝑁 on  𝑛 is the number of measured  events when the beam is on, 𝑁 off  𝑛 is the number of measured events when the beam is  off and cosmic-induced background events are collected, 𝐵𝑛 is the non-CC1p0𝜋 beam-  related background estimated from MC, 𝑁target is the number of scattering nuclei, Φ𝜈  is the integrated incoming neutrino flux, ∆𝜇  𝑛 and ∆𝑝  𝑛 are the differential bin widths,  and 𝜖𝑛 is the effective particle detection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We note that the high cos(𝜃𝜇)  bin has large beam-related background corresponding 𝐵𝑛 in equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1, which is  estimated using the GENIE v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 based MC simulation and is presented in figure 3-  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0  cos(  )  0  50  100  150  200  250  300  350  NonCC1p Bkg Event Count  QE  MEC  RES  DIS  Figure 3-2: Interaction breakdown of the cos𝜃𝜇 plot illustrating the dominance of RES  interactions after the application of our selection cuts for the non-CC1p0𝜋 background  part of the MC sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  As the detection efficiency is a multidimensional function of the interaction vertex  and the particle momentum and direction, the data were binned in three-dimensional  momentum, in-plane, and out-of-place angle bins with the effective detection efficiency  calculated for each such bin separately and integrated over the interaction vertex in  the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The efficiency was extracted based on simulation and is defined as the  60  ratio of the number of reconstructed CC1p0𝜋 events to the number of true generated  CC1p0𝜋 events with a vertex inside our fiducial volume in bin n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This procedure  accounts for bin migration effects such that cross-sections are obtained as a function  of true kinematical variables, as opposed to experimentally reconstructed ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The proton and muon efficiencies were extracted independently of each other, such  that, when the cross-section is differential in muon kinematics, the proton kinematics  is integrated over and vise versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This is done due to the limited data and simulation  statistics and is justified since the proton and muon efficiencies are largely indepen-  dent in the region of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The effect of residual correlations is accounted for in the  systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We further note that the missing transverse momentum re-  quirement increases the sensitivity of our efficiency corrections to the meson exchange  current (MEC) and final state interaction (FSI) models used in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We  accounted for the model sensitivity in our systematic studies detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The neutrino flux was predicted using the flux simulation of the MiniBooNE col-  laboration that used the same beamline [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We accounted for the small distance  between MiniBooNE and MicroBooNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Neutrino cross section modeling uncertainties  were estimated using the GENIE framework of event reweighting [103,104] with its  standard reweighting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For both cross section and flux systematics, we  use a multisim technique [105], which consists of generating many MC replicas, each  one called a “universe”, where model parameters are varied within their uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Each universe represents a different reweighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The simultaneous reweighting of all  model parameters allows the correct treatment of their correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  A different model is followed for detector model systematic uncertainties, that  are dominated by individual detector parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Unisim samples [105] were gener-  ated, where one detector parameter was varied each time by 1𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We then examined  the impact of each parameter variation on the extracted cross sections, by obtain-  ing the differences with respect to the central value on a bin-by-bin basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We note  that the detection efficiency used for the cross section extraction is re-evaluated for  each variation separately, including bin migration corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This procedure there-  fore accounted for the systematic uncertainty in these corrections due to both the  61  cross-section and detector response modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We then defined the total detector  1𝜎 systematic uncertainty by summing in quadrature the effect of each individual  variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  A dedicated MC simulation was used to estimate possible background from events  in which a neutrino interacts outside the MicroBooNE cryostat but produce parti-  cles that enter the TPC and pass the event selection cuts [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' No such events were  found in that study, which is also supported by our observation that the 𝑧-vertex  distributions for the measured events follows a uniform distribution, as can be seen  in figure 3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The measured 𝑧-vertex distribution, after the beam related MC back-  ground has been subtracted, does not show an excess at low-𝑧, which indicates that  background events from interactions upstream of the detector are not accidentally  entering our selection, which would show up as a small-𝑧 enhancement in our vertex  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The deficit at 𝑧 = 700 cm is due to dead wires in our detector and its  effect has been incorporated in our simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  0  200  400  600  800  1000  vertex z [cm]  0  10  20  30  40  Beam On Events  0  200  400  600  800  1000  vertex z [cm]  0  25  50  75  100  125  150  175  Efficiency Corrected Beam On Events  Figure 3-3: Vertex 𝑧 distribution for the measured events, after the beam related MC  background has been subtracted, before (left) and after (right) detection efficiency  corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' No small-𝑧 enhancement is observed and, with efficiency corrections, the  measured distribution is consistent with that of a uniform neutrino interaction vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The MC simulation used to estimate the backgrounds and effective efficiency con-  tains real cosmic data overlayed onto a neutrino interaction simulation that uses  GENIE [103,104] to simulate both the signal events and the beam backgrounds [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  For the simulated portion, the particle propagation is based on GEANT4 [58], while  62  the simulation of the MicroBooNE detector is performed in the LArSoft frame-  work [106,107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The beam-related background subtracted from the candidate CC1p0𝜋  events in the data sample is simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  Quasielastic-like Cross-Section Results  Figure 3-4 shows the flux integrated single differential CC1p0𝜋 cross section as a func-  tion of the cosine of the measured muon scattering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The data were compared to  several theoretical calculations and to our GENIE-based MC prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This predic-  tion is the result of analyzing a sample of MC events produced using our “nominal”  GENIE model and propagated through the full detector simulation in the same way as  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This model (GENIE v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2) [103,104] treats the nucleus as a the Bodek-Ritchie  Fermi Gas (RFG), used the Llewellyn-Smith CCQE scattering prescription [108], the  empirical MEC model [109], the Rein-Sehgal resonance (RES) model, the coherent  (COH) scattering model [110], and a data driven FSI model denoted as “hA” [111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  In addition, theoretical predictions by several other event generators are shown  at the cross-section level without any detector effects [112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These include GENIE  v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 and v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6 [103, 104], NuWro 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 [113], and NEUT v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0 [114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The  agreement between the “nominal” GENIE calculation (v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2) and the MC predic-  tion constitutes a closure test for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The other generators all improve on  GENIE v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 by using updated nuclear interaction models, among which is the use  of a Local Fermi Gas model (LFG) [37] and Random Phase Approximation (RPA)  correction [115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' GENIE v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6 and NEUT also include Coulomb corrections for the  outgoing muon [116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The theoretical models implemented in these event generators  include free parameters that are typically fit to data, with different generators using  different data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We also consider the GiBUU 2019 [117] event generator which  fundamentally differs from the others due to its use of a transport equation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  A brief discussion of the underlying model configuration used in the different event  generator predictions included in this analysis is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  GENIE Nominal: Uses the aforementioned GENIE v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  63  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0  cos(  )  0  3  6  9  12  15  18  d  dcos(  ) [10 38cm2  Ar ]  Simulation  MicroBooNE Data   4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='59×1019 POT  MC  GENIE Nominal  GENIE v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  NuWro 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  NEUT  v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0  GiBUU 2019  Figure 3-4: The flux integrated single differential CC1p0𝜋 cross sections as a function  of the cosine of the measured muon scattering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Inner and outer error bars show  the statistical and total (statistical and systematic) uncertainty at the 1𝜎, or 68%,  confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Colored lines show the results of theoretical absolute cross section  calculations using different event generators (without passing through a detector sim-  ulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The blue band shows the extracted cross section obtained from analyzing  MC events propagated through our full detector simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The width of the band  denotes the simulation statistical uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  NuWro 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 [113]: Using the LFG ground state model [37], the Llewellyn-  Smith CCQE scattering prescription [108], the Transverse Enhancement model  for two–body currents [118], the Adler-Rarita-Schwinger formalism to calculate  the ∆ resonance explicitly [119], the BS COH [120] scattering model and an  intranuclear cascade model for FSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  NEUT v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0 [114]: Using the LFG ground state model [37], the Nieves CCQE  scattering prescription [121], the Nieves MEC model [122], the BS RES [123–  126] and Rein-Sehgal COH [110] scattering models, and FSI with Oset medium  correction for pions [103,104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  GiBUU 2019: Using somewhat similar models, but unlike other generators,  those are implemented in a coherent way, by solving the Boltzmann-Uehling-  64  Uhlenbeck transport equation [117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The models include: Local Fermi Gas  model [37], standard CCQE expression [127], empirical MEC model and a dedi-  cated spin dependent resonance amplitude calculation following the MAID anal-  ysis [128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The DIS model is as in PYTHIA [129] and the FSI treatment is  different as the hadrons propagate through the residual nucleus in a nuclear  potential which is consistent with the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  As can be seen in figure 3-4, all models are in overall good agreement with our  data, except for the highest cos(𝜃𝜇) bin with cos(𝜃𝜇)> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8, where the measured cross  section is significantly lower than the theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This discrepancy cannot  be explained by the systematic uncertainties and is therefore indicative of an issue  with the theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Specifically, high cos(𝜃𝜇) correspond to low momentum  transfer events which were previously observed to not be well reproduced by theory  in inclusive reactions [84,85] and is now also seen in exclusive reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  As the differential cross sections in proton kinematics and muon momentum in-  clude contributions from all muon scattering angles, their agreement with the the-  oretical calculation is affected by this disagreement in the forward muon scattering  angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Therefore, for the results presented below, we repeated the cross-section ex-  traction exercise twice, where the first time we included all the events that satisfy  our selection criteria, and the second time we excluded those events with cos(𝜃𝜇)>  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The corresponding integrated measured CC1p0𝜋 cross sections are summarized  in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The same table also lists the 𝜒2 for the agreement of the different models  with the data for differential cross sections for the full available phase-space and for  cos(𝜃𝜇) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The values reported in the table are the simple sum of those 𝜒2 values  obtained for each distribution separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Systematic uncertainties and correlations  were accounted for using covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  As can be seen in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2, GENIE v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6 is the only model that reaches a 𝜒2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  close to unity for the full phase-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' It is also the closest model to the data at the  highest cos(𝜃𝜇) bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For all other models, the 𝜒2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' in the cos(𝜃𝜇) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8 sample  is reduced by a factor of ∼ 2 as compared to the full phase-space sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' GENIE  v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6 shows a smaller reduction in this case, and GiBUU 2019 obtains a consistently  65  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2: Integrated cross section values and 𝜒2 values for the agreement between  the measured cross sections and various event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Results are listed for the  full measured phase space and for a limited one of cos(𝜃𝜇) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Integrated Cross Section [10−38cm2]  (Differential Cross Section 𝜒2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='f)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='75 < cos(𝜃𝜇) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='95  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='75 < cos(𝜃𝜇) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  Data CC1𝑝0𝜋  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='93 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='55  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='05 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='40  Generators  GENIE Nominal  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='18 (63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2/28)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='04 (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1/27)  GENIE v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='45 (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6/28)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='66 (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4/27)  NuWro 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='67 (76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7/28)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='39 (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9/27)  NEUT v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='64 (78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5/28)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='39 (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2/27)  GiBUU 2019  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='00 (82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='/28)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='78 (40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0/27)  higher 𝜒2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' for both the full and limited phase-space samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  p  [GeV/c]  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='65 < cos(  ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='65 < cos(  ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='95  d  dp  [10 38  cm2  GeV/c Ar ]  d  dp  [10 38  cm2  GeV/c Ar ]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  cos( p)  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='65 < cos(  ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  5  10  15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='65 < cos(  ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='95  Simulation  GENIE Nominal  GENIE v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  NuWro 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  NEUT  v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0  GiBUU 2019  MicroBooNE Data  MC  d  dcos( p) [10 38cm2  Ar ]  d  dcos( p) [10 38cm2  Ar ]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0  pp [GeV/c]  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='65 < cos(  ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  0  5  10  15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='65 < cos(  ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='95  d  dpp  [10 38  cm2  GeV/c Ar ]  d  dpp  [10 38  cm2  GeV/c Ar ]  Figure 3-5: As figure 3-4, but for the differential cross sections as a function of mea-  sured muon momentum (left) and measured proton scattering angle (middle) and  momentum (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Cross sections are shown for the full measured phase-space (top)  and for events with cos(𝜃𝜇) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Inner and outer error bars show the  statistical and total (statistical and systematic) uncertainty at the 1𝜎, or 68%, con-  fidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Colored lines show the results of theoretical absolute cross section  calculations using different event generators (without passing through a detector sim-  ulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The blue band shows the extracted cross section obtained from analyzing  MC events passed through our full detector simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 3-5 shows this comparison between the relevant cross sections in the full  available phase-space (top) and in the case where events with cos(𝜃𝜇) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8 are  excluded (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Removing this part of the phase-space significantly improves the  agreement between data and theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The improved agreement with the data observed  66  for GENIE v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6, especially for the full phase-space sample, is intriguing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Specifically,  GENIE v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6 and NEUT v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0 are quite similar, using the same nuclear, QE, and  MEC models, which are the most significant processes in our energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' They  do differ in the Coulomb corrections that only GENIE v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6 and NEUT have, their  free parameter tuning process, and the implementation of RPA correction, that are  known to be important at low momentum transfer [115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Our data indicates that  these seemingly small differences can have a highly significant impact, as seen in  table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  Q2  CCQE[GeV2/c2]  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='65 < cos(  ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  0  5  10  15  Simulation  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='65 < cos(  ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='95  GENIE Nominal  GENIE v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  NuWro 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  NEUT  v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0  GiBUU 2019  MicroBooNE Data  MC  d  dQ2  CCQE  [10 38  cm2  GeV2/c2 Ar ]  d  dQ2  CCQE  [10 38  cm2  GeV2/c2 Ar ]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  Ecal [GeV]  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='65 < cos(  ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='65 < cos(  ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='95  d  dEcal  [10 38  cm2  GeV Ar ]  d  dEcal  [10 38  cm2  GeV Ar ]  Figure 3-6: The flux integrated single differential CC1p0𝜋 cross sections as a function  of 𝑄2  𝐶𝐶𝑄𝐸 = (𝐸𝑐𝑎𝑙  𝜈  − 𝐸𝜇)2 − (⃗𝑝𝜈 − ⃗𝑝𝜇)2 and 𝐸𝑐𝑎𝑙  𝜈  = 𝐸𝜇 + 𝑇𝑝 + 𝐵𝐸, where 𝐵𝐸 = 40  MeV and ⃗𝑝𝜈 = (0, 0, 𝐸𝑐𝑎𝑙  𝜈 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Colored lines show the results of theoretical absolute  cross section calculations using different event generators (without passing through a  detector simulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The blue band shows the extracted cross section obtained from  analyzing MC events passed through our full detector simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Lastly, figure 3-6 shows the flux-integrated single differential cross sections as a  function of calorimetric measured energy and reconstructed momentum transfer, with  and without events with cos(𝜃𝜇) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The former is defined as 𝐸𝑐𝑎𝑙  𝜈  = 𝐸𝜇+𝑇𝑝+𝐵𝐸,  and the latter as 𝑄2  𝐶𝐶𝑄𝐸 = (⃗𝑝𝜈 − ⃗𝑝𝜇)2 − (𝐸𝑐𝑎𝑙  𝜈  − 𝐸𝜇)2, where E𝜇 is the muon energy,  T𝑝 is the proton kinetic energy, BE = 40 MeV is the effective nucleon binding energy  for argon, and ⃗𝑝𝜈 = (0, 0, 𝐸𝑐𝑎𝑙  𝜈 ) is the reconstructed interacting neutrino momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  67  𝐸𝑐𝑎𝑙  𝜈  is often used as a proxy for the true neutrino energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Overall, good agreement is  observed between data and calculations for these complex variables, even for the full  event sample without the cos(𝜃𝜇) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8 requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The systematic uncertainty of our measurement summed up to 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2% and included  contributions from the neutrino flux prediction and POT estimation (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7%), detector  response modeling (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4%), imperfect proton and muon efficiency decoupling (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7%),  and neutrino interaction cross section modeling (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  Quasielastic-like Cross-Section Analysis Conclusions  In summary, the first measurement of 𝜈𝜇 CCQE-like differential cross sections on ar-  gon was reported for event topologies with a single muon and a single proton detected  in the final state using data sets from the MicroBooNE LArTPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The data are in  good agreement with simulation predictions, except at small muon scattering angles  that correspond to low-momentum-transfer reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This measurement confirmed  and constrained calculations essential for the extraction of oscillation parameters and  highlights kinematic regimes where improvement of theoretical models is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The benchmarking of exclusive CC1p0𝜋 cross sections on argon presented here sug-  gests that measurements of CC1p0𝜋 interactions are a suitable choice for use in pre-  cision neutrino oscillation analyses, especially after theoretical models are reconciled  with the small scattering angle data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  First Multidimensional Measurement Of Kine-  matic Imbalance Cross Sections On Argon  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  Kinematic Imbalance Neutrino Data Analysis  Over the course of two years (2019-2021), the MicroBooNE collaboration made sig-  nificant improvements to the pre-existing analysis framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These improvements  provided high statistics neutrino-argon data sets, improved signal processing [74],  reduced detector systematics [130], a theory-driven interaction modeling [79], and  68  the creation of the first MicroBooNE tune [131].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure 3-7 illustrates the improved  data-MC agreement after the implementation of these changes as a function of cos𝜃𝜇,  where the disagreement in the forward direction is longer observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The improved  picture at cos𝜃𝜇 ≈ 1 is primarily driven by the improved modeling of the MC beam  related backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  1  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  µ  θ  cos  0  5  10  15  20  Ar  2  cm  38  10  µ  θ  dcos  σ  d  (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='86/18)  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='51/18)  (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='38/18)  (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='36/18)  NuWro 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  GiBUU 2021  NEUT v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0  GENIE v3 G18 Tune  MicroBooNE Data  Shape Unc)  ⊕  (Stat  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='79e+20 POT  Norm Unc  Figure 3-7: Muon angular distribution after the implementation of the analysis frame-  work improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' No data-MC disagreement is observed in the forward direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Motivated by these improvements, in this analysis the first study of kinematic  imbalance variables on argon, which are sensitive to nuclear effects, is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These  variables are studied using CC1p0𝜋 events within a neutrino slice, as defined by the  Pandora reconstruction framework and detailed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Exactly one muon  with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 < 𝑝𝜇 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV/c, exactly one proton with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3 < 𝑝𝑝 < 1 GeV/c, no charged  pions above the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='07 GeV/c threshold, and no other mesons of any momenta are  required in the final state originating from charged-current 𝜈𝜇-Ar scattering events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The existence of any number of neutrons, electrons or photons is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Such kinematic imbalance variables of interest include the transverse variables  (TVs), namely 𝛿𝑝𝑇, 𝛿𝛼𝑇 and 𝛿𝜑𝑇 [132, 133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These are built specifically to charac-  terize and minimize the degeneracy between the nuclear effects most pertinent to  long-baseline oscillation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In particular, the TVs facilitate the possible  69  identification of the Fermi motion of the initial state nucleon, the final state re-  interactions of the nucleons in the nucleus and the multi-nucleon interactions (2p2h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  As shown in figure 3-8, the TVs are defined by projecting the lepton and proton  momentum on the plane perpendicular to the neutrino direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 3-8: Schematic illustration of the single transverse variables 𝛿𝑝𝑇, 𝛿𝛼𝑇 and 𝛿𝜑𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure adapted from [134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  In the absence of any nuclear effects, the proton and muon momenta are equal and  opposite in this plane and therefore the measured difference between their projections  is a direct probe of nuclear effects in quasi-elastic (QE) events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' 𝛿⃗𝑝𝑇 can be fully  characterized in terms of the vector magnitude (𝛿𝑝𝑇) and the two angles (𝛿𝛼𝑇 and  𝛿𝜑𝑇):  𝛿𝑝𝑇 = |⃗𝑝𝑇 ℓ + ⃗𝑝𝑇 𝑝|  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2)  𝛿𝛼𝑇 = 𝑎𝑟𝑐𝑐𝑜𝑠( −⃗𝑝𝑇 ℓ·𝛿⃗𝑝𝑇  𝑝𝑇 ℓ·𝛿𝑝𝑇 )  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3)  𝛿𝜑𝑇 = 𝑎𝑟𝑐𝑐𝑜𝑠( −⃗𝑝𝑇 ℓ·⃗𝑝𝑇 𝑝  𝑝𝑇 ℓ·𝑝𝑇 𝑝 )  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4)  where ⃗𝑝𝑇 ℓ and ⃗𝑝𝑇 𝑝 are, respectively, the projections of the momentum of the  outgoing lepton and proton on the transverse plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Different nuclear effects alter  the distributions of the TVs in different and predictable ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Measurements of the  70  pi  b!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  b  pP  bp  pl  p  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  SpT  saTTVs therefore have a unique sensitivity to identify nuclear effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This allows cross  sections extracted using these observables to act as a powerful tool to tune and dis-  tinguish nuclear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Furthermore, in case of disagreement, the TV distributions  provide useful hints on the possible causes of the discrepancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Another kinematic variable of interest corresponds to the total momentum of  the struck nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The formalism introduced in [132] is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This formalism  provides an approximation for the longitudinal component of the struck nucleon mo-  mentum shown in equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5, which is derived in appendix 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1,  𝛿𝑝𝐿 = 1  2𝑅 − 𝑚2  𝐴−1 + 𝛿𝑝2  𝑇  2𝑅  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5)  For simplicity, we defined  𝑅 ≡ 𝑚𝐴 + 𝑝𝜇  𝐿 + 𝑝𝑝  𝐿 − 𝐸𝜇 − 𝐸𝑝  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6)  Combining information from both the longitudinal and the transverse components  gave us access to an approximation for the total struck nucleon momentum  𝑝𝑛,𝑝𝑟𝑜𝑥𝑦 =  √︁  𝛿𝑝2  𝐿 + 𝛿𝑝2  𝑇  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7)  The muon-proton momentum imbalances introduced in [135] parallel and trans-  verse to 𝛿⃗𝑝𝑇, as shown in figure 3-9, are explored,  𝛿𝑝𝑇𝑥 = (ˆ𝑝𝜈 × ˆ𝑝𝜇  𝑇) · 𝛿⃗𝑝𝑇  𝛿𝑝𝑇𝑦 = −ˆ𝑝𝜇  𝑇 · 𝛿⃗𝑝𝑇,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8)  71  and, in terms of the magnitudes,  𝛿𝑝𝑇𝑥 = 𝛿𝑝𝑇 · 𝑠𝑖𝑛𝛿𝛼𝑇  𝛿𝑝𝑇𝑦 = 𝛿𝑝𝑇 · 𝑐𝑜𝑠𝛿𝛼𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9)  Figure 3-9: Schematic illustration of 𝛿𝑝𝑇𝑥 and 𝛿𝑝𝑇𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The measured 𝛿𝑝𝑇𝑥 event distribution shown in figure 3-10 (left) using the “com-  bined” MicroBooNE runs 1-3 exhibit a QE peak near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' If the interaction had occurred  on a free nucleon, then a delta function would be expected at 0 because the muon  and proton final states must balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The width of the QE peak mostly results from  the Fermi motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' If no significant deviation is assumed in the non-QE distributions  originating from MEC and RES/DIS events, then data-MC discrepancies could im-  ply an overestimation of the argon Fermi momentum, and/or a difference in the total  fraction of the FSI contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Unlike the 𝛿𝑝𝑇𝑥 distribution, a non-QE tail is observed towards the negative 𝛿𝑝𝑇𝑦  values shown in figure 3-10 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Inelastic events such as 2p2h, resonance, and DIS  are inefficient at transferring the lepton momentum to the final state nucleons, since  multiple initial states particles are often involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Therefore, the protons tagged in  72  OpT  qT =  pp  pTy  SPTc  py  N  P  Puthe non-QE events will in general have less momenta then the muons and the 𝛿𝑝𝑇𝑦  distribution is shifted to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  BeamOn (9051)  CCQE (4744)  CCMEC (1547)  ExtBNB (642)  CCRES (1035)  CCDIS (181)  [GeV/c]  T,x  p  δ  0  500  1000  1500  2000  2500  # Events / 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='79e+20  Combined  CCQE = 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7 %  Cosmics = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8 %  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  [GeV/c]  T,x  p  δ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  MC+ExtBNB  BeamOn  BeamOn (9051)  CCQE (4744)  CCMEC (1547)  ExtBNB (642)  CCRES (1035)  CCDIS (181)  [GeV/c]  T,y  p  δ  0  500  1000  1500  2000  2500  # Events / 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='79e+20  Combined  CCQE = 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7 %  Cosmics = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8 %  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  [GeV/c]  T,y  p  δ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  MC+ExtBNB  BeamOn  Figure 3-10: Interaction breakdown of the CC1p0𝜋 events as a function of 𝛿𝑝𝑇𝑥 (left)  and 𝛿𝑝𝑇𝑦 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The data correspond to the “combined” MicroBooNE runs 1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  To avoid multiple cosmic contributions and tracks from trajectories that exit the  detector but their end-points are incorrectly reconstructed around its edges, a fiducial  volume of  10 < 𝑥 < 246, −105 < 𝑦 < 105, 10 < 𝑧 < 1026 cm  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='10)  is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Candidate muon and proton tracks that were fully contained in this  region were considered and their momenta were obtained based on their range [136,  137].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The log-likelihood ratio particle identification (LLR PID) score method [138] is  used to obtain our muon and proton candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The candidate track with the greater  LLR PID score was assigned the label of the candidate muon, while the one with the  smaller 3-plane loglikelihood was our candidate proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  To minimize the contribution of misreconstructed tracks, we took advantage of  the fact that we had two muon momentum reconstruction methods available for con-  tained tracks, namely the momentum from range [137] and the one from Multiple  Coulomb Scattering (MCS) [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A quality cut was applied on the contained muons  by requiring the range and MCS momenta to be in agreement within 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  In order to avoid flipped tracks, it was further required that the distance between  73  the track start points and the vertex is smaller than the corresponding distance be-  tween the track end points and the vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' It was also required that the distance  between the start points of the two candidate tracks is smaller than the one between  the two end points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The maximal possible signal contribution was ensured, while the majority of the  cosmic contamination and the beam related MC backgrounds were rejected, by re-  quiring that the proton LLP PID score is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The application of our event selection resulted in 9051 candidate events in our data  sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Using the MC, it was estimated that our event selection yielded a purity of  ≈ 70% and an efficiency of ≈ 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  There was also some contribution from the  remaining cosmic contamination (≈ 8%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' After the application of the event selection,  topological and interaction breakdowns for the kinematic variables of interest, such  as the ones shown in figure 3-11 for 𝛿𝑝𝑇, were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  BeamOn (9051)  Overlay CC1p (5673)  Overlay NonCC1p (1837)  ExtBNB (642)  Dirt CC1p (0)  Dirt NonCC1p (62)  [GeV/c]  T  p  δ  0  500  1000  1500  # Events / 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='79e+20  Combined  CC1p = 69 %  Cosmics = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8 %  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  [GeV/c]  T  p  δ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  MC+ExtBNB  BeamOn  BeamOn (9051)  CCQE (4744)  CCMEC (1547)  ExtBNB (642)  CCRES (1035)  CCDIS (181)  [GeV/c]  T  p  δ  0  500  1000  1500  # Events / 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='79e+20  Combined  CCQE = 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7 %  Cosmics = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8 %  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  [GeV/c]  T  p  δ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  MC+ExtBNB  BeamOn  Figure 3-11: Topological (left) and interaction (right) breakdown after the application  of the event selection for 𝛿𝑝𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  Cross-Section Extraction Technique  The unfolded cross-section results reported in this analysis took advantage of the  Wiener-SVD unfolding [139].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This method combines the use of the singular value  decomposition (SVD) unfolding and a Wiener filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' SVD unfolding [140], such as  the Tikhonov regularisation [141], unfolds a distribution by minimising a 𝜒2 function  comparing a prediction to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' To avoid the large variance introduced, a penalty  74  term is added to regularise the curvature (second derivative) of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The  strength of such a term is determined by finding an appropriate trade-off between the  bias and the variance between the data and the MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' More details on the technique  are included in appendix 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In order to report the cross-section results, two key  ingredients are required, namely the response and covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The construction of the response matrices uses the selected MC CC1p0𝜋 events  to construct a two-dimensional (2D) object, where each entry in true bin i and re-  constructed bin j (𝑁 𝑡𝑟𝑢𝑒 𝑖,𝑟𝑒𝑐𝑜 𝑗) is divided by the true number of events generated in  bin i (𝑆𝑡𝑟𝑢𝑒 𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These response matrices serve as “2D local efficiencies”, as defined in  equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='11 and can be seen in figure 3-12 for 𝛿𝑝𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  𝑀𝑖𝑗 = 𝑁 𝑡𝑟𝑢𝑒 𝑖,𝑟𝑒𝑐𝑜 𝑗  𝑆𝑡𝑟𝑢𝑒 𝑖  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='11)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='03  Combined Response Matrix  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='8  [GeV/c]  T  p  δ  True  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='8  [GeV/c]  T  p  δ  Reco  Combined Response Matrix  Figure 3-12: Response matrices of 𝛿𝑝𝑇 using the selected CC1p0𝜋 MC events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The method also uses a covariance matrix constructed from the MC flux nor-  malised event rate as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The total covariance matrix incorporates information  related to the systematic and statistical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The flux-normalized MC event  75  rates in reconstructed space were obtained as  ˜𝜎𝑟𝑒𝑐𝑜 𝑖 =  𝑁 𝑟𝑒𝑐𝑜 𝑖  Φ𝐶𝑉  𝜈  × 𝑁𝑡𝑎𝑟𝑔𝑒𝑡𝑠  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='12)  where 𝑁 𝑟𝑒𝑐𝑜 𝑖 = 𝑀𝑖𝑗 ×𝑆𝑡𝑟𝑢𝑒 𝑗 +𝐵𝑟𝑒𝑐𝑜 𝑖 is the total number of reconstructed events in  bin i, 𝑀𝑖𝑗 is the response matrix corresponding to reco bin i and true bin j as defined in  equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='11, 𝑆𝑡𝑟𝑢𝑒 𝑗 is the true signal without any detector or reconstruction effects  in bin j, and 𝐵𝑟𝑒𝑐𝑜 𝑖 is the total number of reconstructed beam-related MC background  events in bin i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Substituting 𝑁 𝑟𝑒𝑐𝑜 𝑖 into equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='12 yields  ˜𝜎𝑟𝑒𝑐𝑜 𝑖 = 𝑀 𝑢𝑛𝑖𝑣  𝑖𝑗  × 𝑆𝑡𝑟𝑢𝑒 𝑗 𝐶𝑉 + 𝐵𝑟𝑒𝑐𝑜 𝑖 𝑢𝑛𝑖𝑣  Φ𝐶𝑉  𝜈  × 𝑁𝑡𝑎𝑟𝑔𝑒𝑡𝑠  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='13)  For each systematic variation, each term in equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='13 labelled with “univ” is  reweighted/modified and each term labelled with “CV” is fixed to the central value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The integrated flux, Φ𝐶𝑉  𝜈  , remains fixed for each variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  In the case of the cross section variations, the calculation of the response matrix  in each universe is slightly modified via the normalization to the true signal in a given  universe 𝑆𝑡𝑟𝑢𝑒 𝑗 𝑢𝑛𝑖𝑣,  𝑀 𝑢𝑛𝑖𝑣  𝑖𝑗  = 𝑁 𝑡𝑟𝑢𝑒 𝑗,𝑟𝑒𝑐𝑜 𝑖 𝑢𝑛𝑖𝑣  𝑆𝑡𝑟𝑢𝑒 𝑗 𝑢𝑛𝑖𝑣  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='14)  This treatment of the systematic uncertainties addresses both the signal and the  beam related background uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Using these flux-normalized event rates ˜𝜎, a covariance matrix, 𝐸𝑖𝑗, can be calcu-  lated using the central value and 𝑁𝑢𝑛𝑖𝑣 multisims with the covariance formalism,  𝐸𝑖𝑗 =  1  𝑁𝑢𝑛𝑖𝑣  𝑁𝑢𝑛𝑖𝑣  ∑︁  𝑠=0  (˜𝜎𝑢𝑛𝑖𝑣  𝑖  − ˜𝜎𝐶𝑉  𝑖  )(˜𝜎𝑢𝑛𝑖𝑣  𝑗  − ˜𝜎𝐶𝑉  𝑗  )  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15)  where 𝑁𝑢𝑛𝑖𝑣 is number of alternative universes, ˜𝜎𝑢𝑛𝑖𝑣  𝑖  corresponds to the variation  and ˜𝜎𝐶𝑉  𝑖  to the central-value prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The unfolding model uncertainty is accessed by comparing the data spectra un-  folded with G18 to the data spectra unfolded with two alternative G18 configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  76  The former one did not include the effect of the MicroBooNE tune and the latter in-  cluded an additional weight of 2 on the MEC events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The spread between the three  configurations on a bin-by-bin basis normalized to  √︀  (2) is assigned as an additional  uncertainty [142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The statistical uncertainty of our measurement is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The total uncertainty  sums to 13% and includes contributions from the neutrino flux prediction (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3%),  unfolding model uncertainty (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3%), neutrino interaction cross section modeling (5%),  detector response modeling (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9%), POT estimation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3%), number-of-scattering-  targets (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15%), reinteractions (1%), and out-of-cryostat interaction modeling (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 3-13 shows the total covariance matrix due to the aforementioned sources  of uncertainty for 𝛿𝑝𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  3  −  10  ×  Combined Total Covariance Matrix  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  [GeV/c]  T  p  δ  i bin  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  [GeV/c]  T  p  δ  j bin  Combined Total Covariance Matrix  Figure 3-13: Total covariance matrix for 𝛿𝑝𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The Wiener SVD unfolding machinery returns an unfolded data cross section  along with an unfolded covariance matrix and an additional smearing matrix, 𝐴𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The corresponding 𝐴𝑐 matrix for 𝛿𝑝𝑇 is shown in figure 3-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The smearing matrix  𝐴𝑐 contains information about the regularisation of the measurement and is applied  to the true model cross section predictions when compared to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Therefore, the  result of our measurement lives in a “regularized” phase-space, which is not identical  to the true phase-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  [GeV/c]  T  p  δ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  [GeV/c]  T  p  δ  Combined  Figure 3-14: Additional smearing matrix 𝐴𝑐 for 𝛿𝑝𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  Event Generator Modeling And Configurations  The extracted cross sections were compared to GENIE v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6 G18_10a_02_11a  (G18) and the theory-driven GiBUU 2021 (GiBUU) event generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Additional com-  parisons to the corresponding events generators when FSI are turned off were also  included (G18 No FSI and GiBUU No FSI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' G18 uses the Local Fermi Gas model [37],  the Nieves CCQE scattering prescription [121] which includes Coulomb corrections  for the outgoing muon [116] and Random Phase Approximation correction [115], the  Nieves MEC model [122], the KLN-BS RES [123–126] and Berger-Sehgal COH [120]  scattering models, the hA2018 FSI model [143], and the T2K tune weights [131].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  GiBUU uses somewhat similar models, but, unlike GENIE, those are imple-  mented in a coherent way, by solving the Boltzmann-Uehling-Uhlenbeck transport  equation [117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The modeling includes the Local Fermi Gas model [37], a standard  CCQE expression [127], an empirical MEC model and a dedicated spin dependent  resonances amplitude calculation following the MAID analysis [128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The DIS model  is as in PYTHIA [129] and the FSI treatment is different as the hadrons propagate  through the residual nucleus in a nuclear potential which is consistent with the initial  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  78  Apart from the nominal G18 prediction, we further included a comparison to the  recently added theory driven GENIE v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6 G21_11b_00_000 configuration (G21  hN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The latter uses the SuSAv2 model for QE and MEC interactions [28], and the  hN2018 FSI model [144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The modeling options for RES, DIS, and COH interactions  are the same as for G18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We investigated the effect of the FSI modeling choice by  comparing the G21hN results to the ones obtained with G21 hA, where the hA2018  FSI model was used instead, and to G21 G4 with the recently coupled Geant4 FSI  framework [145].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Lastly, our results present the comparison between the nominal G18 LFG model  and predictions using the same G18 modeling configuration but different nuclear  model options available in the GENIE event generator, namely the Bodek-Ritchie  Fermi Gas (G18 RFG) [36] and an Effective Spectral Function (G18 EffSF) [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Furthermore, the prediction without Random Phase Approximation (RPA) effects  was used for comparison (G18 No RPA) [115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  Kinematic Imbalance Differential Cross-Section Results  The single- and double- in 𝛿𝛼𝑇 bins differential unfolded cross sections as a func-  tion of 𝛿𝑝𝑇 are presented in figure 3-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The single-differential results as a function  of 𝛿𝑝𝑇 using all the events that satisfy our selection are shown in the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The peak height of both generator predictions is ≈ 30% higher when FSI effects are  turned off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Yet, all distributions illustrate a transverse missing momentum tail that  extends beyond the Fermi momentum whether FSI effects are activated or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  ratio between the generator predictions with and without FSI is shown in the insert  and illustrates significant shape variations across the range of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The double-  differential result using events with 𝛿𝛼𝑇 < 45𝑜 shown in the bottom left panel of  figure 3-15 is dominated by events that primarily occupy the region up to the Fermi  momentum and do not exhibit a high momentum tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The corresponding ratio in-  sert illustrates a fairly uniform behavior indicative of transparency effects ranging  between 50-70%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The double-differential results using events with 135𝑜 < 𝛿𝛼𝑇 < 180𝑜  is shown in the bottom right panel of figure 3-15 and illustrate the high transverse  79  missing momentum up to 1 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The case without FSI effects is strongly disfa-  vored and the ratio insert illustrates strong shape variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Therefore, the high 𝛿𝛼𝑇  region is an appealing candidate for neutrino experiments to benchmark and tune the  FSI modeling in event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='6  2  4  6  FSI/No FSI  Figure 3-15: The flux-integrated single- (top) and double- in 𝛿𝛼𝑇 bins (bottom)  differential CC1p0𝜋 cross sections as a function of the transverse missing momentum  𝛿𝑝𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Inner and outer error bars show the statistical and total (statistical and shape  systematic) uncertainty at the 1𝜎, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The gray band shows the  normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Colored lines show the results of theoretical  absolute cross section calculations with and without FSI based on the GENIE and  GiBUU event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The single-differential results as a function of 𝛿𝛼𝑇 using all the events that satisfy  our selection are shown in top panel of figure 3-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The result without FSI illustrates  a uniform behavior across the whole distribution and is disfavored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The addition of  FSI effects leads to a ≈ 30% asymmetry around 𝛿𝛼𝑇 = 90𝑜 due to the fact that the  proton in our selection undergoes FSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The three FSI models used here for comparison  result in a comparable performance, also shown in terms of the ratio plot of the  80  different FSI options to the prediction without FSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The double-differential result  using events with 𝛿𝑝𝑇 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV/c shown in the bottom left panel of figure 3-16  illustrates a uniform distribution indicative of the suppressed FSI impact in that part  of the phase-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The double-differential result using events with 𝛿𝑝𝑇 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV/c  is shown in the bottom right panel of figure 3-16 and illustrates the presence of strong  FSI effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The case without FSI effects is disfavored and the asymmetry around 90𝑜  is significantly enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Therefore, the high 𝛿𝛼𝑇 region is an appealing candidate for  neutrino experiments to benchmark and tune the FSI modeling in event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  0  20  40  60  80  100 120 140 160 180  [deg]  T  α  δ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='2/7)  G21 hA (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7/7)  G21 hN (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0/7)  G21 G4 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2/7)  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV/c, MicroBooNE Preliminary  T  p  δ  (c)  0  50  100  150  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  FSI/No FSI  Figure 3-16: The flux-integrated single- (top) and double- in 𝛿𝑝𝑇 bins (bottom) dif-  ferential CC1p0𝜋 cross sections as a function of the angle 𝛿𝛼𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Inner and outer error  bars show the statistical and total (statistical and shape systematic) uncertainty at  the 1𝜎, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The gray band shows the normalization systematic  uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Colored lines show the results of theoretical absolute cross section calcu-  lations with a number of FSI modeling options based on the GENIE event generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Lastly, figure 3-17 shows the single- (top) and double- in 𝛿𝑝𝑇,𝑦 bins (bottom)  differential unfolded cross sections as a function of 𝛿𝑝𝑇,𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The event distributions of  81  𝛿𝑝𝑇,𝑥 and 𝛿𝑝𝑇,𝑦 have already been presented in figure 3-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The single differential  result (top panel) illustrates a fairly broad symmetric distribution centered around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The double-differential result for events where 𝛿𝑝𝑇,𝑦 < -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c (bottom left panel)  illustrates an even broader distribution where all predictions yield comparable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Unlike the asymmetric part of the 𝛿𝑝𝑇,𝑦 tail, the double-differential result for events  with -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 < 𝛿𝑝𝑇,𝑦 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c (bottom right panel) shows a much narrower peak  which strongly depends on the choice of the underlying model and the addition or  not of nuclear effects, such as RPA ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The G18 LFG and G18 No RPA predictions  are favored in that part of the phase-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  [GeV/c]  T,x  p  δ  0  10  20  30  40  50  GeV/c Ar  2  cm  38  10  T,x  p  δ  d  σ  d  G18 LFG (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8/11)  G18 No RPA (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5/11)  G18 RFG (37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5/11)  G18 EffSF (57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3/11)  Shape)  ⊕  MicroBooNE Data (Stat  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='79e+20 POT  Norm  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='17  Data  σ  (a) All events  MicroBooNE Preliminary  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  [GeV/c]  T,x  p  δ  0  2  4  6  8  10  12  14  Ar  2  /c  2  GeV  2  cm  38  10  T,y  p  δ  d  T,x  p  δ  d  σ  2  d  G18 LFG (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9/11)  G18 No RPA (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2/11)  G18 RFG (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8/11)  G18 EffSF (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3/11)  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='25  Data  σ  < -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c  T,y  p  δ  (b)  MicroBooNE Preliminary  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='4  [GeV/c]  T,x  p  δ  0  20  40  60  80  Ar  2  /c  2  GeV  2  cm  38  10  T,y  p  δ  d  T,x  p  δ  d  σ  2  d  G18 LFG (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='1/11)  G18 RFG (39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3/11)  G18 EffSF (52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3/11)  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='14  Data  σ  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c  T,y  p  δ  (c) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 <  MicroBooNE Preliminary  Figure 3-17: The flux-integrated single- (top) and double- in 𝛿𝑝𝑇,𝑦 bins (bottom) dif-  ferential CC1p0𝜋 cross sections as a function of the angle 𝛿𝑝𝑇,𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Inner and outer error  bars show the statistical and total (statistical and shape systematic) uncertainty at  the 1𝜎, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The gray band shows the normalization system-  atic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Colored lines show the results of theoretical absolute cross section  calculations with a number of event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The 𝜒2 per degree of freedom (d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=') data comparison for each prediction shown  82  on the results in figures 3-15, 3-16, and 3-17 takes into account the total covariance  matrix including the off-diagonal elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figures 3-18 - 3-26 show in the interaction  breakdown of the aforementioned results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='8  [GeV/c]  T  p  δ  0  10  20  30  40  50  GeV/c Ar  2  cm  38  10  T  p  δ  d  σ  d  QE  MEC  RES  DIS  MicroBooNE Data  Shape Unc)  ⊕  (Stat  Norm Unc  G18    All events  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  [GeV/c]  T  p  δ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  deg GeV/c Ar  2  cm  38  10  T  p  δ  d  T  α  δ  d  σ  2  d  o  < 45  T  α  δ  GiBUU  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  [GeV/c]  T  p  δ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  deg GeV/c Ar  2  cm  38  10  T  p  δ  d  T  α  δ  d  σ  2  d  o  < 45  T  α  δ  GiBUU No FSI  Figure 3-19: Cross-section interaction breakdown for events with 𝛿𝛼𝑇 < 45𝑜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  breakdown is shown for (top left) the G18 configuration with FSI effects, (top right)  the G18 configuration without FSI effects, (bottom left) GiB with FSI effects, and  (bottom right) GiB without FSI effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  83  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  [GeV/c]  T  p  δ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  deg GeV/c Ar  2  cm  38  10  T  p  δ  d  T  α  δ  d  σ  2  d  QE  MEC  RES  DIS  MicroBooNE Data  Shape Unc)  ⊕  (Stat  Norm Unc  o  < 180  T  α  δ  <  o  G18    135  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  [GeV/c]  T  p  δ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  deg GeV/c Ar  2  cm  38  10  T  p  δ  d  T  α  δ  d  σ  2  d  o  < 180  T  α  δ  <  o  G18 No FSI    135  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  [GeV/c]  T  p  δ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  deg GeV/c Ar  2  cm  38  10  T  p  δ  d  T  α  δ  d  σ  2  d  o  < 180  T  α  δ  <  o  GiBUU    135  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  [GeV/c]  T  p  δ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  deg GeV/c Ar  2  cm  38  10  T  p  δ  d  T  α  δ  d  σ  2  d  o  < 180  T  α  δ  <  o  GiBUU No FSI    135  Figure 3-20: Cross-section interaction breakdown for events with 135𝑜 < 𝛿𝛼𝑇 < 180𝑜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The breakdown is shown for (top left) the G18 configuration with FSI effects, (top  right) the G18 configuration without FSI effects, (bottom left) GiB with FSI effects,  and (bottom right) GiB without FSI effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  0  20  40  60  80  100 120 140 160 180  [deg]  T  α  δ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='12  deg Ar  2  cm  38  10  T  α  δ  d  σ  d  QE  MEC  RES  DIS  MicroBooNE Data  Shape Unc)  ⊕  (Stat  Norm Unc  G21 hA    All events  0  20  40  60  80  100 120 140 160 180  [deg]  T  α  δ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='12  deg Ar  2  cm  38  10  T  α  δ  d  σ  d  G21 hN    All events  0  20  40  60  80  100 120 140 160 180  [deg]  T  α  δ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='12  deg Ar  2  cm  38  10  T  α  δ  d  σ  d  G21 G4    All events  0  20  40  60  80  100 120 140 160 180  [deg]  T  α  δ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='12  deg Ar  2  cm  38  10  T  α  δ  d  σ  d  G21 No FSI    All events  Figure 3-21: Cross-section interaction breakdown for all the selected events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  breakdown is shown for (top left) the G21 hA configuration with the hA2018 FSI  model, (top right) the G21 hN configuration with the hN FSI model, (bottom left)  the G21 G4 configuration with the G4 FSI model, and (bottom right) the G21 No  FSI configuration without FSI effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  84  0  20  40  60  80  100 120 140 160 180  [deg]  T  α  δ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='35  deg GeV/c Ar  2  cm  38  10  T  p  δ  d  T  α  δ  d  σ  2  d  QE  MEC  RES  DIS  MicroBooNE Data  Shape Unc)  ⊕  (Stat  Norm Unc  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV/c  T  p  δ  G21 hA  0  20  40  60  80  100 120 140 160 180  [deg]  T  α  δ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='35  deg GeV/c Ar  2  cm  38  10  T  p  δ  d  T  α  δ  d  σ  2  d  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV/c  T  p  δ  G21 hN  0  20  40  60  80  100 120 140 160 180  [deg]  T  α  δ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='35  deg GeV/c Ar  2  cm  38  10  T  p  δ  d  T  α  δ  d  σ  2  d  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV/c  T  p  δ  G21 G4  0  20  40  60  80  100 120 140 160 180  [deg]  T  α  δ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='35  deg GeV/c Ar  2  cm  38  10  T  p  δ  d  T  α  δ  d  σ  2  d  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV/c  T  p  δ  G21 No FSI  Figure 3-22: Cross-section interaction breakdown for events with 𝛿𝑝𝑇 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The breakdown is shown for (top left) the G21 hA configuration with the hA2018  FSI model, (top right) the G21 hN configuration with the hN FSI model, (bottom  left) the G21 G4 configuration with the G4 FSI model, and (bottom right) the G21  No FSI configuration without FSI effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  0  20  40  60  80  100 120 140 160 180  [deg]  T  α  δ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='05  deg GeV/c Ar  2  cm  38  10  T  p  δ  d  T  α  δ  d  σ  2  d  QE  MEC  RES  DIS  MicroBooNE Data  Shape Unc)  ⊕  (Stat  Norm Unc  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV/c  T  p  δ  G21 hA  0  20  40  60  80  100 120 140 160 180  [deg]  T  α  δ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='05  deg GeV/c Ar  2  cm  38  10  T  p  δ  d  T  α  δ  d  σ  2  d  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV/c  T  p  δ  G21 hN  0  20  40  60  80  100 120 140 160 180  [deg]  T  α  δ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='05  deg GeV/c Ar  2  cm  38  10  T  p  δ  d  T  α  δ  d  σ  2  d  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV/c  T  p  δ  G21 G4  0  20  40  60  80  100 120 140 160 180  [deg]  T  α  δ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='05  deg GeV/c Ar  2  cm  38  10  T  p  δ  d  T  α  δ  d  σ  2  d  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV/c  T  p  δ  G21 No FSI  Figure 3-23: Cross-section interaction breakdown for events with 𝛿𝑝𝑇 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The breakdown is shown for (top left) the G21 hA configuration with the hA2018  FSI model, (top right) the G21 hN configuration with the hN FSI model, (bottom  left) the G21 G4 configuration with the G4 FSI model, and (bottom right) the G21  No FSI configuration without FSI effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='4  [GeV/c]  T,x  p  δ  0  10  20  30  40  50  GeV/c Ar  2  cm  38  10  T,x  p  δ  d  σ  d  QE  MEC  RES  DIS  MicroBooNE Data  Shape Unc)  ⊕  (Stat  Norm Unc  G18 LFG    All events  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='4  [GeV/c]  T,x  p  δ  0  10  20  30  40  50  GeV/c Ar  2  cm  38  10  T,x  p  δ  d  σ  d  G18 RFG    All events  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='2  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='4  [GeV/c]  T,x  p  δ  0  10  20  30  40  50  GeV/c Ar  2  cm  38  10  T,x  p  δ  d  σ  d  G18 No RPA    All events  Figure 3-24: Cross-section interaction breakdown for all the selected events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  breakdown is shown for (top left) the G18 LFG configuration, (top right) the G18  RFG configuration, (bottom left) the G18 EffSF configuration, and (bottom right)  the G18 No RPA configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='4  [GeV/c]  T,x  p  δ  0  2  4  6  8  10  12  14  Ar  2  /c  2  GeV  2  cm  38  10  T,y  p  δ  d  T,x  p  δ  d  σ  2  d  QE  MEC  RES  DIS  MicroBooNE Data  Shape Unc)  ⊕  (Stat  Norm Unc  < -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c  T,y  p  δ  G18 LFG  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='4  [GeV/c]  T,x  p  δ  0  2  4  6  8  10  12  14  Ar  2  /c  2  GeV  2  cm  38  10  T,y  p  δ  d  T,x  p  δ  d  σ  2  d  < -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c  T,y  p  δ  G18 RFG  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  [GeV/c]  T,x  p  δ  0  2  4  6  8  10  12  14  Ar  2  /c  2  GeV  2  cm  38  10  T,y  p  δ  d  T,x  p  δ  d  σ  2  d  < -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c  T,y  p  δ  G18 EffSF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  [GeV/c]  T,x  p  δ  0  2  4  6  8  10  12  14  Ar  2  /c  2  GeV  2  cm  38  10  T,y  p  δ  d  T,x  p  δ  d  σ  2  d  < -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c  T,y  p  δ  G18 No RPA  Figure 3-25: Cross-section interaction breakdown for events with 𝛿𝑝𝑇,𝑦 < -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The breakdown is shown for (top left) the G18 LFG configuration, (top right) the G18  RFG configuration, (bottom left) the G18 EffSF configuration, and (bottom right)  the G18 No RPA configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='4  [GeV/c]  T,x  p  δ  0  20  40  60  80  Ar  2  /c  2  GeV  2  cm  38  10  T,y  p  δ  d  T,x  p  δ  d  σ  2  d  QE  MEC  RES  DIS  MicroBooNE Data  Shape Unc)  ⊕  (Stat  Norm Unc  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='15 <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  [GeV/c]  T,x  p  δ  0  20  40  60  80  Ar  2  /c  2  GeV  2  cm  38  10  T,y  p  δ  d  T,x  p  δ  d  σ  2  d  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c  T,y  p  δ  G18 RFG    -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  [GeV/c]  T,x  p  δ  0  20  40  60  80  Ar  2  /c  2  GeV  2  cm  38  10  T,y  p  δ  d  T,x  p  δ  d  σ  2  d  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c  T,y  p  δ  G18 EffSF    -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  [GeV/c]  T,x  p  δ  0  20  40  60  80  Ar  2  /c  2  GeV  2  cm  38  10  T,y  p  δ  d  T,x  p  δ  d  σ  2  d  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c  T,y  p  δ  G18 No RPA    -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 <  Figure 3-26: Cross-section interaction breakdown for events with -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 < 𝛿𝑝𝑇,𝑦 <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The breakdown is shown for (top left) the G18 LFG configuration, (top  right) the G18 RFG configuration, (bottom left) the G18 EffSF configuration, and  (bottom right) the G18 No RPA configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  Kinematic Imbalance Cross-Section Analysis Conclusions  The first measurement of 𝜈𝜇 CC1p0𝜋 single and double differential cross sections on  argon as a function of kinematic imbalance variables for event topologies with a single  muon and a single proton detected in the final state was reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The unfolded data  results were compared to a number of event generators, available model configurations  and FSI modeling options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This measurement identified regions of the phase-space  which are ideal to provide constraints for nuclear and final state interaction effects in  generator predictions essential for the extraction of oscillation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  Prospects With Future Neutrino Experiments  The MicroBooNE experiment is a crucial step in the understanding of the underlying  neutrino-argon interactions that will be used to drastically reduce the uncertainties of  forthcoming high precision neutrino oscillation measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Furthermore, Micro-  BooNE is the first LArTPC in a neutrino beam with automated event reconstruction  and selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  87  This work paved the path towards precision 𝜈𝜇 CC cross section measurements  with a single proton and no pions in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  However, only three out of  the five available run periods are used in the results presented in this thesis, as  shown in figure 3-27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Within the next year, runs 4 and 5 will also become available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Therefore, the statistical uncertainties will be further reduced, and the path to further  multidimensional analyses will be further explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 3-27: MicroBooNE total Protons on Target (POT) collected with the Booster  Neutrino Beam (BNB) during the five run periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In this thesis, the first three run  periods were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Beyond MicroBooNE, experiments of the Short-Baseline Neutrino (SBN) Pro-  gram at Fermilab, namely ICARUS and SBND, have already turned (or will very  soon turn) on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The progress of SBN, which shares the same technology and beam  as MicroBooNE, will largely benefit from the LArTPC expertise developed by the  MicroBooNE collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' With these experiments, the largest neutrino-argon scat-  tering data sets will be collected and will be used to test the performance of our  theory predictions in multiple variables simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Despite the abundance of neutrino-argon cross sections that will be reported at the  SBN program, the relevant energy range is significantly lower than the DUNE energy  spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' However, there is a wealth of cross-section opportunities that will become  available once the ArgonCube demonstrator, a 2×2 grid engineering prototype for  the LArTPC DUNE Near Detector (ND) module design, is commissioned in Fall  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' ArgonCube will provide sufficiently high statistics to measure cross sections  88  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0E21  DeliveredPOT  POTontape  Run 1  Run 2  Run 3l  Run 4  Run5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5E21  Cumulative POT  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0E21  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0E20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0E00  20151  2016  2017  2018  2019  2020and to evaluate the corresponding systematic uncertainties using both multi-particle  channels and multidimensional analyses and will pave the path towards the final  design of the DUNE ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  All these neutrino experimental efforts will be complemented by the continuous  benchmarking of the neutrino event generators predictions against external data sets,  such as against electron scattering data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A major step in this direction is made  in chapter 4 with the unification of the event generation process and of the modeling  across the two particle species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Furthermore, chapters 5 and 6 detail the analysis of  electron scattering data sets from the CLAS detector at Thomas Jefferson Laboratory  following neutrino data analysis methods and the testing against the performance of  commonly used GENIE event generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  89  90  Chapter 4  Inclusive Electron Scattering And  The GENIE Event Generator  [Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' D 103, 113003 (2021)]  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  Electron-Nucleus Modeling Development  As already discussed in section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2, the extraction of neutrino mixing parameters  from neutrino oscillation experiments [23, 146, 147] relies on comparing the energy-  dependent neutrino event distribution for a particular neutrino flavor near the neu-  trino production point with the corresponding event distribution at a significant dis-  tance away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  In practice, the yield at each neutrino energy is extracted from the  measured neutrino-nucleus interactions in a detector, as reconstructed from the mea-  sured particles ejected in the neutrino-nucleus interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  This requires detailed  knowledge of the neutrino-nucleus interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Unfortunately, measuring the neutrino-nucleus interaction is difficult due to the  wide energy spread of accelerator-produced neutrino beams [148], as can be seen in  figure 4-1, and the tiny neutrino-nucleus cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A relatively small body of data  has been published [24], which suffers from poor statistics and is flux-averaged over  a wide range of neutrino energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This data is then supplemented with theoretical  models and implemented into event generator codes such as GENIE [33], which is  extensively used across US-based neutrino experiments, to simulate the neutrino-  nucleus interactions across a wide range of energies and targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' GENIE simulations  are then used to aid in extraction of the incident neutrino flux as a function of  91  energy from the neutrino-nucleus scattering events measured in neutrino detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  However, the theoretical models need to describe many different interaction processes  for medium to heavy nuclei (typically C, O, or Ar) where nuclear effects complicate the  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' As a result, the uncertainties in the extraction of oscillation parameters  are often dominated by the lack of knowledge of the neutrino-nucleus interactions [23,  147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  1  2  3  4  5  0  10  20  30  DIS  RES  MEC  QE  (32%)  (37%)  (7%)  (22%)  1  2  3  4  5  0  20  40  60  DIS  RES  MEC  QE  (28%)  (36%)  (8%)  (26%)  [Arb Units]  ν  dE  σ  d  [GeV]  ν  E  [Arb Units]  ν  dE  σ  d  [GeV]  ν  E  Figure 4-1: Charged-current cross sections as a function of neutrino energy obtained  using GENIE for muon neutrino scattering using the DUNE near detector (left) and  far detector (right) oscillated fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The shaded bands show the fractional contri-  bution for each interaction mechanism, quasielastic scattering (QE), meson-exchange  currents (MEC), resonance excitation (RES), and deep inelastic scattering (DIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' See  text for details of the interaction mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The numbers in parentheses indicate  the percentage of the cross section due to each interaction mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 4-1 shows such a wide energy spectrum for the DUNE near detector flux-  averaged cross sections (left) and the far detector oscillated flux-averaged cross sec-  tions (right) using one model configuration in GENIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' All four neutrino-nucleus re-  action mechanisms contribute significantly and all four need to be well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  This is especially true because different reaction mechanisms contribute differently in  the different oscillation peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Understanding one reaction mechanism better than  the others could have significant implications for oscillation analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  To improve our understanding of neutrino-nucleus interactions, we can take ad-  vantage of the fact that neutrinos and electrons are both leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Thus, they interact  92  with atomic nuclei in similar ways via the same reaction mechanisms, as illustrated  in figure 4-2 and detailed in sections 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 4-2: (Left) electron-nucleus inclusive scattering via one-photon exchange and  (right) charged current neutrino-nucleus inclusive scattering via 𝑊 exchange with a  final state charged lepton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The most common lepton-nucleus interaction mechanisms are shown in figure 4-  3 include: (a) quasielastic (QE) scattering from individual moving nucleons in the  nucleus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (b) two-nucleon knockout, due to interactions with a meson being exchanged  between two nucleons referred to two-particle two-hole excitations, 2p2h or its major  component, meson exchange currents (MEC);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (c) interactions which leave the struck  nucleon in an excited resonance state (RES);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' and (d) nonresonant interactions with  a quark within the nucleon (DIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 4-3: Reaction mechanisms for lepton-nucleus scattering (a) quasielastic scat-  tering (QE) where one nucleon is knocked out of the nucleus, (b) 2p2h where two  nucleons are knocked out of the nucleus, (c) RES resonance production where a nu-  cleon is excited to a resonance which decays to a nucleon plus meson(s), and (d) DIS  where the lepton interacts with a quark in the nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  All these interaction processes have to be described in detailed in event generators  like GENIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' GENIE started as an event generator that could exclusively handle  93  V1  SSneutrino interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In recognition of the importance of electron scattering, the  latter was added as a new option in close conjunction with the neutrino scattering  section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' As much as possible, the neutrino cross section references vector and axial  contributions separately and uses the same modeling for vector interactions as the  electron section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Some models were developed separately for electrons and others  were developed for both applications in tandem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  An earlier electron version of GENIE (v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='10) had already been tested by com-  paring with inclusive (𝑒, 𝑒′) data [149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Although the QE peak was well-described for  a variety of energies and nuclei, the RES region was poorly described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' However, the  establishment of full compatibility between the electron and neutrino versions was  then still in its early stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  With this work, we significantly improved both neutrino and electron versions of  GENIE to address these and other issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We fixed significant errors in the previous  version, including an error in the Mott cross section in the electron QE Rosenbluth  interaction, a missing Lorentz boost in the MEC interaction affecting both electron  and neutrino interactions, and incorrect electron couplings used in the RES interac-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We worked to better integrate the electron and neutrino codes for QE and  MEC models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We also added more up-to-date models such as SuSAv2 [150].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These  changes have been incorporated in the latest GENIE version (v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We refer to  the electron-scattering component of the widely-used GENIE [33] event generator as  eGENIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The GENIE improvements can be seen in figure 4-4 [151].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The QE peak (at  𝜔 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c) predicted by the older GENIE v2 is too large and is slightly shifted  to higher energy transfer than the data, while the first simulated resonance peak  is at a larger energy transfer than the one observed in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The QE peak  predicted by the updated GENIE v3 has about the correct integral and is at the  correct energy transfer (but is slightly too narrow) and the first resonance peak is  located at 𝑚Δ − 𝑚 ≈ 300 MeV beyond the QE peak, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Details of the  calculations and of the discrepancies between GENIE v3 and the data are discussed  in detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0  2  4  6  3  10  ×  o  = 60  θ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='56 GeV,  v2  v3  b/sr/GeV]  µ  [  dE  Ω  d  σ  2  d  Energy Transfer [GeV]  Figure 4-4: Comparison between GENIE v2 and v3 descriptions of inclusive C(𝑒, 𝑒′)  scattering cross sections at 𝐸0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='56 GeV, 𝜃𝑒 = 60∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='24 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Black  points show the data, solid black line shows the GENIE v3 results and dashed black  line shows the GENIE v2 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  In this analysis, we specifically focused on testing our knowledge of the electron-  nucleus cross section by benchmarking eGENIE against existing inclusive electron  scattering data for different target nuclei, beam energies and scattering angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  goals are very similar to the ones in reference [149], but we test a much more modern  version of eGENIE and we also compare different models within eGENIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In addition,  if eGENIE describes electron-nucleus scattering well, then it would be an improvement  on the former empirical fit [152] and would be valuable for helping simulate a variety  of electron experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  For fixed incident beam energy and scattered electron angle, the dominant process  changes from QE at low energy transfer (𝜔 ≈ 𝑄2/2𝑚) through MEC to RES and to  DIS at high energy transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Therefore, examining the agreement of eGENIE with  data as a function of energy transfer can provide valuable insight into the specific  shortcomings of the eGENIE models and their implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  This separation  according to the underlying physics interactions gives valuable insights which are not  presently possible with neutrino cross sections, because only broad-energy beams are  available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The GENIE simulation framework offers several models of the nuclear ground  state, multiple models for each of the electron- or neutrino-nucleus scattering mecha-  95  nisms accompanied by various tunable model parameters, and a number of models for  hadronic final state interactions (FSI), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=', intranuclear rescattering of the outgoing  hadrons [33, 104, 153].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We describe the different models relevant for this work and  the electron-specific effects that we accounted for during the eGENIE development  below [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Since our goal is to use electron scattering data to validate neutrino interaction  modeling in GENIE, the GENIE code for electron and neutrino interactions are unified  in many places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The neutrino interacts with a nucleus via the weak interaction and  massive 𝑊 or 𝑍 exchange, whereas the electron interacts mostly electromagnetically  via massless photon exchange, as shown in figure 4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This causes the cross sections  to differ by an overall factor of  8𝜋2𝛼2  𝐺2  𝐹  1  𝑄4  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1)  when equations 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='18 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='19 are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In the code, both interactions use  the same nuclear ground state and many of the nuclear reaction effects, such as FSI,  are very similar or identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Except for mass effects and form factors, the electron  nucleus cross section can be obtained by setting the axial part of the interaction to  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We also accounted for isoscalar and isovector terms appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Many of the models reported in this work, except for SuSAv2, use the GENIE  implementation of the Local Fermi gas (LFG) model to describe the nuclear ground  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In the simplest Fermi gas model, nucleons occupy all momentum states up to  the global Fermi momentum 𝑘𝐹 with equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In the LFG model, the Fermi  momentum at a given radial position depends on the local nuclear density obtained  from measurements of nuclear charge densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' To account for this radial depen-  dence, GENIE selects an initial momentum for the struck nucleon by first sampling  an interaction location 𝑟 inside the nucleus according to the nuclear density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  nucleon momentum is then drawn from a Fermi distribution using the local Fermi  momentum 𝑘𝐹(𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Another commonly used nuclear model is the Relativistic Fermi Gas (RFG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Here  96  a global momentum distribution is used for the entire nucleus, independent of the  interaction location in the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' However, a high-momentum tail of nucleons with  momenta above the Fermi-momentum is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This tail is meant to approximately  account for the effects of two-nucleon short-range correlations [154,155] and follows  a 1/𝑘4 distribution, where 𝑘 is the nucleon momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  We consider two distinct sets of eGENIE configurations:  G2018, which uses the LFG nuclear model, the Rosenbluth cross section for  QE scattering, and the empirical MEC model [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This model set is formally  marked as the G18_10a_02_11a configuration of GENIE v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  GSuSAv2 [28], which follows the universal SuSAv2 super-scaling approach  to lepton scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  This new model set is included in the latest GENIE  v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0 release as the GEM21_11b_00_000 configuration for electron scattering  and G21_11b_00_000 for neutrino interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  In both model sets, RES is modeled using the Berger-Sehgal model [125] and DIS  reactions are modeled using Bodek and Yang [156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The models are described in more  detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  In QE interactions, a lepton scatters on a single nucleon, removing it from the  spectator 𝐴 − 1 nucleus unless final-state interactions lead to reabsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The  electron QE interaction in the G2018 configuration of GENIE uses the Rosenbluth  cross section with the vector structure function parametrization of reference [157].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  We corrected the implementation of this model for eGENIE and modified the cross  section to account for the identified issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This electron QE cross section differs in  important ways from the Valencia CCQE model [158] used in the G2018 configuration  for neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Most notably, the Rosenbluth treatment lacks medium polarization  corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  A new QE model in GENIE, based on the SuSAv2 approach [28, 159, 160], uses  superscaling to write the inclusive cross section in terms of a universal function inde-  pendent of momentum transfer and nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For EM scattering, the scaling function  may be expressed in the form  97  𝑓(𝜓′) = 𝑘𝐹  𝑑2𝜎  𝑑Ω𝑒𝑑𝜈  𝜎𝑀𝑜𝑡𝑡(𝑣𝐿𝐺𝑒𝑒′  𝐿 + 𝑉𝑇𝐺𝑒𝑒′  𝑇 ) ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2)  where 𝜓′ is a dimensionless scaling variable, 𝑘𝐹 is the nuclear Fermi momentum,  the denominator is the single-nucleon elastic cross section, 𝑣𝐿 and 𝑣𝑇 are known  functions of kinematic variables, and 𝐺𝑒𝑒′  𝐿 (𝑞, 𝜔) and 𝐺𝑒𝑒′  𝑇 (𝑞, 𝜔) are the longitudinal  and transverse nucleon structure functions linearly related to 𝐹 𝑒  1 and 𝐹 𝑒  2 [161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For  eGENIE, we extended the original neutrino implementation [28] to the electron case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The original SuSAv2 QE cross section calculations used a Relativistic Mean Field  (RMF) model of the nuclear ground state [162, 163].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  This approach includes the  effects of the real part of the nucleon-nucleus potential on the outgoing nucleons  which creates a “distorted” nucleon momentum distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Although GENIE lacks the option to use an RMF nuclear model directly, we  achieve approximate consistency with the RMF-based results by using a two-step  strategy for QE event generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' First, an energy and scattering angle for the out-  going lepton are sampled according to the inclusive double-differential cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  This cross section is computed by interpolating precomputed values of the nuclear  responses 𝐺𝑒𝑒′  𝐿 (𝑞, 𝜔) and 𝐺𝑒𝑒′  𝐿 (𝑞, 𝜔) which are tabulated on a two-dimensional grid in  (𝑞, 𝜔) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The responses were obtained using the original RMF-based SuSAv2  calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Second, the outgoing nucleon kinematics are determined by choosing its  initial momentum from an LFG distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The default nucleon binding energy used  in GENIE for the LFG model is replaced for SuSAv2 with an effective value tuned to  most closely duplicate the RMF distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The outgoing nucleon kinematics are  not needed for the comparisons to inclusive (𝑒, 𝑒′) data shown in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  MEC describes an interaction that results in the ejection of two nucleons from  the nucleus, thus is often referred to as 2p2h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' It typically proceeds via lepton inter-  action with a pion being exchanged between two nucleons or by interaction with a  nucleon in an Short Range Correlated (SRC) pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' MEC is far less understood than  other reaction mechanisms because, unlike the others, it involves scattering from two  98  nucleons simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' GENIE has several models for MEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The G2018 configuration of eGENIE uses the empirical MEC model [109] that  is useable for both electron- and neutrino-nucleus scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' It assumes that the  MEC peak for inclusive scattering has a Gaussian distribution in 𝑊 and is located  between the QE and first RES peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Although both versions of the model use the  same effective form factors, the amplitude of the MEC peak was tuned separately to  electron and neutrino scattering data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This model was developed in the context of  empirically fitting GENIE to MiniBooNE inclusive neutrino scattering data and is still  used for neutral-current interactions [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For charged-current neutrino interactions,  𝜈GENIE G2018 uses the very different Valencia 2p2h model [158,164] instead of the  empirical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  For the description of the 2p2h MEC contributions, the SuSAv2 model uses the  fully relativistic calculations from [165].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This treatment allows for a proper separation  of neutron-proton and proton-proton pairs in the final state via the analysis of the  direct-exchange interference terms [166].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This approach is capable of reproducing the  nuclear dynamics and superscaling properties observed in inclusive electron-nucleus  scattering reactions [167–169].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The latter serves as a robust test for nuclear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  It further provides an accurate description of existing neutrino data [169–173].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' As in  the case for the SuSAv2 QE model, we extended the original GENIE implementation  of SuSAv2 MEC for neutrinos to the electron case for eGENIE [28,159,174,175].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The SuSAv2 MEC approach is the only fully relativistic model that can be ex-  tended without approximations to the full-energy range of interest for neutrino scat-  tering events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Therefore, it is a very promising modeling choice for present and future  neutrino experiments for one of the least understood interaction channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  RES production in GENIE is simulated using the Berger-Sehgal model [125], in  which the lepton interacts with a single moving nucleon and excites it to one of  16 resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The cross sections are calculated based on the Feynman-Kislinger-  Ravndal (FKR) model [176], without any interferences between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Form factors  are derived separately for vector and axial probes [110] but have not been updated  to include recent electron scattering results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  99  The GENIE treatment of DIS used in this work is based on that of Bodek and  Yang [156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Hadronization is modeled using an approach which transitions gradually  as a function of the hadronic invariant mass 𝑊 between the AGKY model [177] and  the PYTHIA 6 model [178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  At low 𝑊 values, the Bodek-Yang differential cross  section is scaled by tunable parameters that depend on the multiplicity of hadrons in  the final-state [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Integration of the RES and DIS contributions is complicated by the need for a  model of nonresonant meson production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' There is no definite separation of RES and  DIS contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' GENIE makes a sharp cutoff at 𝑊 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='93 GeV in the latest tune  and uses a suppression factor to enable usage of the Bodek-Yang cross section at low  𝑊 in place of a true nonresonant model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These features were recently retuned by the  GENIE collaboration using measurements of charged-current 𝜈𝜇 and ¯𝜈𝜇 scattering on  deuterium [179].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The W cutoff and suppression factors apply to both electron- and  neutrino-nucleus models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Final state interactions of outgoing hadrons with the residual nuclei are calcu-  lated in eGENIE using the INTRANUKE [153,180] package and one of two options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The first, hA, an empirical data-driven method, uses the cross-section of pions and  nucleons with nuclei as a function of energy up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV and the CEM03 [181]  calculation for higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The second, hN, is a full intra-nuclear cascade cal-  culation of the interactions of pions, kaons, photons, and nucleons with nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In  the hN model, each outgoing particle can interact successively with any or all the  nucleons it encounters on its path leaving the nucleus, and any particles created in  those interactions can also subsequently reinteract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The ability of the two models to  describe hadron-nucleus data is very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The eGENIE G2018 configuration uses  the hA FSI model, while GSuSAv2 uses hN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' However, the choice of FSI model has no  effect on the inclusive cross sections considered in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  100  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  Inclusive Electron Scattering Data Comparisons  To test eGENIE, we compared inclusive electron scattering data to theoretical pre-  dictions made using two different program configurations which differ in their choice  of QE and MEC models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Namely, G2018 adopts the Rosenbluth model for QE in-  teractions and the empirical Dytman model for MEC events, while GSuSAv2 uses  SuSAv2 for both QE and MEC interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figures 4-5, 4-6 and 4-7 show the inclusive C(𝑒, 𝑒′) cross sections for a wide range  of beam energies and scattering angles compared to the G2018 and GSuSAv2 mod-  els [151,182–189].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The QE peak is the one at lowest energy transfer (𝜈 ≈ 𝑄2/2𝑚) in each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The next peak at about 300 MeV larger energy transfer corresponds to the ∆(1232)  excitation and the “dip-region” is between the two peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The ∆ peak in the data  is separated from the QE peak by less than the 300 MeV ∆-nucleon mass difference,  indicating that it is shifted in the nuclear medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This shift is more visible at lower  momentum transfer where the ∆ peak is more prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  GSuSAv2 clearly describes the QE and dip-regions much better than G2018, es-  pecially at the three lowest momentum transfers, as shown in figure 4-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' G2018 has  particular difficulty describing the data for 𝐸0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='24 GeV and 𝜃𝑒 = 60∘, where  𝑄2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='05 GeV2 at the QE peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' G2018 also predicts too small a width for the QE  peak and too small a MEC contribution for 𝐸0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='56 GeV and 𝜃𝑒 = 60∘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' GSuSAv2  describes both features far better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  At intermediate momentum transfers shown in figure 4-6, GSuSAv2 describes  the data somewhat better than G2018, although it overpredicts the dip-region cross  section at 𝐸0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='299 GeV and 𝜃𝑒 = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5∘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The MEC contribution for G2018 appears  to be much too small for 𝐸0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='222 GeV and 𝜃𝑒 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='54∘ (𝑄2  𝑄𝐸 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='33 GeV2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Both model sets significantly disagree with the data in the resonance region, where  they use the same RES and DIS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='961 GeV, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5∘ and the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='222 GeV,  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='54∘ data are taken at almost identical 𝑄2  𝑄𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The lower beam-energy data is more  transverse, since it is at larger scattering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The GSuSAv2 MEC contribution is  101  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0  50  3  10  ×  o  = 60  θ  C, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='24 GeV,  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0  50  3  10  ×  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0  50  3  10  ×  o  = 36  θ  C, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='56 GeV,  12  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0  50  3  10  ×  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0  2  4  6  3  10  ×  o  = 60  θ  C, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='56 GeV,  12  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0  2  4  6  3  10  ×  GSuSav2  G2018  b/sr/GeV]  µ  [  dE  Ω  d  σ  2  d  Energy Transfer [GeV]  Energy Transfer [GeV]  Figure 4-5: Comparison of inclusive C(𝑒, 𝑒′) scattering cross sections for data and for  GENIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (left) data vs GSuSAv2 and (right) data vs G2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (top) 𝐸0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='24 GeV,  𝜃𝑒 = 60∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='05 GeV2 (middle) 𝐸0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='56 GeV, 𝜃𝑒 = 36∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='11  GeV2, and (bottom) 𝐸0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='56 GeV, 𝜃𝑒 = 60∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='24 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Black points  show the data, solid black lines show the total GENIE prediction, colored lines show  the contribution of the different reaction mechanisms: (blue) QE, (red) MEC, (green)  RES and (orange) DIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  similar for both data sets, but the G2018 MEC contribution is far smaller for the  higher beam-energy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The GSuSAv2 MEC contribution describes the dip-region  better in the higher beam-energy data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The RES model appears to agree with the  102  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0  5  3  10  ×  o  = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  θ  C, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='961 GeV,  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0  5  3  10  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  0  1  2  3  10  ×  o  = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  θ  C, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='299 GeV,  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  0  1  2  3  10  ×  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  0  20  40  3  10  ×  o  = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='54  θ  C, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='222 GeV,  12  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  0  20  40  3  10  ×  GSuSav2  G2018  b/sr/GeV]  µ  [  dE  Ω  d  σ  2  d  Energy Transfer [GeV]  Energy Transfer [GeV]  Figure 4-6: Comparison of inclusive C(𝑒, 𝑒′) scattering cross sections for data and  for GENIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (left) data vs GSuSAv2 and (right) data vs G2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (top) 𝐸0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='96  GeV, 𝜃𝑒 = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='32 GeV2, (middle) 𝐸0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='30 GeV, 𝜃𝑒 = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5∘ and  𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='54 GeV2, and (bottom) 𝐸0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='22 GeV, 𝜃𝑒 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='33 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Black points show the data, solid black lines show the total GENIE prediction, colored  lines show the contribution of the different reaction mechanisms: (blue) QE, (red)  MEC, (green) RES and (orange) DIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  data slightly better for the lower beam-energy, more transverse, data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  At the highest momentum transfers (𝑄2 ≈ 1 GeV2) shown in figure 4-7, the dis-  agreement at the larger energy transfers is far greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The empirical G2018 MEC  103  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  3  10  ×  o  = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  θ  C, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='501 GeV,  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  3  10  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  0  2  4  3  10  ×  o  = 16  θ  C, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='595 GeV,  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  0  2  4  3  10  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  o  = 20  θ  C, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='595 GeV,  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  GSuSav2  G2018  b/sr/GeV]  µ  [  dE  Ω  d  σ  2  d  Energy Transfer [GeV]  Energy Transfer [GeV]  Figure 4-7: Comparison of inclusive C(𝑒, 𝑒′) scattering cross sections for data and for  GENIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (left) data vs GSuSAv2 and (right) data vs G2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (top) 𝐸0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='501 GeV,  𝜃𝑒 = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='92 GeV2, (middle) 𝐸0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='595 GeV, 𝜃𝑒 = 16∘ and 𝑄2  𝑄𝐸 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='04  GeV2, and (bottom) 𝐸0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='595 GeV, 𝜃𝑒 = 20∘ and 𝑄2  𝑄𝐸 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Black points  show the data, solid black lines show the total GENIE prediction, colored lines show  the contribution of the different reaction mechanisms: (blue) QE, (red) MEC, (green)  RES and (orange) DIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  model contributions are negligible, in marked contrast to the GSuSAv2 MEC con-  tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The RES and DIS contributions are very significant at high 𝑄2 and in  general the GENIE model is larger than the data in the region dominated by RES  104  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0  10  20  3  10  ×  o  = 60  θ  Fe, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='56 GeV,  56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0  10  20  3  10  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0  20  40  3  10  ×  o  = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  θ  Fe, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='961 GeV,  56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0  20  40  3  10  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  0  5  10  3  10  ×  o  = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  θ  Fe, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='299 GeV,  56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  0  5  10  3  10  ×  GSuSav2  G2018  b/sr/GeV]  µ  [  dE  Ω  d  σ  2  d  Energy Transfer [GeV]  Energy Transfer [GeV]  Figure 4-8: Comparison of inclusive Fe(𝑒, 𝑒′) scattering cross sections for data and for  GENIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (left) data vs GSuSAv2 and (right) data vs G2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (top) Fe(𝑒, 𝑒′), 𝐸0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='56  GeV, 𝜃𝑒 = 60∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='24 GeV2, (middle) Fe(𝑒, 𝑒′), 𝐸0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='96 GeV, 𝜃𝑒 = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5∘  and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='32 GeV2, (bottom) Fe(𝑒, 𝑒′), 𝐸0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='30 GeV, 𝜃𝑒 = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='54  GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Black points show the data, solid black lines show the total GENIE prediction,  colored lines show the contribution of the different reaction mechanisms: (blue) QE,  (red) MEC, (green) RES and (orange) DIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  interactions [149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In addition, GENIE does not include the nuclear medium depen-  dent ∆-peak shift, so that the predicted location of the ∆-peak is at larger energy  transfer than that of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  105  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  0  50  100  3  10  ×  o  = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='54  θ  Ar, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='222 GeV,  40  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  0  50  100  3  10  ×  GSuSav2  G2018  b/sr/GeV]  µ  [  dE  Ω  d  σ  2  d  Energy Transfer [GeV]  Energy Transfer [GeV]  Figure 4-9: Comparison of inclusive Ar(𝑒, 𝑒′) scattering cross sections for data and for  GENIE at 𝐸0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='22 GeV, 𝜃𝑒 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='33 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (left) data vs GSuSAv2  and (right) data vs G2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Black points show the data, solid black lines show the  total GENIE prediction, colored lines show the contribution of the different reaction  mechanisms: (blue) QE, (red) MEC, (green) RES and (orange) DIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 4-8 shows the inclusive Fe(𝑒, 𝑒′) cross sections for several beam energies and  scattering angles compared to the G2018 and GSuSAv2 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The GSuSAv2 model  describes the QE region better for all three data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' As described previously, the  GSuSAv2 MEC model is independently calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The empirical G2018 MEC model  was fit using GENIE v2 QE and RES models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The fit will have to be redone once the  QE and RES models stabilize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The GSuSAv2 MEC contributions are significantly  larger than the empirical G2018 MEC contributions and match the dip-region data  far better at 𝑄2  𝑄𝐸 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='24 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='32 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' However, it overpredicts the dip-region  cross section at 𝑄2  𝑄𝐸 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='54 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The RES and DIS models describe the Fe data  better than the C data at large energy transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 4-9 shows the inclusive Ar(𝑒, 𝑒′) cross sections for 𝐸0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='222 GeV and  𝜃𝑒 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='54∘ [185] compared to the G2018 and GSuSAv2 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The GSuSAv2 model  reproduces the data very well in the QE-peak region and the G2018 reproduces the  data moderately well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The GSuSAv2 MEC model describes the dip-region much  better than the G2018 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Again, there is significant disagreement with the RES  and DIS models at larger energy transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The quality of the agreement between data and GENIE depends more on the  beam energy and angle than on the target mass from C to Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' There is a possible  106  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  0  500  1000  o  = 20  θ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='445 GeV,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  0  500  1000  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  0  50  o  = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='98  θ  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='245 GeV,  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  0  50  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  4  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  o  = 41  θ  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5 GeV,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  4  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  Proton  Deuterium  [nb/sr/GeV]  dE  Ω  d  σ  2  d  Energy Transfer [GeV]  Energy Transfer [GeV]  Figure 4-10: Comparison of inclusive proton (left) and deuterium (right) (𝑒, 𝑒′) scat-  tering cross sections for data and for GENIE using G2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (top) 𝐸0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='445 GeV  and 𝜃𝑒 = 20∘, (middle) 𝐸0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='245 GeV and 𝜃𝑒 = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='98∘, and (bottom) 𝐸0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5 GeV  and 𝜃𝑒 = 41∘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Black points show the data, solid black lines show the total GENIE  prediction, colored lines show the contribution of the different reaction mechanisms:  (green) RES and (orange) DIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The first peak at lowest energy transfer is the ∆(1232)  resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  momentum-transfer dependent shift in the location of the SuSAv2 QE peak in Fe due  to the extrapolation via scaling from C to Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The GSuSAv2 QE model generally  describes the data as well as or better than the G2018 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  107  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  0  100  200  3  10  ×  o  = 4  θ  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='499 GeV,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  0  100  200  3  10  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  0  50  100  3  10  ×  o  = 4  θ  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='999 GeV,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  0  50  100  3  10  ×  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  0  20  40  3  10  ×  o  = 4  θ  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='993 GeV,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  0  20  40  3  10  ×  Proton  Deuterium  b/sr/GeV]  µ  [  dE  Ω  d  σ  2  d  Energy Transfer [GeV]  Energy Transfer [GeV]  Figure 4-11: Comparison of inclusive proton (left) and deuterium (right) (𝑒, 𝑒′) scat-  tering cross sections for data and for GENIE using G2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (top) 𝐸0 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='499 GeV  and 𝜃𝑒 = 4∘, (middle) 𝐸0 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='699 GeV and 𝜃𝑒 = 4∘, and (bottom) 𝐸0 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='993 GeV  and 𝜃𝑒 = 4∘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Black points show the data, solid black lines show the total GENIE  prediction, colored lines show the contribution of the different reaction mechanisms:  (green) RES and (orange) DIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The first peak at lowest energy transfer is the ∆(1232)  resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The GSuSAv2 MEC model appears to be significantly superior to the empirical  MEC model, especially at 𝑄2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5 GeV2 or at smaller scattering angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The empir-  ical MEC contribution is often much smaller than needed to explain the dip-region  108  cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' However, as an empirical model, it can be tuned to better describe the  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  eGENIE dramatically overpredicts the large-energy transfer data at higher mo-  mentum transfers (𝑄2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5 GeV2), indicating issues with the RES (Berger-Sehgal)  and DIS (Bodek and Yang) models used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  This discrepancy at larger momentum and energy transfers is due to the elemen-  tary electron-nucleon cross section in the RES and DIS regions, rather than to the  nuclear models, since eGENIE also significantly overpredicts the proton and deuteron  cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' That is the case especially above the ∆ peak, as shown in figures 4-10  and 4-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The discrepancy becomes even more pronounced due to the double counting  of processes common across the two interaction channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This shows that tuning the  RES and DIS models to neutrino data [179] is not sufficient to constrain the vector  part of the cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  Implications For Neutrinos  Electron-scattering data can be a very effective tool for testing neutrino event gener-  ators due to the similarity between the interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure 4-12 shows the remark-  ably similar cross-section shapes for electron-nucleus and neutrino-nucleus scattering  for semi-exclusive 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='16 GeV lepton-carbon scattering with exactly one proton with  𝑄2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 GeV2 and 𝑃𝑝 ≥ 300 MeV/c, no charged pions with 𝑃𝜋 ≥ 70 MeV/c and  no neutral pions or photons of any momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This corresponds approximately to  the JLab CLAS detector thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' When comparing electron and neutrino distri-  butions, the electron events are each weighted by 𝑄4 to reflect the difference in the  electron and neutrino elementary interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Exploiting these similarities within the same code is invaluable for minimizing the  systematic uncertainties of future high-precision neutrino-oscillation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Os-  cillation analysis uncertainties exceeding 1% for signal and 5% for backgrounds may  substantially degrade the experimental sensitivity to CP violation and mass hierar-  chy [146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Such uncertainties already include the relevant neutrino-nucleus interaction  109  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  Energy Transfer [GeV]  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  Area Normalized  ν  e  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  ]  2  /c  2  [GeV  2  Q  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  Area Normalized  ν  e  Figure 4-12:  Comparison of semi-exclusive 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='16 GeV lepton-carbon scattering for  𝑄2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The number of generated events is plotted versus energy transfer  (left) and 4-momuntm transfer squared (right) for events with exactly one proton  with 𝑃𝑝 ≥ 300 MeV/c, no charged pions with 𝑃𝜋 ≥ 70 MeV/c and no neutral pions or  photons of any momentum for eGENIE electrons (orange) and GENIE CC 𝜈𝜇 (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The electron events have been weighted by 𝑄4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Both curves are area normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These uncertainties are driven by the choices of the nuclear models and  cross-section configurations available in event generators like GENIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 4-13 shows that there is a larger difference among QE scattering models  than there is between QE electron and neutrino scattering using the same nuclear  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' All six panels show a “ridge”, a maximum in the cross section as a function of  energy transfer and momentum transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The length of the ridge, namely the decrease  in intensity as the energy and momentum transfers increase, reflects the momentum  transfer dependence of the nucleon form factors used in the cross section model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  width of the distribution perpendicular to the ridge reflects the width of the nuclear  momentum distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The momentum distribution of the LFG model cuts off at about 260 MeV/c for  C, whereas the RMF and the RFG models have “tails” that extend to much larger  momenta, as shown in figure 4-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The Nieves cross section decreases more slowly  with momentum transfer than the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For GSuSAv2, the electron cross section  appears to decrease slightly faster with momentum transfer than the neutrino cross  section, possibly reflecting differences in the axial and vector nucleon form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Our ability to use the GENIE code to transfer knowledge gained from electron  110  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  [GeV/c]  3  q  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  [GeV]  ω  Relativistic MF  SuSav2  ν  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  [GeV/c]  3  q  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  [GeV]  ω  Local FG  Nieves  ν  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  [GeV/c]  3  q  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  [GeV]  ω  Relativistic FG  Llewellyn-Smith  ν  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='6  [GeV/c]  3  q  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='8  1  [GeV]  ω  Relativistic MF  e SuSav2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  [GeV/c]  3  q  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='8  1  [GeV]  ω  Local FG  e Rosenbluth  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='6  [GeV/c]  3  q  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  [GeV]  ω  Relativistic FG  e Rosenbluth  Figure 4-13:  Number of simulated events for QE scattering on 12C at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='161 GeV  with 𝑄2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 shown as a function of the energy transfer 𝜔 and the momentum  transfer 𝑞3 = |⃗𝑞 | for all the available nuclear models in GENIE for neutrinos (top)  and for electrons (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (left) the GSuSAv2 model which uses a Relativistic Mean  Field momentum distribution, (middle) the Nieves or Rosenbluth cross section with  the Local Fermi Gas momentum distribution, and (right) the Llewellyn-Smith or  Rosenbluth cross section with the Relativistic Fermi Gas momentum distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The electron events have been weighted by 𝑄4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  scattering depends critically on the implementation of its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Because of  its modular design, all reaction models in GENIE use the same nuclear model, for  instance RFG or LFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Although the electron scattering capability was added after  the initial code release, many of the reaction models used electron scattering data  to construct the vector components of neutrino interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This was true for the  RES [110,125] and the DIS [156] interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The difference between vector neutrino-  and electron-scattering is an overall factor illustrated in equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 and an appro-  priate change in the form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Similarly, figure 4-15 shows that the distribution of MEC events is very similar for  electrons and for neutrinos within the same model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Thus, measurements of electron  scattering will be able to significantly constrain models of neutrino scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Both QE and MEC models use the same vector form factors for neutrino and for  111  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  [GeV/c]  p  p  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='08  Probability Density  LFG  RFG  Figure 4-14:  Initial momentum distribution of protons in simulated QE C(𝑒, 𝑒′𝑝)  events at 𝐸 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='161 GeV for the Local Fermi Gas (solid histogram) and Relativistic  Fermi Gas (dotted histogram) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The two curves are normalized to have the  same area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  [GeV/c]  3  q  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  [GeV]  ω  SuSav2  MEC  ν  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  [GeV/c]  3  q  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  [GeV]  ω  e  Figure 4-15: Number of simulated events as a function of the energy transfer 𝜔 and  of the momentum transfer 𝑞3 = |⃗𝑞 | for neutrinos (left) and for electrons (right) using  GSuSav2 for MEC interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The electron events have been scaled by 𝑄4 and all  the samples have been generated with 𝑄2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  electron scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' QE models can use nucleon form factors from electron scatter-  ing [157], but MEC models must calculate the form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The GENIE DIS cross section comes from the Bodek-Yang model [156] for the  full cross section which extends to 𝜋N threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The cross section is scaled in the  RES region so that it agrees with neutrino-deuterium data [179].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Since a single factor  112  is used to fit the model to the neutrino data, the high-quality electron-proton and  electron-deuterium data will be poorly described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' While the total neutrino cross sec-  tion and some of the hadronic content of the final state are loosely constrained by the  neutrino-deuterium data, the vector component of the models is poorly constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The QE models describe the data reasonably well in the low energy-transfer region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Similarly, the largest energy-transfer portions of figures 4-7 and 4-10 show a reasonable  agreement between GENIE and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' However, at intermediate energy transfer, the  RES modeling disagrees with the data for both nuclear and nucleon targets, also  observed in [149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This is due to the use of RES form factors that are not up-to-date  and the way the nonresonant contribution was modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Improvements are in progress but are not simple and therefore not available at this  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A possible short-term fix would be to include the electron-proton and electron-  deuterium inclusive electron-scattering models of Bosted and Christy [190,191].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Alter-  natively, the vector resonant form factors could be updated using electroproduction  data from JLab and elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A fit to that data is available [192] and partially  implemented in GENIE, but it does not include nonresonant scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  A more  comprehensive solution would be to use the recent DCC model [193, 194] to simul-  taneously describe both resonant and nonresonant scattering of both electrons and  neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The comparisons presented in this work are focused on inclusive electron cross-  section measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Yet, forthcoming neutrino oscillation experiments like DUNE  will use 4𝜋-coverage liquid argon time projection chamber tracking detectors in order  to investigate exclusive interaction channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Until 2021, no prior tests of the event  generator performance against such exclusive channels existed with electron scattering  events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Chapters 5 and 6 present the work that alleviated this shortage using wide  phase-space electron scattering data sets from Hall B and the e2a experiment at JLab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  113  114  Chapter 5  The CLAS Electrons-For-Neutrinos  Experiment At Thomas Jefferson  National Laboratory  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  Continuous Electron Beam Accelerator Facility  To test the event generator performance against wide phase-space exclusive interac-  tion channels with electron-nucleus scattering events, data sets from Hall B and the  e2a experiment at Thomas Jefferson Laboratory (JLab) in Newport News VA were  used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The Continuous Electron Beam Accelerator Facility (CEBAF) electron accelera-  tor [195] provided electron beams up to 6 GeV to three experimental halls, referred  to as halls A, B, and C, until 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' CEBAF has since been upgraded to 12 GeV and  a new hall (Hall D) was constructed in the context of the 12 GeV upgrade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A laser  shone on a gallium arsenide cathode to emit photo-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The electrons were then  accelerated by cavities operating at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='497 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Each of the halls received electron  bunches every 2 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Those bunches can have different electron densities so that each  hall could be supplied with a different beam current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A schematic of the facility is  shown in figure 5-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  115  Figure 5-1: Schematic view of the accelerator facility and the experimental halls at  Jefferson Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [196].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The electrons are accelerated through the beamline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The latter consists of two  parallel linacs that are connected with two arcs with curvature radii of 80 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Each  one of the linacs could increase the electron beam energy by 550 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The beam was  circulated up to 5 times, for a maximum energy of 6 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The beam energy spread  was ≈ 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  The CEBAF Large Acceptance Spectrometer  The e2a data sets were collected at Hall B using the CEBAF Large Acceptance  Spectrometer (CLAS) spectrometer shown in figure 5-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The spectrometer had ≈  50% angular coverage and ran at luminosities up to 1034 cm−2 sec−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  CLAS used a toroidal magnet for momentum reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The magnet di-  vided CLAS into six almost identical sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Each sector had three regions of drift  chambers (DC) for charged particle trajectory measurements, threshold Cherenkov  counters (CC) for electron identification, scintillator counters (SC) for timing and  for charged hadron identification, and electromagnetic calorimeters (EC) for electron  identification and for photon and neutron detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The 𝜃-angle coverage for the DC  is 8𝑜 − 140𝑜, for the SC 9𝑜 − 140𝑜, and for the EC 8𝑜 − 45𝑜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  116  Recirculation  Arcs  FELFacility  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4-GeV Linac  (20 Cryomodules)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4-GeV Linac  45-MeVInjector  (20 Cryomodules)  Helium  (2 1/4 Cryomodules)  Refrigerator  Extraction  Elements  End  Stationse– beam  Scintillators (timing)  Drift chambers  (tracking)  Calorimeters  (energy)  Cherenkov (e– ID)  Target  ≈8m  Figure 5-2:  Drawing of the CLAS detector showing the sector structure and the  different detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The beam enters from the upper left side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The target is located  at the center of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  Toroidal Magnet  The torus magnet consisted of six sectors and was made of iron-free super conducting  coils oriented around the beam axis [197].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The torus had a 2 T maximum magnetic  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The coils had a length of 5 m, and a width of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The magnetic field was 5  times smaller at large angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The central region of the spectrometer had no magnetic  field allowing for polarized target operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  Drift Chambers  The Drift Chambers (DC) are used for tracking and momentum measurements of the  charged particles produced out of the interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The DC have the structure shown  117  Figure 5-3: The CLAS superconducting toroidal magnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [197].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  in figure 5-4 (left) and more details can be found in [198].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 5-4: (Left) Illustration of the region 3 drift chamber structure for one of the  CLAS sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (Right) Schematic representation of the thee drift chamber regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figures adapted from [198].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Each one of the modules is filled with a gas mixture of 90% Argon and 10% CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The charged particles that transverse the DC ionize the gas mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The produced  electrons drift towards the anode wires and the corresponding ions drift towards the  cathode wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The relevant electrical signal is used to determine the distance from  the charged particle trajectory to the wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  As the ionization electrons approach the wires, the increased electric field results in  the multiplication of the electron-atom collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The latter results in the production  of more electrons by a factor of ≈ 104 and results into an “avalanche”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The CO2  molecule serves as a quenching gas and prevents the creation of secondary avalanches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  118  Coil  Beam  DirectionBeam  Sector-plate  Back-plate  Gas window  Endplate  Wire direction  Endplate  Circuit boardRegion 3  Region 1  beam  Region 2  mThe DC consist of three radial regions as shown in figure 5-4 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Regions 1 and  3 are located in a low magnetic field region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Each one of these regions is made of two  superlayers, one axial and one stereo (at a stereo angle of 6𝑜), to allow for complete  coordinate reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The layers consist of six layers of 192 hexagonal drift cells  with the 20-micron sense wire at the center and six shared field wires creating the  electric field, with the exception of the innermost layer in region 1, which is made of  four layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The DC structure provides a position resolution ≈ 400 𝜇m, ≈ 5 mrad for  the angular variables and ≈ 1% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5%) for the hadron (electron) momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  Electromagnetic Calorimeter  The electromagnetic calorimeter (EC) [199] was used to trigger on and to identify  the electons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' It further had a high momentum neutron detection efficiency of ≈ 50%  and a high photon detection efficiency for energies above 300 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' It was also used  to measure neutral particles with an efficiency of ≈ 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' It consists of 6 modules  corresponding to each one of the detector sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Each one of the modules has a  triangular shape with a projected vertex at the CLAS target location ≈ 5 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Each  element has a thickness corresponding to 16 radiation lengths and is made of 39 layers  of alternating 10 mm thick scintillator strips and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 mm of lead sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' All the layers  have triangular shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In each layer, the scintillator strips are parallel to a given  side of the triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Each subsequent layer is rotated by 120𝑜 with respect to the  previous layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' That results in the formation of three views, namely u,v, and w, as  can be seen in figure 5-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Each view consists of 13 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The scintillator material thickness for each EC  module is 39 cm and the lead one is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This proportionality between the scin-  tillator and the lead layers leads to one third of the shower energy being deposited  in the scintillator part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The time resolution for electrons and neutrons is 200 and  600 ps, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The 13 layers are combined into an inner (5 layers) and an outer  (8 layer) stack in order to provide longitudinal sampling of the showers and a hadron  identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  119  Figure 5-5: Schematic view of a CLAS electromagnetic calorimeter module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figures  adapted from [199].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The EC energy resolution is given by the expression  ∆𝐸  𝐸  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='003 +  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='093  √︀  𝐸 [𝐺𝑒𝑉 ]  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  Cherenkov Counters  There are six Cherenkov Counters (CC) [200], which are used to trigger on the elec-  trons and to separate between pions and electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A schematic of the CC is shown  in figure 5-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Each CC covers the polar angular range of 𝜃 = 8 - 45𝑜 and consists of eighteen  regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Each region is made of two modules called segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The CC is filled with  perfluorobutane (𝐶4𝐹10) gas with an index of refraction of n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='00153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' That yields  a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5 GeV energy threshold for pions and a 10 MeV threshold for electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' When  the particle velocity (v) is greater than the speed of light (c) in the detector medium  (c/n), Cherenkov light is emitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The produced light is then directed to the light  collections cone using elliptical and hyperbolic mirrors in order to get focused to the  PMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="  120  Scintillatorbars  U-plane  Leadsheets  V - plane  W-plane  Fiber Light Guides  (front)  Fiber Light Guides  (rear)  PMT'sFigure 5-6: Optical arrangement of one of the optical modules of the CLAS Cherenkov  detector, showing the optical and light collection components." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figures adapted  from [200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7  Time-Of-Flight Detector  The time-of-flight detectors (TOF) [201] were made of scintillator paddles and are  shown in figure 5-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Each scintillator was 5 cm thick and 10-15 cm wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' They ranged  in length between 25-450 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Using the time-of-flight (t) from the TOF subsystem  and the distance (d) based on tracking information from the DC, the velocity of the  charged particles would be obtained (v = d/t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The particle mass would be calculated  by combining the information of the particle momentum from the DC, as shown in  equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  𝑚 = 𝑝  √︀  1 − 𝛽2  𝛽  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2)  There exist 57 TOF paddles in each sector of the detectororiented perpendicular  to the beamline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Each one covers ≈ 2𝑜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The overall coverage in 𝜃 starts at 8𝑜 and  goes up to 142𝑜, organized in four panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' There is a PMT attached at the end of  each paddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The time resolution for electrons in the TOF is 163 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  121  elliptical mirror  sectorcenterline  PMT  magnetic  shield  lightcollection  cone  cylindrical  mirror  electron  hyperbolicmirror  trackFigure 5-7: Schematic view of the TOF counters in one sector illustrating the grouping  into four panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [201].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  E2a Targets  The Hall B e2a experiment ran between April 15-May 27 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' During the data taking  period, data were collected using the energies and targets presented in table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1: Target areal densities and integrated charges for the 𝑒4𝜈 data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Target  Energy  Areal Density [g/cm2]  Integrated Charge [mC]  3He  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2770  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='08  3He  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2770  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='08  3He  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2770  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='11  4He  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5375  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='17  4He  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5375  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='98  12C  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='08  12C  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='79  12C  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1786  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='84  56Fe  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1181  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='22  56Fe  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1181  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='31  The 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV data sets used a torus current of 2250 A, while the  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV ones were obtained with a 750 A torus current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The beam current ranged  122  between 3-18 nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The solid targets (12C and 56Fe) were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9 cm2 square plates  with a thickness of 1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The liquid targets (3He and 4He) were stored in cylindrical-  shaped vessel with a diameter of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' More details on the used data sets can be  found in the already published analyses [202–207].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  123  124  Chapter 6  Electrons-For-Neutrinos Results  [Nature 599, 565–570 (2021)]  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  Electron Data Mining Analysis  As discussed in section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5, the incomplete lepton-nucleus knowledge can be lever-  aged using the fact that electrons and neutrinos interact similarly with nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' While  previous work has compared these interaction models with inclusive electron scatter-  ing [42,149], this analysis is the first comparison of electron scattering data with these  interaction models where events with one or more detected hadrons are used [208].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The data presented here can therefore test and constrain neutrino-nucleus interaction  models to be used in analysis of neutrino oscillation measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  For the purposes of the “Electrons-For-Neutrinos” (e4𝜈) analysis, electron scatter-  ing data sets from the CEBAF Large Acceptance Spectrometer (CLAS) [209] at the  Thomas Jefferson National Accelerator Facility (JLab) were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The components  of the detector are described in detail in chapter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The e2a data sets detailed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8 were used, which were measured in 1999  and reported in many published analyses [202–207].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These include electron scattering  events on 4He, 12C, and 56Fe nuclei at beam energies of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The beam energy equaled the injector energy plus the pass number times the linac  energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The three-pass beam energy was measured using the Hall A arc measurement  and the four pass energy was measured using the Hall C arc measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These  gave a central linac energy of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0979 GeV and the Hall B one-, two-, and four-pass  125  beam energies of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' An uncertainty of 2×10−3  was assigned to these energies, based on the difference between the Hall A and Hall C  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The incident energies used in this analysis span the range of typical  accelerator-based neutrino beam energies [20,210], as can be seen in figure 6-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  carbon data are relevant for scintillator-based experiments such as MINER𝜈A and  NO𝜈A [211] and similar to the oxygen in water-based Cherenkov detectors such as  Super-Kamiokande (SK) [22, 23] and HK [212].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The iron is similar to the argon in  the liquid argon time projection chambers of MicroBooNE [93], the Fermilab short-  baseline oscillation program [94] and DUNE [213].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Many nuclear interaction processes  are mass dependent, so it is important to measure a range of target nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  0  1  2  3  4  5  [GeV]  ν  E  Flux [arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=']  ν  (a)  1  2  3  4  5  [GeV]  ν  E  Flux [arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=']  ν  DUNE  T2K  NOvA  (b)  Figure 6-1: The expected energy distribution of different 𝜈𝜇 beams, (left) before  oscillation at the near detector and (right) after oscillation at the far detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  vertical lines show the three electron beam energies of this measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The NO𝜈A  far-detector beam flux is calculated using the near detector flux and the neutrino  oscillation parameters from [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Electrons with energies 𝐸𝑒 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='55 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 GeV for 𝐸𝑏𝑒𝑎𝑚 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257,  and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV respectively, and angles 15∘ ≤ 𝜃𝑒 ≤ 45∘ were detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Protons  with momenta 𝑝𝑝 ≥ 300 MeV/𝑐 and angles 𝜃𝑝 ≥ 10∘, charged pions with momenta  𝑝𝜋 ≥ 150 MeV/𝑐 and angles 𝜃𝜋+ ≥ 10∘ and 𝜃𝜋− ≥ 22∘, and photons with energy  𝐸𝛾 ≥ 300 MeV and 8 ≤ 𝜃𝛾 ≤ 45∘ were detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Separate fiducial cuts were applied  for electrons, negatively-charged pions, positively-charged particles, and photons, to  select momentum-dependent regions of CLAS where the detection efficiency was con-  126  stant and close to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These hadron detection thresholds are similar to those of  neutrino detectors [87], however neutrino detectors have full angular coverage and  lower lepton energy thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Apart from the aforementioned momentum and angular cuts, additional angular  outlines are applied to account for the detector acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The minimum electron  angle as a function of electron momentum 𝑝 for each beam energy was determined as  𝜃1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  𝑒  ≥ 17∘ +  7∘  𝑝 [GeV]  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1)  𝜃2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  𝑒  ≥ 16∘ + 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5∘  𝑝 [GeV]  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2)  𝜃4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  𝑒  ≥ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5∘ +  15∘  𝑝 [GeV]  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3)  and the minimum 𝜋− angle as  𝜃1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  𝜋− ≥ 17∘ +  4∘  𝑝 [GeV]  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4)  and  𝜃2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  𝜋−  ≥ 25∘ +  7∘  𝑝 [GeV]  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5)  for 𝑝𝜋− < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='35 GeV/c and  𝜃2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  𝜋−  ≥ 16∘ +  10∘  𝑝 [GeV]  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6)  for 𝑝𝜋− ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='35 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The minimum 𝜋+ and proton angle was 𝜃 > 12∘ for all  data sets and momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The momentum and charge of the outgoing charged particles were obtained from  their measured positions in the drift chambers and the curvature of their trajectories  in the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Electrons were identified by requiring that the track originated  in the target, produced a time-correlated signal in the Cherenkov counter, and de-  127  posited enough energy in the electromagnetic calorimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The charged pions and  protons were separated by requiring that the track originated in the target and that  the measured time of flight agreed with that calculated from the particle’s momentum  and assumed mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Photons were identified by requiring a signal in the electromag-  netic calorimeter which implied a velocity greater than ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='96 c [214].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Elastic electron scattering from hydrogen was used to correct the electron momen-  tum as a function of angle for uncertainties in the CLAS magnetic field and in the  tracking chamber locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These corrections also significantly narrowed the elastic  peak width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Typical correction factors were less than 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The momentum correction  factors at lower scattered electron energies were tested using the H(𝑒, 𝑒′𝜋+)𝑋 and  3He(𝑒, 𝑒′𝑝𝑝)𝑋 reactions and found that they gave the correct missing mass for the  undetected neutron [208].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Low momentum protons were corrected for energy losses traversing the target  and detector material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The CLAS GEANT Monte Carlo (MC) was used to simulate  the proton energy loss in CLAS as a function of proton momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The maximum  correction was about 20 MeV/c for a proton momentum of 300 MeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The correction  was negligible for protons with momenta greater than 600 MeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The results from the e2a data sets were compared to predictions from the GE-  NIE [33] simulation, which is used by most neutrino experiments in the USA and has  an electron-scattering version (eGENIE) that was recently overhauled to be consistent  with the neutrino counterpart (𝜈GENIE), as detailed in [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' GENIE includes quasi-  elastic lepton scattering (QE), meson exchange currents (MEC), resonance production  (RES) and deep inelastic scattering (DIS), as well as rescattering via final state in-  teractions (FSI) of the outgoing hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The two GENIE configurations already  presented in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 were compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These include the significantly improved  G2018 setup which reproduces measured neutrino [215] and electron inclusive cross  sections, and newly implemented SuSAv2 that uses modern, theoretically-inspired,  recently-implemented QE and MEC models [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The resulting simulated events  were then analyzed using the same code as the data and the two were compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Electrons, unlike neutrinos, radiate bremsstrahlung photons in the electric field  128  of the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Events where the photons from scattered-electron radiation were  detected in CLAS were vetoed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' It was assumed that the photons came from either  radiation by the outgoing electron approximately parallel to its motion or from 𝜋0  decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The radiated photons were identified by requiring that they be detected within  ∆𝜑𝛾,𝑒′ ≤ 30∘ and ∆𝜃𝛾,𝑒′ ≤ 40∘ of the scattered electron and removed them from the  data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The events that were removed are indicated by the red box in figure 6-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  1  10  2  10  3  10  Fe @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="2 GeV  56  Data  150  −  100  −  50  −  0  50  100 150  [deg]  ,e'  γφ  ∆  0  20  40  60  80  100  [deg]  ,e'  γθ  ∆  from radiation  γ  0  π  from  γ  0  π  from  γ  Fe @ 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV  56  Data  Figure 6-2: ∆𝜃𝛾,𝑒′ as a function of ∆𝜑𝛾,𝑒′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The red box indicated the region with  radiated photons which was removed in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The incoming and outgoing electrons can each radiate a real photon, which changes  the kinematics of the interaction or the detected particles, and there can be vertex  or propagator corrections that change the cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' When comparing electron  scattering data to models, either the data or the model needs to be corrected for ra-  diative effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Published electron scattering cross sections are typically corrected for  radiative effects, but this correction is complicated and somewhat model-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  A framework for electron radiative corrections in GENIE was implemented for the  first time to allow comparisons to nonradiatively corrected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The framework al-  lows electron radiation, which can change the kinematics of the event by changing  either the incident or scattered electron energy through radiation of a real photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  129  We modeled external radiation in the same way as the JLab SIMC event genera-  tor [216, 217].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The implementation takes advantage of the peaking approximation  that greatly simplifies the calculation of the angular distribution of the emitted pho-  ton radiation by making the assumption that radiation along the direction of a given  particle can be interpreted as radiation due to that particle [218].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Future versions  of eGENIE will incorporate cross section changes due to vertex and propagator cor-  rections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The radiative correction procedure was validated by comparing a simulated  sample to electron scattering from protons at JLab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure 6-3 shows the data com-  pared to the GENIE simulation with and without radiative corrections [219].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  radiatively corrected calculation is clearly much closer to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This correction  can be used for comparisons with nonradiatively corrected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="  [GeV]  cal  H(e,e'p) E  1  4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='305  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='31  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='315  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='32  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='325  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='33  # events  0  2  4  6  8  10  12  3  10  ×  Data  GENIE + radiative correction  GENIE default  Figure 6-3: Number of events vs 𝐸𝑐𝑎𝑙 = 𝐸𝑒′ + 𝑇𝑝 the scattered electron energy plus  proton kinetic energy for 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='32 GeV H(𝑒, 𝑒′𝑝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Black points are data, the blue histogram  shows the unradiated GENIE prediction and the black histogram shows the GENIE  prediction with electron radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The GENIE calculations have been scaled to have  the same integral as the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The primary focus of this analysis was events with one electron and zero pions  or photons from 𝜋0 decay above threshold, which are referred to as (e,e’)0𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' That  choice was made to maximize the contribution of QE events where the incident lepton  scattered from a single nucleon in the nucleus, as is done in many neutrino oscillation  130  analyses [1, 220].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Furthermore, events with one detected electron, one proton, and  zero pions, denoted here as (e,e’p)1𝑝0𝜋 were examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These events were expected to  be dominated by well-understood QE events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Because the CLAS geometrical coverage is incomplete (≈ 50%), undetected pions  and photons needed to be subtracted to obtain true (e,e’)0𝜋 and (e,e’p)1𝑝0𝜋 event sam-  ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The undetected pion, photon, and proton (if applicable) contribution was quan-  tified from the events with detected pions, photons or protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The pion-production  cross section was assumed to be independent of 𝜑𝑞𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The latter is the angle between  the electron-scattering plane (the plane containing the incident and scattered elec-  trons and the virtual photon) and the hadron plane (the plane containing the virtual  photon and pion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The data-driven correction for these undetected hadrons is out-  lined below for the (e,e’p)1𝑝0𝜋 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For each detected (𝑒, 𝑒′𝑝𝜋) event, the proton-pion  pair was rotated around the momentum transfer direction ⃗𝑞 randomly multiple times,  as can be seen in figure 6-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For each rotation, it was determined whether the particle  was within the fiducial region of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The particle acceptance would then  be 𝐴𝜋 = 𝑁𝑑𝑒𝑡/𝑁𝑟𝑜𝑡, where 𝑁𝑟𝑜𝑡 is the number of rotations and 𝑁𝑑𝑒𝑡 is the number of  times the pion would have been detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The corresponding number of undetected  (𝑒, 𝑒′𝑝𝜋) events for that detected (𝑒, 𝑒′𝑝𝜋) event is (𝑁𝑟𝑜𝑡 − 𝑁𝑑𝑒𝑡)/𝑁𝑑𝑒𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' That was used  as a weight to subtract for the undetected pion events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For example, if one specific  (𝑒, 𝑒′𝑝𝜋) event would have been detected 250 times out of 1000 rotations, then it was  inferred that for each detected event, there were three more that were not detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The appropriate variables were calculated for that event and subtracted it from the  corresponding distributions with a weight of three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' That was done separately for  𝜋+, 𝜋− and photons in order to obtain a true (e,e’p)1𝑝0𝜋 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The same process is  repeated for the (e,e’)0𝜋 channel, where the corrections included undetected protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Higher multiplicity events were also accounted for, such as for events with two  detected 𝜋± or photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' When these events were rotated, each rotated event could  have been detected as a 2𝜋 event, a 1𝜋 event, or a 0𝜋 event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' If it appeared as a  0𝜋 event, its contribution was subtracted from the various 0𝜋 spectra as described  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' If it appeared as a 1𝜋 event, it was included in the set of 1𝜋 events with the  131  Figure 6-4: Schematic illustration of the data driven background correction using  detected (e,e’p𝜋) events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  appropriate negative weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' It was then treated as a regular 1𝜋 event, which was  then rotated and added to the 0𝜋 data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Some of the detected 1𝜋 events were  actually 2𝜋 events with an undetected pion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' When the effect of these events was  accounted for, there were fewer true 1𝜋 events left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This reduced the contamination  of the 1𝜋 events in the 0𝜋 channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  In practice, the process was initiated with the highest multiplicity events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Then,  their contributions to each of the detected lower multiplicity channels were subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The process was repeated recursively by rotating the higher multiplicity events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In  this way, their contributions to the lower multiplicity channels were determined and  subtracted, and then each of the lower multiplicity channels in turn were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Event multiplicities up to three pions and photons (total) for the (𝑒, 𝑒′)0𝜋 channel and  up to three protons, pions and photons (total) for the (𝑒, 𝑒′𝑝)1𝑝0𝜋 channel, where the  subtraction converged, were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The effects of the subtraction and its con-  vergence can be seen in figure 6-5 for 𝐸𝑄𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The number of events with an undetected  𝜋± or photon is about equal to the number of events with a detected 𝜋± or photon,  consistent with the ≈ 50% CLAS geometrical acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The effect of including two  𝜋± or photon events is much less than that of the one 𝜋± or photon events and the  132  effect of including three 𝜋± or photon events is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  2  3  4  5  [GeV]  QE  E  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  Weighted Events / GeV  γ/  ±  π  Detected 1  (-)  γ/  ±  π  Undetected 1  (+)  γ/  ±  π  Undetected 2  Fe  56  (a)  2  3  4  5  [GeV]  QE  E  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  Weighted Events / GeV  No cuts  γ/  ±  π  No detected  γ/  ±  π  Subtract undetected  Fe  56  (b)  Figure 6-5:  The effect of undetected pion subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The number of weighted  events as a function of reconstructed energy 𝐸𝑄𝐸 for 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV Fe(𝑒, 𝑒′) events for  (left) events with a detected 𝜋± or photon (blue), events with one (red) or two (light  brown) undetected 𝜋± or photons and (right) all (𝑒, 𝑒′𝑋) events with detected or  undetected 𝜋± or photon (blue), (𝑒, 𝑒′) events with no detected 𝜋± or photon (red),  and (𝑒, 𝑒′) events after subtraction for undetected 𝜋± or photon (light brown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The subtraction method was tested by applying it to eGENIE events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The result-  ing subtracted spectra agreed reasonably with the true 1𝑝0𝜋 spectra as can be seen  in figure 6-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The method diverges for total hadron multiplicities greater than four  due to the proton and pion multiplicity differences shown in figure 6-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' It is clear  that eGENIE dramatically overpredicts the number of events with large proton and  pion multiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The CLAS acceptance maps were used to determine the probability that each  particle produced by eGENIE was detected as a function of the momentum, the  angular orientation, and the particle species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure 6-9 shows the electron acceptance  map for 12C at 𝐸𝑏𝑒𝑎𝑚 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV as a function of (left) cos𝜃𝑒 vs 𝜑𝑒 and (right) cos𝜃𝑒  vs momentum 𝑝𝑒 illustrating an acceptance greater than 90% across the majority of  the detector fiducial volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The particle momenta were smeared with an effective CLAS resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Namely,  electrons and proton momentum resolutions of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5% and 1%, respectively, for the  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV data and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5% and 3% for the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV data, which was taken  with a lower torus magnetic field, were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  133  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="4  [GeV]  cal  E  π  1p0  (e,e'p)  0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="1  Normalized Yield  π  1p0  True (e,e'p)  π  1p0  Subtracted (e,e'p)  π  1p0  Unsubtracted (e,e'p)  C @ 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV  12  Figure 6-6: Illustration of the successful closure test of the data driven correction  for undetected particles as a function of 𝐸𝐶𝑎𝑙 using the (e,e’p)1𝑝0𝜋 channel on 12C at  𝐸𝑏𝑒𝑎𝑚 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The contribution of the unsubtracted (e,e’p)1𝑝0𝜋 spectrum (black)  is reduced to the subtracted (e,e’p)1𝑝0𝜋 spectrum (magenta), which is in reasonable  agreement with the true (e,e’p)1𝑝0𝜋 spectrum (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  0  1  2  3  4  Multiplicities  4  10  5  10  6  10  7  10  # Events  SuSav2  G2018  Protons  ±  π  Figure 6-7:  The proton (black) and charged pion (blue) multiplicities for data  (points), SuSav2 (solid histogram) and G2018 (dashed histogram) for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV  carbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Error bars show the 68% (1𝜎) confidence limits for the statistical and point-  to-point systematic uncertainties added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Error bars are not shown  when they are smaller than the size of the data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Normalization uncertainties of  3% not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The cross section as a function of variables of interest for particles above the  minimum angles shown in equations 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6 was determined in several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  All  134  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  Radiation Correction  (g)  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  Radiation Correction  (h)  2  3  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  Radiation Correction  (i)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0  5  10  15  Acceptance Uncertainty [%]  (d)  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  0  5  10  15  Acceptance Uncertainty [%]  (e)  2  3  4  0  5  10  15  Acceptance Uncertainty [%]  (f)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  2  4  6  8  10  12  Acceptance Correction  (a)  C  12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  2  4  6  8  10  12  Acceptance Correction  (b)  Fe  56  He  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV  2  3  4  2  4  6  8  10  12  Acceptance Correction  (c)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="453 GeV  [GeV]  cal  E  π  1p0  (e,e'p)  Figure 6-8:  (Top row) Acceptance correction factors, (middle row) acceptance cor-  rection factor uncertainties, and (bottom row) electron radiation correction factors  plotted vs E𝑐𝑎𝑙 for the three incident beam energies." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Results for carbon are shown  in black, helium in green and iron in magenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The left column (a,d,g) shows the  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 GeV results, the middle column (b,e,h) shows the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV results and the right  column (c,f,i) shows the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  90  92  94  96  98  100  Acceptance (%)  0  50 100 150 200 250 300  [deg]  e  φ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9  eθ  cos  90  92  94  96  98  100  Acceptance (%)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9  eθ  cos  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  [GeV/c]  ep  Figure 6-9: Electron acceptance maps for 12C at 𝐸𝑏𝑒𝑎𝑚 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV as a function of  (left) cos𝜃𝑒 vs 𝜑𝑒 and (right) cos𝜃𝑒 vs momentum 𝑝𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  events were first weighted by a factor of 𝑄4 to account for the major difference in  electron- and neutrino-nucleus scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The number of weighted events was then  135  determined and corrected (if appropriate) for events with undetected pions, photons,  and extra protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The background-subtracted event distribution was divided by the  number of target nuclei per area and the number of incident beam electrons to get  the normalized yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The delivered integrated beam charge was measured using the  CLAS Faraday Cup [209].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Electron radiation effects were corrected for by multiplying  the resulting spectra by the ratio of eGENIE without electron radiation divided by  eGENIE with electron radiation, as shown in panels g-i of figure 6-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This includes  a multiplicative factor to account for the effects of internal radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Electron and  proton acceptance and other detector effects were corrected for using eGENIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  acceptance correction factor is the ratio of the number of true signal events without  detector effects to the number of true signal events with detector effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The detector  effects included momentum resolution, fiducial cuts, and acceptance map effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  fiducial cuts determine the useful areas of the detector as a function of particle mo-  menta and angles, and the acceptance maps describe the efficiency of the detector as  a function of particle momenta and angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This factor corrects the effective electron  and proton solid angles to almost 4𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' It excludes all electrons, pions and protons  below their minimum angles defined in equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The acceptance correction  factor was obtained using both G2018 and SuSav2 shown in panels a-c of figure 6-8  as the bin-by-bin average of the two configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The G2018 results were shifted  so that the energy reconstruction peaks lined up at the correct beam energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Finally,  the bin width division was taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  In order to perform a sanity check of our cross-section extraction procedure, the  inclusive 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5∘ GeV C(𝑒, 𝑒′) cross section was determined as follows:  𝑑𝜎  𝑑Ω𝑑𝜔 =  𝑁𝑒  ∆Ω𝑁𝑖𝑁𝑡  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7)  where 𝑁𝑒 is the number of detected electrons in Sector 1 within 36∘ ≤ 𝜃𝑒 ≤ 39∘  and a 12∘ range in 𝜑𝑒, ∆Ω = sin 𝜃𝑒𝑑𝜃𝑒𝑑𝜑𝑒 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='68 msr, 𝑁𝑖 is the number of incident  electrons, and 𝑁𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='179 g/cm2 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='95×10−9 nuclei/𝜇b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The extracted cross section  136  was further compared to measurements from SLAC [182], as can be seen in figure 6-  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The measured JLab cross section is in reasonable agreement with the GENIE  predictions and also consistent with the SLAC measurements at lower and higher  energies [182].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 6-10:  Comparison between the inclusive C(𝑒, 𝑒′) cross sections measured at  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5∘ for data (points) and SuSav2 (lines) for the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='961 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='299 GeV SLAC data  and our 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV CLAS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Several major sources of systematic uncertainties were considered, including the  angular dependence of the pion-production cross section for the undetected-pion sub-  traction, the effects of fiducial cuts on undetected particle subtraction, photon iden-  tification cuts, the sector-to-sector variation of the data to eGENIE ratio, the model-  dependence of the acceptance correction, and uncertainties in the normalization mea-  surement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 shows the summary of the total systematic uncertainties used in  the e4𝜈 analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  When events containing pions were rotated around the momentum transfer vec-  tor, the cross section was assumed to not depend on 𝜑𝑞𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The 𝜑𝑞𝜋 independence of  the pion-production cross section was tested by weighting the subtraction using the  measured 𝜑𝑞𝜋-dependent 𝐻(𝑒, 𝑒′𝑝𝜋) cross sections of reference [221].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This changed  the subtracted spectra by about 1% and was included as a systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  137  0 = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5°  SLAC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='961 GeV  GeV  JLab  A  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV  S  dQdE  2  d  SLAC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='299 GeV  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  Energy Transfer [GeV]Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1: Summary of the total systematic uncertainties used in the e4𝜈 analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Source  Uncertainty (%)  Detector acceptance  Identification cuts  Number of rotations  𝜑𝑞𝜋 cross-section dependence  2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7  (@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV)  Sector dependence  6  Acceptance correction  2-15  Overall normalization  3  Electron inefficiency  2  The subtraction of events with undetected pions depends on the CLAS acceptance  for such particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The final spectrum should be independent of the CLAS pion  acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The effect of varying the CLAS acceptance on the undetected particle  subtraction was estimated by comparing the results using the nominal fiducial cuts  and using fiducial cuts with the 𝜑 acceptance in each CLAS sector reduced by 6∘  or about 10-20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This changed the resulting subtracted spectra by about 1% at  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV and by 4% at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This difference was included as a  point-to-point systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The photon identification cuts were also varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Photons were also identified  as neutral particle hits in the calorimeter with a velocity greater than 2𝜎 (3𝜎 at  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV) below the mean of the photon velocity peak at 𝑣 = 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This limit was  varied by ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='25𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This gave an uncertainty in the resulting subtracted spectra of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5% and 2% at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  CLAS had six almost identical sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The primary difference among the sectors  is the distribution of dead detector channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These dead channels were accounted for  in our fiducial cuts and in our acceptance maps, where the effect of the dead detectors  on the particle detection efficiency was measured and applied that efficiency to the  particles generated in the eGENIE simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' If our fiducial cuts and acceptance  maps completely accounted for the effect of the dead and inefficient detector channels,  then the ratio of data to eGENIE should be the same for all six sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Sectors with  anomalous data to eGENIE ratios were discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' More precisely, sectors 3/5 were  138  discarded at 𝐸𝑏𝑒𝑎𝑚 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV, as well as sectors 3/4/5 at 𝐸𝑏𝑒𝑎𝑚 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' All the  sectors were used at 𝐸𝑏𝑒𝑎𝑚 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The variance of the ratios for the remaining  sectors was used as a measure of the uncertainty in the measured normalized yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  This gave a point-to-point systematic uncertainty of 6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The acceptance correction factor uncertainty was obtained using both G2018 and  SuSav2 and their bin-by-bin difference divided by  √  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The uncertainty was averaged  over the entire peak to avoid large uncertainties due to small misalignments, as shown  in panels d-f of figure 6-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The overall normalization was determined using inclusive 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV H(𝑒, 𝑒′) mea-  surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The measured and simulated H(𝑒, 𝑒′) cross sections agreed to within an  uncertainty of 3%, which is used as a normalization uncertainty [222].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The statistical uncertainty and the point-to-point systematic uncertainties were  added in quadrature and displayed on the data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The total point-to-point  systematic uncertainties ranged between 7-25%, with the largest uncertainties for the  smallest cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  Incident Energy Reconstruction Results  There are two general approaches for reconstructing the incident neutrino energy  based on the particle detection capabilities of the neutrino detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Water Cherenkov detectors only measure charged leptons and pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' If the neu-  trino scattered quasi-elastically from a stationary nucleon in the nucleus, its energy  can be reconstructed from the measured lepton as:  𝐸𝑄𝐸 = 2𝑀𝑁𝜖 + 2𝑀𝑁𝐸𝑙 − 𝑚2  𝑙  2(𝑀𝑁 − 𝐸𝑙 + 𝑘𝑙 cos 𝜃𝑙),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8)  where 𝜖 ≈ 20 MeV is the average nucleon separation energy, 𝑀𝑁 is the nucleon  mass, and (𝑚𝑙, 𝐸𝑙, 𝑘𝑙, 𝜃𝑙) are the scattered lepton mass, energy, momentum, and angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 6-11 shows the 𝐸𝑄𝐸 distribution for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV C(𝑒, 𝑒′)0𝜋 events, which are  most relevant for T2K and HK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A broad peak is observed centered at the real beam  energy with a large tail extending to lower energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The peak is doppler-broadened  139  by the motion of the nucleons in the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The tail is caused by non-QE reactions  that pass the (𝑒, 𝑒′)0𝜋 selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The tail is cut off at the lowest energies by the CLAS  minimum detected electron energy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The SuSAv2 eGENIE peak has the  correct width, but is somewhat larger than the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' It overestimates the tail by  about 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The G2018 eGENIE peak also exceeds the data, but is too narrow, with  a Gaussian width of 𝜎 = 76 MeV, compared to 89 MeV for the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This is due to  inexact modeling of the nuclear ground state momentum distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The tail dips  below the data at around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9 GeV, and is larger than the data at lower reconstructed  energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Neither model describes the data quantitatively well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="4  [GeV]  QE  E  π  0  C(e,e')  0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  GeV  b  µ  QE  dE  σ  d  Data  SuSav2 (Total)  QE  MEC  RES  DIS  G2018  beam  E  ⇓  Figure 6-11: The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV C(𝑒, 𝑒′)0𝜋 cross section plotted as a function of the  reconstructed energy 𝐸𝑄𝐸 for data (black points), GENIE SuSAv2 (solid black curve)  and GENIE G2018 (dotted black curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The colored lines show the contributions  of different processes to the GENIE SuSAv2 cross section: QE (blue), MEC (red),  RES (green) and DIS (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Error bars show the 68% (1𝜎) confidence limits  for the statistical and point-to-point systematic uncertainties added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Error bars are not shown when they are smaller than the size of the data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Normalization uncertainty of 3% not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 6-12 shows the cross section as a function of 𝐸𝑄𝐸 for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 and  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV C(𝑒, 𝑒′)0𝜋 events and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV Fe(𝑒, 𝑒′)0𝜋 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' It is clear  that the mismodeling already observed in the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV C(𝑒, 𝑒′)0𝜋 sample becomes  140  even more pronounced for higher energies and heavier nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  2  3  4  5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  2  3  4  5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  C  12  Fe  56  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV (x1/2)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="453 GeV (x5)  [GeV]  QE  E  π  0  (e,e')  Data  SuSav2 (Total)  QE  MEC  RES DIS  G2018  GeV  b  µ  QE  dE  σ  d  GeV  b  µ  QE  dE  σ  d  Figure 6-12: The 𝐴(𝑒, 𝑒′𝑝)0𝜋 cross section plotted as a function of the reconstructed  quasielastic energy 𝐸𝑄𝐸 for data (black points), SuSAv2 (black solid curve) and G2018  (black dotted curve)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Different panels show results for different beam energy and  target nucleus combinations: (top row) Carbon target at (left to right) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257  and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV, and (bottom) Iron target at (left) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 and (right) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV incident  beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV yields have been scaled by 1/2 and the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV yields have  been scaled by 5 to have the same vertical scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Colored lines show the contributions  of different processes to the SuSAv2 GENIE simulation: QE (blue), MEC (red),  RES (green) and DIS (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Error bars show the 68% (1𝜎) confidence limits  for the statistical and point-to-point systematic uncertainties added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Error bars are not shown when they are smaller than the size of the data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Normalization uncertainties of 3% not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Tracking detectors measure all charged particles above their detection thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The “calorimetric” incident neutrino energy is then the sum of all the detected particle  energies:  𝐸𝑐𝑎𝑙 =  ∑︁  𝐸𝑖 + 𝜖,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9)  where 𝐸𝑖 are the detected nucleon kinetic energies and the lepton and meson total  energies and 𝜖 is the average total removal energy for the detected particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This  quantity 𝜖, used in reconstructing the incident energies in equations 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9, was  141  determined from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' It is defined using the difference in the binding energies  for knocking a proton out of nucleus 𝐴 as 𝜖 = |𝑀𝐴 − 𝑀𝐴−1 − 𝑚𝑝| + ∆𝜖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The removal  energy correction ∆𝜖 was adjusted so that the peaks in the 𝐸𝑐𝑎𝑙 spectrum for low  transverse missing momentum events reconstructed to the correct beam energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' It  was found that ∆𝜖 = 5 and 11 MeV for 12C and 56Fe, respectively, which are consistent  with average excitation energies from single-nucleon knockout from nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 6-13 shows the cross section as a function of 𝐸𝑐𝑎𝑙 for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 and  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV C(𝑒, 𝑒′𝑝)1𝑝0𝜋 events and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV Fe(𝑒, 𝑒′𝑝)1𝑝0𝜋 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  All  spectra show a sharp peak at the real beam energy, followed by a large tail at lower  energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' For carbon, only 30-40% of the events reconstruct to within 5% of the real  beam energy, as illustrated in table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2: (𝑒, 𝑒′𝑝)1𝑝0𝜋 events reconstructed to the correct beam energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Peak Fraction  refers to the fraction of events reconstructed to the correct beam energy and Peak Sum  refers to the integrated weighted cross section (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' 6-13) reconstructed  to the correct beam energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The peak integration windows are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 ≤ 𝐸𝑐𝑎𝑙 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='22  GeV, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='19 ≤ 𝐸𝑐𝑎𝑙 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='34 GeV, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='35 ≤ 𝐸𝑐𝑎𝑙 ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='60 GeV, respectively, for the three  incident beam energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' SuSAv2 is not intended to model nuclei lighter than 12C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV  Peak  Peak  Peak  Peak  Peak  Peak  Fraction  Sum [𝜇b]  Fraction  Sum [𝜇b]  Fraction  Sum [𝜇b]  12C  Data  39  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='13  31  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='26  32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='34  SuSAv2  44  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='33  27  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='76  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='20  G2018  51  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='53  37  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='44  23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='43  56Fe  Data 20  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='73  23  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='01  SuSAv2 21  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='28  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='58  G2018 30  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='22  19  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='48  For iron this fraction is only 20-25%, highlighting the crucial need to well model  the low-energy tail of these distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' eGENIE overpredicts the fraction of events  in the peak at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV and significantly underpredicts it at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' eGENIE  using SuSAv2 dramatically overpredicts the peak cross section at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257  GeV, and significantly underestimates the peak cross section at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV, as shown  142  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  Cross Section  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  Cross Section  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  0  2  4  6  Cross Section  (d)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  2  3  4  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  Cross Section  (c)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  2  3  4  0  2  4  6  Cross Section  (e)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  C  12  Fe  56  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV (x1/2)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="453 GeV (x5)  [GeV]  cal  E  π  1p0  (e,e'p)  Data  SuSav2 (Total)  QE  MEC  RES DIS  G2018  GeV  b  µ  cal  dE  σ  d  GeV  b  µ  cal  dE  σ  d  Figure 6-13: The 𝐴(𝑒, 𝑒′𝑝)1𝑝0𝜋 cross section plotted as a function of the reconstructed  calorimetric energy 𝐸𝑐𝑎𝑙 for data (black points), SuSAv2 (black solid curve) and G2018  (black dotted curve)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Different panels show results for different beam energy and  target nucleus combinations: (top row) Carbon target at (left to right) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257  and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV, and (bottom) Iron target at (left) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 and (right) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV incident  beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV yields have been scaled by 1/2 and the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV yields  have been scaled by 5 to have the same vertical scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The insets show the cross  sections with the same horizontal scale and an expanded vertical scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Colored lines  show the contributions of different processes to the SuSAv2 GENIE simulation: QE  (blue), MEC (red), RES (green) and DIS (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Error bars show the 68% (1𝜎)  confidence limits for the statistical and point-to-point systematic uncertainties added  in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Error bars are not shown when they are smaller than the size of the  data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Normalization uncertainties of 3% not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  in table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' eGENIE using the older G2018 models overestimates the peak cross  section at all three incident energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' It also reconstructs the peak position (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' the  incident energy) to be 10, 25 and 36 MeV too low for 4He, C and Fe, respectively,  at all three beam energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  This is due to an error in the G2018 QE modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  This beam-energy dependence of the data-GENIE discrepancy could have significant  implications for the neutrino flux reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  At 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV, eGENIE using SuSAv2 slightly overpredicts the low energy tail  and eGENIE using G2018 is reasonably close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Both models dramatically overpredict  143  the low energy tail at the higher beam energies shown in the insets of figure 6-  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The tail seems to be dominated by RES and DIS at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV that did not  result in the production of other charged particles above detection threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This  overprediction has already been observed in inclusive electron scattering from the  proton and deuteron, and thus appears to be due to the electron-nucleon interaction,  rather than to the nuclear modeling [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  SuSAv2 describes the peak cross section - the part of the cross section that re-  constructs to the correct beam energy - equally well for C and for Fe, while G2018  over estimates the peak cross section more for Fe than for C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Both models predict a  greater peak fraction relative to the data for Fe than for C, particularly at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV,  as shown in table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' While the (𝑒, 𝑒′)0𝜋 QE reconstruction of equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8 gives a  much broader peak at the true beam energy than the calorimetric energy 𝐸𝑐𝑎𝑙 due to  the effects of nucleon motion, as shown in figures 6-14 and 6-15, it has the same tail  of lower energy events for the same (𝑒, 𝑒′𝑝)1𝑝0𝜋 data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="2  −  0  Feeddown  cal  E  π  1p0  (e,e'p)  2  −  10  1  −  10  1  b  µ  Feed  cal  dE  σ  d  (a)  /  1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV  /  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV  /  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV(x4)  C  12  Data/SuSav2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="2  Feeddown  QE  E  π  0  (e,e')  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  b  µ  Feed  QE  dE  σ  d  (b)  /  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV  /  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV  /  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV(x4)  C  12  Data/SuSav2  Figure 6-14:  Energy feed-down cross-sections (𝐸𝑟𝑒𝑐 − 𝐸𝑡𝑟𝑢𝑒)/𝐸𝑡𝑟𝑢𝑒 for data (points)  and SuSav2 (lines) for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV (red triangles and dotted lines), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV (green  squares and dashed lines) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV (blue dots and solid lines) on carbon for (a)  𝐸𝑐𝑎𝑙, and (b) 𝐸𝑄𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  Kinematic Imbalance Results  Neutrino experiments use the “transverse variables” (TVs) outlined in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  to enhance their sensitivity to different aspects of the reaction mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These  144  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="2  −  0  Feeddown  cal  E  π  1p0  (e,e'p)  2  −  10  1  −  10  1  10  b  µ  Feed  cal  dE  σ  d  (c)  /  2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV  /  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV(x4)  Fe  56  Data/SuSav2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="2  Feeddown  QE  E  π  0  (e,e')  2  4  6  b  µ  Feed  QE  dE  σ  d  (d)  /  2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV  /  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV(x4)  Fe  56  Data/SuSav2  Figure 6-15:  Energy feed-down cross-sections (𝐸𝑟𝑒𝑐 − 𝐸𝑡𝑟𝑢𝑒)/𝐸𝑡𝑟𝑢𝑒 for data (points)  and SuSav2 (lines) for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV (red triangles and dotted lines), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV (green  squares and dashed lines) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV (blue dots and solid lines) on iron for (c)  𝐸𝑐𝑎𝑙, and (d) Fe 𝐸𝑄𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  TVs are independent of the neutrino energy and use the momentum of the detected  particles transverse to the incident lepton [132, 223, 224] as shown in equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='12,  where ⃗𝑃 𝑒′  𝑇 and ⃗𝑃 𝑝  𝑇 are the three-momenta of the detected lepton and proton perpen-  dicular to the direction of the incident lepton, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The ⃗𝑃𝑇 vector is intended  to characterize the nuclear ground state, 𝛿𝛼𝑇 the FSI and ∆𝜑𝑇 is intended to probe  regions where MEC events dominate [132,223,224].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  ⃗𝑃𝑇  =  ⃗𝑃 𝑒′  𝑇 + ⃗𝑃 𝑝  𝑇  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='10)  𝛿𝛼𝑇  =  arccos(−  ⃗𝑃 𝑒′  𝑇 · ⃗𝑃𝑇  𝑃 𝑒′  𝑇 𝑃𝑇  )  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='11)  𝛿𝜑𝑇  =  arccos(−  ⃗𝑃 𝑒′  𝑇 · ⃗𝑃 𝑝  𝑇  𝑃 𝑒′  𝑇 𝑃 𝑝  𝑇  )  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='12)  Purely QE events without final state interactions, where the lepton scattered  from a bound moving proton, will have small 𝑃𝑇, consistent with the motion of the  struck nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Events with small 𝑃𝑇 should thus reconstruct to the correct incident  energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Non-QE events, where neutral or sub-detection-threshold charged particles  were produced, will have larger 𝑃𝑇 and will not reconstruct to the correct incident  energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  𝑃𝑇 is thus an ideal observable for tuning reaction models to ensure they  145  correctly account for non-QE processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The 𝑃𝑇 distribution for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV C(𝑒, 𝑒′𝑝)1𝑝0𝜋 is shown in figure 6-16 and the  other targets and energies are shown in figure 6-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Both data and eGENIE peak  at relatively low momenta, as expected, and both have a large tail extending out to  1 GeV/𝑐 and containing about half of the measured events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The high-𝑃𝑇 tail is pre-  dominantly due to resonance production that did not result in an additional pion or  nucleon above the detection threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' eGENIE using SuSAv2 reproduces the shape  of the data moderately well, suggesting adequate reaction modeling, including the  contribution of non-QE processes such as resonance production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' As expected, both  data and eGENIE/SuSAv2 events with 𝑃𝑇  < 200 MeV/𝑐 almost all reconstruct to  the correct incident energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' However, events with 𝑃𝑇≥ 400 MeV/𝑐 do not reconstruct  to the correct energy and are poorly reproduced by eGENIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This disagreement be-  comes even more pronounced at higher energies and heavier nuclei, and indicates that  including high-𝑃𝑇 data in oscillation analyses could bias the extracted parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  As high-𝑃𝑇 data accounts for 25 − 50% of the measured events, care must be taken  to improve the models implemented in GENIE, so that they can reproduce the high-  𝑃𝑇 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This will be especially true at the higher incident neutrino energies expected  for DUNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The opening angle 𝛿𝛼𝑇 measures the angle between 𝑃𝑇 and the transverse mo-  mentum transfer (⃗𝑞𝑇 = −⃗𝑃 𝑒′  𝑇 ) in the transverse plane and is isotropic in the absence  of final state interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' 𝛿𝜑𝑇 measures the opening angle between the detected pro-  ton momentum and the transverse momentum transfer and is forward peaked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  𝛿𝛼𝑇 distributions become progressively less isotropic at higher energies and heavier  targets, indicating the increasing importance of FSI and of non-QE reaction mech-  anisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' GENIE agrees best with data at the lowest beam energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' At the higher  beam energies GENIE describes the relatively flat smaller angles much better than  the back-angle peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' GENIE also describes the lowest energy 𝛿𝜑𝑇 distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' At  higher energies, GENIE overestimates the height of the forward peak, as shown in  figure 6-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  In line with the neutrino-based MicrobooNE analysis presented in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4,  146  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="8  1  [GeV/c]  T  P  π  1p0  C(e,e'p)  0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  GeV/c  b  µ  T  dP  σ  d  Data  SuSav2 (Total)  QE  MEC  RES  DIS  G2018  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='05  > 400 [MeV/c]  T  P  (c)  > 400 [MeV/c]  T  P  (c)  > 400 [MeV/c]  T  P  (c)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  < 400 [MeV/c]  T  200 < P  (b)  < 400 [MeV/c]  T  200 < P  (b)  < 400 [MeV/c]  T  200 < P  (b)  0  1  2  < 200 [MeV/c]  T  0 < P  (a)  < 200 [MeV/c]  T  0 < P  (a)  < 200 [MeV/c]  T  0 < P  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="08  [GeV]  cal  E  π  1p0  (e,e'p)  GeV  b  µ  cal  dE  σ  d  Figure 6-16: (Left) the 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV C(𝑒, 𝑒′𝑝)1𝑝0𝜋 cross section plotted versus missing  transverse momentum, 𝑃𝑇, for data (black points), SuSav2 (black solid line) and  G2018 (black dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The vertical lines at 200 MeV/𝑐 and at 400 MeV/𝑐 separate  the three bins in 𝑃𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Colored lines show the contributions of different processes to the  SuSAv2 GENIE simulation: QE (blue), MEC (red), RES (green) and DIS (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (Right) The cross section plotted versus the calorimetric energy 𝐸𝑐𝑎𝑙 for different bins  in 𝑃𝑇: (top) 𝑃𝑇< 200 MeV/𝑐, (middle) 200 MeV/𝑐 ≤ 𝑃𝑇≤ 400 MeV/𝑐, and (bottom)  𝑃𝑇> 400 MeV/𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  these kinematic variables were further investigated in the form of multidimensional  cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure 6-19 shows the data-simulation cross-section comparisons for  12C at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='261 GeV as a function of 𝑃𝑇 for (top) all the events, (bottom left) events with  𝛿𝛼𝑇 < 45𝑜 dominated by QE interactions and no reinteractions, and (bottom right)  events with 135𝑜 < 𝛿𝛼𝑇 < 180𝑜 maximally affected by FSI and multi-hadron channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Using all the events (top) that satisfy our selection yielded a QE-rich region up to  ≈ 300 MeV/c and a RES-dominated tail that extended to ≈ 1 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Slicing the  available 𝑃𝑇 phase-space in 𝛿𝛼𝑇 regions revealed regions with specific features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' More  precisely, the region with 𝛿𝛼𝑇 < 45𝑜 can be used to isolate primarily QE events and  to test nuclear models in event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Even in this region though, differences  of 20-30% in the QE strength are observed by both model configurations used for  147  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='8  (a)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  (d)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='8  (c)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  (e)  C  12  Fe  56  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="453 GeV (x4)  [GeV/c]  T  P  π  1p0  (e,e'p)  Data  SuSav2 (Total)  QE  MEC  RES DIS  G2018  GeV/c  b  µ  T  dP  σ  d  GeV/c  b  µ  T  dP  σ  d  Figure 6-17: The cross section plotted vs transverse missing momentum 𝑃𝑇 for data  (black points), SuSAv2 (black solid curve) and G2018 (black dotted curve)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Different  panels show results for different beam energy and target nucleus combinations: (top  row) Carbon target at (left to right) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV, and (bottom) Iron  target at (left) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 and (right) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV yields have been scaled by  four to have the same vertical scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Colored lines show the contributions of different  processes to the SuSAv2 GENIE simulation: QE (blue), MEC (red), RES (green) and  DIS (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  comparison in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Meanwhile, the 135𝑜 < 𝛿𝛼𝑇 < 180𝑜 region is dominated  by RES interactions and could be used to tune FSI parameters and to improve the  RES modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 6-20 shows the data-simulation comparisons for 12C at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='261 GeV as a  function of 𝛿𝛼𝑇 for (left) all the events, (middle) events with 𝑃𝑇 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV/c, and  (right) events with 𝑃𝑇 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The 𝑃𝑇 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV/c slice is dominated by QE  events with minimal FSI effects that result in a fairly uniform distribution with a  slight enhancement in the forward direction due to the RES contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The  𝑃𝑇 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV/c slice is dominated by multi-hadron and enhanced-FSI events that  result in a sharp peak in the region close to 150𝑜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Therefore, this investigation of the  𝛿𝛼𝑇 phase-space in slices of 𝑃𝑇 is complimentary to the one illustrated in figure 6-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Using the exact same 𝑒4𝜈 data sets and cross-section extraction technique, the  148  0  50  100  150  0  1  2  3  (a)  0  50  100  150  0  1  2  3  (b)  0  50  100  150  0  5  10  (d)  0  50  100  150  0  1  2  3  (c)  0  50  100  150  0  5  10  (e)  C  12  Fe  56  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV (x2)  [deg]  T  α  δ  π  1p0  (e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="e'p)  Data  SuSav2 (Total)  QE  MEC  RES DIS  G2018  deg  nb  T  α  d  σ  d  deg  nb  T  α  d  σ  d  0  50  100  150  0  2  4  6  8  (f)  0  50  100  150  0  2  4  6  8  (g)  0  50  100  150  0  5  10  15  20  (i)  0  50  100  150  0  2  4  6  8  (h)  0  50  100  150  0  5  10  15  20  (j)  C  12  Fe  56  (x2)  [deg]  T  φ  δ  π  1p0  (e," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="e'p)  deg  nb  T  φ  d  σ  d  deg  nb  T  φ  d  σ  d  Figure 6-18: The cross section plotted vs 𝛿𝛼𝑇 (a-e) and vs 𝛿𝜑𝑇 (f-j) for data (black  points)," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' SuSAv2 (black solid curve) and G2018 (black dotted curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Different panels  show results for different beam energy and target nucleus combinations: (top row)  Carbon target at (left to right) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV, and (bottom) Iron target  at (left) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 and (right) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV yields have been scaled by two  to have the same vertical scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Colored lines show the contributions of different  processes to the SuSAv2 GENIE simulation: QE (blue), MEC (red), RES (green)  and DIS (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  cross section as a function of the total struck nucleon momentum approximation,  derived following the approach in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1, was reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The only difference  compared to the already outlined formalism on 40Ar is that B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='09216 GeV is the  149  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="8  1  [GeV/c]  T  P  π  1p0  (e,e'p)  0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  GeV/c  b  µ  T  dP  σ  d  12C @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, All events  Data  SuSav2 (Total)  QE  MEC  RES  DIS  G2018  12C @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, All events  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="8  1  [GeV/c]  T  P  π  1p0  (e,e'p)  0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  3  −  10  ×  deg GeV/c  b  µ  T  dP  T  α  δ  d  σ  2  d  o  < 45  T  α  δ  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV,  o  < 45  T  α  δ  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV,  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="8  1  [GeV/c]  T  P  π  1p0  (e,e'p)  0  1  2  3  4  3  −  10  ×  deg GeV/c  b  µ  T  dP  T  α  δ  d  σ  2  d  o  < 180  T  α  δ  <  o  12C @2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, 135  o  < 180  T  α  δ  <  o  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, 135  Figure 6-19: Data-simulation comparisons for 12C at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='261 GeV showing the cross  section results as a function of 𝑃𝑇 for (top) all the events, (bottom left) events with  𝛿𝛼𝑇 < 45𝑜 dominated by QE interactions and no reinteractions, and (bottom right)  events with 135𝑜 < 𝛿𝛼𝑇 < 180𝑜 maximally affected by FSI and multi-hadron channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Colored lines show the contributions of different processes to the SuSAv2 GENIE  simulation: QE (blue), MEC (red), RES (green) and DIS (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  binding energy for 12C, and 𝜖𝑁 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='024 GeV is the corresponding removal energy [225].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  This approximation, refered to as 𝑃𝑛,𝑝𝑟𝑜𝑥𝑦, is compared to the true missing momentum  𝑃𝑀𝑖𝑠𝑠 calculated as shown in equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='13,  𝑃𝑀𝑖𝑠𝑠 = |⃗𝑞 − ⃗𝑝|,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='13)  where ⃗𝑞 is the 3-vector for the momentum transfer based on the difference between  the kinematics of the incoming and the outgoing lepton, and ⃗𝑝 is the 3-vector of the  outgoing proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 6-21 shows the data-simulation comparisons for 12C at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="261 GeV as a  function of (top left) the true missing momentum 𝑃𝑀𝑖𝑠𝑠 and (top right) the missing  150  0  50  100  150  [deg]  T  α  δ  π  1p0  (e,e'p)  0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  3  −  10  ×  deg  b  µ  T  α  δ  d  σ  d  12C @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, All events  Data  SuSav2 (Total)  QE  MEC  RES  DIS  G2018  12C @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="257 GeV, All events  0  50  100  150  [deg]  T  α  δ  π  1p0  (e,e'p)  0  1  2  3  3  −  10  ×  deg GeV/c  b  µ  T  dP  T  α  δ  d  σ  2  d  < 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV/c  T  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, P  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV/c  T  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="257 GeV, P  0  50  100  150  [deg]  T  α  δ  π  1p0  (e,e'p)  0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  2  3  −  10  ×  deg GeV/c  b  µ  T  dP  T  α  δ  d  σ  2  d  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV/c  T  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, P  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV/c  T  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, P  Figure 6-20: Data-simulation comparisons for 12C at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='261 GeV showing the cross  section results as a function of 𝛿𝛼𝑇 for (top) all the events, (bottom left) events  with 𝑃𝑇 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV/c dominated by QE interactions and no reinteractions, and (bot-  tom right) events with 𝑃𝑇 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV/c maximally affected by FSI and multi-hadron  channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Colored lines show the contributions of different processes to the SuSAv2  GENIE simulation: QE (blue), MEC (red), RES (green) and DIS (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  momentum approximation 𝑃𝑛,𝑝𝑟𝑜𝑥𝑦 commonly used by neutrino experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The bot-  tom panel shows the overlay of the two extracted data cross sections illustrating that  𝑃𝑛,𝑝𝑟𝑜𝑥𝑦 fails to reproduce 𝑃𝑀𝑖𝑠𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' As can be seen in the interaction breakdown plots,  the differences are primarily driven by the RES interactions which are reconstructed  at lower values when the QE-like assumption deployed for 𝑃𝑛,𝑝𝑟𝑜𝑥𝑦 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figures 6-22 and 6-23 also illustrate the equivalent of the multidimensional Mi-  croBooNE analysis using the 𝑃𝑇,𝑥 cross sections in slices of 𝑃𝑇,𝑦 (figure 6-22) and vice  versa (figure 6-23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' As can be seen in the interaction breakdown plots, the regions  close to 0 for both kinematic variables (bottom left panels) are the ideal place to iso-  late QE events with small contributions from more complex events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Yet, even in this  151  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="5  [GeV/c]  Miss  P  π  1p0  (e,e'p)  0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  ]  GeV/c  b  µ  [  Miss  dP  σ  d  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, All events  Data  SuSav2 (Total)  QE  MEC  RES  DIS  G2018  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, All events  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="5  [GeV/c]  n,proxy  P  π  1p0  (e,e'p)  0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  ]  GeV/c  b  µ  [  n,proxy  dP  σ  d  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, All events  Data  SuSav2 (Total)  QE  MEC  RES  DIS  G2018  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, All events  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="5  [GeV/c]  Miss  P  π  1p0  (e,e'p)  0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='25  ]  GeV/c  b  µ  [  Miss  dP  σ  d  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, All events  Miss  P  n,proxy  P  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, All events  Figure 6-21: (Top) data-simulation comparisons on 12C at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='261 GeV showing the  cross section results as a function of (top left) the true missing momentum 𝑃𝑀𝑖𝑠𝑠 and  (top right) the missing momentum approximation 𝑃𝑛,𝑝𝑟𝑜𝑥𝑦 commonly used by neutrino  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (Bottom) overlay of the two extracted data cross sections illustrating  the differences between 𝑃𝑀𝑖𝑠𝑠 and 𝑃𝑛,𝑝𝑟𝑜𝑥𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  very QE-dominated region, significant data-simulation differences are observed, most  likely due to the already-observed RES overestimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' On the other hand, events  occupying the phase-space outside that QE-rich band (bottom right) are dominated  primarily by most complicated interactions (RES and DIS) and QE events with strong  FSI effects, as indicated by the relevant pointy shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These populations result in  much more broader and smeared distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Of particular interest is the asymmetric behavior observed in figure 6-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  As  discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1, this clearly-observed asymmetry is caused by multi-nucleon  and FSI effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' More precisely, the asymmetry in the QE-enhanced region (bottom  left) is entirely driven by the overpredicted RES events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In the non-QE-dominated  slice (bottom right), data-simulation disagreements by a factor of ∼ 2 are further  152  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="5  [GeV/c]  T,x  P  π  1p0  (e,e'p)  0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  ]  GeV/c  b  µ  [  T,x  dP  σ  d  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, All events  Data  SuSav2 (Total)  QE  MEC  RES  DIS  G2018  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, All events  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="5  [GeV/c]  T,x  P  π  1p0  (e,e'p)  0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  2  /c  2  GeV  b  µ  T,y  dP  T,x  dP  σ  2  d  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c  T,y  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 < P  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c  T,y  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 < P  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="5  [GeV/c]  T,x  P  π  1p0  (e,e'p)  0  5  10  15  20  25  3  −  10  ×  2  /c  2  GeV  b  µ  T,y  dP  T,x  dP  σ  2  d  < -0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c  T,y  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, P  < -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c  T,y  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, P  Figure 6-22: Data-simulation comparisons for 12C at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='261 GeV showing the cross  section results as a function of 𝑃𝑇,𝑥 for (top) all the events, (bottom left) events  with -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 < 𝑃𝑇,𝑦 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c dominated by QE interactions and no reinteractions,  and (bottom right) events with 𝑃𝑇,𝑦 < -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c maximally affected by FSI and  multi-hadron channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  observed and are driven by RES/DIS events and FSI-enhanced QE interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  Electrons-For-Neutrinos Conclusions  In this electron-based analysis, the similarities between electron- and neutrino-nucleus  interactions were exploited, and electron scattering data with known beam energies  to test energy reconstruction methods and interaction models were used [208].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Even  in simple interactions where no pions are detected, only a small fraction of events was  found to reconstruct to the correct incident energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' More importantly, widely-used  interaction models reproduced the reconstructed energy distribution only qualita-  tively and the quality of the reproduction varied strongly with beam energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This  153  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  −  1  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="5  −  0  [GeV/c]  T,y  P  π  1p0  (e,e'p)  0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  ]  GeV/c  b  µ  [  T,y  dP  σ  d  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, All events  Data  SuSav2 (Total)  QE  MEC  RES  DIS  G2018  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, All events  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="5  [GeV/c]  T,y  P  π  1p0  (e,e'p)  0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  1  2  /c  2  GeV  b  µ  T,y  dP  T,x  dP  σ  2  d  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c  T,x  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 < P  < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c  T,x  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 < P  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content="5  [GeV/c]  T,y  P  π  1p0  (e,e'p)  0  2  4  6  8  10  3  −  10  ×  2  /c  2  GeV  b  µ  T,y  dP  T,x  dP  σ  2  d  < -0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c  T,x  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, P  < -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c  T,x  12C @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV, P  Figure 6-23: Data-simulation comparisons for 12C at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='261 GeV showing the cross  section results as a function of 𝑃𝑇,𝑦 for (top) all the events, (bottom left) events  with -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 < 𝑃𝑇,𝑥 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c dominated by QE interactions and no reinteractions,  and (bottom right) events with 𝑃𝑇,𝑥 < -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c maximally affected by FSI and  multi-hadron channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  shows both the need and the pathway to improve current models to meet the re-  quirements of next-generation, high-precision experiments such as DUNE [146] and  Hyper-Kamiokande (HK) [226].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  Prospects With CLAS12  Jefferson Lab Experiment E12-17-006, “Electrons for Neutrinos: Addressing Critical  Neutrino–Nucleus Issues” (scientific rating: A) [227] has already taken further data on  more targets with a greater kinematical range using the upgraded CLAS12 detector  shown in figure 6-24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The approved experiment includes measurements on 4He, C, Ar  and Sn with 1-, 2-, 4-, and 6-GeV electron beams, as well as measurements on O with  154  1- and 2-GeV electron beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The 1- and 2-GeV measurements will be performed  with a minimum electron scattering angle of 5𝑜, compared to a minimum CLAS angle  of about 15𝑜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This will extend the measurements down to the much lower momentum  transfers, typical of some neutrino experiments, and to multi-hadron topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' It  will therefore allow comparisons with the lower beam-energy data of T2K and HK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The first part of the experiment ran in the second half of 2021 and the second one in  the beginning of 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The majority of the 2022 data sets have already been collected,  with the exception of the 1 GeV and 16O data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Cherenkov  Time-of-Flight  BAND  Calorimeter  e– beam  Target  Tracker  Figure 6-24: Schematic view of the upgraded CLAS12 detector components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  155  LTCC  FTOF  EC & PCAL  HTCC  CTOF + CND  BAND156  Chapter 7  Summary  The findings presented in this thesis report on both neutrino and electron cross-  section modeling and analysis results in order to critically improve the understanding  of lepton-nucleus interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This knowledge will be used to significantly reduce the  cross-section related uncertainties of forthcoming experiments that aim to extract the  neutrino oscillation parameters with high-accuracy measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The outlined work took a significant step towards this high-precision era with the  use of neutrino data sets from the MicroBooNE liquid argon time projection chamber  detector at Fermi National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The reported neutrino analyses isolated parts  of the phase space where significant model improvements are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Furthermore,  valuable kinematic variables were identified and established as tools to probe specific  nuclear effects with multi-differential measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  This thesis further realized the connections between electron and neutrino inter-  actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This realization resulted in significant improvements of the modeling used  in neutrino oscillation analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Yet, these improved predictions failed to reproduce  exclusive electron scattering results from the CLAS detector at Thomas Jefferson  Laboratory using high-statistics data sets and monoenergetic beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' However, such  comparisons against electron scattering data sets can definitively constrain the vector  part and the nuclear effects in lepton-nucleus interactions in event generators that  will be used for the forthcoming oscillation analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  157  158  Chapter 8  Appendices  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1  Total Struck Nucleon Momentum Derivation  The total momentum of the struck nucleon can be obtained following the derivation  outlined below for two-body interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We followed the approach introduced by  the Minerva collaboration to reconstruct the longitudinal and total nucleon momenta  in [132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We focused on quasielastic-like (QE-like) CC1p0𝜋 processes,  𝜈𝜇 + 𝐴 → 𝜇− + 𝑝 + (𝐴 − 1),  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1)  while using the formalism detailed in [228] for the nuclear masses,  𝑚𝐴 = 22 × 𝑀𝑛 + 18 × 𝑀𝑝 − 𝐵 [𝐺𝑒𝑉 ]  𝑚𝐴−1 = 𝑚𝐴 − 𝑀𝑛 + 𝜖𝑁 [𝐺𝑒𝑉 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2)  Here 𝑀𝑝 and 𝑀𝑛 denote the proton and neutron masses, respectively, B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='34381  GeV is the argon binding energy (obtained from page 3 in [228] for 40 nucleons with  an average binding energy of 9 MeV) and 𝜖𝑁 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='0309 GeV is the removal energy,  159  with  𝜖𝑁 = 𝑆𝑁 + 𝐸𝑁  𝑥 + ⟨𝑇𝐴−1⟩  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3)  where 𝑆𝑁 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9 MeV is the neutron separation energy obtained for argon from  table 7 in [225], 𝐸𝑁  𝑥 is the excitation energy, and ⟨𝑇𝐴−1⟩ is the average kinetic energy  of the remnant system, which is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  In equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1, the incident neutrino energy 𝐸𝜈 is unknown, but the dependence  of 𝛿⃗𝑝 (struck nucleon momentum before the interaction) on 𝐸𝜈 can be removed under  a QE-like approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This was achieved with the process detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  First, we decomposed 𝛿⃗𝑝 into longitudinal and transverse components with respect  to the neutrino direction, used energy/momentum conservation equations and that  𝑝𝜈 = 𝐸𝜈,  𝛿⃗𝑝 ≡ (𝛿⃗𝑝𝑇, 𝛿𝑝𝐿)  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4)  ⃗𝑝𝜈 = ⃗𝑝𝜇 + ⃗𝑝𝑝 + ⃗𝑝𝐴−1  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5)  𝐸𝜈 + 𝛿𝑝𝐿 = 𝑝𝜇  𝐿 + 𝑝𝑝  𝐿  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6)  𝛿⃗𝑝𝑇 = ⃗𝑝𝜇  𝑇 + ⃗𝑝𝑝  𝐿  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7)  𝐸𝜈 + 𝑚𝐴 = 𝐸𝜇 + 𝐸𝑝 + 𝐸𝐴−1  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8)  where ⃗𝑝𝜇 and ⃗𝑝𝑝 are the muon and proton momenta, respectively, 𝐸𝜇 = √︀𝑝2  𝜇 + 𝑚2  𝜇  for the muon candidate and 𝐸𝑝 = √︀𝑝2  𝑝 + 𝑚2  𝑝 for the proton candidate, where the  corresponding momenta are obtained based on the particles’ ranges [137], and 𝐸𝐴−1  is the energy of remnant nuclear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Under the assumption that no final state interactions (FSI) take place,  ⃗𝑝𝜈 + 𝛿⃗𝑝 = ⃗𝑝𝜇 + ⃗𝑝𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9)  160  If we combine equations 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9, 𝛿⃗𝑝 gives the magnitude of its recoil momen-  tum, 𝛿⃗𝑝 = −⃗𝑝𝐴−1 and 𝛿𝑝 = 𝑝𝐴−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Combining equations 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8 to eliminate 𝐸𝜈  yields  𝛿𝑝𝐿 = 𝑚𝐴 + 𝑝𝜇  𝐿 + 𝑝𝑝  𝐿 − 𝐸𝜇 − 𝐸𝑝 − 𝐸𝐴−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='10)  Using the fact that 𝐸𝐴−1 =  √︁  𝑚2  𝐴−1 + 𝑝2  𝐴−1, 𝛿𝑝 = 𝑝𝐴−1 and the decomposition of  𝛿⃗𝑝 into longitudinal and transverse components in equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4,  𝛿𝑝𝐿 = 𝑚𝐴 + 𝑝𝜇  𝐿 + 𝑝𝑝  𝐿 − 𝐸𝜇 − 𝐸𝑝 −  √︁  𝑚2  𝐴−1 + 𝛿𝑝2  𝑇 + 𝛿𝑝2  𝐿  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='11)  For simplicity, we defined  𝑅 ≡ 𝑚𝐴 + 𝑝𝜇  𝐿 + 𝑝𝑝  𝐿 − 𝐸𝜇 − 𝐸𝑝  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='12)  Using R to simplify equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='11,  𝛿𝑝𝐿 = 𝑅 −  √︁  𝑚2  𝐴−1 + 𝛿𝑝2  𝑇 + 𝛿𝑝2  𝐿  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='13)  Rearranging the terms, squaring each side and solving for 𝛿𝑝𝐿 yields  𝛿𝑝𝐿 = 1  2𝑅 − 𝑚2  𝐴−1 + 𝛿𝑝2  𝑇  2𝑅  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='14)  Finally, combining the longitudinal and the transverse components, we obtain the  total struck nucleon momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Given that this momentum is an approximation  following the procedure and the assumptions mentioned above (QE-like scattering  161  and no FSI), we will be referring to it as 𝑝𝑛,𝑝𝑟𝑜𝑥𝑦 as opposed to 𝛿𝑝, where  𝑝𝑛,𝑝𝑟𝑜𝑥𝑦 =  √︁  𝛿𝑝2  𝐿 + 𝛿𝑝2  𝑇  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15)  162  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  Wiener SVD Regularization Technique  The procedure detailed below relies on [139].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We choose to work with a 𝜒2(𝑠) metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  𝜒2(𝑠) = (m − 𝑟 · 𝑠)𝑇𝐶𝑜𝑣−1(m − 𝑟 · 𝑠)  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='16)  Notation  Explanation  Dimension  And Format  m  measured spectrum in data (signal and background events)  m × 1 vector  s  free variable in 𝜒2(𝑠) function  n × 1 vector  ˆ𝑠  estimator of true signal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' obtained after minimizing 𝜒2  n × 1 vector  𝑠𝑡𝑟𝑢𝑒  true signal  n × 1 vector  𝑠  nominal MC signal prediction  n × 1 vector  r  response matrix  m × n matrix  Cov  symmetric covariance matrix  with statistical and systematic uncertainties  m × m matrix  Note that we use 𝑠𝑡𝑟𝑢𝑒 to represent the true signal in order to differentiate from s  which is a variable in the function 𝜒2(𝑠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We use ˆ𝑠 to represent the estimator of the  true signal 𝑠𝑡𝑟𝑢𝑒, which is obtained after minimizing the 𝜒2(𝑠) function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We further  restrict ourselves to the m ≥ n case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Our objective is to minimize 𝜒2(𝑠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Since the covariance matrix Cov is symmetric,  the inverse of it 𝐶𝑜𝑣−1 is also symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Hence, 𝐶𝑜𝑣−1 can be decomposed with  Cholesky decomposition [229] into  𝐶𝑜𝑣−1 = 𝑄𝑇 · 𝑄  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='17)  where Q is a uniquely defined lower triangular matrix and 𝑄𝑇 is its transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  163  We further define  𝑀 ≡ 𝑄 · m  𝑅 ≡ 𝑄 · 𝑟  (𝑝𝑟𝑒 − 𝑠𝑐𝑎𝑙𝑖𝑛𝑔)  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='18)  Notation  Explanation  Dimension  And Format  M  measured spectrum after pre-scaling  m × 1 vector  R  response matrix after pre-scaling  m × n matrix  Q  Lower triangular matrix from Cholesky decomposition of 𝐶𝑜𝑣−1  m × m matrix  𝑀  expectation spectrum after pre-scaling (R · 𝑠)  m × 1 vector  Replacing the definitions in equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='18 into equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='16 yields  𝜒2(𝑠) = (𝑀 − 𝑅 · 𝑠)𝑇(𝑀 − 𝑅 · 𝑠) =  ∑︁  𝑖  (𝑀𝑖 −  ∑︁  𝑗  𝑅𝑖𝑗 · 𝑠𝑗)2  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='19)  Ideally, we want to achieve 𝜒2(𝑠) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' That is obtained when  𝑀 − 𝑅 · 𝑠 = 0 ⇒ 𝑀 = 𝑅 · 𝑠 ⇒ 𝑅𝑇 · 𝑀 = 𝑅𝑇 · 𝑅 · 𝑠  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='20)  We use the fact that 𝑅𝑇 · 𝑅 is a square n×n invertable matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Thus, the exact  solution ˆ𝑠 is uniquely defined as  ˆ𝑠 = (𝑅𝑇 · 𝑅)−1 · 𝑅𝑇 · 𝑀  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='21)  We decompose the measured spectrum into the signal (R · s) part and the noise  / background (N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  𝑀 = 𝑅 · 𝑠𝑡𝑟𝑢𝑒 + 𝑁  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='22)  164  Substituting equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='22 into equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='21 yields  ˆ𝑠 = (𝑅𝑇 · 𝑅)−1 · 𝑅𝑇 · (𝑅 · 𝑠𝑡𝑟𝑢𝑒 + 𝑁)  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='23)  with N representing the “noise” coming from uncertainties (statistical and system-  atic uncertainties associated with both m and r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Each term in the noise vector after  pre-scaling follows a normal distribution with 𝜇 = 0 and 𝜎 = 1, since the denominator  of the 𝜒2 function in equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='19 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' square of error) is unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Given the fact that  each term in the noise vector is independent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' uncorrelated), we refer to the basis  in this domain as orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The response matrix after pre-scaling (R) can be decomposed, using the singular  value decomposition (SVD) approach [230], as  𝑅 = 𝑈 · 𝐷 · 𝑉 𝑇  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='24)  with both 𝑈𝑚×𝑚 and 𝑉𝑛×𝑛 being orthogonal matrices that satisfy 𝑈 𝑇 · 𝑈 = 𝑈 · 𝑈 𝑇  = 𝐼𝑚×𝑚 and 𝑉 𝑇 ·𝑉 = 𝑉 ·𝑉 𝑇 = 𝐼𝑛×𝑛, with U,V and D being uniquely defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' I is the  identity matrix and the subscript represents the dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' D is an m×n diagonal  matrix with positive definite diagonal elements (known as singular values) 𝐷𝑖𝑖 = 𝑑𝑖  arranged in descending order as i increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Notation  Explanation  Dimension  And Format  𝑉 𝑇  right orthogonal matrix from decomposition of R  n × n vector  D  diagonal matrix from decomposition of R  m × n matrix  U  left orthogonal matrix from decomposition of R  m × m matrix  165  Inserting equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='24 into 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='23, we have  ˆ𝑠 = 𝑉 · 𝐷−1 · 𝑈 𝑇 · (𝑅 · 𝑠𝑡𝑟𝑢𝑒 + 𝑁)  = 𝑉 · 𝐷−1 · (𝑅𝑈 · 𝑠𝑡𝑟𝑢𝑒 + 𝑁𝑈)  = 𝑉 · 𝐷−1 · 𝑀𝑈  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='25)  where 𝑅𝑈 ≡ 𝑈 𝑇 · 𝑅, 𝑁𝑈 ≡ 𝑈 𝑇 · 𝑁, and 𝑀𝑈 ≡ 𝑈 𝑇 · 𝑀 are transformations of the  smearing matrix R, the noise N, and the measured signal M, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Note that,  since U is an orthogonal matrix and the elements of the original noise vector N are  uncorrelated, the elements of the new noise vector 𝑁𝑈 are still uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Each  element follows a normal distribution with 𝜇 = 0 and 𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Thus, the basis in this  new domain is still orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  However, there are cases where the unbiased solution to an unfolding problem  via a direct inversion can have catastrophic oscillations via the introduction of huge  variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  A proposed solution is the introduction of a trade-off term between the bias and  the variance to suppress the oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We refer to that term as “regularization” and  is introduced in the form of an n×n matrix 𝐹 that is applied on the exact solution ˆ𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  ˆ𝑠 = 𝐹 · 𝑉 · 𝐷−1 · (𝑅𝑈 · 𝑠𝑡𝑟𝑢𝑒 + 𝑁𝑈)  = 𝐹 · 𝑉 · 𝐷−1 · 𝑈 𝑇 · 𝑀  = 𝐹 · 𝑉 · 𝐷−1 · 𝑀𝑈  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='26)  Using equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='26 (focusing on the diagonal elements ii, using the fact that V  is orthogonal 𝑉 · 𝑉 𝑇 = 𝐼, and that D is diagonal 𝐷−1  𝑖𝑖 = 1/𝑑𝑖),  (𝑉 𝑇 ˆ𝑠)𝑖 = 𝐹𝑖𝑖 · 𝑀𝑈,𝑖  𝑑𝑖  ˆ𝑆(𝜔) = 𝐹(𝜔) · 𝑀(𝜔)  𝑅(𝜔)  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='27)  166  It is easy to see the similarities between the two lines in equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Therefore,  following the same terminology as that for the signal processing, we refer to 𝑀𝑈 after  the SVD transformation as the measurement in the effective frequency domain in  analogy to the frequency domain in the signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The additional matrix F can be decomposed as  𝐹 = 𝑉 · 𝑊 · 𝑉 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='28)  At this point, the F and W matrices are still unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Plugging equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='28 into equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='26 yields  ˆ𝑠 = 𝑉 · 𝑊 · 𝐷−1 · (𝑅𝑈 · 𝑠𝑡𝑟𝑢𝑒 + 𝑁𝑈).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='29)  We consider the expectation value of the signal in the effective frequency domain:  𝑀𝑈 = 𝑈 𝑇 · 𝑀 = 𝑈 𝑇 · 𝑅 · 𝑠  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='30)  In general, the 𝑠𝑡𝑟𝑢𝑒 is unknown, so the expectation signal 𝑠 using the nominal  simulation prediction is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The construction of W is based on the Wiener filter  𝑅2(𝜔)·𝑆2(𝜔)  𝑅2(𝜔)·𝑆2(𝜔)+𝑁2(𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Taking  equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='30 at bin i, we have  ⟨𝑅2 · 𝑆2⟩ = 𝑀 2  𝑈,𝑖 =𝐸𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='27 𝑑2  𝑖 · (  ∑︁  𝑗  𝑉 𝑇  𝑖𝑗 · 𝑠𝑗)2  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='31)  ⟨𝑁 2⟩ = 1  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='32)  resulting in a Wiener filter of  𝑊𝑖𝑘 =  𝑑2  𝑖 · (∑︀  𝑗 𝑉 𝑇  𝑖𝑗 · 𝑠𝑗)2  𝑑2  𝑖 · (∑︀  𝑗 𝑉 𝑇  𝑖𝑗 · 𝑠𝑗)2 + 1 · 𝛿𝑖𝑘  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='33)  At this point, W is uniquely defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Here, equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='32 is obtained, since each  167  element of noise 𝑁𝑈 follows a normal distribution with 𝜇 = 0 and 𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We have  (𝑊 · 𝐷−1)𝑖𝑗 =  𝑑𝑖 · (∑︀  𝑗 𝑉 𝑇  𝑖𝑗 · 𝑠𝑗)2  𝑑2  𝑖 · (∑︀  𝑗 𝑉 𝑇  𝑖𝑗 · 𝑠𝑗)2 + 1 · 𝛿𝑖𝑘  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='34)  The small value of 𝑑𝑖 is balanced by the finite value of the expectation value of  𝑁 2 ≡ 1 and, thus, equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='29 doesn’t suffer from catastrophic oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' From  equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='33, the construction of the Wiener filter takes into account the strengths  of both the signal and noise expectations and is independent of the regularization  strength 𝜏 used in traditional regularization techniques [231].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  As shown in [231], the regularization can be applied on the curvature of the  spectrum instead of the strength of the spectrum, which involves an additional matrix  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This is also be achieved in the Wiener-SVD approach:  𝑀 = 𝑅 · 𝐶−1 · 𝐶 · 𝑠  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='35)  by including an additional matrix C that has the commonly used regularization  forms, such as the first and second order derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Since the effective frequency  domain is determined by the smearing matrix R, the inclusion of C would alter the  basis of the effective frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In this case, the SVD decomposition becomes  (just like shown in equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='24)  𝑅 · 𝐶−1 = 𝑈𝐶 · 𝐷𝐶 · 𝑉 𝑇  𝐶 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='36)  Notation  Explanation  Dimension  And Format  𝑉 𝑇  𝐶  right orthogonal matrix from decomposition of R ·𝐶−1  n × n vector  𝐷𝐶  diagonal matrix from decomposition of R ·𝐶−1  m × n matrix  𝑈𝐶  left orthogonal matrix from decomposition of R ·𝐶−1  m × m matrix  The final solution of the regularisation becomes (just like shown in equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='25  168  by inserting it into equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='23)  ˆ𝑠 = 𝐶−1 · 𝑉𝐶 · 𝑊𝐶 · 𝑉 𝑇  𝐶 · 𝐶 · (𝑅𝑇𝑅)−1 · 𝑅𝑇 · 𝑀  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='37)  or, equivalently,  ˆ𝑠 = 𝐴𝐶 · (𝑅𝑇𝑅)−1 · 𝑅𝑇 · 𝑀  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='38)  where  𝐴𝐶 = 𝐶−1 · 𝑉𝐶 · 𝑊𝐶 · 𝑉 𝑇  𝐶 · 𝐶  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='39)  The corresponding Wiener filter would be (once again, by realizing the similarities  between the two expressions in equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='27 and by using equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='39)  𝑊𝐶,𝑖𝑖 =  𝑑2  𝐶𝑖 · (∑︀  𝑗 𝑉 𝑇  𝐶𝑖𝑗 · (∑︀  𝑙 𝐶𝑗𝑙𝑠𝑙))2  𝑑2  𝐶𝑖 · (∑︀  𝑗 𝑉 𝑇  𝐶𝑖𝑗 · (∑︀  𝑙 𝐶𝑗𝑙𝑠𝑙))2 + 1  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='40)  where 𝐶𝑗𝑙, 𝑉 𝑇  𝐶𝑖𝑗, and 𝑑𝐶𝑖 are matrix elements of matrices C, 𝑉𝐶, and 𝐷𝐶, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Since the unfolded results are a linear transformation of the measurement, we  can easily evaluate the uncertainties associated with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='38 can be  rewritten as  ˆ𝑠 = 𝑅𝑡𝑜𝑡 · 𝑚  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='41)  with  𝑅𝑡𝑜𝑡 = 𝐴𝐶 · (𝑅𝑇𝑅)−1 · 𝑅𝑇 · 𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='42)  Then, the covariance matrix of ˆ𝑠 can be deduced from the covariance matrix of  m as  𝐶𝑜𝑣^𝑠,m = 𝑅𝑡𝑜𝑡 · 𝐶𝑜𝑣m · 𝑅𝑇  𝑡𝑜𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='43)  The variances of the unfolded data can also be easily calculated given that their  169  origin N in equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='37 is uniquely defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Defining N(i) as a vector with the i-th  element being 1 and the rest of elements being 0, we can calculate the variance in s  due to i-th element in N as:  𝑇𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛(𝑖) = 𝐴𝐶 · (𝑅𝑇𝑅)−1 · 𝑅𝑇 · 𝑁(𝑖),  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='44)  with 𝑇𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛(i) being a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The variance of the j-th element of 𝑇𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛,𝑗 can  thus be written as:  𝑇𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛,𝑗 =  √︃∑︁  𝑖  𝑇 2  𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛,𝑗(𝑖),  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='45)  after summing the contribution from each independent noise source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The square  of 𝑇𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛,𝑗 corresponds to the j-th diagonal element of the covariance matrix 𝐶𝑜𝑣𝑠  in equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Given equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='37, we can understand the entire process of unfolding as to  “remove” the effect of R through multiplying (𝑅𝑇𝑅)−1 · 𝑅𝑇 and then replace it with  a new smearing matrix 𝐴𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Therefore, it is straightforward to estimate the bias on  the unfolded results:  𝑇𝑏𝑖𝑎𝑠 = (𝐴𝐶 − 𝐼) · (𝑅𝑇𝑅)−1 · 𝑅𝑇 · 𝑀  = (𝐴𝐶 − 𝐼) · 𝑠  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='46)  with I being identity matrix and 𝑠 being the expectation of the nominal MC signal  prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  170  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3  Electrons-For-Neutrinos Fiducials  Fiducial cuts for e2a have been defined and used by several analyses [232–234] and  follow the same general procedure for all charged particle species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A series of event  selection cuts are applied to data to produce event samples suitable for defining the  fiducial regions of CLAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' From these event samples, regions of good acceptance are  found, defined as flat regions of 𝜑 space in bins of momentum and 𝜃, which are then  parameterised into functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The specific procedures for each particle species for  which fiducial cuts have been defined are summarised in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (a)  (b)  Figure 8-1: The 𝜃 vs 𝜑 distributions for 𝜋−, before (a) and after (b) the application  of fiducial cuts as defined in [235] for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV analysis of 4He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Electron fiducial cuts were defined in [236], using electron candidates identified by  the CLAS triggering requirements for e2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These electron samples are then subjected  to geometric cuts in the 𝑢, 𝑣, and 𝑤 “views” corresponding to the orientation of the  scintillator layers of the electromagnetic calorimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This cut accounts for poorly  understood electron acceptance at the edges of the calorimeter, and the difference in  acceptance between the electromagnetic calorimeter and the Cerenkov Counter, which  can be seen in data as a characteristic “smile” feature in the uncut 𝜃 vs 𝜑 distributions  for electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Additionally, a cut is applied on the ratio of energy deposition by an  electron in the calorimeter to electron momentum as measured by the drift chambers,  𝐸𝑡𝑜𝑡/𝑝𝑒′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This ratio is known as the ‘sampling fraction’ and is fixed by the design of  the detector to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' At the detector edges, this value can decrease due to shower  171  , (degrees)  180  160  450  140  400  350  120  300  100  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='.  250  80  200  60  150  40  100  20  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  50  100  150  200  250  300  350  0  (degrees) (degrees)  180  400  160  350  140  300  120  250  100  200  80  150  60  100  40  20  50  0  50  100  150  200  250  300  350  0  (degrees)leakage, where the electron energy is not fully deposited in the calorimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The electrons distributions for each beam energy and torus field setting are then  split into sectors and momentum bins 50 𝑀𝑒𝑉/𝑐 wide, spanning the electron momen-  tum range of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A plot of 𝜃𝑒 against 𝜑𝑒 is produced for each momentum bin,  and for each degree in 𝜃𝑒, the plot was projected onto the 𝜑𝑒 axis and fitted with a  trapezoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (a)  (b)  Figure 8-2: The 𝜃 vs 𝜑 distributions for electrons, before (left) and after (right) the  application of fiducial cuts for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV analysis of 4He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Positively charged hadrons, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' protons and 𝜋+, have their fiducial cuts defined as  a single species, under the assumption that their identical charge means their fiducial  regions will be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The positive hadron fiducial cuts were defined in [237].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Event samples are identified as tracks with good drift chamber status, a good hit  in the time-of-flight system, and particle identification via a 𝜒2 cut on the DCPB  bank  In order to eliminate protons in quasi-free reactions, which are not uniformly  distributed in 𝜑, an energy transfer cut is applied  The positively charged hadron distributions for each beam energy and torus field  setting are then split into sectors and momentum bins 50 𝑀𝑒𝑉/𝑐 wide, spanning the  momentum range of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A plot of 𝜃 against 𝜑 is produced for each momentum  bin, and for each degree in 𝜃, the plot was projected onto the 𝜑 axis and fitted with  a trapezoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  172  120  5000  100  4000  80  60  3000  40  2000  20  1000  0  0  50  100  150  200  250  300  350(degrees)  120  3500  100  3000  80  2500  60  2000  1500  40  1000  20  5000  0  50  100  150  200  250  300  350 (degrees)(a)  (b)  Figure 8-3: The 𝜃 vs 𝜑 distributions for protons, before (left) and after (right) the  application of fiducial cuts for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV analysis of 4He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (a)  (b)  Figure 8-4: The 𝜃 vs 𝜑 distributions for 𝜋+, before (left) and after (right) the appli-  cation of fiducial cuts for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV analysis of 4He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  As neutral particles, the fiducial cuts for photons take the from of a cut on the  detection area of the electromagnetic calorimeter, with no momentum dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Photon events are selected after applying the same 𝑢, 𝑣 and 𝑤 calorimeter cuts as  used for electrons, and the fiducial regions for each energy and target defined on  distributions of 𝑐𝑜𝑠𝜃 vs 𝜑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Two first order polynomials are used to describe the  outline of the sides of the sector and two second order polynomials to describe the  top and bottom edges, as shown in figure 8-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Additionally, two ‘hot spots’ in the  corners of sector four were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The originally defined fiducial cut functions for e2a were implemented in C++ and  FORTRAN versions for the ROOT and PAW data analysis frameworks respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  173  180  160  4000  140  3500  120  3000  100  2500  80  2000  60  1500  40  1000  20  500  50  100  150  200  250  350  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  300180  160  4000  140  3500  120  3000  100  2500  80  2000  60  1500  40  1000  20  500  50  100  150  200  250  300  350  0  0  (degrees)  proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='- (degrees)  180  160  006  800  140  700  120  600  100  500  80  400  60  300  40  200  20  100  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  50  100  150  200  250  300  350  0  (degrees)  T* (degrees)  180  900  160  800  140  700  120  600  100  500  80  400  60  人  300  40  200  20  100  50  100  150  200  250  300  350  0  0  (degrees)  T(a) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 8-5: The cos 𝜃 vs 𝜑 distributions for photons, with fiducial cut outline indicated  by red for 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV analysis for 3He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  They were checked into the CLAS CVS repository for e2 analysis software, and have  been applied as part of the various analyses performed on this data since it was  collected in 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The e4nu analyses of e2a data have used the C++/ROOT implementation of the  fiducial cuts, incorporating them into their analysis software, with minor modifica-  tions to ensure compatibility with contemporary software environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  In the analysis of [235], 𝜋− fiducial cuts were defined, initially by direct reuse of  the electron fiducial cut parameters to cut fiducial regions for 𝜋−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This was considered  to offer a reasonable approximation for 𝜋−, having the same charge and thus similar  behaviour in the torus field of CLAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' However, because the electron fiducial cuts are  only defined to a minimum momentum of 350 MeV/c, and the minimum momentum  threshold used for pion detection in CLAS is 150 MeV/c, a new set of cut parameters  had to be defined for the 𝜋− in the momentum range 150 − 350 MeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' These used  50 MeV/c momentum bins on 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV 12C data, and follow the same procedure to  obtain the 𝜃 vs 𝜑 outline cuts and the 𝜃 gaps corresponding to malfunctioning TOF  paddles at this low momentum as described in [234] for the 𝑝 > 350 MeV/c region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  At 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 GeV beam energy, this procedure was used to define 𝜋− fiducial cuts for  the whole momentum range, as the electron cuts offer minimal coverage for 𝜋−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The theta gap functions for dead channels in 𝜋− were updated to account for  previously missed gaps, and to skip electron gaps from the CC, a detector not used  174  160  140  120  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='9  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  80  60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='7  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5  0  50  100  150  200  250  300  350  o[Deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' ]in determining the 𝜋−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Several functions were refit to extend to the lower momentum  range of the pions, removing the need to apply them only in specific momentum  ranges, and allowing the clumsy box cuts previously used to be eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 8-6: The 𝜃 vs momentum distributions for 𝜋− in sector 3, after the application  of the original fiducial and theta gap cuts for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV analysis of 4He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  At low  momentum, the theta gap cuts fail, and have been replaced box cuts that do not  appropriately describe the gaps at low momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 8-7 shows the polar angle versus momentum distributions for 𝜋− in each  sector of CLAS, after the application of fiducial cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Of the theta gap functions used  for the electrons, several are carried over unchanged, some are updated in order to  appropriately apply to the full momentum range of 𝜋−, while gaps defined for the CC  are omitted, as this subsystem is not used to identify 𝜋−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 8-8 shows the polar angle versus momentum distributions for 𝜋− in each  sector of CLAS, after the application of fiducial cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' As with the 1 GeV case, several  theta gap functions are carried over unchanged from electrons, some are updated in  order to appropriately apply to the full momentum range seen for 𝜋−, and CC gaps  are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Additionally at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV, the maximum polar angle condition  for 𝑝 > 350 MeV/c, imposed by the electron fiducial cut parameters, is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  At 1 GeV beam energy, dedicated fiducial cut parameters for 𝜋− at 750 A torus  current were defined as part of the analysis of [235].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The 1500 A data was not used in  175  rees)  180  180  (degre  160  160  140  140  120  120  100  100  80  80  60  60  40  40  20  20  U  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8  2  元momentum(GeV)Figure 8-7: The 𝜃 vs momentum distributions for 𝜋−, showing the 𝜋− updated theta  gap cuts (green), retained (red) and not used (yellow) electron theta gap cuts for  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 GeV analysis of 3He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  the analysis of [235], therefore no bespoke 𝜋− fiducial cut parameters were produced  for this field setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure 8-9 shows the 𝜃 vs momentum distributions for the 𝜋−  at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 GeV, after the application of the updated fiducial cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure 8-10 shows the 𝜃 vs momentum distributions for the 𝜋− at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV, after  the application of the updated fiducial cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The same cuts are used in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV  analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  As seen in the reuse of electron theta gap cuts on the 𝜋−, where the gap functions  were not defined to the lower momentum of the pions, a similar effect is seen in 𝜋+,  with the gap functions defined for protons directly reused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This has resulted in similar  box cuts being applied at low momentum, as can be seen in figure 8-11, removing  good events rather than low and poorly understood acceptance regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Unlike the 𝜋−, the positive hadron fiducial cut functions, defined for protons, are  for the most part valid at the lower pion momentum range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' We have extended the  application of several of these functions to define cuts at lower 𝜋+ momenta, where  the functions continue to appropriately describe the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' This is shown for sector 3  176  180  35  180  180  160  160  160F  35  30  140  30  140E  140  30  120E  25  120 E  25  120E  25  100E  20  100  20  100  20  80E  415  80  80  15  15  60  60  60E  10  10  40F  10  40  40  20F  5  20  5  20  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='2Figure 8-8: The 𝜃 vs momentum distributions for 𝜋−, showing the 𝜋− updated theta  gap cuts (green), retained (red) and not used (yellow) electron theta gap cuts for  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV analysis of 3He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A new parameterisation of the maximum polar angle cut is  also shown (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The same gap functions are applied at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV  (a)  Figure 8-9: The 𝜃 vs momentum distributions for 𝜋−, after the application of the  updated theta gap cuts for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 GeV analysis of 3He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  177  180  180  180  18C  16C  160  14(  160  160E  16C  14C  140  12(  140  140  14C  12C  120E  120 E  120E  10C  12C  10C  100F  100F  100E  10C  80  80E  80F  80  80  80  60  60  60F  60  60  60  40F  40  40  40  40  40  20F  20  20  20  20  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  178  180  180  180  18C  16C  140  160  160  16C  14C  140  12(  140  140  14C  12C  120  120  120E  100  12C  10C  100E  100  100E  10C  80  80E  80F  80  80  80  60  60  60  60  60  60  40F  40  40  40  40  40  20  20  20  20  20  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  (a)  (b)  Figure 8-12: The 𝜃 vs momentum distributions for 𝜋+ in sector 3, showing the original  (a) and new (b) theta gap cuts for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='5  元*momentum(GeV)180  List of Figures  1-1  The Standard Model of particle physics illustrating the three genera-  tions of fermions, the gauge bosons, and the scalar Higgs boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  12  1-3  The experimental setup of an accelerator-based long-baseline neutrino  oscillation experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  20  1-4  Neutrino energy spectra reconstruction depends on our ability to model  the interaction of neutrinos with atomic nuclei and the propagation  of particles through the atomic nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  This flow chart shows the  process, starting with an oscillated far-detector incident-energy spec-  trum (green), differentiating the physical neutrino interactions (green  arrows) from the experimental analysis (blue arrows), and ending up  with an inferred incident-energy spectrum that hopefully matches the  actual one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  22  1-5  The four main interaction processes for neutrino-nucleus scattering  events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  25  1-6  Neutrino cross-section evolution as a function of the neutrino energy  illustrating the energy range where each one of the four main processes  dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  26  1-7  Feyman diagrams illustrating charged current (CC) and neutral current  (NC) processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  26  181  1-8  Nucleon momentum distribution options available in commonly used  neutrino event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  27  1-9  Schematic illustration the wealth of possible final state interactions  that the final state hadrons can undergo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  27  1-10 Quasielastic lepton-nucleus scattering where the outgoing nucleon does  not reinteract as it leaves the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  30  1-11 Graphic illustration of the three complementary approaches used in  this thesis to improve our understanding of lepton-nucleus interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  31  2-1  The Fermilab accelerator complex showing the accelerator components  and the different beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  34  2-2  The ingredients for a neutrino beam include the accelerated protons,  the target, the magnetic horn, the decay pipe, and the absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Fig-  ure adapted from [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  35  2-3  The BNB neutrino flux prediction through the MicroBooNE detector  for 𝜈𝜇, ¯𝜈𝜇, 𝜈𝑒, and ¯𝜈𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A TPC volume with dimensions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='56 m × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='33 m  × 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='37 m is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  35  2-4  Schematic illustration of the MicroBooNE detector and its dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure adapted from [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  37  2-5  The MicroBooNE coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  37  2-6  The design of CRT planes as part of the MicroBooNE detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Simula-  tion of cosmic rays crossing the CRT, the brown lines represent possible  cosmic ray trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' There are four CRT planes: top plane, bottom  plane, pipe side plane and feedthrough side plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The beam direction  is along the z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  38  2-7  Working principle of a LArTPC detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  39  182  2-8  (Left) bipolar (U and V induction planes) and unipolar (Y collec-  tion plane) signal induction on the three MicroBooNE planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure  adapted from [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (Right) schematic view of the wire planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  vertical collection Y wires are shown in pink, the induction U wires,  angled at +60𝑜 are shown in blue and the induction V wires, angled at  60𝑜, are shown in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  40  2-9  Entry/exit points of cosmic muon tracks with a signal from a muon  counter located outside of the cryostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In the absence of space charge  effects, the points should be located along the TPC boundaries indi-  cated by the dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  41  2-10 The production of scintillation light in liquid argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' A charged particle  can either excite or ionise the argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  42  2-11 The MicroBooNE light collection system with the 32 PMTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure  adapted from [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  44  2-12 (Left) cosmic-induced event that was stored because of the coincidence  of a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='6 𝜇s accelerator BNB signal and light detected by the PMTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (Right) neutrino-induced event where the light was coming from a neu-  trino interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  45  2-13 Optically reconstructed flash object recorded by the MicroBooNE PMT  light collection system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The dark orange regions represent a higher PE  yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  46  2-14 Neutrino-induced tracks (black) are matched to the corresponding light  signals collected by PMTs (red circles) and are clearly separated from  the cosmic-induced ones (dimmed color tracks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' 47  2-15 Candidate neutrino event display from MicroBooNE data on one of the  induction planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (a) The raw waveform image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (b) The image after  noise-filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (c) The image after 2D deconvolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The image qual-  ity near the neutrino interaction vertex significantly improves after the  2D deconvolution and the latter leads to improvements in the pattern  recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  49  183  2-16 Illustration of the hierarchical structure of particles reconstructed for  a simulated charged current 𝜈𝜇 event in MicroBooNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The interaction  includes a muon, proton and charged pion in the visible final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Figure adapted from [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  50  2-17 Illustration of the measured dQ/dx vs dE/dx distribution with the  modified recombination model in the MC simulation with the Ar-  goNeuT parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  51  2-18 MicroBooNE event display of an event in a cosmic overlay sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  A simulated neutrino event (orange box) is overlaid on top of cosmic  events (blue box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  52  3-1  Interaction breakdown of the cos𝜃𝜇 plot illustrating the dominance of  CCQE interactions after the application of our selection cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  59  3-2  Interaction breakdown of the cos𝜃𝜇 plot illustrating the dominance of  RES interactions after the application of our selection cuts for the non-  CC1p0𝜋 background part of the MC sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  60  3-3  Vertex 𝑧 distribution for the measured events, after the beam related  MC background has been subtracted, before (left) and after (right)  detection efficiency corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' No small-𝑧 enhancement is observed  and, with efficiency corrections, the measured distribution is consistent  with that of a uniform neutrino interaction vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  62  3-4  The flux integrated single differential CC1p0𝜋 cross sections as a func-  tion of the cosine of the measured muon scattering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Inner and  outer error bars show the statistical and total (statistical and system-  atic) uncertainty at the 1𝜎, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Colored lines  show the results of theoretical absolute cross section calculations using  different event generators (without passing through a detector simula-  tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The blue band shows the extracted cross section obtained from  analyzing MC events propagated through our full detector simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The width of the band denotes the simulation statistical uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  64  184  3-5  As figure 3-4, but for the differential cross sections as a function of  measured muon momentum (left) and measured proton scattering an-  gle (middle) and momentum (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Cross sections are shown for the  full measured phase-space (top) and for events with cos(𝜃𝜇) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='8 (bot-  tom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Inner and outer error bars show the statistical and total (statis-  tical and systematic) uncertainty at the 1𝜎, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Colored lines show the results of theoretical absolute cross section cal-  culations using different event generators (without passing through a  detector simulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The blue band shows the extracted cross section  obtained from analyzing MC events passed through our full detector  simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  66  3-6  The flux integrated single differential CC1p0𝜋 cross sections as a func-  tion of 𝑄2  𝐶𝐶𝑄𝐸 = (𝐸𝑐𝑎𝑙  𝜈  − 𝐸𝜇)2 − (⃗𝑝𝜈 − ⃗𝑝𝜇)2 and 𝐸𝑐𝑎𝑙  𝜈  = 𝐸𝜇 + 𝑇𝑝 + 𝐵𝐸,  where 𝐵𝐸 = 40 MeV and ⃗𝑝𝜈 = (0, 0, 𝐸𝑐𝑎𝑙  𝜈 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Colored lines show the  results of theoretical absolute cross section calculations using different  event generators (without passing through a detector simulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  blue band shows the extracted cross section obtained from analyzing  MC events passed through our full detector simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  67  3-7  Muon angular distribution after the implementation of the analysis  framework improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' No data-MC disagreement is observed in  the forward direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  69  3-8  Schematic illustration of the single transverse variables 𝛿𝑝𝑇, 𝛿𝛼𝑇 and  𝛿𝜑𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  70  3-9  Schematic illustration of 𝛿𝑝𝑇𝑥 and 𝛿𝑝𝑇𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  72  3-10 Interaction breakdown of the CC1p0𝜋 events as a function of 𝛿𝑝𝑇𝑥 (left)  and 𝛿𝑝𝑇𝑦 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The data correspond to the “combined” MicroBooNE  runs 1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  73  3-11 Topological (left) and interaction (right) breakdown after the applica-  tion of the event selection for 𝛿𝑝𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  74  3-12 Response matrices of 𝛿𝑝𝑇 using the selected CC1p0𝜋 MC events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  75  185  3-13 Total covariance matrix for 𝛿𝑝𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  77  3-14 Additional smearing matrix 𝐴𝑐 for 𝛿𝑝𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  78  3-15 The flux-integrated single- (top) and double- in 𝛿𝛼𝑇 bins (bottom) dif-  ferential CC1p0𝜋 cross sections as a function of the transverse missing  momentum 𝛿𝑝𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Inner and outer error bars show the statistical and  total (statistical and shape systematic) uncertainty at the 1𝜎, or 68%,  confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The gray band shows the normalization systematic  uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Colored lines show the results of theoretical absolute cross  section calculations with and without FSI based on the GENIE and  GiBUU event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  80  3-16 The flux-integrated single- (top) and double- in 𝛿𝑝𝑇 bins (bottom)  differential CC1p0𝜋 cross sections as a function of the angle 𝛿𝛼𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Inner  and outer error bars show the statistical and total (statistical and shape  systematic) uncertainty at the 1𝜎, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The gray  band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Colored lines  show the results of theoretical absolute cross section calculations with  a number of FSI modeling options based on the GENIE event generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' 81  3-17 The flux-integrated single- (top) and double- in 𝛿𝑝𝑇,𝑦 bins (bottom)  differential CC1p0𝜋 cross sections as a function of the angle 𝛿𝑝𝑇,𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Inner and outer error bars show the statistical and total (statistical and  shape systematic) uncertainty at the 1𝜎, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Colored  lines show the results of theoretical absolute cross section calculations  with a number of event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  82  3-18 Cross-section interaction breakdown for all the selected events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  breakdown is shown for (top left) the G18 configuration with FSI ef-  fects, (top right) the G18 configuration without FSI effects, (bottom  left) GiB with FSI effects, and (bottom right) GiB without FSI effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  83  186  3-19 Cross-section interaction breakdown for events with 𝛿𝛼𝑇 < 45𝑜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  breakdown is shown for (top left) the G18 configuration with FSI ef-  fects, (top right) the G18 configuration without FSI effects, (bottom  left) GiB with FSI effects, and (bottom right) GiB without FSI effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  83  3-20 Cross-section interaction breakdown for events with 135𝑜 < 𝛿𝛼𝑇 <  180𝑜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The breakdown is shown for (top left) the G18 configuration  with FSI effects, (top right) the G18 configuration without FSI effects,  (bottom left) GiB with FSI effects, and (bottom right) GiB without  FSI effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  84  3-21 Cross-section interaction breakdown for all the selected events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  breakdown is shown for (top left) the G21 hA configuration with the  hA2018 FSI model, (top right) the G21 hN configuration with the hN  FSI model, (bottom left) the G21 G4 configuration with the G4 FSI  model, and (bottom right) the G21 No FSI configuration without FSI  effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  84  3-22 Cross-section interaction breakdown for events with 𝛿𝑝𝑇 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The breakdown is shown for (top left) the G21 hA configuration with  the hA2018 FSI model, (top right) the G21 hN configuration with the  hN FSI model, (bottom left) the G21 G4 configuration with the G4  FSI model, and (bottom right) the G21 No FSI configuration without  FSI effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  85  3-23 Cross-section interaction breakdown for events with 𝛿𝑝𝑇 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The breakdown is shown for (top left) the G21 hA configuration with  the hA2018 FSI model, (top right) the G21 hN configuration with the  hN FSI model, (bottom left) the G21 G4 configuration with the G4  FSI model, and (bottom right) the G21 No FSI configuration without  FSI effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  85  187  3-24 Cross-section interaction breakdown for all the selected events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  breakdown is shown for (top left) the G18 LFG configuration, (top  right) the G18 RFG configuration, (bottom left) the G18 EffSF con-  figuration, and (bottom right) the G18 No RPA configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  86  3-25 Cross-section interaction breakdown for events with 𝛿𝑝𝑇,𝑦 < -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The breakdown is shown for (top left) the G18 LFG configuration, (top  right) the G18 RFG configuration, (bottom left) the G18 EffSF con-  figuration, and (bottom right) the G18 No RPA configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  86  3-26 Cross-section interaction breakdown for events with -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 < 𝛿𝑝𝑇,𝑦 <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='15 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The breakdown is shown for (top left) the G18 LFG con-  figuration, (top right) the G18 RFG configuration, (bottom left) the  G18 EffSF configuration, and (bottom right) the G18 No RPA config-  uration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  87  3-27 MicroBooNE total Protons on Target (POT) collected with the Booster  Neutrino Beam (BNB) during the five run periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' In this thesis, the  first three run periods were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  88  4-1  Charged-current cross sections as a function of neutrino energy ob-  tained using GENIE for muon neutrino scattering using the DUNE near  detector (left) and far detector (right) oscillated fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The shaded  bands show the fractional contribution for each interaction mechanism,  quasielastic scattering (QE), meson-exchange currents (MEC), reso-  nance excitation (RES), and deep inelastic scattering (DIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' See text  for details of the interaction mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  92  4-2  (Left) electron-nucleus inclusive scattering via one-photon exchange  and (right) charged current neutrino-nucleus inclusive scattering via  𝑊 exchange with a final state charged lepton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  93  188  4-3  Reaction mechanisms for lepton-nucleus scattering (a) quasielastic scat-  tering (QE) where one nucleon is knocked out of the nucleus, (b) 2p2h  where two nucleons are knocked out of the nucleus, (c) RES resonance  production where a nucleon is excited to a resonance which decays to  a nucleon plus meson(s), and (d) DIS where the lepton interacts with  a quark in the nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  93  4-4  Comparison between GENIE v2 and v3 descriptions of inclusive C(𝑒, 𝑒′)  scattering cross sections at 𝐸0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='56 GeV, 𝜃𝑒 = 60∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='24  GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Black points show the data, solid black line shows the GENIE  v3 results and dashed black line shows the GENIE v2 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  95  4-5  Comparison of inclusive C(𝑒, 𝑒′) scattering cross sections for data and  for GENIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (left) data vs GSuSAv2 and (right) data vs G2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (top)  𝐸0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='24 GeV, 𝜃𝑒 = 60∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='05 GeV2 (middle) 𝐸0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='56  GeV, 𝜃𝑒 = 36∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='11 GeV2, and (bottom) 𝐸0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='56 GeV,  𝜃𝑒 = 60∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='24 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Black points show the data, solid  black lines show the total GENIE prediction, colored lines show the  contribution of the different reaction mechanisms: (blue) QE, (red)  MEC, (green) RES and (orange) DIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  102  4-6  Comparison of inclusive C(𝑒, 𝑒′) scattering cross sections for data and  for GENIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (left) data vs GSuSAv2 and (right) data vs G2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (top)  𝐸0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='96 GeV, 𝜃𝑒 = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='32 GeV2, (middle) 𝐸0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='30  GeV, 𝜃𝑒 = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='54 GeV2, and (bottom) 𝐸0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='22 GeV,  𝜃𝑒 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='33 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Black points show the data, solid  black lines show the total GENIE prediction, colored lines show the  contribution of the different reaction mechanisms: (blue) QE, (red)  MEC, (green) RES and (orange) DIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  103  189  4-7  Comparison of inclusive C(𝑒, 𝑒′) scattering cross sections for data and  for GENIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (left) data vs GSuSAv2 and (right) data vs G2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (top)  𝐸0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='501 GeV, 𝜃𝑒 = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='92 GeV2, (middle) 𝐸0 =  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='595 GeV, 𝜃𝑒 = 16∘ and 𝑄2  𝑄𝐸 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='04 GeV2, and (bottom) 𝐸0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='595  GeV, 𝜃𝑒 = 20∘ and 𝑄2  𝑄𝐸 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='3 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Black points show the data,  solid black lines show the total GENIE prediction, colored lines show  the contribution of the different reaction mechanisms: (blue) QE, (red)  MEC, (green) RES and (orange) DIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  104  4-8  Comparison of inclusive Fe(𝑒, 𝑒′) scattering cross sections for data and  for GENIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (left) data vs GSuSAv2 and (right) data vs G2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (top)  Fe(𝑒, 𝑒′), 𝐸0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='56 GeV, 𝜃𝑒 = 60∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='24 GeV2, (middle)  Fe(𝑒, 𝑒′), 𝐸0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='96 GeV, 𝜃𝑒 = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='32 GeV2, (bot-  tom) Fe(𝑒, 𝑒′), 𝐸0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='30 GeV, 𝜃𝑒 = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='54 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Black points show the data, solid black lines show the total GENIE  prediction, colored lines show the contribution of the different reaction  mechanisms: (blue) QE, (red) MEC, (green) RES and (orange) DIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  105  4-9  Comparison of inclusive Ar(𝑒, 𝑒′) scattering cross sections for data and  for GENIE at 𝐸0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='22 GeV, 𝜃𝑒 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5∘ and 𝑄2  𝑄𝐸 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='33 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (left)  data vs GSuSAv2 and (right) data vs G2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Black points show the  data, solid black lines show the total GENIE prediction, colored lines  show the contribution of the different reaction mechanisms: (blue) QE,  (red) MEC, (green) RES and (orange) DIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  106  4-10 Comparison of inclusive proton (left) and deuterium (right) (𝑒, 𝑒′) scat-  tering cross sections for data and for GENIE using G2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (top) 𝐸0 =  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='445 GeV and 𝜃𝑒 = 20∘, (middle) 𝐸0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='245 GeV and 𝜃𝑒 = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='98∘,  and (bottom) 𝐸0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5 GeV and 𝜃𝑒 = 41∘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Black points show the  data, solid black lines show the total GENIE prediction, colored lines  show the contribution of the different reaction mechanisms: (green)  RES and (orange) DIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The first peak at lowest energy transfer is the  ∆(1232) resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  107  190  4-11 Comparison of inclusive proton (left) and deuterium (right) (𝑒, 𝑒′) scat-  tering cross sections for data and for GENIE using G2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  (top)  𝐸0 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='499 GeV and 𝜃𝑒 = 4∘, (middle) 𝐸0 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='699 GeV and 𝜃𝑒 = 4∘,  and (bottom) 𝐸0 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='993 GeV and 𝜃𝑒 = 4∘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Black points show the  data, solid black lines show the total GENIE prediction, colored lines  show the contribution of the different reaction mechanisms: (green)  RES and (orange) DIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The first peak at lowest energy transfer is the  ∆(1232) resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  108  4-12 Comparison of semi-exclusive 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='16 GeV lepton-carbon scattering for  𝑄2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The number of generated events is plotted versus en-  ergy transfer (left) and 4-momuntm transfer squared (right) for events  with exactly one proton with 𝑃𝑝 ≥ 300 MeV/c, no charged pions with  𝑃𝜋 ≥ 70 MeV/c and no neutral pions or photons of any momentum for  eGENIE electrons (orange) and GENIE CC 𝜈𝜇 (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The electron  events have been weighted by 𝑄4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Both curves are area normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  110  4-13 Number of simulated events for QE scattering on 12C at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='161 GeV with  𝑄2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 shown as a function of the energy transfer 𝜔 and the momen-  tum transfer 𝑞3 = |⃗𝑞 | for all the available nuclear models in GENIE for  neutrinos (top) and for electrons (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (left) the GSuSAv2 model  which uses a Relativistic Mean Field momentum distribution, (middle)  the Nieves or Rosenbluth cross section with the Local Fermi Gas mo-  mentum distribution, and (right) the Llewellyn-Smith or Rosenbluth  cross section with the Relativistic Fermi Gas momentum distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The electron events have been weighted by 𝑄4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  111  4-14 Initial momentum distribution of protons in simulated QE C(𝑒, 𝑒′𝑝)  events at 𝐸 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='161 GeV for the Local Fermi Gas (solid histogram)  and Relativistic Fermi Gas (dotted histogram) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The two curves  are normalized to have the same area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  112  191  4-15 Number of simulated events as a function of the energy transfer 𝜔 and  of the momentum transfer 𝑞3 = |⃗𝑞 | for neutrinos (left) and for electrons  (right) using GSuSav2 for MEC interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The electron events have  been scaled by 𝑄4 and all the samples have been generated with 𝑄2 ≥  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  112  5-1  Schematic view of the accelerator facility and the experimental halls  at Jefferson Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [196].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  116  5-2  Drawing of the CLAS detector showing the sector structure and the  different detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The beam enters from the upper left side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The  target is located at the center of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  117  5-3  The CLAS superconducting toroidal magnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [197].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='118  5-4  (Left) Illustration of the region 3 drift chamber structure for one of  the CLAS sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (Right) Schematic representation of the thee drift  chamber regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figures adapted from [198].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  118  5-5  Schematic view of a CLAS electromagnetic calorimeter module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Fig-  ures adapted from [199].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  120  5-6  Optical arrangement of one of the optical modules of the CLAS Cherenkov  detector, showing the optical and light collection components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figures  adapted from [200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  121  5-7  Schematic view of the TOF counters in one sector illustrating the  grouping into four panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Figure adapted from [201].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  122  6-1  The expected energy distribution of different 𝜈𝜇 beams, (left) before  oscillation at the near detector and (right) after oscillation at the far  detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The vertical lines show the three electron beam energies of  this measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The NO𝜈A far-detector beam flux is calculated  using the near detector flux and the neutrino oscillation parameters  from [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  126  6-2  ∆𝜃𝛾,𝑒′ as a function of ∆𝜑𝛾,𝑒′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The red box indicated the region with  radiated photons which was removed in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  129  192  6-3  Number of events vs 𝐸𝑐𝑎𝑙 = 𝐸𝑒′ + 𝑇𝑝 the scattered electron energy plus  proton kinetic energy for 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' Black points are data,  the blue histogram shows the unradiated GENIE prediction and the  black histogram shows the GENIE prediction with electron radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The GENIE calculations have been scaled to have the same integral as  the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  130  6-4  Schematic illustration of the data driven background correction using  detected (e,e’p𝜋) events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  132  6-5  The effect of undetected pion subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The number of weighted  events as a function of reconstructed energy 𝐸𝑄𝐸 for 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV Fe(𝑒, 𝑒′)  events for (left) events with a detected 𝜋± or photon (blue), events with  one (red) or two (light brown) undetected 𝜋± or photons and (right)  all (𝑒, 𝑒′𝑋) events with detected or undetected 𝜋± or photon (blue),  (𝑒, 𝑒′) events with no detected 𝜋± or photon (red), and (𝑒, 𝑒′) events  after subtraction for undetected 𝜋± or photon (light brown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  133  6-6  Illustration of the successful closure test of the data driven correc-  tion for undetected particles as a function of 𝐸𝐶𝑎𝑙 using the (e,e’p)1𝑝0𝜋  channel on 12C at 𝐸𝑏𝑒𝑎𝑚 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The contribution of the un-  subtracted (e,e’p)1𝑝0𝜋 spectrum (black) is reduced to the subtracted  (e,e’p)1𝑝0𝜋 spectrum (magenta), which is in reasonable agreement with  the true (e,e’p)1𝑝0𝜋 spectrum (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  134  6-7  The proton (black) and charged pion (blue) multiplicities for data  (points), SuSav2 (solid histogram) and G2018 (dashed histogram) for  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV carbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Error bars show the 68% (1𝜎) confidence limits  for the statistical and point-to-point systematic uncertainties added in  quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Error bars are not shown when they are smaller than the  size of the data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Normalization uncertainties of 3% not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  134  193  6-8  (Top row) Acceptance correction factors, (middle row) acceptance cor-  rection factor uncertainties, and (bottom row) electron radiation cor-  rection factors plotted vs E𝑐𝑎𝑙 for the three incident beam energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Results for carbon are shown in black, helium in green and iron in ma-  genta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The left column (a,d,g) shows the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 GeV results, the middle  column (b,e,h) shows the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2 GeV results and the right column (c,f,i)  shows the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  135  6-9  Electron acceptance maps for 12C at 𝐸𝑏𝑒𝑎𝑚 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV as a function  of (left) cos𝜃𝑒 vs 𝜑𝑒 and (right) cos𝜃𝑒 vs momentum 𝑝𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  135  6-10 Comparison between the inclusive C(𝑒, 𝑒′) cross sections measured at  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='5∘ for data (points) and SuSav2 (lines) for the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='961 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='299 GeV  SLAC data and our 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV CLAS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  137  6-11 The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV C(𝑒, 𝑒′)0𝜋 cross section plotted as a function of the re-  constructed energy 𝐸𝑄𝐸 for data (black points), GENIE SuSAv2 (solid  black curve) and GENIE G2018 (dotted black curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The colored lines  show the contributions of different processes to the GENIE SuSAv2  cross section: QE (blue), MEC (red), RES (green) and DIS (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Error bars show the 68% (1𝜎) confidence limits for the statistical and  point-to-point systematic uncertainties added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Error  bars are not shown when they are smaller than the size of the data  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Normalization uncertainty of 3% not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  140  194  6-12 The 𝐴(𝑒, 𝑒′𝑝)0𝜋 cross section plotted as a function of the reconstructed  quasielastic energy 𝐸𝑄𝐸 for data (black points), SuSAv2 (black solid  curve) and G2018 (black dotted curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Different panels show results  for different beam energy and target nucleus combinations: (top row)  Carbon target at (left to right) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV, and (bot-  tom) Iron target at (left) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 and (right) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV incident beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV yields have been scaled by 1/2 and the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV yields  have been scaled by 5 to have the same vertical scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Colored lines  show the contributions of different processes to the SuSAv2 GENIE  simulation: QE (blue), MEC (red), RES (green) and DIS (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Error bars show the 68% (1𝜎) confidence limits for the statistical and  point-to-point systematic uncertainties added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Error  bars are not shown when they are smaller than the size of the data  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Normalization uncertainties of 3% not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  141  6-13 The 𝐴(𝑒, 𝑒′𝑝)1𝑝0𝜋 cross section plotted as a function of the reconstructed  calorimetric energy 𝐸𝑐𝑎𝑙 for data (black points), SuSAv2 (black solid  curve) and G2018 (black dotted curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Different panels show results  for different beam energy and target nucleus combinations: (top row)  Carbon target at (left to right) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV, and (bot-  tom) Iron target at (left) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 and (right) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV incident beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV yields have been scaled by 1/2 and the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV yields  have been scaled by 5 to have the same vertical scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The insets show  the cross sections with the same horizontal scale and an expanded verti-  cal scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Colored lines show the contributions of different processes to  the SuSAv2 GENIE simulation: QE (blue), MEC (red), RES (green)  and DIS (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Error bars show the 68% (1𝜎) confidence limits  for the statistical and point-to-point systematic uncertainties added in  quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Error bars are not shown when they are smaller than the  size of the data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Normalization uncertainties of 3% not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  143  195  6-14 Energy feed-down cross-sections (𝐸𝑟𝑒𝑐 − 𝐸𝑡𝑟𝑢𝑒)/𝐸𝑡𝑟𝑢𝑒 for data (points)  and SuSav2 (lines) for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV (red triangles and dotted lines),  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV (green squares and dashed lines) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV (blue dots  and solid lines) on carbon for (a) 𝐸𝑐𝑎𝑙, and (b) 𝐸𝑄𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  144  6-15 Energy feed-down cross-sections (𝐸𝑟𝑒𝑐 − 𝐸𝑡𝑟𝑢𝑒)/𝐸𝑡𝑟𝑢𝑒 for data (points)  and SuSav2 (lines) for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159 GeV (red triangles and dotted lines),  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV (green squares and dashed lines) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV (blue dots  and solid lines) on iron for (c) 𝐸𝑐𝑎𝑙, and (d) Fe 𝐸𝑄𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  145  6-16 (Left) the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 GeV C(𝑒, 𝑒′𝑝)1𝑝0𝜋 cross section plotted versus missing  transverse momentum, 𝑃𝑇, for data (black points), SuSav2 (black solid  line) and G2018 (black dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The vertical lines at 200 MeV/𝑐  and at 400 MeV/𝑐 separate the three bins in 𝑃𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Colored lines show  the contributions of different processes to the SuSAv2 GENIE simula-  tion: QE (blue), MEC (red), RES (green) and DIS (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' (Right)  The cross section plotted versus the calorimetric energy 𝐸𝑐𝑎𝑙 for dif-  ferent bins in 𝑃𝑇: (top) 𝑃𝑇< 200 MeV/𝑐, (middle) 200 MeV/𝑐 ≤ 𝑃𝑇≤  400 MeV/𝑐, and (bottom) 𝑃𝑇> 400 MeV/𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  147  6-17 The cross section plotted vs transverse missing momentum 𝑃𝑇 for data  (black points), SuSAv2 (black solid curve) and G2018 (black dotted  curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Different panels show results for different beam energy and  target nucleus combinations: (top row) Carbon target at (left to right)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV, and (bottom) Iron target at (left) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257  and (right) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='453 GeV yields have been scaled by four  to have the same vertical scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Colored lines show the contributions  of different processes to the SuSAv2 GENIE simulation: QE (blue),  MEC (red), RES (green) and DIS (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  148  196  6-18 The cross section plotted vs 𝛿𝛼𝑇 (a-e) and vs 𝛿𝜑𝑇 (f-j) for data (black  points), SuSAv2 (black solid curve) and G2018 (black dotted curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Different panels show results for different beam energy and target nu-  cleus combinations: (top row) Carbon target at (left to right) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='159,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='257 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='257 and  (right) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' Colored lines show the contributions of  different processes to the SuSAv2 GENIE simulation: QE (blue), MEC  (red), RES (green) and DIS (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='261 GeV showing the cross  section results as a function of 𝑃𝑇 for (top) all the events, (bottom  left) events with 𝛿𝛼𝑇 < 45𝑜 dominated by QE interactions and no  reinteractions, and (bottom right) events with 135𝑜 < 𝛿𝛼𝑇 < 180𝑜  maximally affected by FSI and multi-hadron channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Colored lines  show the contributions of different processes to the SuSAv2 GENIE  simulation: QE (blue), MEC (red), RES (green) and DIS (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  150  6-20 Data-simulation comparisons for 12C at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  177  8-10 The 𝜃 vs momentum distributions for 𝜋−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  178  8-11 The 𝜃 vs momentum distributions for 𝜋−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  178  8-12 The 𝜃 vs momentum distributions for 𝜋+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='  179  198  List of Tables  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' Table adapted  from [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' Re-  sults are listed for the full measured phase space and for a limited one  of cos(𝜃𝜇) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='1  Target areal densities and integrated charges for the 𝑒4𝜈 data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content='1  Summary of the total systematic uncertainties used in the e4𝜈 analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' 138  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='2  (𝑒, 𝑒′𝑝)1𝑝0𝜋 events reconstructed to the correct beam energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Peak Frac-  tion refers to the fraction of events reconstructed to the correct beam  energy and Peak Sum refers to the integrated weighted cross section (as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' 6-13) reconstructed to the correct beam energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' The peak  integration windows are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='1 ≤ 𝐸𝑐𝑎𝑙 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='22 GeV, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='19 ≤ 𝐸𝑐𝑎𝑙 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='34  GeV, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='35 ≤ 𝐸𝑐𝑎𝑙 ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='60 GeV, respectively, for the three incident  beam energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' SuSAv2 is not intended to model nuclei lighter than 12C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' Hernández,  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Mena, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+page_content=' A 372 (1996) 469 [hep-ph/9509307].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  [232] Rustam Niyazov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Measurement of Correlated Pair Momentum Distributions on  3𝐻𝑒(e,e’pp)n with CLAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' PhD thesis, Old Dominion University, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  [233] Dan Protopopescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Fiducial cuts for electrons in the CLAS/E2 data at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  CLAS-NOTE 2000-007 JLAB, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  [234] Bin Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Electron fiducial cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='jlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='org/Hall-B/secure/  e2/bzh/efiducialcut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='html, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  217  [235] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Khachatryan and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Weinstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Validation of neutrino energy estimation  using electron scattering data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Technical report, Old Dominion University,  November 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' CLAS-Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  [236] Steven McLauchlan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  ∆ electroproduction in 12𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  PhD thesis, University of  Glasgow, January 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  [237] Lawrence Weinstein Rustam Niyazov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  Fiducial cut for positive hadrons in  CLAS/E2 data at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content=' CLAS-NOTE 2001-013 JLAB, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
+page_content='  218' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfWgQs/content/2301.03114v1.pdf'}
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+1 
+ 
+SFI-Swin: Symmetric Face Inpainting with Swin 
+Transformer by Distinctly Learning Face Components 
+Distributions 
+ 
+MohammadReza Naderi1*, MohammadHossein Givkashi1*, Nader Karimi1, Shahram Shirani2, Shadrokh Samavi1,2,3 
+1Department of Electrical and Computer Engineering, Isfahan University of Technology, 84156-83111, Iran 
+2Department of Electrical and Computer Engineering, McMaster University, L8S 4L8, Canada 
+3Computer Science Department, Seattle University, Seattle 98122 USA  
+Abstract 
+Image inpainting consists of filling holes or missing parts of an image. Inpainting face images with symmetric 
+characteristics is more challenging than inpainting a natural scene. None of the powerful existing models can fill 
+out the missing parts of an image while considering the symmetry and homogeneity of the picture. Moreover, the 
+metrics that assess a repaired face image quality cannot measure the preservation of symmetry between the rebuilt 
+and existing parts of a face. In this paper, we intend to solve the symmetry problem in the face inpainting task by 
+using multiple discriminators that check each face organ's reality separately and a transformer-based network. 
+We also propose "symmetry concentration score" as a new metric for measuring the symmetry of a repaired face 
+image. The quantitative and qualitative results show the superiority of our proposed method compared to some 
+of the recently proposed algorithms in terms of the reality, symmetry, and homogeneity of the inpainted parts.  
+The code for the proposed method is available at https://github.com/mohammadrezanaderi4/SFI-Swin   
+1. Introduction 
+Removing objects from an image or filling in holes is a typical application of computer vision. With 
+the image inpainting technique, it is possible to either fill in empty regions or remove a few elements 
+from the image. New deep learning models with convolutional neural networks or transformer models 
+are intended to produce realistic-looking inpainted images. Face inpainting is a subset of image 
+inpainting. Its purpose is to fill the missing regions of a face image. Two major concerns should be 
+considered carefully during the inpainting of missing parts of a face: First, the inpainted regions must be 
+homogeneous with the other parts of the face and highly correlated to the available surrounding areas of 
+the input image. Second, facial symmetry must be preserved between the left and right sides. Many 
+inpainting methods have been proposed, and some achieved excellent results in repairing missing areas 
+of natural images. But almost all have difficulty repairing a face image symmetrically and 
+homogeneously. This shortcoming is because the network losses do not convey a general understanding 
+of the facial features to the generator. To further illustrate the main issues of previous works, we will 
+discuss the effect of usual losses that have been used in references [1] to [6]. 
+The loss functions that are mainly used in inpainting are pixel-wise, adversarial, feature-matching, 
+and perceptual loss. We will discuss the effect of each loss on the model training in this section. 
+1.1 Pixel-Wise Loss: As shown in Equation 1, Pixel-wise loss is computed between the inpainted 
+image and ground truth. Its goal is to lead the model to inpaint the missing regions similar to ground 
+truth by considering available parts of the face. However, the available regions cannot completely 
+describe the missing parts of the image. Therefore, this loss only can lead the inpainting network to 
+understand the low-level features of the missing parts. 
+ 
+ℒ������������������������ (������������, �������������) = ‖������������ − ������������� ‖1 or 2 
+ 
+(1) 
+ 
+In Equation 1, ������������ is the inpainted image, �������������  is the ground truth, and ‖ ‖1 or 2 stands for L1 or L2 norm 
+computation. 
+                                                 
+* The first two authors contributed equally to this work. 
+
+2 
+ 
+ 
+1.2 Adversarial and Feature Matching Losses: The adversarial loss [7] (Equations 2, 3, 4) attempts 
+to check the reality of an inpainted image based on the distribution of ground truths and generated images 
+using an extra network called a discriminator. In Equations 2, 3, and 4, ������������, �������������, ������������, ������������, ������������, and ������������, represent 
+the inpainted image, ground truth, discriminator, generator, parameters of the discriminator, and 
+parameters of the generator. Also, ������������������������ is the stop gradient, indicating that the backpropagated gradient 
+stops when it reaches specific parameters. Finally, the feature matching loss (Equation 5) is computed 
+between the features extracted from the discriminator's middle layers for the inpainted image and ground 
+truth. In Equation 5, ������������������������������������ presents middle features of the discriminator. Adversarial and feature 
+matching losses came from the idea that although we cannot reconstruct the missing regions exactly 
+similar to ground truth, at least we can be sure that the inpainted regions look realistic.  
+ 
+ℒ������������ = −������������������������[log ������������ℰ(������������)] − �������������������������log  �1 − ������������������������(�������������)�� 
+(2) 
+ℒ������������ = −������������������������[log ������������������������(�������������)] 
+(3) 
+ℒ������������������������������������ = ������������������������������������(ℒ������������) + ������������������������������������(ℒ������������) 
+(4) 
+ℒ������������������������(������������, �������������) = ‖ ������������������������������������������������(������������) − ������������������������������������������������(�������������)‖1 or 2 
+(5) 
+ 
+The architecture of the discriminator is patch-based. The discriminator tests the reality of the patches 
+of the whole image [7]. 
+1.3 Perceptual Loss: The perceptual loss is shown in Equation 6.  This loss is computed using the 
+encoder of a pre-trained segmentation network to compare the high-level features of the generated and 
+ground truth images. In Equation 6, ������������, �������������, ������������, present inpainted image, ground truth, and encoder of a pre-
+trained segmentation network, respectively.  This loss mostly considered high-level features such as 
+edges in the image. 
+ 
+ℒ������������������������(������������, �������������) = ‖ ������������(������������) −  ������������(�������������)‖1 or 2 
+(6) 
+ 
+The adversarial, feature matching, and perceptual losses mainly focus on the reality of the patches 
+and edges' smoothness. Therefore, these losses do not represent the importance of the homogeneity and 
+symmetry of the image. The perceptual loss causes the inpainting network to occasionally prefer the 
+reality of the patches over the validity of the whole picture. In other words, due to the pressure of the 
+mentioned losses, the generator sacrifices the symmetry of the face to get more realistic patches. This 
+patch-based behavior is one of the apparent drawbacks of available image synthesizing and image 
+inpainting methods. The mentioned shortcoming motivates us to add a new term to the loss function of 
+image inpainting networks to consider symmetry and global features of each part of the face. We can 
+get more realistic faces by assessing an image's symmetry and global facial features. Meanwhile, we use 
+a base inpainting network that balances global and patch-based losses to get the maximum benefit from 
+this additional loss.  
+A large receptive field is used in Swin transformer blocks [8]. We used Swin transformer blocks in 
+this paper and applied a new loss focusing on distinct facial feature distribution. Hence, we increase the 
+inpainting model's concentration on the symmetry problem. Furthermore, current metrics cannot 
+measure the symmetry of the inpainted face. Therefore, we propose a new metric to resolve this 
+shortcoming. 
+ 
+ 
+
+3 
+ 
+Our main contributions are as follows: 
+• Proposing a homogeneity-aware loss to solve the heterogeneity and asymmetricity problems 
+in inpainting methods. 
+• Using transformer base architecture to achieve a wider receptive field and balance the usage 
+of global and local features and input and output image gradients. 
+• Increasing the generalizability of the inpainting model by upgrading its understanding of the 
+semantic face parts using separate discriminators for each part of the face. 
+• Proposing a new metric, symmetry concentration score (SCS), to assess the symmetry and 
+homogeneity of facial organs.  
+The structure of this paper is as follows: In section 2, we will present some previous relevant methods. 
+Then, section 3 offers the proposed method. Finally, experimental results are detailed in section 4, and 
+the conclusion is presented in section 5. 
+2. Related Works 
+Several methods for image inpainting have been proposed, including deep generative base and 
+transformer base methods. This section will discuss these two types of methods in more detail. 
+2.1 Deep Generative Methods 
+In recent years, the generative adversarial network has been used for image completion and 
+inpainting [9]–[12]. In [1], a new approach has been proposed based on conditional and unconditional 
+generative networks. The authors present new metrics for measuring image completeness. These metrics 
+are based on the linear separability of features in a feature space. These metrics demonstrate a measure 
+of the perceptual fidelity of the inpainted image compared to the real image. A method for generating 
+high-quality inpainted image based on aggregated contextual transformations was presented in [2]. 
+Authors of [2] also employed distant contexts, not in the masked region's neighborhood. Inpainting 
+methods are evaluated on different tasks, such as removing logos, editing faces, and removing objects 
+[13]–[16]. In [3], the authors focused on a large missing region and proposed an adversarial method for 
+image inpainting. The model can generate visually realistic images for contiguous and separated large 
+missing areas. Also, a new loss was proposed to measure the non-local correlation between patches and 
+help the model get a better inference result.  
+A method is proposed in [4] that uses edges and masked images to generate inpainted images. They 
+used an edge generator that completed edges in missing regions and a completion network that used an 
+edge map and input image to generate the result. With this approach, they achieved a good result with a 
+focus on the edges of the image. In [17], they introduced a method for human body completion. They 
+used three steps in their work: prior encoding, segmentation completion, and texture completion. A 
+segmentation map can help capture important information from the human body. They also designed a 
+memory module, a dictionary for storing learned latent vectors. These segmentation maps are typically 
+used to segment each part of the human body and assist in extracting high-level information. As a result, 
+it helps the model to learn better. Multi-scale structure discriminators have been used to generate 
+segmentation maps with good information, so the results were interesting in human body completion. In 
+[18], the authors used a generative adversarial network that contains dilated convolution for increasing 
+receptive fields. They tried to achieve a reasonable structure without blurriness in the output. Further, 
+they developed a new self-guided regression loss technique for enhancing semantic features and 
+concentrating on uncertain areas. When a model fills holes in an image, it's essential to generate a good 
+result and have color consistency in all pictures.  
+In [19], the authors proposed a method for image inpainting with attention to color consistency in 
+images. The output was reliable without any artifacts, and the stable colors in different parts of the 
+images provided the ability to create realistic images. In [5], the authors used gated convolutions in the 
+
+4 
+ 
+model to solve the problem of using vanilla convolutions that understand the pixels differently. They 
+also proposed a new loss function named SN-PatchGAN that helps the model generate a high-quality 
+and flexible image compared to previous methods.    
+2.2 Transformer Methods 
+Since transformers are highly effective for tasks related to natural language processing, researchers 
+have used them in vision tasks. A transformer has a global view and uses an attention mechanism to 
+extract information [20–23]. In [24], a transformer was used for image completion. They use the global 
+view of the transformer and achieve better results for large masks than other methods. Recently a 
+combination of transformer and convolution has been used. In [25], the authors proposed a model 
+containing a transformer and convolution for image inpainting. The global view of the transformer and 
+feature extraction in convolution helped the model generate good results in image completion. In [26], 
+a novel inpainting framework has been proposed for high-resolution images. They also proposed 
+modifying transformer blocks to stabilize the training of large masks.  
+With the long-range interaction modeling in the transformer, the model generated high-fidelity 
+images. Therefore, the architecture can understand critical dependencies between different regions on 
+the image with attention-based models. In [27], they proposed an inpainting method that used a 
+transformer and CNN model. The model gets additional input that contains lines and edges on the 
+masked image. As a result, the model can reconstruct edges in a better way. With this approach, the 
+proposed model can learn to reconstruct masked images focusing on the edges and lines. In [28], the 
+blind face inpainting method has been proposed. They used two stages for the blind face inpainting 
+process due to the difficulty in detecting masks of different shapes and sizes, as well as the difficulty of 
+restoring masked images that are realistic. They used a transformer model during the first stage to detect 
+corrupted regions. In the next step, a network has been proposed for restoring features at different levels 
+in a hierarchical manner, thereby producing semantically coherent content based on unmasked regions 
+of the face. 
+3. Proposed method 
+Our proposed method is called Symmetric Face Inpainting with Swin Transformer (SFI-Swin). We 
+will discuss our method from two perspectives. First, we will discuss the generator architecture based 
+on a transformer in subsection 3.1. Then in subsection 3.2, we will concentrate on the loss functions and 
+propose a loss that focuses on the symmetry and homogeneity of the face features. We use six additional 
+discriminators with the same architecture to compute this loss.  
+3.1 Generator architecture 
+As discussed in [6], using a generator architecture with a wider receptive field produces more 
+homogeneous outputs while considering the entire facial features. Thus, they added fast Fourier 
+convolution layers [29] to their models to make a generator with a wider receptive field. Although their 
+work seems to be effective, it sometimes fails to make a balance between using global and local features 
+[6]. We use the Swin-Unet [30] architecture as our generator to create a balance between local and global 
+features. Swin-Unet has a large receptive field because of its self-attention mechanism and could balance 
+local and global feature usage well. As discussed in [31], skip connections downgrade the image 
+inpainting ability of the generator model because it allows the generator to copy the available parts of 
+the input to the output.  
+Skip connections prevent the model from constructing the whole available and missed parts of the 
+input based on high-level features that are extracted in the middle of the generator. Such a network 
+results in totally heterogeneous face features in output. Therefore, we omit the skip connections from 
+the Swin-Unet model, and our final generator architecture is shown in Fig. 1. The patch discriminator 
+[7] architecture is also shown in Fig. 1. 
+ 
+
+5 
+ 
+ 
+Using the Swin transformer [8] layers, which could balance the utilization of local and global 
+features, leads to more homogeneous and realistic facial features. In addition, Swin transformer blocks 
+do not add much computational complexity compared to convolutional models with a similar number of 
+parameters. This behavior is because the Swin transformer computes the correlation between the patches 
+in certain parts of the image. 
+3.2 Homogeneity Loss 
+In this section, we propose a new loss that we will use in addition to the losses that were discussed 
+in Section 1. The idea behind this new loss is that we compute the realness of each part of the face 
+compared to the distribution of that specific part in the whole dataset. This will oblige the generator to 
+pay more attention to symmetric and global features. We consider six additional discriminators with the 
+same architecture to compute the homogeneity loss, as shown in Figure 1. Each discriminator is for a 
+specific part of the face, such as skin, eyes, lips, clothes, and hair. It is noteworthy that each of these 
+parts is extracted from a face image using a pre-trained semantic segmentation module [32]. The 
+segmentation module parameters are frozen during training. While the generator intends to inpaint the 
+missed regions, computing the realness of each facial organ will conclude more symmetry in the output 
+image. The architecture of these discriminators is designed to assess the total realness of a specific face. 
+ 
+Fig. 1. Block diagram of our proposed method. First, the generator takes the masked image as input and attempts to inpaint it. Then 
+the inpainted image is fed to the patch discriminator to check the overall reality of the patches. Meanwhile, the inpainted image is 
+also fed into a semantic segmentation network [32] to separate the semantic parts of the face, such as eyes, and ears. The architecture 
+of these six semantic discriminators is the same. In the next step, six distinct discriminators calculate the total realness of each 
+semantic part of the face. Then, the generator parameters will be updated based on these seven discriminators and the pixel-wise loss 
+gradient signals, which are backpropagated to the generator. 
+
+Patch Partition
+Patch Merging
+Linear Projection
+Linear Embedding
+Swin Transformer
+Swin Transformer
+Block x 2
+Patch Expanding
+Block x 1
+Encoder
+Bottlenec
+Decoder
+Skin
+Discriminator
+F××48
+Wx兴xC
+Wx×2C
+WH
+WH
+WxH
+xC
+W×H×C(4x) W×H×Clas
+Eye
+Discriminator
+Masked
+Hair
+Image
+Discriminator
+Semantic
+Generator
+Segmentor
+Lip
+Discriminator
+Patch
+Discriminator
+Cloth
+Discriminator
+Conv 4 × 4, stride=2
+Linear
+Input Conv
+Conv 4 x 4, stride=2
+Input Conv
+projection
+Ear
+Conv 4 × 4, stride = 1
+Conv 1 × 1, stride = 1
+Conv 4 x 4, stride = 1
+Conv 1 × 1, stride = 1
+Discriminator
+Discriminator
+Discriminator
+Output =
+Vector with
+10 elements6 
+ 
+This characteristic is unlike the patch discriminator, which does not completely understand the face 
+image. One of the main features of the face is its symmetry which the related discriminator will consider. 
+Using these semantic discriminators to check the realness of each face organ increases the generator's 
+knowledge about different semantic parts of the image. Considering facial organs helps the network to 
+repair portraits more realistically. The block diagram of SFI-Swin is shown in Figure 1.  
+For each of these discriminators and the corresponding facial organ, the adversarial and feature-
+matching loss is computed using Equations 4 and 5. Therefore the overall homogeneity loss could be 
+presented as Equations 7 and 8. 
+ 
+⎩
+⎪⎪
+⎨
+⎪⎪
+⎧ ℒ������������������������������������������������(������������, �������������) = ℒ������������������������������������(������������������������������������������������������������, �������������������������������������������������������������) + ℒ������������������������(������������������������������������������������������������, �������������������������������������������������������������)
+ℒ������������������������������������(������������, �������������) = ℒ�������������������������������������������������������������������������������������, �������������������������������������������������� + ℒ�������������������������������������������������������������������������, ��������������������������������������������������
+ℒℎ������������������������������������(������������, �������������) = ℒ������������������������������������(������������ℎ������������������������������������, �������������ℎ������������������������������������) + ℒ������������������������(������������ℎ������������������������������������, �������������ℎ������������������������������������)
+ℒ������������������������������������(������������, �������������) = ℒ�������������������������������������������������������������������������������������, �������������������������������������������������� + ℒ�������������������������������������������������������������������������, ��������������������������������������������������
+ℒ������������������������������������������������ℎ(������������, �������������) = ℒ������������������������������������(������������������������������������������������������������ℎ, �������������������������������������������������������������ℎ) + ℒ������������������������(������������������������������������������������������������ℎ, �������������������������������������������������������������ℎ)
+ℒ������������������������������������(������������, �������������) = ℒ������������������������������������(������������������������������������������������, �������������������������������������������������) + ℒ������������������������(������������������������������������������������, �������������������������������������������������)
+⎭
+⎪⎪
+⎬
+⎪⎪
+⎫
+ 
+ 
+(7) 
+ℒ������������������������(������������, �������������) = ������������1ℒ������������������������������������������������(������������, �������������) + ������������2ℒ������������������������������������(������������, �������������) + ������������3ℒℎ������������������������������������(������������, �������������) + ������������4ℒ������������������������������������(������������, �������������)
++ ������������5ℒ������������������������������������������������ℎ(������������, �������������) + ������������6ℒ������������������������������������(������������, �������������) 
+(8) 
+ 
+ 
+where ������������1, ������������2, ������������3, ������������4, ������������5, and ������������6 are set to be 0.083, 0.25, 0.083, 0.25, 0.083, 0.083, and 0.25, 
+respectively. The coefficients of eyes, ears, and lips are set three times greater than other parts of the 
+face. This is because, during the experiments, we realized that the model is mostly incapable of 
+maintaining the symmetry of these face organs.  
+4. Experimental results 
+In this section, we present the experiment setups and results accomplished using these setups.  
+4.1 Experiments setups 
+In the following, we will discuss the dataset, metrics, hyperparameters, and the total loss function 
+we used to train our models. 
+Data and metrics: We use CelebAHQ [33] dataset. This dataset contains 28k images. We split the 
+data to train, validate, and test, similar to [6]. Learned Perceptual Image Patch Similarity (LPIPS) [34] 
+and Fr'echet inception distance (FID) [35] metrics are used to evaluate our proposed method 
+performance. Compared to pixel-level L1 and L2 distances, LPIPS and FID are more suitable for 
+measuring the performance of large masks when multiple natural completions are plausible. The 
+experimentation pipeline is implemented using PyTorch [36].  
+Training hyperparameters: The learning rate of the generator is 0.001, and that of the seven 
+discriminators is 0.0001. The discrimination process is more straightforward than the generation process, 
+especially in the initial epochs of the training [6]. Hence, a lower initial learning rate for the 
+discriminators allows the generator to converge faster during the initial epochs. This will balance the 
+training procedures between the discriminators and the generator. This balance prevents the training 
+procedure from collapsing. These learning rates also are optimized using a beam search algorithm to get 
+the best results [6]. The batch size is also set to 20, and we train our model for 40 epochs using the Adam 
+optimizer [37]. We trained our model with an NVIDIA GeForce RTX 3090 with 24GB RAM GPU. 
+
+7 
+ 
+Loss function: besides the proposed homogeneity loss, we used the same losses as [6]. Therefore, 
+the total loss function of our model is shown in Equation 9. 
+ℒ(������������, �������������) =  ������������ℒ������������������������(������������, �������������) + ������������ℒ������������������������������������(������������, �������������) +  ������������ℒ������������������������(������������, �������������) + ������������ℒ������������������������(������������, �������������) 
+ 
+(9) 
+ 
+where ������������, ������������, ������������, and ������������ are set to 10, 10, 100, and 20. These values are based on our experiments to get the 
+best results on the test data.  
+ 
+4.2 Qualitative results: Fig. 2 shows SFI-Swin performance to inpaint for different mask sizes, in 
+comparison with [6]. To further illustrate our proposed method's capability to inpaint the face images 
+symmetrically, we generate masks that omit one of the eyes and slowly grow around the omitted region 
+to cover half of the face. We compare our method with [6], and the results are shown in Figure 3. Further, 
+we randomly chose ten images from the test set. Then, we masked (eliminated) one eye and a K×K block 
+of the image. We tested with K set to 16, 32, and 64.  
+We repaired the image with a missing eye and a K×K block. We then calculated the difference 
+between the repaired eye while a K×K patch was masked with the inpainted image that only the eye was 
+missing. We then find the mean of the absolute difference image and assign this value to the 
+corresponding K×K block. The average absolute difference will be large if a block is essential to repair 
+the missing eye. Finally, we built a heatmap to show the blocks that play an important role in the 
+inpainting of the missing eye. The results are shown in Figure 4. As we can see, the non-missing eye 
+and the chick area around the missing eye are the most important patches for reconstructing the missing 
+area. We repeated this experiment for half of the face. The effects of each missing K×K patch on the 
+inpaint eye and half face are shown in Figure 5. The lighter the color of the shown patch, the more effect 
+it has on reconstructing the missing right-side eye.  
+ 
+Fig. 2. a) Original images, b) masks, inpainted images by c) LaMa [6], d) Swin, and e) SFI-Swin. 
+
+Narrow masks
+Medium masks
+Wide masks
+a)
+b)
+c)
+(p
+e)8 
+ 
+ 
+While inpainting a facial part from one side of the face, our method focuses on the same organ on the 
+other side.  This consideration causes symmetry in the repaired image.  
+4.3 Quantitative results: Table 1 shows the performance of our proposed method to repair the 
+missing parts of the face compared with the best recently proposed methods in this field in terms of FID 
+and LPIPS scores. Because previous works trained multiple models for different mask sizes, therefore 
+to compare fairly, we show their performance based on the mask sizes which they used in training. 
+However, our method follows the aggressive mask generation proposed by [6] that helps it handle 
+narrow, medium, and wide masks simultaneously. As a result, our proposed methods, Swin and Swin 
+ 
+Fig. 3. Performances of b) LaMa, c) Swin, d) SFI-Swin in comparison with e) ground-truth to inpaint a) masked images starting from an 
+eye and grown to cover half of the face. A white border is drawn to show the reconstructed region.  
+
+D=0
+D = 12
+D = 24
+D = 36
+D = 48
+D=0
+D = 12
+D = 24
+D= 36
+D = 48
+a)
+b)
+c)
+(p
+e)
+a)
+b)
+d)
+e)9 
+ 
+with multiple semantic discriminators (SFI-Swin), achieved the best results in medium and wide mask 
+inpainting. 
+ 
+FID and LPIPS are patch-based metrics and do not consider homogeneity and symmetricity in the 
+face. Thus, we propose the symmetry concentration score (SCS) to assess the symmetry of the left and 
+right sides of the face. In Figure 5, we presented heatmaps that depict the influence the model takes from 
+a K×K block of the face during the inpainting of an eye or half of the face. Our symmetry concentration 
+score measures the amount of attention the network pays to a part of the face while repairing the same 
+semantic part on the other side. To acquire this metric, we calculate the mean of the overlapping K × K 
+patches with the desired organs while considering three different patch sizes (16×16, 32×32, 64×64). 
+The results are shown in Table 2. Our method using multiple semantic discriminators performed better 
+than the Swin network without these discriminators. Also, we achieved better symmetric results than 
+[6].  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 4. Steps to compute symmetry concentration score (SCS). First, we mask an eye and a K×K patch of the face, and reconstruct the 
+missing eye and the K×K patch. Then the absolute difference between the image with inpainted eye with the image with a missing K×K 
+patch and the missing eye is computed. The difference shows the effect of that K×K patch on inpainting result of the missing eye. The 
+effect of all K×K patches is computed and shown as a heatmap. The face borders are also depicted to investigate the impact of each part of 
+the face on inpainting the missed eye. 
+
+eye mask
+Absolute value of
+Difference between
+generated eye when
+only that eye is masked
+Mask
+Output
+Output x eye mask
+and when an additional
+patch is also masked
+from the Original
+X
+Image
+Overlay the mean of the
+difference image on the
+X
+corresponding patch
+absO, mean
+Original
+X
+Image
+absO, mean
+Effect of each
+patch on
+filling the
+abs(), mean
+empty eye
+X
+X
+absO, mean10 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 5. Influence that the generator gets from each K×K block of the face receives to repair the missing parts of the face. a) Lama, b) 
+Swin, c) SFI-Swin. The missing parts are shown as empty black regions (an eye or half of the face), while the background is not 
+considered in these experiments. 
+
+Eye
+Half face
+Block size = 16x16
+a)
+b)
+c)
+a)
+Block size = 32x32
+n
+C
+a)
+Block size = 64x64
+b)
+C11 
+ 
+Table 1. Presenting the performance of our method SFI-Swin to inpaint different types of masks comparing to powerful methods in 
+this field using two popular metrics, FID and LPIPS. The three best models in each column are shown in red, orange, and green. 
+Train 
+masks type 
+Methods 
+CelebA-HQ (256×256) 
+Narrow masks 
+Medium masks 
+Wide masks 
+40-50% masked 
+All samples 
+40-50% masked 
+All samples 
+40-50% masked 
+All samples 
+FID 
+LPIPS 
+FID 
+LPIPS 
+FID 
+LPIPS 
+FID 
+LPIPS 
+FID 
+LPIPS 
+FID 
+LPIPS 
+Aggressive 
+train 
+masks 
+Ours (SFI-Swin) 
+23.7 
+0.157 
+7.44 
+0.101 
+33.43 
+0.161 
+5.54 
+0.088 
+26.81 
+0.102 
+5.97 
+0.104 
+Swin [30] 
+23.2 
+0.151 
+7.31 
+0.098 
+32.88 
+0.160 
+5.60 
+0.086 
+26.69 
+0.101 
+5.98 
+0.102 
+LaMa-Fourier [6] 
+22.7 
+0.132 
+7.26 
+0.085 
+34.1 
+0.145 
+6.13 
+0.080 
+27.8 
+0.168 
+6.96 
+0.098 
+Narrow 
+train 
+masks 
+CoModGAN [1] 
+35.9 
+0.139 
+16.8 
+0.079 
+48.4 
+0.169 
+19.4 
+0.092 
+64.4 
+0.191 
+24.4 
+0.102 
+AOT GAN [2] 
+21.0 
+0.127 
+6.67 
+0.081 
+39.1 
+0.162 
+7.28 
+0.089 
+40.4 
+0.204 
+10.3 
+0.118 
+RegionWise [3] 
+32.5 
+0.188 
+11.1 
+0.124 
+40.4 
+0.179 
+7.52 
+0.101 
+33.9 
+0.205 
+8.54 
+0.121 
+DeepFill v2 [5] 
+37.0 
+0.201 
+12.5 
+0.190 
+45.3 
+0.189 
+9.05 
+0.105 
+43.0 
+0.214 
+11.2 
+0.126 
+EdgeConnect [4] 
+29.2 
+0.156 
+9.61 
+0.099 
+40.5 
+0.174 
+7.56 
+0.095 
+34.7 
+0.205 
+9.02 
+0.120 
+Wide 
+ train 
+masks 
+RegionWise [3] 
+47.5 
+0.246 
+17.9 
+0.164 
+50.9 
+0.220 
+10.3 
+0.124 
+42.6 
+0.233 
+11.2 
+0.140 
+DeepFill v2 [5] 
+30.4 
+0.169 
+9.99 
+0.108 
+40.3 
+0.173 
+7.65 
+0.095 
+34.6 
+0.196 
+8.95 
+0.115 
+EdgeConnect [4] 
+55.5 
+0.248 
+18.3 
+0.152 
+40.2 
+0.174 
+7.79 
+0.097 
+32.7 
+0.196 
+8.43 
+0.116 
+ 
+ 
+Table 2. Comparing the symmetry concentration score (SCS) of our method SFI-Swin to 
+inpaint certain parts of the face compared to Swin and LaMa [6]. 
+Method | Metric + face parts 
+SCS for eye 
+SCS for half face 
+Ours (SFI-Swin) 
+0.7177 
+0.4233 
+Swin [30] 
+0.6319 
+0.3948 
+LaMa [6] 
+0.6225 
+0.3740 
+ 
+5. Conclusion 
+ 
+This paper discussed the effect of using multiple semantic discriminators incorporated with the Swin 
+transformer-based architecture to repair face images. Our proposed method preserved the symmetry and 
+homogeneity of the face parts. Our experimental results show the proposed method's superiority over 
+powerful rivals, especially on the medium and wide masks.  
+We also proposed a new method to assess the concentration of the inpainting network while inpainting 
+a specific face organ.  
+By using multiple discriminators to compute the reality of each facial organ, the generator was guided 
+to preserve the symmetry and homogeneity of the face. This resulted in a generator that resolved one of 
+the most critical inpainting shortcomings.  
+ 
+ 
+ 
+ 
+ 
+
+12 
+ 
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+
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new file mode 100644
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--- /dev/null
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@@ -0,0 +1,733 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf,len=732
+page_content='1  SFI-Swin: Symmetric Face Inpainting with Swin Transformer by Distinctly Learning Face Components Distributions  MohammadReza Naderi1*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' MohammadHossein Givkashi1*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Nader Karimi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Shahram Shirani2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Shadrokh Samavi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='3 1Department of Electrical and Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Isfahan University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' 84156-83111,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Iran 2Department of Electrical and Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' McMaster University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' L8S 4L8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Canada 3Computer Science Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Seattle University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Seattle 98122 USA Abstract Image inpainting consists of filling holes or missing parts of an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Inpainting face images with symmetric characteristics is more challenging than inpainting a natural scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' None of the powerful existing models can fill out the missing parts of an image while considering the symmetry and homogeneity of the picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Moreover, the metrics that assess a repaired face image quality cannot measure the preservation of symmetry between the rebuilt and existing parts of a face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=" In this paper, we intend to solve the symmetry problem in the face inpainting task by using multiple discriminators that check each face organ's reality separately and a transformer-based network." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' We also propose "symmetry concentration score" as a new metric for measuring the symmetry of a repaired face image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The quantitative and qualitative results show the superiority of our proposed method compared to some of the recently proposed algorithms in terms of the reality, symmetry, and homogeneity of the inpainted parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The code for the proposed method is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='com/mohammadrezanaderi4/SFI-Swin 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  Introduction Removing objects from an image or filling in holes is a typical application of computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' With the image inpainting technique, it is possible to either fill in empty regions or remove a few elements from the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' New deep learning models with convolutional neural networks or transformer models are intended to produce realistic-looking inpainted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Face inpainting is a subset of image inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Its purpose is to fill the missing regions of a face image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Two major concerns should be considered carefully during the inpainting of missing parts of a face: First, the inpainted regions must be homogeneous with the other parts of the face and highly correlated to the available surrounding areas of the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Second, facial symmetry must be preserved between the left and right sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Many inpainting methods have been proposed, and some achieved excellent results in repairing missing areas of natural images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' But almost all have difficulty repairing a face image symmetrically and homogeneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' This shortcoming is because the network losses do not convey a general understanding of the facial features to the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' To further illustrate the main issues of previous works, we will discuss the effect of usual losses that have been used in references [1] to [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The loss functions that are mainly used in inpainting are pixel-wise, adversarial, feature-matching, and perceptual loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  We will discuss the effect of each loss on the model training in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='1 Pixel-Wise Loss: As shown in Equation 1, Pixel-wise loss is computed between the inpainted image and ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Its goal is to lead the model to inpaint the missing regions similar to ground truth by considering available parts of the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' However, the available regions cannot completely describe the missing parts of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Therefore, this loss only can lead the inpainting network to understand the low-level features of the missing parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  ℒ������������������������ (������������, �������������) = ‖������������ − ������������� ‖1 or 2  (1)  In Equation 1, ������������ is the inpainted image, �������������  is the ground truth, and ‖ ‖1 or 2 stands for L1 or L2 norm computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  The first two authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='2 Adversarial and Feature Matching Losses: The adversarial loss [7] (Equations 2, 3, 4) attempts to check the reality of an inpainted image based on the distribution of ground truths and generated images using an extra network called a discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' In Equations 2, 3, and 4, ������������, �������������, ������������, ������������, ������������, and ������������, represent the inpainted image, ground truth, discriminator, generator, parameters of the discriminator, and parameters of the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Also, ������������������������ is the stop gradient, indicating that the backpropagated gradient stops when it reaches specific parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=" Finally, the feature matching loss (Equation 5) is computed between the features extracted from the discriminator's middle layers for the inpainted image and ground truth." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' In Equation 5, ������������������������������������ presents middle features of the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Adversarial and feature matching losses came from the idea that although we cannot reconstruct the missing regions exactly similar to ground truth, at least we can be sure that the inpainted regions look realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  ℒ������������ = −������������������������[log ������������ℰ(������������)] − �������������������������log  �1 − ������������������������(�������������)�� (2) ℒ������������ = −������������������������[log ������������������������(�������������)] (3) ℒ������������������������������������ = ������������������������������������(ℒ������������) + ������������������������������������(ℒ������������) (4) ℒ������������������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������) = ‖ ������������������������������������������������(������������) − ������������������������������������������������(�������������)‖1 or 2 (5)  The architecture of the discriminator is patch-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The discriminator tests the reality of the patches of the whole image [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='3 Perceptual Loss: The perceptual loss is shown in Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' This loss is computed using the encoder of a pre-trained segmentation network to compare the high-level features of the generated and ground truth images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' In Equation 6, ������������, �������������, ������������, present inpainted image, ground truth, and encoder of a pre- trained segmentation network, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' This loss mostly considered high-level features such as edges in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content="  ℒ������������������������(������������, �������������) = ‖ ������������(������������) −  ������������(�������������)‖1 or 2 (6)  The adversarial, feature matching, and perceptual losses mainly focus on the reality of the patches and edges' smoothness." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Therefore, these losses do not represent the importance of the homogeneity and symmetry of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The perceptual loss causes the inpainting network to occasionally prefer the reality of the patches over the validity of the whole picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' In other words, due to the pressure of the mentioned losses, the generator sacrifices the symmetry of the face to get more realistic patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' This patch-based behavior is one of the apparent drawbacks of available image synthesizing and image inpainting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The mentioned shortcoming motivates us to add a new term to the loss function of image inpainting networks to consider symmetry and global features of each part of the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=" We can get more realistic faces by assessing an image's symmetry and global facial features." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Meanwhile, we use a base inpainting network that balances global and patch-based losses to get the maximum benefit from this additional loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' A large receptive field is used in Swin transformer blocks [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' We used Swin transformer blocks in this paper and applied a new loss focusing on distinct facial feature distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=" Hence, we increase the inpainting model's concentration on the symmetry problem." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Furthermore, current metrics cannot measure the symmetry of the inpainted face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Therefore, we propose a new metric to resolve this shortcoming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  3  Our main contributions are as follows: • Proposing a homogeneity-aware loss to solve the heterogeneity and asymmetricity problems in inpainting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' • Using transformer base architecture to achieve a wider receptive field and balance the usage of global and local features and input and output image gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' • Increasing the generalizability of the inpainting model by upgrading its understanding of the semantic face parts using separate discriminators for each part of the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' • Proposing a new metric, symmetry concentration score (SCS), to assess the symmetry and homogeneity of facial organs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The structure of this paper is as follows: In section 2, we will present some previous relevant methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Then, section 3 offers the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Finally, experimental results are detailed in section 4, and the conclusion is presented in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Related Works Several methods for image inpainting have been proposed, including deep generative base and transformer base methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' This section will discuss these two types of methods in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='1 Deep Generative Methods In recent years, the generative adversarial network has been used for image completion and inpainting [9]–[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' In [1], a new approach has been proposed based on conditional and unconditional generative networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The authors present new metrics for measuring image completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' These metrics are based on the linear separability of features in a feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  These metrics demonstrate a measure of the perceptual fidelity of the inpainted image compared to the real image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' A method for generating high-quality inpainted image based on aggregated contextual transformations was presented in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=" Authors of [2] also employed distant contexts, not in the masked region's neighborhood." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Inpainting methods are evaluated on different tasks, such as removing logos, editing faces, and removing objects [13]–[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' In [3], the authors focused on a large missing region and proposed an adversarial method for image inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The model can generate visually realistic images for contiguous and separated large missing areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Also, a new loss was proposed to measure the non-local correlation between patches and help the model get a better inference result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' A method is proposed in [4] that uses edges and masked images to generate inpainted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' They used an edge generator that completed edges in missing regions and a completion network that used an edge map and input image to generate the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' With this approach, they achieved a good result with a focus on the edges of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' In [17], they introduced a method for human body completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' They used three steps in their work: prior encoding, segmentation completion, and texture completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' A segmentation map can help capture important information from the human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' They also designed a memory module, a dictionary for storing learned latent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  These segmentation maps are typically used to segment each part of the human body and assist in extracting high-level information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' As a result, it helps the model to learn better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Multi-scale structure discriminators have been used to generate segmentation maps with good information, so the results were interesting in human body completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' In [18], the authors used a generative adversarial network that contains dilated convolution for increasing receptive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' They tried to achieve a reasonable structure without blurriness in the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Further, they developed a new self-guided regression loss technique for enhancing semantic features and concentrating on uncertain areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=" When a model fills holes in an image, it's essential to generate a good result and have color consistency in all pictures." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' In [19], the authors proposed a method for image inpainting with attention to color consistency in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The output was reliable without any artifacts, and the stable colors in different parts of the images provided the ability to create realistic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' In [5], the authors used gated convolutions in the  4  model to solve the problem of using vanilla convolutions that understand the pixels differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' They also proposed a new loss function named SN-PatchGAN that helps the model generate a high-quality and flexible image compared to previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='2 Transformer Methods Since transformers are highly effective for tasks related to natural language processing, researchers have used them in vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' A transformer has a global view and uses an attention mechanism to extract information [20–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' In [24], a transformer was used for image completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' They use the global view of the transformer and achieve better results for large masks than other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Recently a combination of transformer and convolution has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' In [25], the authors proposed a model containing a transformer and convolution for image inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The global view of the transformer and feature extraction in convolution helped the model generate good results in image completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' In [26], a novel inpainting framework has been proposed for high-resolution images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' They also proposed modifying transformer blocks to stabilize the training of large masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' With the long-range interaction modeling in the transformer, the model generated high-fidelity images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Therefore, the architecture can understand critical dependencies between different regions on the image with attention-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' In [27], they proposed an inpainting method that used a transformer and CNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  The model gets additional input that contains lines and edges on the masked image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' As a result, the model can reconstruct edges in a better way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' With this approach, the proposed model can learn to reconstruct masked images focusing on the edges and lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' In [28], the blind face inpainting method has been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' They used two stages for the blind face inpainting process due to the difficulty in detecting masks of different shapes and sizes, as well as the difficulty of restoring masked images that are realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' They used a transformer model during the first stage to detect corrupted regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' In the next step, a network has been proposed for restoring features at different levels in a hierarchical manner, thereby producing semantically coherent content based on unmasked regions of the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Proposed method Our proposed method is called Symmetric Face Inpainting with Swin Transformer (SFI-Swin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' We will discuss our method from two perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' First, we will discuss the generator architecture based on a transformer in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Then in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='2, we will concentrate on the loss functions and propose a loss that focuses on the symmetry and homogeneity of the face features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' We use six additional discriminators with the same architecture to compute this loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='1 Generator architecture As discussed in [6], using a generator architecture with a wider receptive field produces more homogeneous outputs while considering the entire facial features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  Thus, they added fast Fourier convolution layers [29] to their models to make a generator with a wider receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Although their work seems to be effective, it sometimes fails to make a balance between using global and local features [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' We use the Swin-Unet [30] architecture as our generator to create a balance between local and global features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Swin-Unet has a large receptive field because of its self-attention mechanism and could balance local and global feature usage well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' As discussed in [31], skip connections downgrade the image inpainting ability of the generator model because it allows the generator to copy the available parts of the input to the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Skip connections prevent the model from constructing the whole available and missed parts of the input based on high-level features that are extracted in the middle of the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Such a network results in totally heterogeneous face features in output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Therefore, we omit the skip connections from the Swin-Unet model, and our final generator architecture is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The patch discriminator [7] architecture is also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  5  Using the Swin transformer [8] layers, which could balance the utilization of local and global features, leads to more homogeneous and realistic facial features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' In addition, Swin transformer blocks do not add much computational complexity compared to convolutional models with a similar number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' This behavior is because the Swin transformer computes the correlation between the patches in certain parts of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='2 Homogeneity Loss In this section, we propose a new loss that we will use in addition to the losses that were discussed in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The idea behind this new loss is that we compute the realness of each part of the face compared to the distribution of that specific part in the whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' This will oblige the generator to pay more attention to symmetric and global features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' We consider six additional discriminators with the same architecture to compute the homogeneity loss, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Each discriminator is for a specific part of the face, such as skin, eyes, lips, clothes, and hair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' It is noteworthy that each of these parts is extracted from a face image using a pre-trained semantic segmentation module [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The segmentation module parameters are frozen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' While the generator intends to inpaint the missed regions, computing the realness of each facial organ will conclude more symmetry in the output image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The architecture of these discriminators is designed to assess the total realness of a specific face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Block diagram of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' First, the generator takes the masked image as input and attempts to inpaint it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Then the inpainted image is fed to the patch discriminator to check the overall reality of the patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Meanwhile, the inpainted image is also fed into a semantic segmentation network [32] to separate the semantic parts of the face, such as eyes, and ears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The architecture of these six semantic discriminators is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' In the next step, six distinct discriminators calculate the total realness of each semantic part of the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Then, the generator parameters will be updated based on these seven discriminators and the pixel-wise loss gradient signals, which are backpropagated to the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  Patch Partition Patch Merging Linear Projection Linear Embedding Swin Transformer Swin Transformer Block x 2 Patch Expanding Block x 1 Encoder Bottlenec Decoder Skin Discriminator F××48 Wx兴xC Wx×2C WH WH WxH xC W×H×C(4x) W×H×Clas Eye Discriminator Masked Hair Image Discriminator Semantic Generator Segmentor Lip Discriminator Patch Discriminator Cloth Discriminator Conv 4 × 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' stride=2 Linear Input Conv Conv 4 x 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' stride=2 Input Conv projection Ear Conv 4 × 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' stride = 1 Conv 1 × 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' stride = 1 Conv 4 x 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' stride = 1 Conv 1 × 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' stride = 1 Discriminator Discriminator Discriminator Output = Vector with 10 elements6  This characteristic is unlike the patch discriminator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' which does not completely understand the face image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' One of the main features of the face is its symmetry which the related discriminator will consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=" Using these semantic discriminators to check the realness of each face organ increases the generator's knowledge about different semantic parts of the image." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Considering facial organs helps the network to repair portraits more realistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The block diagram of SFI-Swin is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' For each of these discriminators and the corresponding facial organ, the adversarial and feature- matching loss is computed using Equations 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Therefore the overall homogeneity loss could be presented as Equations 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  ⎩ ⎪⎪ ⎨ ⎪⎪ ⎧ ℒ������������������������������������������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������) = ℒ������������������������������������(������������������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������������������������������������������������������) + ℒ������������������������(������������������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������������������������������������������������������) ℒ������������������������������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������) = ℒ�������������������������������������������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������������������������������������������� + ℒ�������������������������������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������������������������������������������� ℒℎ������������������������������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������) = ℒ������������������������������������(������������ℎ������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������ℎ������������������������������������) + ℒ������������������������(������������ℎ������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������ℎ������������������������������������)  ℒ������������������������������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������) = ℒ�������������������������������������������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������������������������������������������� + ℒ�������������������������������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������������������������������������������� ℒ������������������������������������������������ℎ(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������) = ℒ������������������������������������(������������������������������������������������������������ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������������������������������������������������������ℎ) + ℒ������������������������(������������������������������������������������������������ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������������������������������������������������������ℎ) ℒ������������������������������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������) = ℒ������������������������������������(������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������������������������������������������) + ℒ������������������������(������������������������������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������������������������������������������) ⎭ ⎪⎪ ⎬ ⎪⎪ ⎫  (7) ℒ������������������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������) = ������������1ℒ������������������������������������������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������) + ������������2ℒ������������������������������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������) + ������������3ℒℎ������������������������������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������) + ������������4ℒ������������������������������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������) + ������������5ℒ������������������������������������������������ℎ(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������) + ������������6ℒ������������������������������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' �������������) (8)  where ������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' ������������2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' ������������3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' ������������4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' ������������5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' and ������������6 are set to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='083, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='083, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='083, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='083, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='25, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The coefficients of eyes, ears, and lips are set three times greater than other parts of the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' This is because, during the experiments, we realized that the model is mostly incapable of maintaining the symmetry of these face organs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Experimental results In this section, we present the experiment setups and results accomplished using these setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='1 Experiments setups In the following, we will discuss the dataset, metrics, hyperparameters, and the total loss function we used to train our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Data and metrics: We use CelebAHQ [33] dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' This dataset contains 28k images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' We split the data to train, validate, and test, similar to [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=" Learned Perceptual Image Patch Similarity (LPIPS) [34] and Fr'echet inception distance (FID) [35] metrics are used to evaluate our proposed method performance." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Compared to pixel-level L1 and L2 distances, LPIPS and FID are more suitable for measuring the performance of large masks when multiple natural completions are plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The experimentation pipeline is implemented using PyTorch [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Training hyperparameters: The learning rate of the generator is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='001, and that of the seven discriminators is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The discrimination process is more straightforward than the generation process, especially in the initial epochs of the training [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  Hence, a lower initial learning rate for the discriminators allows the generator to converge faster during the initial epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' This will balance the training procedures between the discriminators and the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' This balance prevents the training procedure from collapsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' These learning rates also are optimized using a beam search algorithm to get the best results [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The batch size is also set to 20, and we train our model for 40 epochs using the Adam optimizer [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' We trained our model with an NVIDIA GeForce RTX 3090 with 24GB RAM GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  7  Loss function: besides the proposed homogeneity loss, we used the same losses as [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Therefore, the total loss function of our model is shown in Equation 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' ℒ(������������, �������������) =  ������������ℒ������������������������(������������, �������������) + ������������ℒ������������������������������������(������������, �������������) +  ������������ℒ������������������������(������������, �������������) + ������������ℒ������������������������(������������, �������������)  (9)  where ������������, ������������, ������������, and ������������ are set to 10, 10, 100, and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' These values are based on our experiments to get the best results on the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='2 Qualitative results: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' 2 shows SFI-Swin performance to inpaint for different mask sizes, in comparison with [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=" To further illustrate our proposed method's capability to inpaint the face images symmetrically, we generate masks that omit one of the eyes and slowly grow around the omitted region to cover half of the face." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' We compare our method with [6], and the results are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Further, we randomly chose ten images from the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Then, we masked (eliminated) one eye and a K×K block of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' We tested with K set to 16, 32, and 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' We repaired the image with a missing eye and a K×K block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' We then calculated the difference between the repaired eye while a K×K patch was masked with the inpainted image that only the eye was missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' We then find the mean of the absolute difference image and assign this value to the corresponding K×K block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The average absolute difference will be large if a block is essential to repair the missing eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Finally, we built a heatmap to show the blocks that play an important role in the inpainting of the missing eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The results are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' As we can see, the non-missing eye and the chick area around the missing eye are the most important patches for reconstructing the missing area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' We repeated this experiment for half of the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The effects of each missing K×K patch on the inpaint eye and half face are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  The lighter the color of the shown patch, the more effect it has on reconstructing the missing right-side eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' a) Original images, b) masks, inpainted images by c) LaMa [6], d) Swin, and e) SFI-Swin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  Narrow masks  Medium masks  Wide masks  a)  b)  c)  (p  e)8  While inpainting a facial part from one side of the face, our method focuses on the same organ on the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' This consideration causes symmetry in the repaired image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='3 Quantitative results: Table 1 shows the performance of our proposed method to repair the missing parts of the face compared with the best recently proposed methods in this field in terms of FID and LPIPS scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Because previous works trained multiple models for different mask sizes, therefore to compare fairly, we show their performance based on the mask sizes which they used in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' However, our method follows the aggressive mask generation proposed by [6] that helps it handle narrow, medium, and wide masks simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' As a result, our proposed methods, Swin and Swin  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Performances of b) LaMa, c) Swin, d) SFI-Swin in comparison with e) ground-truth to inpaint a) masked images starting from an eye and grown to cover half of the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' A white border is drawn to show the reconstructed region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  D=0  D = 12  D = 24  D = 36  D = 48  D=0  D = 12  D = 24  D= 36  D = 48  a)  b)  c)  (p  e)  a)  b)  d)  e)9  with multiple semantic discriminators (SFI-Swin), achieved the best results in medium and wide mask inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  FID and LPIPS are patch-based metrics and do not consider homogeneity and symmetricity in the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Thus, we propose the symmetry concentration score (SCS) to assess the symmetry of the left and right sides of the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' In Figure 5, we presented heatmaps that depict the influence the model takes from a K×K block of the face during the inpainting of an eye or half of the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Our symmetry concentration score measures the amount of attention the network pays to a part of the face while repairing the same semantic part on the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' To acquire this metric, we calculate the mean of the overlapping K × K patches with the desired organs while considering three different patch sizes (16×16, 32×32, 64×64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The results are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Our method using multiple semantic discriminators performed better than the Swin network without these discriminators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Also, we achieved better symmetric results than [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Steps to compute symmetry concentration score (SCS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' First, we mask an eye and a K×K patch of the face, and reconstruct the missing eye and the K×K patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Then the absolute difference between the image with inpainted eye with the image with a missing K×K patch and the missing eye is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The difference shows the effect of that K×K patch on inpainting result of the missing eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The effect of all K×K patches is computed and shown as a heatmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The face borders are also depicted to investigate the impact of each part of the face on inpainting the missed eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  eye mask Absolute value of Difference between generated eye when only that eye is masked Mask Output Output x eye mask and when an additional patch is also masked from the Original X Image Overlay the mean of the difference image on the X corresponding patch absO, mean Original X Image absO, mean Effect of each patch on filling the abs(), mean empty eye X X absO, mean10  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Influence that the generator gets from each K×K block of the face receives to repair the missing parts of the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' a) Lama, b) Swin, c) SFI-Swin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The missing parts are shown as empty black regions (an eye or half of the face), while the background is not considered in these experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content='  Eye  Half face  Block size = 16x16  a)  b)  c)  a)  Block size = 32x32  n  C  a)  Block size = 64x64  b)  C11  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Presenting the performance of our method SFI-Swin to inpaint different types of masks comparing to powerful methods in this field using two popular metrics, FID and LPIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' The three best models in each column are shown in red, orange, and green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Train masks type Methods CelebA-HQ (256×256) Narrow masks Medium masks Wide masks 40-50% masked All samples 40-50% masked All samples 40-50% masked All samples FID LPIPS FID LPIPS FID LPIPS FID LPIPS FID LPIPS FID LPIPS Aggressive train masks Ours (SFI-Swin) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
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+page_content=' Comparing the symmetry concentration score (SCS) of our method SFI-Swin to inpaint certain parts of the face compared to Swin and LaMa [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Method | Metric + face parts SCS for eye SCS for half face Ours (SFI-Swin) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
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+page_content='3948 LaMa [6] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
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+page_content='3740  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Conclusion  This paper discussed the effect of using multiple semantic discriminators incorporated with the Swin transformer-based architecture to repair face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' Our proposed method preserved the symmetry and homogeneity of the face parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=" Our experimental results show the proposed method's superiority over powerful rivals, especially on the medium and wide masks." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' We also proposed a new method to assess the concentration of the inpainting network while inpainting a specific face organ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' By using multiple discriminators to compute the reality of each facial organ, the generator was guided to preserve the symmetry and homogeneity of the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
+page_content=' This resulted in a generator that resolved one of the most critical inpainting shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfXwRb/content/2301.03130v1.pdf'}
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diff --git a/INE0T4oBgHgl3EQfhwHS/content/tmp_files/2301.02437v1.pdf.txt b/INE0T4oBgHgl3EQfhwHS/content/tmp_files/2301.02437v1.pdf.txt
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@@ -0,0 +1,1353 @@
+Valid P-Value for Deep Learning-Driven Salient Region
+Daiki Miwa∗
+Nagoya Institute of Technology
+miwa.daiki.mllab.nit@gmail.com
+Vo Nguyen Le Duy∗
+RIKEN
+duy.mllab.nit@gmail.com
+Ichiro Takeuchi†
+Nagoya University and RIKEN
+ichiro.takeuchi@mae.nagoya-u.ac.jp
+January 9, 2023
+Abstract
+Various saliency map methods have been proposed to interpret and explain predictions of
+deep learning models. Saliency maps allow us to interpret which parts of the input signals have a
+strong influence on the prediction results. However, since a saliency map is obtained by complex
+computations in deep learning models, it is often difficult to know how reliable the saliency map
+itself is. In this study, we propose a method to quantify the reliability of a salient region in the
+form of p-values. Our idea is to consider a salient region as a selected hypothesis by the trained
+deep learning model and employ the selective inference framework. The proposed method can
+provably control the probability of false positive detections of salient regions. We demonstrate
+the validity of the proposed method through numerical examples in synthetic and real datasets.
+Furthermore, we develop a Keras-based framework for conducting the proposed selective inference
+for a wide class of CNNs without additional implementation cost.
+∗Equal contribution
+†Corresponding author
+1
+arXiv:2301.02437v1  [stat.ML]  6 Jan 2023
+
+1
+Introduction
+Deep neural networks (DNNs) have exhibited remarkable predictive performance in numerous practical
+applications in various domains owing to their ability to automatically discover the representations
+needed for prediction tasks from the provided data. To ensure that the decision-making process of
+DNNs is transparent and easy to understand, it is crucial to effectively explain and interpret DNN
+representations.
+For example, in image classification tasks, obtaining salient regions allows us to
+explain which parts of the input image strongly influence the classification results.
+Several saliency map methods have been proposed to explain and interpret the predictions of DNN
+models [Ribeiro et al., 2016, Bach et al., 2015, Doshi-Velez and Kim, 2017, Lundberg and Lee, 2017,
+Zhou et al., 2016, Selvaraju et al., 2017]. However, the results obtained from saliency methods are
+fragile [Kindermans et al., 2017, Ghorbani et al., 2019, Melis and Jaakkola, 2018, Zhang et al., 2020,
+Dombrowski et al., 2019, Heo et al., 2019]. It is important to develop a method for quantifying the
+reliability of DNN-driven salient regions.
+Our idea is to interpret salient regions as hypotheses driven by a trained DNN model and employ
+a statistical hypothesis testing framework. We use the p-value as a criterion to quantify the statistical
+reliability of the DNN-driven hypotheses. Unfortunately, constructing a valid statistical test for DNN-
+driven salient regions is challenging because of the selection bias. In other words, because the trained
+DNN selects the salient region based on the provided data, the post-selection assessment of importance
+is biased upwards.
+To correct the selection bias and compute valid p-values for DNN-driven salient regions, we intro-
+duce a conditional selective inference (SI) approach. The selection bias is corrected by conditional SI
+in which the test statistic conditional on the event that the hypotheses (salient regions) are selected
+using the trained DNNs. Our main technical contribution is to develop a computational method for
+explicitly deriving the exact (non-asymptotic) conditional sampling distribution of the salient region
+for a wide class convolutional neural networks (CNNs), which enables us to conduct conditional SI
+and compute valid p-values. Figure 1 presents an example of the problem setup.
+Related works.
+In this study, we focus on statistical hypothesis testing for post-hoc analysis, i.e.,
+quantifying the statistical significance of the salient regions identified in a trained DNN model when
+a test input instance is fed into the model. Several methods have been developed to visualize and
+understand trained DNNs. Many of these post-hoc approaches [Mahendran and Vedaldi, 2015, Zeiler
+and Fergus, 2014, Dosovitskiy and Brox, 2016, Simonyan et al., 2013] have focused on developing
+visualization tools for saliency maps given a trained DNN. Other methods have aimed to identify
+the discriminative regions in an input image given a trained network [Selvaraju et al., 2017, Fong
+2
+
+Input Image
+Saliency Map
+Salient Region
+Reference Image
+Two-sample 
+Test
+(a) Image without tumor region. The naive-p = 0.00 (wrong detection) and selective-p = 0.43 (true negative)
+Input Image
+Saliency Map
+Salient Region
+Reference Image
+Two-sample 
+Test
+(b) Image with tumor region. The naive-p = 0.00 (true positive) and selective-p = 0.00 (true positive)
+Figure 1: Examples of the problem setup and the proposed method on brain tumor dataset. By
+applying a saliency method called CAM [Zhou et al., 2016] on a query input image, we obtain the
+salient region. Our goal is to provide the statistical significance of the salient region in the form of
+p-value by considering two-sample test between the salient region and the corresponding region in
+the a reference image. Note that, since the salient region is selected based on the data, the degree
+of saliency in the selected region is biased upward. In the upper image where there is no true brain
+tumor, the naive p-value which is obtained without caring the selection bias is nearly zero, indicating
+the false positive finding of the salient region.
+On the other hand, the selective p-value which is
+obtained by the proposed conditional SI approach is 0.43, indicating that the selected saliency region
+is not statistically significant. In the lower figure where there is a true brain tumor, both the naive p-
+value and the selective p-value are very small, indicating true positive finding. These results illustrate
+that naive p-value cannot be used to quantify the reliability of DNN-based salient region. In contrast,
+with the selective p-values, we can successfully identify false positive and true positive detections with
+a desired error rate.
+and Vedaldi, 2017, Zhou et al., 2016, Lundberg and Lee, 2017]. Furthermore, some recent studies
+have shown that many popular methods for explanation and interpretation are not stable against a
+perturbation or adversarial attack on the input data and model [Kindermans et al., 2017, Ghorbani
+et al., 2019, Melis and Jaakkola, 2018, Zhang et al., 2020, Dombrowski et al., 2019, Heo et al.,
+2019]. However, to the best of our knowledge, no study to date has quantitatively evaluated and
+reproducibility of DNN-driven salient regions with a rigorous statistical inference framework.
+Recently, conditional SI has been recognized as a promising new approach for evaluating the
+3
+
+27statistical significance of data-driven hypotheses. Conditional SI has been mainly studied for inference
+of linear model features selected by a feature selection method such as Lasso [Lee et al., 2016, Liu
+et al., 2018, Hyun et al., 2018, Le Duy and Takeuchi, 2021] and stepwise feature selection [Tibshirani
+et al., 2016, Sugiyama et al., 2021a]. The main idea of conditional SI study is to make inferences
+conditional on selection events, which allows us to derive exact sampling distributions of test statistics.
+In addition, conditional SI has been applied to various problems [Fithian et al., 2015, Tian and Taylor,
+2018, Yang et al., 2016, Hyun et al., 2021, Duy et al., 2020, Sugiyama et al., 2021b, Chen and Bien,
+2019, Panigrahi et al., 2016, Tsukurimichi et al., 2021, Hyun et al., 2018, Tanizaki et al., 2020, Duy
+and Takeuchi, 2021, Tibshirani et al., 2016, Sugiyama et al., 2021a, Suzumura et al., 2017, Das et al.,
+2021, Duy and Takeuchi, 2022].
+Most relevant existing work of this study is Duy et al. [2022], where the authors provide a framework
+for computing valid p-values for DNN-based image segmentation results. In this paper, we generalized
+this work so that hypotheses characterized by any internal nodes of the network can be considered,
+enabling us to quanfity the statistical significance of salient regions.
+This is in contrast to Duy
+et al. [2022]’s work, which only considered the inference of the DNN’s output in a segmentation task.
+Furthermore, we introduce a Keras-based implementation framework that enables us to conduct SI for
+a wide class of CNNs without additional implementation costs. This is in contrast to Duy et al. [2022]’s
+work, where the selection event must be implemented whenever the network architecture is changed.
+In another direction, Burns et al. [2020] considered the black box model interpretability as a multiple-
+hypothesis testing problem. They aimed to deduce important features by testing the significance of the
+difference between the model prediction and what would be expected when replacing the features with
+their counterfactuals. The difficulty of this multiple-hypothesis testing approach is that the number
+of hypotheses to be considered is large (e.g., in the case of an image with n pixels, the number of
+possible salient regions is 2n). Multiple testing correction methods, such as the Bonferroni correction,
+are highly conservative when the number of hypotheses is large. To circumvent this difficulty, they
+only considered a tractable number of regions selected by a human expert or object detector, which
+causes selection bias because these candidate regions are selected based on the data.
+Contribution.
+Our main contributions are as follows:
+• We provide an exact (non-asymptotic) inference method for salient regions based on the SI
+concept. To the best of our knowledge, this is the first method that proposes to provide valid p-values
+to statistically quantify the reliability of DNN-driven salient regions.
+• We propose a novel algorithm and its implementation. Specifically, we propose Keras-based
+implementation enables us to conduct conditional SI for a wide class of CNNs without additional
+implementation costs.
+4
+
+• We conducted experiments on both synthetic and real-world datasets, through which we show
+that our proposed method can successfully control the false positive rate, has good performance in
+terms of computational efficiency, and provides good results in practical applications. We provide the
+detailed description of our implementation in the supplementary document. Our code is available at
+https://github.com/takeuchi-lab/selective inference dnn salient region.
+2
+Problem Formulation
+In this paper, we consider the problem of quantifying the statistical significance of the salient regions
+identified by a trained DNN model when a test input instance is fed into the model. Consider an
+n-dimensional query input vector
+X = (X1, ..., Xn)⊤ = s + ε,
+ε ∼ N(0, σ2In)
+and an n-dimensional reference input vector,
+Xref = (Xref
+1 , ..., Xref
+n )⊤ = sref + εref,
+εref ∼ N(0, σ2In),
+where s, sref ∈ Rn are the signals and ε, εref ∈ Rn are the noises for query and reference input vectors,
+respectively. We assume that the signals, s and sref are unknown, whereas the distribution of noises
+ε and εref are known (or can be estimated from external independent data) to follow N(0, σ2In),
+an n-dimensional normal distribution with a mean vector 0 and covariance matrix σ2In, which are
+mutually independent. In the illustrative example presented in §1, X is a query brain image for a
+potential patient (we do not know whether she/he has a brain tumor), whereas Xref is a brain image
+of a healthy person known to be without brain tumors.
+Consider a saliency method for a trained CNN. We denote the saliency method as a function
+A : Rn → Rn that takes a query input vector X ∈ Rn and returns the saliency map A(X) ∈ Rn. We
+define a salient region MX for the query input vector X as the set of elements whose saliency map
+value is greater than a threshold
+MX = {i ∈ [n] : Ai(X) ≥ τ} ,
+(1)
+where τ ∈ R denotes the given threshold. In this study, we consider CAM [Zhou et al., 2016] as an
+example of saliency method and threshold-based definition of the salient region. Our method can be
+applied to other saliency methods and other definition of salient region.
+Statistical inference.
+To quantify the statistical significance of the saliency region MX, we con-
+sider such two-sample test to quantify the statistical significance of the difference between the salient
+5
+
+regions of the query input vector XMX and corresponding region of the reference input vector Xref
+MX.
+As concrete examples of the two-sample test, we consider the mean null test:
+H0 :
+1
+|MX|
+�
+i∈MX
+si =
+1
+|MX|
+�
+i∈MX
+sref
+i
+v.s.
+H1 :
+1
+|MX|
+�
+i∈MX
+si ̸=
+1
+|MX|
+�
+i∈MX
+sref
+i .
+(2)
+and global null test:
+H0 : si = sref
+i , ∀i ∈ MX,
+v.s.
+H1 : si ̸= sref
+i , ∃i ∈ MX,
+(3)
+In the mean null test depicted in Eq. (2), we consider a null hypothesis that the average signals in the
+salient region MX are the same between X and Xref. In contrast, in the global null test in Eq. (3),
+we consider a null hypothesis that all elements of the signals in the salient region MX are the same
+between X and Xref. The p-values for these two-sample tests can be used to quantify the statistical
+significance of the salient region MX.
+Test-statistic.
+For a two-sample test conducted between XMX and Xref
+MX, we consider a class of
+test statistics called conditionally linear test-statistic, which is expressed as
+T(X, Xref) = η⊤
+MX
+� X
+Xref
+�
+,
+(4)
+and conditionally χ test-statistic, which is expressed as
+T(X, Xref) = σ−1
+����PMX
+� X
+Xref
+����� ,
+(5)
+where ηMX ∈ R2n is a vector and PMX ∈ R2n×2n is a projection matrix that depends on saliency
+region MX.
+The test statistics for the mean null tests and the global null test can be written in the form of Eq.
+(4) and (5), respectivery. For the mean null test in Eq. (2), we consider the following test-statistic
+T(X, Xref) = η⊤
+MX
+� X
+Xref
+�
+=
+1
+|MX|
+�
+i∈MX
+Xi −
+1
+|MX|
+�
+i∈MX
+Xref
+i
+,
+where ηMX =
+1
+|MX|
+�
+� 1n
+MX
+−1n
+MX
+�
+� ∈ R2n. For the gloabl null test in Eq. (3), we consider the following
+test-statistic
+T(X, Xref) = σ−1
+����PMX
+� X
+Xref
+����� =
+�
+�
+�
+� �
+i∈MX
+�Xi − Xref
+i
+√
+2σ
+�2
+,
+where
+PMX = 1
+2
+�
+� diag(1n
+MX)
+−diag(1n
+MX)
+−diag(1n
+MX)
+diag(1n
+MX)
+�
+� .
+(6)
+6
+
+To obtain p-values for these two-sample tests we need to know the sampling distribution of the
+test-statistics. Unfortunately, it is challenging to derive the sampling distributions of test-statistics
+because they depend on the salient region MX, which is obtained through a complicated calculation
+in the trained CNN.
+3
+Computing Valid p-value by Conditional Selective Inference
+In this section, we introduce an approach to compute the valid p-values for the two-sample tests for
+the salient region MX between the query input vector X and the reference input vector Xref based
+on the concept of conditional SI [Lee et al., 2016].
+3.1
+Conditional Distribution and Selective p-value
+Conditional distribution.
+The basic idea of conditional SI is to consider the sampling distribution
+of the test-statistic conditional on a selection event. Specifically, we consider the sampling property
+of the following conditional distribution
+T(X, Xref)
+��� {MX = MXobs} ,
+(7)
+where Xobs is the observation (realization) of random vector X. The condition in Eq.(7) indicates the
+randomness of X conditional on the event that the same salient region MX as the observed MXobs
+is obtained. By conditioning on the salient region MX, derivation of the sampling distribution of the
+conditionally linear and χ test-statistic T(X, Xref) is reduced to a derivation of the distribution of
+linear function and quadratic function of (X, Xref), respectively.
+Selective p-value.
+After considering the conditional sampling distribution in (7), we introduce the
+following selective p-value:
+pselective = PH0
+� ��T(X, Xref)
+�� ≥
+��T(Xobs, Xref
+obs)
+��
+��� MX = MXobs, QX,Xref = Qobs
+�
+,
+(8)
+where
+QX,Xref = ΩX,Xref,
+Qobs = QXobs,Xref
+obs
+with
+ΩX,Xref =
+�
+I2n − ηMXη⊤
+MX
+∥ηMX∥2
+� � X
+Xref
+�
+∈ R2n
+in the case of mean null test, and
+QX,Xref =
+�
+VX,Xref, UX,Xref
+�
+,
+Qobs = QXobs,Xref
+obs
+7
+
+with
+VX,Xref = σPMX
+� X
+Xref
+������PMX
+� X
+Xref
+����� ∈ R2n,
+UX,Xref = P ⊥
+MX
+� X
+Xref
+�
+∈ R2n
+in the case of global null test. The QX,Xref is the sufficient statistic of the nuisance parameter that
+needs to be conditioned on in order to tractably conduct the inference 1.
+The selective p-value in Eq.(8) has the following desired sampling property
+PH0
+�
+pselective ≤ α | MX = MXobs
+�
+= α,
+∀α ∈ [0, 1].
+(9)
+This means that the selective p-values pselective can be used as a valid statistical significance measure
+for the salient region MX.
+3.2
+Characterization of the Conditional Data Space
+To compute the selective p-value in (8), we need to characterize the conditional data space whose
+characterization is described introduced in the next section. We define the set of (X Xref)⊤ ∈ R2n
+that satisfies the conditions in Eq. (8) as
+D =
+�
+(X Xref)⊤ ∈ R2n �� MX = MXobs, QX,Xref = Qobs
+�
+.
+(10)
+According to the second condition, the data in D is restricted to a line in R2n as stated in the following
+Lemma.
+Lemma 1. Let us define let us define,
+a = ΩXobs,Xref
+obs
+and
+b =
+ηMX
+∥ηMX∥2 ∈ R2n.
+(11)
+in the mean null test, and
+a = UXobs,Xref
+obs
+and
+b = VXobs,Xref
+obs
+(12)
+in the case of global null test. Then, the set D in (10) can be rewritten as D =
+��
+X Xref�⊤ = a+bz |
+z ∈ Z
+�
+by using the scalar parameter z ∈ R, where
+Z = {z ∈ R | Ma1:n+b1:nz = MXobs} .
+(13)
+x1:n represents a vector of elements 1 through n of x.
+1This nuisance parameter QX,Xref corresponds to the component z in the seminal conditional SI paper [Lee et al.,
+2016] (see Sec. 5, Eq. 5.2 and Theorem 5.2) and z, w in [Chen and Bien, 2019](see Sec. 3, Theorem 3.7). We note that
+additional conditioning on QX,Xref is a standard approach in the conditional SI literature and is used in almost all
+the conditional SI-related studies. Here, we would like to note that the selective p-value depend on QX,Xref , but the
+property in (9) is satisfied without this additional condition because we can marginalize over all values of QX,Xref (see
+the lower part of the proof of Theorem 5.2 in Lee et al. [2016] and the proof of Theorem 3.7 in Chen and Bien [2019] ).
+8
+
+Proof. The proof is deferred to Appendix A.1
+■
+Lemma 1 indicates that we do not need to consider the 2n-dimensional data space. Instead, we
+only need to consider the one-dimensional projected data space Z in (13).
+Now, let us consider
+a random variable Z ∈ R and its observation Zobs ∈ R that satisfies (X Xref)⊤ = a + bZ and
+(Xobs Xref
+obs)⊤ = a + bZobs. The selective p-value (8) is rewritten as
+pselective = PH0 (|Z| ≥ |Zobs| | Z ∈ Z) .
+(14)
+Because the variable Z ∼ N(0, σ2∥η∥2) in the case of mean null test and Z ∼ χ (Trace(P)) in the case
+of global null test under the null hypothesis, Z | Z ∈ Z follows a truncated normal distribution and a
+truncated χ distribution, respectively. Once the truncation region Z is identified, computation of the
+selective p-value in (14) is straightforward. Therefore, the remaining task is to identify Z.
+In general, computation of Z in (13) is difficult because we need to identify the selection event
+Ma1:n+b1:nz for all values of z ∈ R, which is computationally challenging. In the next section, we
+show that the challenge can be resolved under a wide class of problems.
+4
+Piecewise Linear Network
+The problem of computing selective p-values for the selected salient region is casted into the problem
+of identifying a set of intervals Z = {z ∈ R | MX(z) = MXobs}. Given the complexity of saliency
+computation in a trained DNN, it seems difficult to obtain Z. In this section, however, we explain
+that this is feasible for a wide class of CNNs.
+Piecewise linear components in CNN
+The key idea is to note that most of basic operations and
+common activation functions used in a trained CNN can be represented as piecewise linear functions
+in the following form:
+Definition 1. (Piecewise Linear Function) A piecewise linear function f : Rn �→ Rm is defined as:
+f(X) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Ψf
+1X + ψf
+1 ,
+if X ∈ Pf
+1 := {X′ ∈ Rn | ∆f
+1X′ ≤ δf
+1 },
+Ψf
+2X + ψf
+2 ,
+if X ∈ Pf
+2 := {X′ ∈ Rn | ∆f
+2X′ ≤ δf
+2 },
+...
+Ψf
+K(f)X + ψf
+K(f),
+if X ∈ Pf
+K(f) := {X′ ∈ Rn | ∆f
+K(f)X′ ≤ δf
+K(f)}
+where Ψf
+k, ψf
+k, ∆f
+k and δf
+k for k ∈ [K(f)] are certain matrices and vectors with appropriate dimensions,
+Pf
+k := {x ∈ Rn | ∆f
+kx ≤ δf
+k} is a polytope in Rn for k ∈ [K(f)], and K(f) is the number of polytopes
+for the function f.
+9
+
+Examples of piecewise linear components in a trained CNN are shown in Appendix A.2.
+Piecewise Linear Network
+Definition 2. (Piecewise Linear Network) A network obtained by concatenations and compositions
+of piecewise linear functions is called piecewise linear network.
+Since the concatenation and the composition of piecewise linear functions is clearly piecewise linear
+function, the output of any node in the piecewise linear network is written as a piecewise linear function
+of an input vector X. This is also true for the saliency map function Ai(X), i ∈ [n]. Furthermore,
+as discussed in §4, we can focus on the input vector in the form of X(z) = a1:n + b1:nz which is
+parametrized by a scalar parameter z ∈ R. Therefore, the saliency map value for each element is
+written as a piecewise linear function of the scalar parameter z, i.e.,
+Ai(X(z)) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+κAi
+1 z + ρAi
+1 ,
+if z ∈ [LAi
+1 , U Ai
+1 ],
+κAi
+2 z + ρAi
+2 ,
+if z ∈ [LAi
+2 , U Ai
+2 ],
+...
+κAi
+K(Ai)z + ρf
+K(Ai),
+if z ∈ [LAi
+K(Ai), U Ai
+K(Ai)],
+(15)
+where K(Ai) is the number of linear pieces of the piecewise linear function, κAi
+k , ρAi
+k
+are certain scalar
+parameters, [LAi
+k , U Ai
+k ] are intervals for k ∈ [K(Ai)] (note that a polytope in Rn is reduced to an
+interval when it is projected onto one-dimensional space).
+This means that, for each piece of the piecewise linear function, we can identify the interval of z
+such that Ai(X(z)) ≥ τ as follows 2
+z ∈
+�
+�
+�
+�
+max
+�
+LAi
+k ,
+�
+τ − ρAi
+k
+�
+/κAi
+k
+�
+, U Ai
+k
+�
+if κAi
+k
+> 0
+�
+LAi
+k , min
+�
+U Ai
+k ,
+�
+τ − ρAi
+k
+�
+/κAi
+k
+�
+,
+�
+if κAi
+k
+< 0
+⇒
+Ai(X(z)) ≥ τ.
+(16)
+With a slight abuse of notation, let us collectively denote the finite number of intervals on z ∈ R that
+are defined by LAi
+k , U Ai
+k , (τ − ρAi
+i /κAi
+k ) for all (k, i) ∈ [K(Ai)] × [n] as
+[z0, z1], [z1, z2], . . . , [zt−1, zt], [zt, zt+1], . . . , [zT −1, zT ],
+where zmin = z0 and zmax = zT are defined such that the probability mass of z < zmin and z > zmax
+are negligibly small.
+2For simplicity, we omit the description for the case of κAi
+k
+= 0. In this case, if ρAi
+k
+≥ τ, then z ∈ [LAi
+k , UAi
+k
+] ⇒ i ∈
+MX(z).
+10
+
+Algorithm 1 SI DNN Saliency
+Input: Xobs, zmin, zmax, T ← ∅
+1: Obtain Eobs, compute η as well as a and b ← Eq. (12), and initialize: t = 1, zt = zmin
+2: for t ≤ T do
+3:
+Compute zt+1 by Auto-Conditioning (see §5)
+4:
+if EX(z),Xref (z) = Eobs in z ∈ [zt, zt+1] (by using Eq.(16)) then
+5:
+T ← T + {t}
+6:
+end if
+7:
+t = t + 1
+8: end for
+9: Identify Z ← �
+t∈T [zt, zt+1]
+10: pselective ← Eq. (14)
+Output: pselective
+Algorithm
+Algorithm 1 shows how we identify Z = {z ∈ R | MX(z),Xref(z) = Mobs}. We simply
+check the intervals of z in the order of [z0, z1], [z1, z2], ..., [zT −1, zT ] to see whether MX(z) = MX(zobs)
+or not in the interval by using Eq.(16). Then, the truncation region Z in Eq.(13) is given as Z =
+�
+t∈[T ]|EX(z),Xref (z)=Eobs for z∈[zt,zt+1][zt, zt+1].
+5
+Implementation: Auto-Conditioning
+The bottleneck of our algorithm is Line 3 in Algorithm 1, where zt+1 must be found by considering
+all relevant piecewise linear components in a complicated trained CNN. The difficulty lies not only in
+the computational cost but also in the implementation cost. To implement conditional SI in DNNs
+naively, it is necessary to characterize all operations at each layer of the network as selection events and
+implement each of the specifically[Duy et al., 2022] To circumvent this difficulty, we introduce a mod-
+ular implementation scheme called auto-conditioning, which is similar to auto-differentiation [Baydin
+et al., 2018] in concept. This enables us to conduct conditional SI for a wide class of CNNs without
+additional implementation cost.
+The basic idea in auto-conditioning is to add a mechanism to compute and maintain the interval
+z ∈ [Lf
+k, U f
+k ] for each piecewise linear component f in the network (e.g., layer API in the Keras
+framework).
+This enables us to automatically compute the interval [Lf
+k, U f
+k ] of a piecewise linear
+function f when it is obtained as concatenation and/or composition of multiple piecewise linear
+components. If f is obtained by concatenating two piecewise linear functions f1 and f2, we can easily
+obtain [Lf
+k, U f
+k ] = [Lf1
+k1, U f1
+k1 ] ∩ [Lf2
+k2, U f2
+k2 ]. However, if f is obtained as a composition of two piecewise
+linear functions f1 and f2, the calculation of the interval is given by the following lemma.
+Lemma 2. Consider the composition of two piecewise linear functions, that is, f(X(z)) = (f2 ◦
+11
+
+f1)(X(z)). Given a real value of z, the interval [Lf2
+k , U f2
+k ] in the input domain of f2 can be computed
+as
+Lf2
+k2 =
+max
+j:(∆f2
+k2γf1)j<0
+(δf2
+k2)j − (∆f2
+k2βf1)j
+(∆f2
+k2γf1)j
+,
+U f2
+k2 =
+min
+j:(∆f2
+k2γf1)j>0
+(δf2
+k2)j − (∆f2
+k2βf1)j
+(∆f2
+k2γf1)j
+,
+where βf1 + γf1z is the output of f1 (i.e., the input of f2). Moreover, ∆f2
+k2 and δf2
+k2 are obtained by
+verifying the value of βf1 + γf1z. Then, the interval of the composite function is obtained as follows:
+[Lf
+k, U f
+k ] = [Lf1
+k1, U f1
+k1 ] ∩ [Lf2
+k2, U f2
+k2 ]
+The proof is provided in Appendix A.3.
+Here, the variables βfk and γfk can be recursively
+computed through layers as
+βfk+1 = Ψfk
+k βfk + ψfk
+k
+and
+γfk+1 = Ψfk
+k γfk.
+Lemma 2 indicates that the intervals in which X(z) decreases can be forwardly propagated through
+these layers. This means that the lower bound LAi
+k
+and upper bound U Ai
+k
+of the current piece in the
+piecewise linear function in Eq. (15) can be automatically computed by forward propagation of the
+intervals of the relevant piecewise linear components.
+6
+Experiment
+We only highlight the main results. More details (methods for comparison, network structure, etc.)
+can be found in the Appendix A.4.
+Experimental setup.
+We compared our proposed method with the naive method, over-conditioning
+(OC) method, and Bonferroni correction. To investigate the false positive rate (FPR) we consid-
+erd, 1000 null images X = (X1, ..., Xn) and 1000 reference images Xref = (Xref
+1 , ..., xref
+n ), where
+s = sref = 0 and ε, εref ∼ N(0, In), for each n ∈ {64, 256, 1024, 4096}. To investigate the true positive
+rate (TPR), we set n = 256 and generated 1,000 images, in which si = signal for any i ∈ S where
+S is the “true” salient region whose location is randomly determined. si = 0 for any i ̸∈ S and
+ε ∼ N(0, In). We set ∆ ∈ {1, 2, 3, 4}. Reference images were generated in the same way as in the case
+of FPR. In all experiments, we set the threshold for selecting the salient region τ = 0 in the mean
+null test and τ = 5 in the global null test . We set the significance level α = 0.05. We used CAM as
+the saliency method in all experiments.
+Numerical results.
+The results of FPR control are presented in Fig. 2. The proposed method, OC,
+and Bonferroni successfully controlled the FPR in both the mean and global null test cases, whereas
+12
+
+64
+256
+1024
+4096
+n
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+False Positive Rate(FPR)
+Proposed
+OC
+Bonferroni
+Naive
+(a) Mean null test
+64
+256
+1024
+4096
+n
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+False Positive Rate(FPR)
+Proposed
+OC
+Bonferroni
+Naive
+(b) Global null test
+Figure 2: False Positive Rate (FPR) comparison.
+1
+2
+3
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+True Positive Rate(TPR)
+Proposed
+OC
+Bonferroni
+(a) Mean null test
+1
+2
+3
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+True Positive Rate(TPR)
+Proposed
+OC
+Bonferroni
+(b) Global null test
+Figure 3: True Positive Rate (FPR) comparison.
+Input Image
+Saliency Map
+Salient Region
+Reference Image
+Reference Region
+Figure 4: Mean null test for image without tumor (pnaive = 0.00, pselective = 0.78).
+the others could not. Because naive methods failed to control the FPR, we no longer considered their
+TPR. The results of the TPR comparison are shown in Fig. 3. The proposed method has the highest
+TPR in all cases. The Bonferroni method has the lowest TPR because it is conservative owing to
+considering the number of all possible hypotheses. The OC method also has a low TPR than the
+proposed method because it considers several extra conditions, which causes the loss of TPR.
+Real data experiments.
+We examined the brain image dataset extracted from the dataset used
+in Buda et al. [2019], which included 939 and 941 images with and without tumors, respectively. The
+results of the mean null test are presented in Figs. 4 and 5. The results of the global null test are
+presented in Figs. 6 and 7. The naive p-value remains small even when the image has no tumor region,
+which indicates that naive p-values cannot be used to quantify the reliability of DNN-based salient
+regions. The proposed method successfully identified false positive and true positive detections.
+7
+Conclusion
+In this study, we proposed a novel method to conduct statistical inference on the significance of
+DNN-driven salient regions based on the concept of conditional SI. We provided a novel algorithm for
+efficiently and flexibly conducting conditional SI for salient regions. We conducted experiments on
+13
+
+Input Image
+Saliency Map
+Salient Region
+Reference Image
+Reference Region
+Figure 5: Mean null test for image with a tumor (pnaive = 0.00, pselective = 1.92 × 10−4).
+Input Image
+Saliency Map
+Salient Region
+Reference Image
+Reference Region
+Figure 6: Global null test for image without tumor (pnaive = 0.03, pselective = 0.46)
+Input Image
+Saliency Map
+Salient Region
+Reference Image
+Reference Region
+Figure 7: Global null test for image with a tumor (pnaive = 0.00, pselective = 1.51 × 10−3).
+both synthetic and real-world datasets to demonstrate the performance of the proposed method.
+Acknowledgements
+This work was partially supported by MEXT KAKENHI (20H00601), JST CREST (JPMJCR21D3),
+JST Moonshot R&D (JPMJMS2033-05), JST AIP Acceleration Research (JPMJCR21U2), NEDO
+(JPNP18002, JPNP20006), and RIKEN Center for Advanced Intelligence Project.
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+A
+Appendix
+A.1
+Proof of Lemma 1
+In the mean null test, according to the second condition in (10), we have
+ΩX,Xref = ΩXobs,Xref
+obs
+⇔
+�
+I2n − ηMXη⊤
+MX
+η⊤
+MXηMX
+� � X
+Xref
+�
+= ΩXobs,Xref
+obs
+⇔
+� X
+Xref
+�
+= ΩXobs,Xref
+obs +
+ηMX
+∥ηMX∥2 η⊤
+MX
+� X
+Xref
+�
+.
+By defining a = qXobs,Xref
+obs, b =
+ηMX
+∥ηMX ∥2 , z = η⊤
+MX
+� X
+Xref
+�
+, we obtain the result in Lemma 1.
+In the global null test, according to the second condition in (10),
+UX,Xref = UXobs,Xref
+obs
+⇔ P ⊥
+MX
+� X
+Xref
+�
+= UXobs,Xref
+obs
+⇔ (I2n − PMX)
+� X
+Xref
+�
+= UXobs,Xref
+obs
+⇔
+� X
+Xref
+�
+= UXobs,Xref
+obs + VXobs,Xref
+obsσ−1
+����PMX
+� X
+Xref
+����� .
+By defining a = UXobs,Xref
+obs, b = VXobs,Xref
+obs, z = σ−1 ���PMX
+� X
+Xref
+���� , we obtain the result in Lemma
+1.
+A.2
+Examples of piecewise linear functions
+Examples of piecewise linear components in a trained CNN with X ∈ R2 are provided as follows:
+18
+
+ReLU: Consider f is ReLU function. Then, K(f) = 4, ψk = (0 0)⊤ for any k ∈ [4],
+Ψf
+1 =
+�
+�0
+0
+0
+0
+�
+� , Pf
+1 =
+�
+�
+�X : X1 < 0,
+X2 < 0
+�
+�
+� ,
+Ψf
+2 =
+�
+�0
+0
+0
+1
+�
+� , Pf
+2 =
+�
+�
+�X : X1 < 0,
+X2 ≥ 0
+�
+�
+� ,
+Ψf
+3 =
+�
+�1
+0
+0
+0
+�
+� , Pf
+3 =
+�
+�
+�X : X1 ≥ 0,
+X2 < 0
+�
+�
+� ,
+Ψf
+4 =
+�
+�1
+0
+0
+1
+�
+� , Pf
+4 =
+�
+�
+�X : X1 ≥ 0,
+X2 ≥ 0
+�
+�
+� .
+This can be similarly extended to the case of Leaky ReLU.
+Max-pooling: Consider f(X) = max{X1, X2}. Then, it is represented as a piecewise linear function
+with K(f) = 2, ψk = (0) for any k ∈ [2],
+Ψf
+1 =
+�
+1
+0
+�
+, Pf
+1 = {X : X1 ≥ X2} ,
+Ψf
+2 =
+�
+0
+1
+�
+, Pf
+2 = {X : X1 < X2} .
+Convolution and matrix-vector multiplication: In a neural network, the multiplication results between
+the weight matrix and the output of the previous layer and its summation with the bias vector is a
+linear function. In a CNN, the convolution operation is obviously a linear function.
+Upsampling: Consider f is the upsampling operation on X ∈ R2, then it can be represented as a
+piecewise linear function with K(f) = 1, ψ1 = (0 0 0 0)⊤,
+Ψf
+1 =
+�
+�1
+1
+0
+0
+0
+0
+1
+1
+�
+�
+⊤
+,
+Pf
+1 = R2.
+Sigmoid and hyperbolic tangent: If there is any specific demand to use non-piecewise linear activation
+functions, we can apply a piecewise-linear approximation approach to these functions.
+A.3
+Proof of Lemma 2
+At f1, given a a real value z, the input is βf0 + γf0z = a1:n + b1:nz. By checking the value of this
+input, we can easily obtain the polytope
+{∆f1
+k1(βf0 + γf0z) ≤ δf1
+k1},
+k1 ∈ [K(f1)],
+that βf0 + γf0z belongs to. Based on the obtained polytope, we can calculate the interval [Lf1
+k1, U f1
+k1 ],
+Lf1
+k1 =
+max
+j:(∆f1
+k1γf0)j<0
+(δf1
+k1)j − (∆f1
+k1βf0)j
+(∆f1
+k1γf0)j
+and
+U f1
+k1 =
+min
+j:(∆f1
+k1γf0)j>0
+(δf1
+k1)j − (∆f1
+k1βf0)j
+(∆f1
+k1γf0)j
+.
+Moreover, based on the obtained polytope, we can easily obtain Ψf1
+k1 and ψf1
+k1, k1 ∈ [K(f1)]. Therefore,
+the output of the first layer at z can be defined as
+f1(z) = Ψf1
+k1(βf0 + γf0z) + ψf1
+k1
+= βf1 + γf1z,
+19
+
+where βf1 = Ψf1
+k1βf0 + ψf1
+k1 and γf1 = Ψf1
+k1γf0. Next, we input βf1, γf1 to f2.
+At the 2nd layer, similarly, the input is βf1 + γf1z. By checking the value of this input, we can
+easily obtain the polytope
+{∆f2
+k2(βf1 + γf1z) ≤ δf2
+k2},
+k2 ∈ [K(f2)],
+that βf1 + γf1z belongs to. Based on the obtained polytope, we can calculate the interval [Lf2
+k2, U f2
+k2 ],
+Lf2
+k2 =
+max
+j:(∆f2
+k2γf1)j<0
+(δf2
+k2)j − (∆f2
+k2βf1)j
+(∆f2
+k2γf1)j
+and
+U f2
+k2 =
+min
+j:(∆f2
+k2γf1)j>0
+(δf2
+k2)j − (∆f2
+k2βf1)j
+(∆f2
+k2γf1)j
+.
+Moreover, based on the obtained polytope, we can easily obtain Ψf2
+k2 and ψf2
+k2, k2 ∈ [K(f2)]. Therefore,
+the output of the first layer at z can be defined as
+f2(z) = Ψf2
+k2(βf1 + γf1z) + ψf2
+k2
+= βf2 + γf2z,
+where βf2 = Ψf2
+k2βf1 + ψf2
+k2 and γf2 = Ψf2
+k2γf1.
+A.4
+Experimental details.
+Methods for comparison.
+We compared our proposed method with the following approaches:
+• Naive: the classical z-test is used to calculate the naive p-value.
+• Bonferroni: the number of all possible hypotheses are considered to account for the selection
+bias. The p-value is computed by pbonferroni = min(1, pnaive ∗ 2n)
+• Over-conditioning (OC): additionally conditioning on the observed activeness and inactiveness
+of all the nodes. The limitation of this method is its low statistical power due to over-conditioning.
+Network structure.
+In all the experiments, we used the network structure shown in Fig. 8.
+Experimental setting on brain image dataset.
+We examine the brain image dataset extracted
+from the dataset used in Buda et al. [2019], which includes 941 images without tumors (C1) and 939
+images with tumors (C2). We selected 50 images from C1 as reference images. We used 841 images
+from C1 and 889 images from C2 for DNN training. The remaining images from C1 and C2 are used
+for demonstrating the advantages of the proposed selective p-value.
+More results on brain image dataset.
+Additional results are shown in Figs. 11, 12, 9 and 10
+20
+
+Conv
+MaxPooling
+GAP
+UpSampling
+Image
+Saliency Map
+Prediction
+FC
+Weight
+( 𝑛, 𝑛, 1)
+( 𝑛, 𝑛, 4)
+( 𝑛/2, 𝑛/2,4)
+(4)
+(1)
+( 𝑛, 𝑛, 4)
+( 𝑛/2, 𝑛/2,4)
+CAM
+( 𝑛, 𝑛, 4)
+( 𝑛, 𝑛, 4)
+Figure 8: Network structure.
+21
+
+Input Image
+Saliency Map
+Salient Region
+Reference Image
+Reference Region
+(a) pnaive = 0.01, pselective = 0.47
+Figure 9: Inference on salient regions for images without tumor (mean null test).
+Input Image
+Saliency Map
+Salient Region
+Reference Image
+Reference Region
+(a) pnaive = 0.00, pselective = 2.82 × 10−4
+Figure 10: Inference on salient regions for images where there exists a tumor (mean null test).
+Input Image
+Saliency Map
+Salient Region
+Reference Image
+Reference Region
+(a) pnaive = 3.00 × 10−4, pselective = 0.29
+Figure 11: Inference on salient regions for images without tumor (global null test).
+Input Image
+Saliency Map
+Salient Region
+Reference Image
+Reference Region
+(a) pnaive = 0.00, pselective = 2.66 × 10−20
+Figure 12: Inference on salient regions for images where there exists a tumor (global null test).
+22
+
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+page_content='Valid P-Value for Deep Learning-Driven Salient Region  Daiki Miwa∗  Nagoya Institute of Technology  miwa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='daiki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='mllab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='nit@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='com  Vo Nguyen Le Duy∗  RIKEN  duy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='mllab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='nit@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='com  Ichiro Takeuchi†  Nagoya University and RIKEN  ichiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='takeuchi@mae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='nagoya-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='jp  January 9, 2023  Abstract  Various saliency map methods have been proposed to interpret and explain predictions of  deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Saliency maps allow us to interpret which parts of the input signals have a  strong influence on the prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' However, since a saliency map is obtained by complex  computations in deep learning models, it is often difficult to know how reliable the saliency map  itself is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' In this study, we propose a method to quantify the reliability of a salient region in the  form of p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Our idea is to consider a salient region as a selected hypothesis by the trained  deep learning model and employ the selective inference framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The proposed method can  provably control the probability of false positive detections of salient regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' We demonstrate  the validity of the proposed method through numerical examples in synthetic and real datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Furthermore, we develop a Keras-based framework for conducting the proposed selective inference  for a wide class of CNNs without additional implementation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  ∗Equal contribution  †Corresponding author  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='02437v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='ML]  6 Jan 2023  1  Introduction  Deep neural networks (DNNs) have exhibited remarkable predictive performance in numerous practical  applications in various domains owing to their ability to automatically discover the representations  needed for prediction tasks from the provided data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' To ensure that the decision-making process of  DNNs is transparent and easy to understand, it is crucial to effectively explain and interpret DNN  representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  For example, in image classification tasks, obtaining salient regions allows us to  explain which parts of the input image strongly influence the classification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Several saliency map methods have been proposed to explain and interpret the predictions of DNN  models [Ribeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2016, Bach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2015, Doshi-Velez and Kim, 2017, Lundberg and Lee, 2017,  Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2016, Selvaraju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' However, the results obtained from saliency methods are  fragile [Kindermans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2017, Ghorbani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2019, Melis and Jaakkola, 2018, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2020,  Dombrowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2019, Heo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' It is important to develop a method for quantifying the  reliability of DNN-driven salient regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Our idea is to interpret salient regions as hypotheses driven by a trained DNN model and employ  a statistical hypothesis testing framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' We use the p-value as a criterion to quantify the statistical  reliability of the DNN-driven hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Unfortunately, constructing a valid statistical test for DNN-  driven salient regions is challenging because of the selection bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' In other words, because the trained  DNN selects the salient region based on the provided data, the post-selection assessment of importance  is biased upwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  To correct the selection bias and compute valid p-values for DNN-driven salient regions, we intro-  duce a conditional selective inference (SI) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The selection bias is corrected by conditional SI  in which the test statistic conditional on the event that the hypotheses (salient regions) are selected  using the trained DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Our main technical contribution is to develop a computational method for  explicitly deriving the exact (non-asymptotic) conditional sampling distribution of the salient region  for a wide class convolutional neural networks (CNNs), which enables us to conduct conditional SI  and compute valid p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Figure 1 presents an example of the problem setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  In this study, we focus on statistical hypothesis testing for post-hoc analysis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=',  quantifying the statistical significance of the salient regions identified in a trained DNN model when  a test input instance is fed into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Several methods have been developed to visualize and  understand trained DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Many of these post-hoc approaches [Mahendran and Vedaldi, 2015, Zeiler  and Fergus, 2014, Dosovitskiy and Brox, 2016, Simonyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2013] have focused on developing  visualization tools for saliency maps given a trained DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Other methods have aimed to identify  the discriminative regions in an input image given a trained network [Selvaraju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2017, Fong  2  Input Image  Saliency Map  Salient Region  Reference Image  Two-sample  Test  (a) Image without tumor region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The naive-p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='00 (wrong detection) and selective-p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='43 (true negative)  Input Image  Saliency Map  Salient Region  Reference Image  Two-sample  Test  (b) Image with tumor region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The naive-p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='00 (true positive) and selective-p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='00 (true positive)  Figure 1: Examples of the problem setup and the proposed method on brain tumor dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' By  applying a saliency method called CAM [Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2016] on a query input image, we obtain the  salient region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Our goal is to provide the statistical significance of the salient region in the form of  p-value by considering two-sample test between the salient region and the corresponding region in  the a reference image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Note that, since the salient region is selected based on the data, the degree  of saliency in the selected region is biased upward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' In the upper image where there is no true brain  tumor, the naive p-value which is obtained without caring the selection bias is nearly zero, indicating  the false positive finding of the salient region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  On the other hand, the selective p-value which is  obtained by the proposed conditional SI approach is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='43, indicating that the selected saliency region  is not statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' In the lower figure where there is a true brain tumor, both the naive p-  value and the selective p-value are very small, indicating true positive finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' These results illustrate  that naive p-value cannot be used to quantify the reliability of DNN-based salient region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' In contrast,  with the selective p-values, we can successfully identify false positive and true positive detections with  a desired error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  and Vedaldi, 2017, Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2016, Lundberg and Lee, 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Furthermore, some recent studies  have shown that many popular methods for explanation and interpretation are not stable against a  perturbation or adversarial attack on the input data and model [Kindermans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2017, Ghorbani  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2019, Melis and Jaakkola, 2018, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2020, Dombrowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2019, Heo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=',  2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' However, to the best of our knowledge, no study to date has quantitatively evaluated and  reproducibility of DNN-driven salient regions with a rigorous statistical inference framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Recently, conditional SI has been recognized as a promising new approach for evaluating the  3  27statistical significance of data-driven hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Conditional SI has been mainly studied for inference  of linear model features selected by a feature selection method such as Lasso [Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2016, Liu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2018, Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2018, Le Duy and Takeuchi, 2021] and stepwise feature selection [Tibshirani  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2016, Sugiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2021a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The main idea of conditional SI study is to make inferences  conditional on selection events, which allows us to derive exact sampling distributions of test statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  In addition, conditional SI has been applied to various problems [Fithian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2015, Tian and Taylor,  2018, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2016, Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2021, Duy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2020, Sugiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2021b, Chen and Bien,  2019, Panigrahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2016, Tsukurimichi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2021, Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2018, Tanizaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2020, Duy  and Takeuchi, 2021, Tibshirani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2016, Sugiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2021a, Suzumura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2017, Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=',  2021, Duy and Takeuchi, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Most relevant existing work of this study is Duy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' [2022], where the authors provide a framework  for computing valid p-values for DNN-based image segmentation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' In this paper, we generalized  this work so that hypotheses characterized by any internal nodes of the network can be considered,  enabling us to quanfity the statistical significance of salient regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  This is in contrast to Duy  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' [2022]’s work, which only considered the inference of the DNN’s output in a segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Furthermore, we introduce a Keras-based implementation framework that enables us to conduct SI for  a wide class of CNNs without additional implementation costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' This is in contrast to Duy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' [2022]’s  work, where the selection event must be implemented whenever the network architecture is changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  In another direction, Burns et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' [2020] considered the black box model interpretability as a multiple-  hypothesis testing problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' They aimed to deduce important features by testing the significance of the  difference between the model prediction and what would be expected when replacing the features with  their counterfactuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The difficulty of this multiple-hypothesis testing approach is that the number  of hypotheses to be considered is large (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', in the case of an image with n pixels, the number of  possible salient regions is 2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Multiple testing correction methods, such as the Bonferroni correction,  are highly conservative when the number of hypotheses is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' To circumvent this difficulty, they  only considered a tractable number of regions selected by a human expert or object detector, which  causes selection bias because these candidate regions are selected based on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Our main contributions are as follows:  We provide an exact (non-asymptotic) inference method for salient regions based on the SI  concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' To the best of our knowledge, this is the first method that proposes to provide valid p-values  to statistically quantify the reliability of DNN-driven salient regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  We propose a novel algorithm and its implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Specifically, we propose Keras-based  implementation enables us to conduct conditional SI for a wide class of CNNs without additional  implementation costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  4  We conducted experiments on both synthetic and real-world datasets, through which we show  that our proposed method can successfully control the false positive rate, has good performance in  terms of computational efficiency, and provides good results in practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' We provide the  detailed description of our implementation in the supplementary document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Our code is available at  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='com/takeuchi-lab/selective inference dnn salient region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  2  Problem Formulation  In this paper, we consider the problem of quantifying the statistical significance of the salient regions  identified by a trained DNN model when a test input instance is fed into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Consider an  n-dimensional query input vector  X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', Xn)⊤ = s + ε,  ε ∼ N(0, σ2In)  and an n-dimensional reference input vector,  Xref = (Xref  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', Xref  n )⊤ = sref + εref,  εref ∼ N(0, σ2In),  where s, sref ∈ Rn are the signals and ε, εref ∈ Rn are the noises for query and reference input vectors,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' We assume that the signals, s and sref are unknown, whereas the distribution of noises  ε and εref are known (or can be estimated from external independent data) to follow N(0, σ2In),  an n-dimensional normal distribution with a mean vector 0 and covariance matrix σ2In, which are  mutually independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' In the illustrative example presented in §1, X is a query brain image for a  potential patient (we do not know whether she/he has a brain tumor), whereas Xref is a brain image  of a healthy person known to be without brain tumors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Consider a saliency method for a trained CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' We denote the saliency method as a function  A : Rn → Rn that takes a query input vector X ∈ Rn and returns the saliency map A(X) ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' We  define a salient region MX for the query input vector X as the set of elements whose saliency map  value is greater than a threshold  MX = {i ∈ [n] : Ai(X) ≥ τ} ,  (1)  where τ ∈ R denotes the given threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' In this study, we consider CAM [Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2016] as an  example of saliency method and threshold-based definition of the salient region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Our method can be  applied to other saliency methods and other definition of salient region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Statistical inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  To quantify the statistical significance of the saliency region MX, we con-  sider such two-sample test to quantify the statistical significance of the difference between the salient  5  regions of the query input vector XMX and corresponding region of the reference input vector Xref  MX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  As concrete examples of the two-sample test, we consider the mean null test:  H0 :  1  |MX|  �  i∈MX  si =  1  |MX|  �  i∈MX  sref  i  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  H1 :  1  |MX|  �  i∈MX  si ̸=  1  |MX|  �  i∈MX  sref  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  (2)  and global null test:  H0 : si = sref  i , ∀i ∈ MX,  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  H1 : si ̸= sref  i , ∃i ∈ MX,  (3)  In the mean null test depicted in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' (2), we consider a null hypothesis that the average signals in the  salient region MX are the same between X and Xref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' In contrast, in the global null test in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' (3),  we consider a null hypothesis that all elements of the signals in the salient region MX are the same  between X and Xref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The p-values for these two-sample tests can be used to quantify the statistical  significance of the salient region MX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Test-statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  For a two-sample test conducted between XMX and Xref  MX, we consider a class of  test statistics called conditionally linear test-statistic, which is expressed as  T(X, Xref) = η⊤  MX  � X  Xref  �  ,  (4)  and conditionally χ test-statistic, which is expressed as  T(X, Xref) = σ−1  ����PMX  � X  Xref  ����� ,  (5)  where ηMX ∈ R2n is a vector and PMX ∈ R2n×2n is a projection matrix that depends on saliency  region MX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  The test statistics for the mean null tests and the global null test can be written in the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  (4) and (5), respectivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' For the mean null test in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' (2), we consider the following test-statistic  T(X, Xref) = η⊤  MX  � X  Xref  �  =  1  |MX|  �  i∈MX  Xi −  1  |MX|  �  i∈MX  Xref  i  ,  where ηMX =  1  |MX|  �  � 1n  MX  −1n  MX  �  � ∈ R2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' For the gloabl null test in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' (3), we consider the following  test-statistic  T(X, Xref) = σ−1  ����PMX  � X  Xref  ����� =  �  �  �  � �  i∈MX  �Xi − Xref  i  √  2σ  �2  ,  where  PMX = 1  2  �  � diag(1n  MX)  −diag(1n  MX)  −diag(1n  MX)  diag(1n  MX)  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  (6)  6  To obtain p-values for these two-sample tests we need to know the sampling distribution of the  test-statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Unfortunately, it is challenging to derive the sampling distributions of test-statistics  because they depend on the salient region MX, which is obtained through a complicated calculation  in the trained CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  3  Computing Valid p-value by Conditional Selective Inference  In this section, we introduce an approach to compute the valid p-values for the two-sample tests for  the salient region MX between the query input vector X and the reference input vector Xref based  on the concept of conditional SI [Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='1  Conditional Distribution and Selective p-value  Conditional distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  The basic idea of conditional SI is to consider the sampling distribution  of the test-statistic conditional on a selection event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Specifically, we consider the sampling property  of the following conditional distribution  T(X, Xref)  ��� {MX = MXobs} ,  (7)  where Xobs is the observation (realization) of random vector X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' (7) indicates the  randomness of X conditional on the event that the same salient region MX as the observed MXobs  is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' By conditioning on the salient region MX, derivation of the sampling distribution of the  conditionally linear and χ test-statistic T(X, Xref) is reduced to a derivation of the distribution of  linear function and quadratic function of (X, Xref), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Selective p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  After considering the conditional sampling distribution in (7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' we introduce the  following selective p-value:  pselective = PH0  � ��T(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Xref)  �� ≥  ��T(Xobs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Xref  obs)  ��  ��� MX = MXobs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' QX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='Xref = Qobs  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  (8)  where  QX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='Xref = ΩX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='Xref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Qobs = QXobs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='Xref  obs  with  ΩX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='Xref =  �  I2n − ηMXη⊤  MX  ∥ηMX∥2  � � X  Xref  �  ∈ R2n  in the case of mean null test,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' and  QX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='Xref =  �  VX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='Xref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' UX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='Xref  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Qobs = QXobs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='Xref  obs  7  with  VX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='Xref = σPMX  � X  Xref  ������PMX  � X  Xref  ����� ∈ R2n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  UX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='Xref = P ⊥  MX  � X  Xref  �  ∈ R2n  in the case of global null test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The QX,Xref is the sufficient statistic of the nuisance parameter that  needs to be conditioned on in order to tractably conduct the inference 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  The selective p-value in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' (8) has the following desired sampling property  PH0  �  pselective ≤ α | MX = MXobs  �  = α,  ∀α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  (9)  This means that the selective p-values pselective can be used as a valid statistical significance measure  for the salient region MX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='2  Characterization of the Conditional Data Space  To compute the selective p-value in (8), we need to characterize the conditional data space whose  characterization is described introduced in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' We define the set of (X Xref)⊤ ∈ R2n  that satisfies the conditions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' (8) as  D =  �  (X Xref)⊤ ∈ R2n �� MX = MXobs, QX,Xref = Qobs  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  (10)  According to the second condition, the data in D is restricted to a line in R2n as stated in the following  Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Let us define let us define,  a = ΩXobs,Xref  obs  and  b =  ηMX  ∥ηMX∥2 ∈ R2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  (11)  in the mean null test, and  a = UXobs,Xref  obs  and  b = VXobs,Xref  obs  (12)  in the case of global null test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Then, the set D in (10) can be rewritten as D =  ��  X Xref�⊤ = a+bz |  z ∈ Z  �  by using the scalar parameter z ∈ R, where  Z = {z ∈ R | Ma1:n+b1:nz = MXobs} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  (13)  x1:n represents a vector of elements 1 through n of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  1This nuisance parameter QX,Xref corresponds to the component z in the seminal conditional SI paper [Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=',  2016] (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' 5, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='2 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='2) and z, w in [Chen and Bien, 2019](see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' 3, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' We note that  additional conditioning on QX,Xref is a standard approach in the conditional SI literature and is used in almost all  the conditional SI-related studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Here, we would like to note that the selective p-value depend on QX,Xref , but the  property in (9) is satisfied without this additional condition because we can marginalize over all values of QX,Xref (see  the lower part of the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='2 in Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' [2016] and the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='7 in Chen and Bien [2019] ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  8  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The proof is deferred to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='1  ■  Lemma 1 indicates that we do not need to consider the 2n-dimensional data space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Instead, we  only need to consider the one-dimensional projected data space Z in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Now, let us consider  a random variable Z ∈ R and its observation Zobs ∈ R that satisfies (X Xref)⊤ = a + bZ and  (Xobs Xref  obs)⊤ = a + bZobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The selective p-value (8) is rewritten as  pselective = PH0 (|Z| ≥ |Zobs| | Z ∈ Z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  (14)  Because the variable Z ∼ N(0, σ2∥η∥2) in the case of mean null test and Z ∼ χ (Trace(P)) in the case  of global null test under the null hypothesis, Z | Z ∈ Z follows a truncated normal distribution and a  truncated χ distribution, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Once the truncation region Z is identified, computation of the  selective p-value in (14) is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Therefore, the remaining task is to identify Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  In general, computation of Z in (13) is difficult because we need to identify the selection event  Ma1:n+b1:nz for all values of z ∈ R, which is computationally challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' In the next section, we  show that the challenge can be resolved under a wide class of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  4  Piecewise Linear Network  The problem of computing selective p-values for the selected salient region is casted into the problem  of identifying a set of intervals Z = {z ∈ R | MX(z) = MXobs}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Given the complexity of saliency  computation in a trained DNN, it seems difficult to obtain Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' In this section, however, we explain  that this is feasible for a wide class of CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Piecewise linear components in CNN  The key idea is to note that most of basic operations and  common activation functions used in a trained CNN can be represented as piecewise linear functions  in the following form:  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' (Piecewise Linear Function) A piecewise linear function f : Rn �→ Rm is defined as:  f(X) =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  Ψf  1X + ψf  1 ,  if X ∈ Pf  1 := {X′ ∈ Rn | ∆f  1X′ ≤ δf  1 },  Ψf  2X + ψf  2 ,  if X ∈ Pf  2 := {X′ ∈ Rn | ∆f  2X′ ≤ δf  2 },  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Ψf  K(f)X + ψf  K(f),  if X ∈ Pf  K(f) := {X′ ∈ Rn | ∆f  K(f)X′ ≤ δf  K(f)}  where Ψf  k, ψf  k, ∆f  k and δf  k for k ∈ [K(f)] are certain matrices and vectors with appropriate dimensions,  Pf  k := {x ∈ Rn | ∆f  kx ≤ δf  k} is a polytope in Rn for k ∈ [K(f)], and K(f) is the number of polytopes  for the function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  9  Examples of piecewise linear components in a trained CNN are shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Piecewise Linear Network  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' (Piecewise Linear Network) A network obtained by concatenations and compositions  of piecewise linear functions is called piecewise linear network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Since the concatenation and the composition of piecewise linear functions is clearly piecewise linear  function, the output of any node in the piecewise linear network is written as a piecewise linear function  of an input vector X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' This is also true for the saliency map function Ai(X), i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Furthermore,  as discussed in §4, we can focus on the input vector in the form of X(z) = a1:n + b1:nz which is  parametrized by a scalar parameter z ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Therefore, the saliency map value for each element is  written as a piecewise linear function of the scalar parameter z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=',  Ai(X(z)) =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  κAi  1 z + ρAi  1 ,  if z ∈ [LAi  1 , U Ai  1 ],  κAi  2 z + ρAi  2 ,  if z ∈ [LAi  2 , U Ai  2 ],  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  κAi  K(Ai)z + ρf  K(Ai),  if z ∈ [LAi  K(Ai), U Ai  K(Ai)],  (15)  where K(Ai) is the number of linear pieces of the piecewise linear function, κAi  k , ρAi  k  are certain scalar  parameters, [LAi  k , U Ai  k ] are intervals for k ∈ [K(Ai)] (note that a polytope in Rn is reduced to an  interval when it is projected onto one-dimensional space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  This means that, for each piece of the piecewise linear function, we can identify the interval of z  such that Ai(X(z)) ≥ τ as follows 2  z ∈  �  �  �  �  max  �  LAi  k ,  �  τ − ρAi  k  �  /κAi  k  �  , U Ai  k  �  if κAi  k  > 0  �  LAi  k , min  �  U Ai  k ,  �  τ − ρAi  k  �  /κAi  k  �  ,  �  if κAi  k  < 0  ⇒  Ai(X(z)) ≥ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  (16)  With a slight abuse of notation, let us collectively denote the finite number of intervals on z ∈ R that  are defined by LAi  k , U Ai  k , (τ − ρAi  i /κAi  k ) for all (k, i) ∈ [K(Ai)] × [n] as  [z0, z1], [z1, z2], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' , [zt−1, zt], [zt, zt+1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' , [zT −1, zT ],  where zmin = z0 and zmax = zT are defined such that the probability mass of z < zmin and z > zmax  are negligibly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  2For simplicity, we omit the description for the case of κAi  k  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' In this case, if ρAi  k  ≥ τ, then z ∈ [LAi  k , UAi  k  ] ⇒ i ∈  MX(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  10  Algorithm 1 SI DNN Saliency  Input: Xobs, zmin, zmax, T ← ∅  1: Obtain Eobs, compute η as well as a and b ← Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' (12), and initialize: t = 1, zt = zmin  2: for t ≤ T do  3:  Compute zt+1 by Auto-Conditioning (see §5)  4:  if EX(z),Xref (z) = Eobs in z ∈ [zt, zt+1] (by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' (16)) then  5:  T ← T + {t}  6:  end if  7:  t = t + 1  8: end for  9: Identify Z ← �  t∈T [zt, zt+1]  10: pselective ← Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' (14)  Output: pselective  Algorithm  Algorithm 1 shows how we identify Z = {z ∈ R | MX(z),Xref(z) = Mobs}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' We simply  check the intervals of z in the order of [z0, z1], [z1, z2], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', [zT −1, zT ] to see whether MX(z) = MX(zobs)  or not in the interval by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='(16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Then, the truncation region Z in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' (13) is given as Z =  �  t∈[T ]|EX(z),Xref (z)=Eobs for z∈[zt,zt+1][zt, zt+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  5  Implementation: Auto-Conditioning  The bottleneck of our algorithm is Line 3 in Algorithm 1, where zt+1 must be found by considering  all relevant piecewise linear components in a complicated trained CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The difficulty lies not only in  the computational cost but also in the implementation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' To implement conditional SI in DNNs  naively, it is necessary to characterize all operations at each layer of the network as selection events and  implement each of the specifically[Duy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2022] To circumvent this difficulty, we introduce a mod-  ular implementation scheme called auto-conditioning, which is similar to auto-differentiation [Baydin  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', 2018] in concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' This enables us to conduct conditional SI for a wide class of CNNs without  additional implementation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  The basic idea in auto-conditioning is to add a mechanism to compute and maintain the interval  z ∈ [Lf  k, U f  k ] for each piecewise linear component f in the network (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', layer API in the Keras  framework).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  This enables us to automatically compute the interval [Lf  k, U f  k ] of a piecewise linear  function f when it is obtained as concatenation and/or composition of multiple piecewise linear  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' If f is obtained by concatenating two piecewise linear functions f1 and f2, we can easily  obtain [Lf  k, U f  k ] = [Lf1  k1, U f1  k1 ] ∩ [Lf2  k2, U f2  k2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' However, if f is obtained as a composition of two piecewise  linear functions f1 and f2, the calculation of the interval is given by the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Consider the composition of two piecewise linear functions, that is, f(X(z)) = (f2 ◦  11  f1)(X(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Given a real value of z, the interval [Lf2  k , U f2  k ] in the input domain of f2 can be computed  as  Lf2  k2 =  max  j:(∆f2  k2γf1)j<0  (δf2  k2)j − (∆f2  k2βf1)j  (∆f2  k2γf1)j  ,  U f2  k2 =  min  j:(∆f2  k2γf1)j>0  (δf2  k2)j − (∆f2  k2βf1)j  (∆f2  k2γf1)j  ,  where βf1 + γf1z is the output of f1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', the input of f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Moreover, ∆f2  k2 and δf2  k2 are obtained by  verifying the value of βf1 + γf1z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Then, the interval of the composite function is obtained as follows:  [Lf  k, U f  k ] = [Lf1  k1, U f1  k1 ] ∩ [Lf2  k2, U f2  k2 ]  The proof is provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Here, the variables βfk and γfk can be recursively  computed through layers as  βfk+1 = Ψfk  k βfk + ψfk  k  and  γfk+1 = Ψfk  k γfk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Lemma 2 indicates that the intervals in which X(z) decreases can be forwardly propagated through  these layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' This means that the lower bound LAi  k  and upper bound U Ai  k  of the current piece in the  piecewise linear function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' (15) can be automatically computed by forward propagation of the  intervals of the relevant piecewise linear components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  6  Experiment  We only highlight the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' More details (methods for comparison, network structure, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=')  can be found in the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  We compared our proposed method with the naive method, over-conditioning  (OC) method, and Bonferroni correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' To investigate the false positive rate (FPR) we consid-  erd, 1000 null images X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', Xn) and 1000 reference images Xref = (Xref  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=', xref  n ), where  s = sref = 0 and ε, εref ∼ N(0, In), for each n ∈ {64, 256, 1024, 4096}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' To investigate the true positive  rate (TPR), we set n = 256 and generated 1,000 images, in which si = signal for any i ∈ S where  S is the “true” salient region whose location is randomly determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' si = 0 for any i ̸∈ S and  ε ∼ N(0, In).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' We set ∆ ∈ {1, 2, 3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Reference images were generated in the same way as in the case  of FPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' In all experiments, we set the threshold for selecting the salient region τ = 0 in the mean  null test and τ = 5 in the global null test .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' We set the significance level α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' We used CAM as  the saliency method in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  The results of FPR control are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The proposed method, OC,  and Bonferroni successfully controlled the FPR in both the mean and global null test cases, whereas  12  64  256  1024  4096  n  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='0  False Positive Rate(FPR)  Proposed  OC  Bonferroni  Naive  (a) Mean null test  64  256  1024  4096  n  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='35  False Positive Rate(FPR)  Proposed  OC  Bonferroni  Naive  (b) Global null test  Figure 2: False Positive Rate (FPR) comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  1  2  3  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='8  True Positive Rate(TPR)  Proposed  OC  Bonferroni  (a) Mean null test  1  2  3  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='0  True Positive Rate(TPR)  Proposed  OC  Bonferroni  (b) Global null test  Figure 3: True Positive Rate (FPR) comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Input Image  Saliency Map  Salient Region  Reference Image  Reference Region  Figure 4: Mean null test for image without tumor (pnaive = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='00, pselective = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  the others could not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Because naive methods failed to control the FPR, we no longer considered their  TPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The results of the TPR comparison are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The proposed method has the highest  TPR in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The Bonferroni method has the lowest TPR because it is conservative owing to  considering the number of all possible hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The OC method also has a low TPR than the  proposed method because it considers several extra conditions, which causes the loss of TPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Real data experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  We examined the brain image dataset extracted from the dataset used  in Buda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' [2019], which included 939 and 941 images with and without tumors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The  results of the mean null test are presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The results of the global null test are  presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The naive p-value remains small even when the image has no tumor region,  which indicates that naive p-values cannot be used to quantify the reliability of DNN-based salient  regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The proposed method successfully identified false positive and true positive detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  7  Conclusion  In this study, we proposed a novel method to conduct statistical inference on the significance of  DNN-driven salient regions based on the concept of conditional SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' We provided a novel algorithm for  efficiently and flexibly conducting conditional SI for salient regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' We conducted experiments on  13  Input Image  Saliency Map  Salient Region  Reference Image  Reference Region  Figure 5: Mean null test for image with a tumor (pnaive = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='00, pselective = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='92 × 10−4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Input Image  Saliency Map  Salient Region  Reference Image  Reference Region  Figure 6: Global null test for image without tumor (pnaive = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='03, pselective = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='46)  Input Image  Saliency Map  Salient Region  Reference Image  Reference Region  Figure 7: Global null test for image with a tumor (pnaive = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='00, pselective = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='51 × 10−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  both synthetic and real-world datasets to demonstrate the performance of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Acknowledgements  This work was partially supported by MEXT KAKENHI (20H00601), JST CREST (JPMJCR21D3),  JST Moonshot R&D (JPMJMS2033-05), JST AIP Acceleration Research (JPMJCR21U2), NEDO  (JPNP18002, JPNP20006), and RIKEN Center for Advanced Intelligence Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
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+page_content=' Fergus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Visualizing and understanding convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' In European  conference on computer vision, pages 818–833.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Springer, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  17  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Zhang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Shen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Ji, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Luo, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Interpretable deep learning under fire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' In  29th {USENIX} Security Symposium ({USENIX} Security 20), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Zhou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Khosla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Lapedriza, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Oliva, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Torralba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Learning deep features for discriminative  localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition,  pages 2921–2929, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  A  Appendix  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='1  Proof of Lemma 1  In the mean null test, according to the second condition in (10), we have  ΩX,Xref = ΩXobs,Xref  obs  ⇔  �  I2n − ηMXη⊤  MX  η⊤  MXηMX  � � X  Xref  �  = ΩXobs,Xref  obs  ⇔  � X  Xref  �  = ΩXobs,Xref  obs +  ηMX  ∥ηMX∥2 η⊤  MX  � X  Xref  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  By defining a = qXobs,Xref  obs, b =  ηMX  ∥ηMX ∥2 , z = η⊤  MX  � X  Xref  �  , we obtain the result in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  In the global null test, according to the second condition in (10),  UX,Xref = UXobs,Xref  obs  ⇔ P ⊥  MX  � X  Xref  �  = UXobs,Xref  obs  ⇔ (I2n − PMX)  � X  Xref  �  = UXobs,Xref  obs  ⇔  � X  Xref  �  = UXobs,Xref  obs + VXobs,Xref  obsσ−1  ����PMX  � X  Xref  ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  By defining a = UXobs,Xref  obs, b = VXobs,Xref  obs, z = σ−1 ���PMX  � X  Xref  ���� , we obtain the result in Lemma  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='2  Examples of piecewise linear functions  Examples of piecewise linear components in a trained CNN with X ∈ R2 are provided as follows:  18  ReLU: Consider f is ReLU function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Then, K(f) = 4, ψk = (0 0)⊤ for any k ∈ [4],  Ψf  1 =  �  �0  0  0  0  �  � , Pf  1 =  �  �  �X : X1 < 0,  X2 < 0  �  �  � ,  Ψf  2 =  �  �0  0  0  1  �  � , Pf  2 =  �  �  �X : X1 < 0,  X2 ≥ 0  �  �  � ,  Ψf  3 =  �  �1  0  0  0  �  � , Pf  3 =  �  �  �X : X1 ≥ 0,  X2 < 0  �  �  � ,  Ψf  4 =  �  �1  0  0  1  �  � , Pf  4 =  �  �  �X : X1 ≥ 0,  X2 ≥ 0  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  This can be similarly extended to the case of Leaky ReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Max-pooling: Consider f(X) = max{X1, X2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Then, it is represented as a piecewise linear function  with K(f) = 2, ψk = (0) for any k ∈ [2],  Ψf  1 =  �  1  0  �  , Pf  1 = {X : X1 ≥ X2} ,  Ψf  2 =  �  0  1  �  , Pf  2 = {X : X1 < X2} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Convolution and matrix-vector multiplication: In a neural network, the multiplication results between  the weight matrix and the output of the previous layer and its summation with the bias vector is a  linear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' In a CNN, the convolution operation is obviously a linear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Upsampling: Consider f is the upsampling operation on X ∈ R2, then it can be represented as a  piecewise linear function with K(f) = 1, ψ1 = (0 0 0 0)⊤,  Ψf  1 =  �  �1  1  0  0  0  0  1  1  �  �  ⊤  ,  Pf  1 = R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Sigmoid and hyperbolic tangent: If there is any specific demand to use non-piecewise linear activation  functions, we can apply a piecewise-linear approximation approach to these functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='3  Proof of Lemma 2  At f1, given a a real value z, the input is βf0 + γf0z = a1:n + b1:nz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' By checking the value of this  input, we can easily obtain the polytope  {∆f1  k1(βf0 + γf0z) ≤ δf1  k1},  k1 ∈ [K(f1)],  that βf0 + γf0z belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Based on the obtained polytope, we can calculate the interval [Lf1  k1, U f1  k1 ],  Lf1  k1 =  max  j:(∆f1  k1γf0)j<0  (δf1  k1)j − (∆f1  k1βf0)j  (∆f1  k1γf0)j  and  U f1  k1 =  min  j:(∆f1  k1γf0)j>0  (δf1  k1)j − (∆f1  k1βf0)j  (∆f1  k1γf0)j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Moreover, based on the obtained polytope, we can easily obtain Ψf1  k1 and ψf1  k1, k1 ∈ [K(f1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Therefore,  the output of the first layer at z can be defined as  f1(z) = Ψf1  k1(βf0 + γf0z) + ψf1  k1  = βf1 + γf1z,  19  where βf1 = Ψf1  k1βf0 + ψf1  k1 and γf1 = Ψf1  k1γf0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Next, we input βf1, γf1 to f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  At the 2nd layer, similarly, the input is βf1 + γf1z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' By checking the value of this input, we can  easily obtain the polytope  {∆f2  k2(βf1 + γf1z) ≤ δf2  k2},  k2 ∈ [K(f2)],  that βf1 + γf1z belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Based on the obtained polytope, we can calculate the interval [Lf2  k2, U f2  k2 ],  Lf2  k2 =  max  j:(∆f2  k2γf1)j<0  (δf2  k2)j − (∆f2  k2βf1)j  (∆f2  k2γf1)j  and  U f2  k2 =  min  j:(∆f2  k2γf1)j>0  (δf2  k2)j − (∆f2  k2βf1)j  (∆f2  k2γf1)j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Moreover, based on the obtained polytope, we can easily obtain Ψf2  k2 and ψf2  k2, k2 ∈ [K(f2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' Therefore,  the output of the first layer at z can be defined as  f2(z) = Ψf2  k2(βf1 + γf1z) + ψf2  k2  = βf2 + γf2z,  where βf2 = Ψf2  k2βf1 + ψf2  k2 and γf2 = Ψf2  k2γf1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='4  Experimental details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Methods for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  We compared our proposed method with the following approaches:  Naive: the classical z-test is used to calculate the naive p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Bonferroni: the number of all possible hypotheses are considered to account for the selection  bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The p-value is computed by pbonferroni = min(1, pnaive ∗ 2n)  Over-conditioning (OC): additionally conditioning on the observed activeness and inactiveness  of all the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The limitation of this method is its low statistical power due to over-conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  In all the experiments, we used the network structure shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Experimental setting on brain image dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  We examine the brain image dataset extracted  from the dataset used in Buda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' [2019], which includes 941 images without tumors (C1) and 939  images with tumors (C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' We selected 50 images from C1 as reference images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' We used 841 images  from C1 and 889 images from C2 for DNN training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' The remaining images from C1 and C2 are used  for demonstrating the advantages of the proposed selective p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  More results on brain image dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Additional results are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content=' 11, 12, 9 and 10  20  Conv  MaxPooling  GAP  UpSampling  Image  Saliency Map  Prediction  FC  Weight  ( 𝑛, 𝑛, 1)  ( 𝑛, 𝑛, 4)  ( 𝑛/2, 𝑛/2,4)  (4)  (1)  ( 𝑛, 𝑛, 4)  ( 𝑛/2, 𝑛/2,4)  CAM  ( 𝑛, 𝑛, 4)  ( 𝑛, 𝑛, 4)  Figure 8: Network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  21  Input Image  Saliency Map  Salient Region  Reference Image  Reference Region  (a) pnaive = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='01, pselective = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='47  Figure 9: Inference on salient regions for images without tumor (mean null test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Input Image  Saliency Map  Salient Region  Reference Image  Reference Region  (a) pnaive = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='00, pselective = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='82 × 10−4  Figure 10: Inference on salient regions for images where there exists a tumor (mean null test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Input Image  Saliency Map  Salient Region  Reference Image  Reference Region  (a) pnaive = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='00 × 10−4, pselective = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='29  Figure 11: Inference on salient regions for images without tumor (global null test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  Input Image  Saliency Map  Salient Region  Reference Image  Reference Region  (a) pnaive = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='00, pselective = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='66 × 10−20  Figure 12: Inference on salient regions for images where there exists a tumor (global null test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
+page_content='  22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE0T4oBgHgl3EQfhwHS/content/2301.02437v1.pdf'}
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+arXiv:2301.12007v1  [math.OC]  27 Jan 2023
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC
+OPTIMIZATION PROBLEMS
+POUYA SAMPOURMAHANI∗, MOHAMMADHOSSEIN MOHAMMADISIAHROUDI, TAM ´AS TERLAKY
+Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA, USA, 18015
+Abstract. Second-order conic optimization (SOCO) can be considered as a special case of semidefinite optimiza-
+tion (SDO). In the literature it has been advised that a SOCO problem can be embedded in an SDO problem using
+the arrow-head matrix transformation. However, a primal-dual solution pair cannot be mapped simultaneously
+using the arrow-head transformation as we might lose complementarity and duality in some cases. To address this
+issue, we investigate the relationship between SOCO problems, and their SDO counterpart. Through derivation of
+standard semidefinite representations of SOCO problems, we introduce admissible mappings. We show that the
+proposed mappings preserve both feasibility and optimality. Further, we discuss how the optimal partition of a
+SOCO problem maps to the optimal partition of its SDO counterpart.
+Keywords. Second-order conic Optimization; Semidefinite Optimization; Semidefinite Representation; Mapping;
+Optimal Partition.
+2020 Mathematics Subject Classification. 90C25, 90C22, 90C99.
+1. INTRODUCTION
+In the hierarchy of convex optimization problems, second-order conic optimization (SOCO)
+problems can be seen as a special case of semidefinite optimization (SDO) problems. SOCO
+problems minimize a linear function over the intersection of an affine space with the Cartesian
+product of second-order cones, also known as Lorentz cones. An SDO problem consists of
+minimizing a linear objective function over the intersection of the cone of positive semidefinite
+matrices with an affine space. SDO encompasses other subclasses of conic optimization prob-
+lems namely linear optimization (LO), and SOCO, in the hierarchy. This means that each one
+can be represented as a special case of SDO [1].
+In this paper, we focus on the relationship of SOCO and SDO. We investigate their relation-
+ship in order to gain theoretical insight and realize how these problems get mapped to each
+other. Only a few papers [9, 11] were devoted to study this relationship from a theoretical point
+of view. Sim and Zhao [9], in particular, studied the relationship between a SOCO problem
+and its counterpart SDO problem. They provided a mapping based on the direct correspon-
+dence between the dual problems of SOCO and SDO. Their SDO representation is defined on
+the product of some cones of positive semidefinite matrices, which is a special case of standard
+∗Corresponding author.
+E-mail address: pos220@lehigh.edu (P. Sampourmahani), mom219@lehigh.edu (M. Mohammadisiahroudi),
+terlaky@lehigh.edu (T. Terlaky).
+Received xx, x, xxxx; Accepted xx, x, xxxx.
+©2023 Communications in Optimization Theory
+1
+
+2
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+SDO and needs further analysis. In this paper, we extend their analysis by considering the actual
+standard case which returns an SDO representation through a large positive semidefinite cone.
+Furthermore, we propose a framework that allows full description of the point-to-set map
+from SOCO to its SDO counterpart. Then, we analyze how the optimal partition of a SOCO
+problem is mapped to that of SDO, and vice versa. This is important in understanding the
+relationship between these two problems as we are mapping between an index-based partition
+and a subspace-based partition.
+Throughout this paper, the following notation is used. The Lorentz cone of dimension ni
+is denoted by L ni, and Rn denotes the n-dimensional Euclidean space. Superscripts are used
+to represent cone-related information, and subscripts are used for matrix and vector entries.
+For a given matrix A, Ai j represents the (i, j)-th entry, while Ai denotes the i-th matrix. The
+notation (.;.;...;.) denotes the concatenation of the column vectors. The set of all p×q matrices
+with real entries is denoted by Rp×q. For a symmetric matrix X, X ⪰ 0 (X ≻ 0) means X is
+positive semidefinite (positive definite). Furthermore, the trace operator is denoted by Tr(.).
+The remaining notations will be introduced at appropriate places.
+This paper is structured as follows. Section 2 reviews the preliminaries required for this pa-
+per. Section 3 studies the relationship between SOCO and SDO relying on the correspondence
+of dual problems. Section 4 takes the other direction and proposes mappings focusing on corre-
+spondence of primal problems as the starting point. Section 5 analyzes how the optimal partition
+of SOCO maps to that of it’s SDO counterpart. Section 6 concludes the paper, summarizing our
+results.
+2. PRELIMINARIES
+Let L ni denote the Lorentz cone of dimension ni, and L n = L n1 ×L n2 ×...×L nr, where
+n = ∑r
+i=1 ni. Then, the primal and dual SOCO problems are defined as follows
+z∗
+PSOCO := min (c1)Tx1 +...+(cr)Txr
+s.t. A1x1 +...+Arxr = b,
+xi ∈ L ni,
+for i = 1,...,r,
+(PSOCO)
+z∗
+DSOCO := max bTy
+s.t. (Ai)Ty+si = ci,
+for i = 1,...,r,
+si ∈ L ni,
+for i = 1,...,r,
+(DSOCO)
+where ci ∈ Rni, Ai ∈ Rm×ni, b ∈ Rm. We define the feasible sets of the primal-dual problems as
+follows,
+FPSOCO = {(x1;x2;...;xr) ∈ L n : A1x1 +...+Arxr = b},
+FDSOCO = {(y;s1;s2;...;sr) ∈ Rm ×L n : (Ai)Ty+si = ci for i = 1,...,r},
+and the sets of optimal solutions as
+P∗
+SOCO = {x∗ = (x1;x2;...;xr) ∈ FPSOCO : cTx∗ = z∗
+PSOCO},
+D∗
+SOCO = {(y∗,s∗) = (y;s1;s2;...;sr) ∈ FDSOCO : bTy∗ = z∗
+DSOCO},
+respectively. An optimal solution of SOCO, if there exists any, is denoted by (x∗;y∗;s∗).
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+3
+Let Arw(·) denote the arrow-head (Lorentz) transformation [8, 10], with the structure of
+Arw(xi) :=
+� xi
+1
+(xi
+2:ni)T
+xi
+2:ni
+xi
+1Ini−1
+�
+,
+where (xi
+2:ni)T denotes the vector (xi
+2,...,xi
+ni). Then, the Jordan product is defined as
+xi ◦si = Arw(xi)si = Arw(si)xi =
+�
+(xi)Tsi
+xi
+1si
+2:ni +si
+1xi
+2:ni
+�
+,
+i = 1,...,r.
+Any feasible solutions satisfying x◦s = 0 is called complementary. Here, we have
+x◦s := (x1 ◦s1,x2 ◦s2,...,xr ◦sr).
+Feasible solution are complementary if and only if they are optimal with zero duality gap.
+Definition 2.1. An optimal solution (x∗;y∗;s∗) ∈ P∗
+SOCO × D∗
+SOCO is called maximally com-
+plementary if x∗ ∈ ri(P∗
+SOCO) and (y∗;s∗) ∈ ri(D∗
+SOCO). Further, (x∗;y∗;s∗) is called strictly
+complementary if x∗ +s∗ ∈ int(L n).
+Next, we define the primal and dual SDO problems.
+z∗
+PSDO := min Tr(CX)
+s.t. Tr(AiX) = bi
+for all i = 1,...,m,
+X ⪰ 0,
+(PSDO)
+z∗
+PSDO := max bTy
+s.t.
+m
+∑
+i=1
+yiAi +S = C,
+S ⪰ 0,
+(DSDO)
+where X,S,C, and Ai for i = 1,...,m are n×n symmetric matrices, and b,y ∈ Rm. We define
+the feasible sets of SDO problems as
+FPSDO = {X ∈ Sn : Tr(AiX) = bi,i = 1,...,m,X ⪰ 0},
+FDSDO = {(y,S) ∈ Rm ×Sn :
+m
+∑
+i=1
+yiAi +S = C,S ⪰ 0},
+where Sn denotes the set of n ×n symmetric matrices. The sets of optimal solutions for a pair
+of SDO problems are
+P∗
+SDO = {X∗ ∈ FPSDO : Tr(CX∗) = z∗
+PSDO},
+D∗
+SDO = {(y∗,S∗) ∈ FDSDO : bTy∗ = z∗
+DSDO}.
+A feasible and an optimal solution of SDO are denoted as (X,y,S), and (X∗,y∗,S∗), respectively.
+Any feasible solution (X,y,S) satisfying XS = 0 is called complementary. Similar to SOCO, a
+feasible solution is optimal and yields zero duality gap if and only if it is complementary.
+Definition 2.2. A primal-dual optimal solution (X∗,y∗,S∗) ∈ P∗
+SDO × D∗
+SDO is called maxi-
+mally complementary if X∗ ∈ ri(P∗
+SDO) and (y∗,S∗) ∈ ri(D∗
+SDO). A maximally complementary
+optimal solution (X∗,y∗,S∗) is called strictly complementary if X∗ +S∗ ≻ 0.
+
+4
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+Next, we define the optimal partitions of SOCO and SDO. The notion of the optimal partition
+of LO can be extended to SOCO [8]. Even though a SOCO problem can be embedded in SDO,
+the optimal partition in SOCO may be more nuanced when it is defined and analyzed directly
+in the SOCO setting. In SOCO, the index set {1,...,r} of the second-order cones is partitioned
+into four subsets
+¯
+B,
+¯
+N , ¯
+R, and
+¯
+T , where
+¯
+T is further partitioned to
+¯
+T := ( ¯T1, ¯T2, ¯T3) as
+follows,
+¯
+B := { i | xi
+1 > ||xi
+2:ni||2, for some x ∈ P∗
+SOCO},
+¯
+N := { i | si
+1 > ||si
+2:ni||2, for some s ∈ D∗
+SOCO},
+¯
+R := { i | xi
+1 = ||xi
+2:ni||2 > 0,si
+1 = ||si
+2:ni||2 > 0, for some (x;y;s) ∈ P∗
+SOCO ×D∗
+SOCO},
+¯
+T1 := { i | xi = si = 0, for all (x;y;s) ∈ P∗
+SOCO ×D∗
+SOCO},
+¯
+T2 := { i | si = 0, for all (y;s) ∈ D∗
+SOCO, and xi
+1 = ||xi
+2:ni||2 > 0, for some x ∈ P∗
+SOCO},
+¯
+T3 := { i | xi = 0, for all x ∈ P∗
+SOCO, and si
+1 = ||si
+2:ni||2 > 0, for some (y;s) ∈ D∗
+SOCO}.
+It should be highlighted that, due to the convexity of the optimal set,
+¯
+B,
+¯
+N , ¯
+R, and
+¯
+T are
+mutually disjoint and their union is the index set {1,...,r}. Therefore, it follows from the
+complementarity condition that for all (˜x∗; ˜y∗; ˜s∗) ∈ P∗
+SOCO ×D∗
+SOCO, ˜xi = 0 for all i ∈
+¯
+N , and
+˜si = 0 for all i ∈ ¯
+B [8].
+For SDO, let B := R(X∗) and N := R(S∗), where (X∗,y∗,S∗) is a maximally comple-
+mentary optimal solution, meaning that we have R(X) ⊆ B and R(S) ⊆ N for all (X,y,S) ∈
+P∗
+SDO × D∗
+SDO. By the complementarity condition, the subspaces B and N are orthogonal.
+Moreover, let subspace T , be the orthogonal complement to B+N . The partition (B,N ,T )
+of Rn is called the optimal partition of an SDO problem. We can represent X∗ and S∗ using
+a common eigenvector basis, Q∗, as X∗ = Q∗Λ(X∗)(Q∗)T, and S∗ = Q∗Λ(S∗)(Q∗)T, where
+Λ(X∗) and Λ(S∗) corresponds to the diagonal matrices containing the eigenvalues of X∗ and S∗,
+respectively. Thus, we have R(X∗) = R(Q∗Λ(X∗)), and R(S∗) = R(Q∗Λ(S∗)). In particular,
+the columns of Q∗ corresponding to the positive eigenvalues of X∗ and S∗ can be chosen as an
+orthonormal basis for B and N , respectively [8].
+Current literature [1, 2, 3, 4, 5, 7, 9, 8, 10] suggests that a SOCO problem can be embedded
+in an SDO problem using the arrow-head matrix transformation,
+Arw(xi) :=
+� xi
+1
+(xi
+2:ni)T
+xi
+2:ni
+xi
+1Ini−1
+�
+⪰ 0 ⇔ xi ∈ L ni.
+(2.1)
+However, this transformation cannot be used to map both primal and dual solutions at the same
+time. Upon using the arrow-head representation of vectors xi and si simultaneously, we might
+lose duality and complementarity. The following example illustrates that we may lose comple-
+mentarity.
+Example 2.3. Let (x;y;s) be an optimal solution of SOCO and assume that there exist at least
+one index i ∈ R. For all i ∈ R, we can represent a solution as
+xi = ζ i
+
+
+1
+xi
+2:ni
+||xi
+2:ni||2
+
+,
+si = ξ i
+
+
+1
+si
+2:ni
+||si
+2:ni||2
+
+,
+xi
+2:ni
+||xi
+2:ni||2
+= −
+si
+2:ni
+||si
+2:ni||2
+,
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+5
+where ζ i = xi
+1 ≥ 0, ξ i = si
+1 ≥ 0, and for at least one (x∗;y∗;s∗) we have both ζ i,ξ i > 0. Without
+loss of generality, and for the sake of simplicity, assume that ζ i = ξ i = 1. Moreover, let
+uj =
+xi
+j
+||xi
+2:ni||2
+= −
+si
+j
+||si
+2:ni||2
+,
+j = 2,...,ni,
+and u = (u2;...;uni). Then, using the arrow-head matrix transformation, we have
+Xi = Arw(xi) =
+�1
+uT
+u
+Ini−1
+�
+,
+Si = Arw(si) =
+� 1
+−uT
+−u
+Ini−1
+�
+.
+While xi ◦si = 0, this transformation does not preserve complementarity as we have
+XiSi =
+�1
+uT
+u
+Ini−1
+�� 1
+−uT
+−u
+Ini−1
+�
+=
+�
+0
+[0]1×(n−1)
+[0](n−1)×1
+Ini−1 −uuT
+�
+̸= 0.
+Example 2.3 shows that the arrow-head matrix transformation is not sufficient to represent a
+primal-dual pair of SOCO problems as an SDO problem. Thus, it seems worth exploring the
+actual relationship between an instance of SOCO and it’s SDO counterpart.
+To address this issue, Sim and Zhao [9] started from a SOCO dual problem and exploited the
+arrowhead representation (2.1) of the dual SOCO problem, to obtain the SDO dual as follows,
+max bTy
+s.t.
+m
+∑
+i=1
+yiArw(aj
+(i))+S j = Arw(cj) for all j = 1,...,r,
+S j ⪰ 0 for all j = 1,...,r,
+(DSZ)
+where aj
+(i) denotes ith row of the matrix A corresponding to Lorentz cone j. Observe that S j as a
+linear combination of arrow-head matrices is an arrow-head matrix, too. Using this dual model,
+we get the SDO primal problem as
+min
+r
+∑
+j=1
+Tr(Arw(cj)X j)
+s.t.
+r
+∑
+j=1
+Tr(Arw(aj
+(i))X j) = bi for all i = 1,...,m,
+X j ⪰ 0 for all j = 1,...,r.
+(PSZ)
+They showed that X j = Arw(xj) is not a feasible solution for the SDO primal problem (PSZ).
+In fact, primal feasible solutions of (PSZ) are fully dense, and do not have an arrow-head
+structure. To fix this issue, they proposed the mapping
+MR(xj) =
+� 1
+4θ j
+1
+2(xj
+2:n)T
+1
+2xj
+2:n
+x j
+1−∥x j
+2:n∥
+2(n−1) I + x j
+2:n(x j
+2:n)T
+θ j
+�
+,
+(2.2)
+where θ j = xj
+1 +∥x j
+2:n∥+
+�
+(xj
+1 +∥xj
+2:n∥)2 −4∥xj
+2:n∥2.
+In our study, a key concept is the notion of admissible map which is defined next.
+
+6
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+Definition 2.4. A mapping M is called admissible if it preserves feasibility and objective func-
+tion value, i.e.
+(x,y,s) ∈ FPSOCO ×FDSOCO ⇒ M (x,y,s) ∈ FPSDO ×FDSDO,
+(X,y,S) ∈ FPSDO ×FDSDO ⇒ M −1(X,y,S) ∈ FPSOCO ×FDSOCO,
+cTx = Tr(CX), bTy = bTy.
+The mapping of Sim and Zhao [9] is admissible, and they proved that it maps a solution from
+the boundary (interior) of the Lorentz cone to a solution on the boundary (interior) of the cone
+of semidefinite matrices. In this paper, we seek to extend their approach and explore mappings
+that satisfy the definition of admissible mapping. Although the mapping of [9] is a point to
+point map, the image of (x,y,s) might be a point or a set. In addition, the SDO representation
+of SOCO of [9], (PSZ) and (DSZ), is defined using the product of multiple cones of positive
+semidefinite matrices, but we use a more general approach to get an SDO representation in
+standard form. The major goal of this paper is clarifying more the relationship between SOCO
+and the related SDO by developing different mappings and exploring the relationship between
+the optimal partitions of these problems.
+Without loss of generality, in Sections 3 and 4, we first present the results in case of a single
+second-order cone, and then we generalize the results to the multiple cone case. To this end, we
+consider the following primal and dual problems,
+z∗
+P1
+SOCO = min
+�
+cTx : Ax = b, x ∈ L n�
+,
+(P1
+SOCO)
+z∗
+D1
+SOCO = max
+�
+bTy : ATy+s = c, (y,s) ∈ Rm ×L n�
+,
+(D1
+SOCO)
+with feasible sets FP1
+SOCO = {x ∈ L n : Ax = b} and FD1
+SOCO = {(y,s) ∈ Rm×L n : ATy+s = c},
+and optimal solution sets P1
+SOCO
+∗ = {x ∈ FP1
+SOCO : cTx = z∗
+P1
+SOCO} and D1
+SOCO
+∗ = {(y,s) ∈
+FD1
+SOCO : bTy = z∗
+D1
+SOCO}, respectively.
+3. FROM SOCO TO SDO: STARTING FROM THE DUAL SIDE
+One can derive the SDO counterpart of a SOCO problem starting with either the primal or
+dual SOCO problem. In this section, similar to [9], we initiate the derivation from the dual
+side of SOCO. Thus, as mentioned earlier, we preserve the arrow-head structure of the matrix S
+corresponding to dual solution s.
+3.1. Derivation and Solution Mapping. We utilize the arrow-head transformation to the vec-
+tor c and the rows of matrix A,
+⃗C = Arw(c),
+⃗Ai = Arw(a(i)),
+i = 1,2,...,m.
+Since in the SOCO dual s = c−ATy, by applying the arrow-head structure to A and c, we have
+that S = ⃗C −∑m
+i=1 yi⃗Ai has the arrow-head structure, as it is a linear combination of arrow-head
+matrices. Therefore, the SDO counterpart of the SOCO dual problem (D1
+SOCO) is as follows,
+z∗
+DD
+SDO := max
+�
+bTy :
+m
+∑
+i=1
+yi⃗Ai +S = ⃗C, S ⪰ 0
+�
+,
+(DD
+SDO)
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+7
+which has the following dual,
+z∗
+PD
+SDO := min
+�
+Tr(⃗CX) : Tr(⃗AiX) = bi,
+i = 1,...,m,
+X ⪰ 0
+�
+,
+(PD
+SDO)
+as it’s SDO primal problem. For the SDO problems (PD
+SDO) and (DD
+SDO), let
+FDD
+SDO = {(y,S) ∈ Rm ×Sn :
+m
+∑
+i=1
+yi⃗Ai +S = ⃗C,S ⪰ 0},
+FPD
+SDO = {X ∈ Sn : Tr(⃗AiX) = bi,i = 1,...,m,X ⪰ 0}
+represent the feasible sets, and
+DD
+SDO
+∗ = {(y,S) ∈ FDD
+SDO : bTy = z∗
+DD
+SDO},
+PD
+SDO
+∗ = {X ∈ FPD
+SDO : Tr(⃗CX) = z∗
+PD
+SDO},
+represent the optimal solution sets, respectively. The following theorem provides a point to set
+admissible mapping, see Definition 2.4 for r = 1, based on the (D1
+SOCO) and (P1
+SOCO), and their
+representations (DD
+SDO), (PD
+SDO).
+Theorem 3.1. Consider the SOCO problem pairs (P1
+SOCO) and (D1
+SOCO) with (x,y,s) ∈ FP1
+SOCO ×
+FD1
+SOCO, and the SDO problem pairs (PD
+SDO) and (DD
+SDO) with (X,y,S) ∈ FPD
+SDO × FDD
+SDO.
+Then, mapping (X,y,S) = M (x,y,s) with
+S = Arw(s),
+y = y,
+X =
+
+
+X11
+X12
+...
+X1n
+X12
+X22
+...
+X2n
+...
+...
+...
+...
+X1n
+X2n
+...
+Xnn
+
+ ⪰ 0 with
+
+
+∑n
+i=1Xii
+X12
+...
+X1n
+
+ =
+
+
+x1
+x2
+2...
+xn
+2
+
+,
+is a point-to-set admissible mapping. In addition, the inverse mapping denoted by (x,y,s) =
+M −1(X,y,S), with
+s = Arw−1(S),
+y = y,
+x =
+�
+∑n
+i=1 Xii,2X12,...,2X1n
+�T ,
+is a point-to-point admissible mapping.
+Proof. The proof of this theorem is presented in Appendix A.
+□
+The following corollaries restate that the provided mapping preserves the objective function
+value.
+Corollary 3.2. We have z∗
+P1
+SOCO = z∗
+PD
+SDO, and z∗
+D1
+SOCO = z∗
+DD
+SDO.
+Corollary 3.3. A feasible solution (x,y,s) ∈ FP1
+SOCO ×FD1
+SOCO is optimal for a pair of SOCO
+problems (P1
+SOCO), and (D1
+SOCO) with optimal value (z∗
+P,z∗
+D) if and only if the mapped solution
+(X,y,S) is optimal for the SDO problems (PD
+SDO), and (DD
+SDO) with optimal value (z∗
+P,z∗
+D) .
+Corollary 3.4. A feasible solution (x,y,s) ∈ FP1
+SOCO ×FD1
+SOCO is optimal for a pair of SOCO
+problems (P1
+SOCO), and (D1
+SOCO) with zero duality gap if and only if the mapped solution
+(X,y,S) is optimal for the SDO problems (PD
+SDO), and (DD
+SDO) with zero duality gap, i.e.,
+x◦s = 0 ⇐⇒ Tr(XS) = 0
+
+8
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+Note that these results are valid regardless of duality (strong / weak with gap), status of the
+SOCO problems. Furthermore, observe that one can propose different admissible mappings
+that satisfies the conditions presented in Theorem 3.1. In Section 3.2, we propose a rank-one
+mapping, which is the simplest option. In Section 3.3, we show that when x ∈ int(L n), full
+rank mappings can also be obtained, which map a solution in the interior of SOCO to a solution
+in the interior of the SDO cone.
+3.2. Rank-one Mapping. In this section, we construct a rank-one matrix X for vector x that
+satisfies the conditions in Theorem 3.1. Thus, we introduce the vector β ∈ Rn, and define the
+rank-one matrix.
+X = ββ T,
+with
+n
+∑
+i=1
+β 2
+i = x1,
+β1β j = xj
+2 for all j = 2,...,n.
+We need to solve this n-variable-n-equation system. The solution of this system is β = 0 if
+x = 0, and if x ̸= 0 then
+β =
+1
+�
+2(x1 +δ)
+(x1 +δ,x2,...,xn)T,
+where δ =
+�
+(x1)2 −||x2:n||2. It is easy to see that if we are on the boundary of the second-
+order cone, then δ = 0. Otherwise, we have δ ̸= 0. Using this vector, we can construct a
+suitable matrix X.
+Theorem 3.5. Consider the rank-one mapping with
+X =
+�
+[0]n×n
+if x = (0,0,...,0),
+DMR1(x)
+otherwise.
+(3.1)
+where
+DMR1(x) = ββ T =
+
+
+x1+δ
+2
+x2
+2
+...
+xn
+2
+x2
+2
+x2
+2
+2[x1+δ]
+...
+x2xn
+2[x1+δ]
+...
+...
+...
+...
+xn
+2
+x2xn
+2[x1+δ]
+...
+x2n
+2[x1+δ]
+
+
+.
+Then, (3.1) together with (y,S) = (y,Arw(s)) is a point-to-point admissible mapping.
+Proof. The proof is straightforward as it is enough to show that matrix DMR1(x) satisfies the
+conditions in Theorem 3.1.
+□
+3.3. Higher Rank Mapping. In this section, we show that when a SOCO solution is in the
+interior of the cone, i.e. x ∈ int(L n), then we can use a full rank mapping, i.e.
+DMRn(x) =
+n
+∑
+i=1
+β i(β i)T.
+Theorem 3.6. There exist mappings DMR where rank(DMR(¯x)) = n for x ∈ int(L n).
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+9
+Proof. The proof of this theorem is presented in Appendix B.
+□
+One can easily modify Algorithm 1 (presented in proof of Theorem 3.6) to map a solution
+x ∈ int(L n) to a matrix DMRk(x) with rank 1 ≤ k ≤ n. This means that the primal feasible set
+of a SOCO can be mapped to different subsets of the SDO primal feasible region, e.g., rank-one
+mapping maps the primal feasible set of a SOCO to a one-dimensional face of the SDO primal
+feasible set.
+Recall that Theorem 1 of [9] proposes the mapping
+MR(x) =
+� 1
+4θ
+1
+2xT
+2:n
+1
+2x2:n
+x1−∥x2:n∥
+2(n−1) I + x2:nxT
+2:n
+θ
+�
+,
+where θ = x1 +∥x2:n∥+
+�
+(x1 +∥x2:n∥)2 −4∥x2:n∥2. They showed that this map is admissible.
+Moreover, it has full rank when x1 > ∥x2:n∥, i.e. x ∈ int(L n), and it is a rank-one matrix when
+x1 = ∥x2:n∥. This map also proves Theorem 3.6, while our proof follows a different approach.
+In our approach, we can generate different mappings and explore the feasible set of SDO by
+changing parameter ε in Algorithm 1. However, to prove that the set of all maps with different
+rank produced by our approach can build the whole feasible set of the SDO representation is an
+ongoing research.
+Given the mapping of Sim and Zhao [9], consider the case in which the solution is on the
+boundary of the second-order cone, i.e. x1 = ∥x2:n∥. Then, θ = 2x1, and
+MR1(x) =
+� 1
+2x1
+1
+2xT
+2:n
+1
+2x2:n
+x2:nxT
+2:n
+2x1
+�
+.
+We can see that this is exactly identical to the rank one mapping we presented in Theorem 3.5.
+We can write it as
+MR1(x) = ν1(ν1)T,
+where
+ν1 =
+1
+√
+θ
+�1
+2θ,x2,...,xn
+�T
+.
+In order to construct MR(x) as the sum of rank-one matrices, we define
+ν j =
+�
+x1 −∥x2:n∥
+2(n−1)
+ej
+for j = 2,...,n,
+where ej is a unit vector with 1 in element j. By this setting, we have
+MR(x) =
+n
+∑
+j=1
+ν j(ν j)T.
+If the solution x of the SOCO primal problem is on the boundary of the cone, i.e. x1 = ∥x2:n∥,
+then we get
+ν1 =
+1
+√2x1
+(x1,x2,...,xn)T ,
+ν j = 0,
+for j = 2,...,n.
+
+10
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+This results in the rank one mapping MR1(x). On the other hand, if the solution is in the interior
+of the cone, i.e. x1 > ∥x2:n∥, then ν j ̸= 0 for j = 2,...,n, and we can construct the full rank
+mapping MR(x). Note that one cannot take a combination of first k vectors ν j to construct a
+rank-k mapping as it would not be admissible since it violates the conditions given in Theorem
+3.1, i.e. sum of diagonals will not be equal to x1. To overcome this issue and construct a rank-k
+mapping, let N ⊆ {2,...,n} with |N | = k. One can take the following definition of ν j,
+ν j =
+�
+x1 −∥x2:n∥
+2(k −1)
+ej
+for j ∈ N , and ν j = 0 for j /∈ N .
+The resulting matrix MRk(x) has rank k, and one can easily see that it satisfies the conditions
+of Theorem 3.1. Considering k = n, this choice of ν j results in identical mapping to MR(x).
+Next theorem shows that we can have mappings with different ranks when we are mapping a
+solution from interior of the Lorentz cone.
+Theorem 3.7. Let ρ(x) = max{rank(M (x)) : M is admissible map}. We have
+• ρ(x) = n if x ∈ int(L n).
+• ρ(x) = 1 if x ∈ ∂(L n).
+Proof. Proof of this theorem is similar to Theorem 1 of [9].
+□
+In the next section, we generalize our result for the case there are multiple Lorentz cones.
+3.4. Generalization to Multiple SOCs. In this section, we extend our rank-one mapping to
+the case with multiple second-order cones. We can use similar conditions as in Theorem 3.1 to
+extend our mapping to the case of multiple second-order cones. Although, instead of the “Arw”
+operator, we need to introduce a new operator called “DArw” which constructs a block diagonal
+matrix with arrow-head matrices of input vectors. First, we transform the objective coefficient
+vectors and coefficient matrices into proper block-diagonal structure as
+˜C = DArw(c1,c2,...,cr) =
+
+
+⃗C1
+⃗C2
+...
+⃗Cr
+
+,
+where ⃗Ci is the arrow-head matrix corresponding to vector ci for all i = 1,...,r. Moreover, we
+define a similar block diagonal matrix
+˜Aj = DArw(A1
+j,A2
+j,...,Ar
+j) =
+
+
+⃗A1
+j
+⃗A2
+j
+...
+⃗Ar
+j
+
+
+for all j = 1,...,m,
+where Ai
+j corresponds to the row j of matrix Ai for all i = 1,...,r. Now, the SDO representation
+of the SOCO problem can be derived as (PSDO) and (DSDO). Compared to the SDO represen-
+tation of [9], (PSZ) and (DSZ), our representation is a standard SDO over one cone of positive
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+11
+semidefinite matrices of dimension n×n. Given the introduced notations ˜C and ˜A, we have
+z∗
+P ˜D
+SDO := min Tr( ˜C ˜X)
+s.t. Tr( ˜Ai ˜X) = bi
+for all i = 1,...,m,
+˜X ⪰ 0,
+(P ˜D
+SDO)
+and the dual problem is
+z∗
+D ˜D
+SDO := max bTy
+s.t.
+m
+∑
+i=1
+yi ˜Ai + ˜S = ˜C,
+˜S ⪰ 0.
+(D ˜D
+SDO)
+Analogously, we can define the sets of feasible and optimal solutions for the problems (P ˜D
+SDO)
+and (D ˜D
+SDO).
+We can present the following theorem to specify admissible mappings for the general form.
+Theorem 3.8. Consider the SOCO problem pairs (PSOCO) and (DSOCO) with
+(x1;x2;...;xr;y;s1;s2;...;sr) ∈ FPSOCO ×FDSOCO,
+and SDO problem pairs (P ˜D
+SDO) and (D ˜D
+SDO) with ( ˜X, ˜y, ˜S) ∈ FP ˜D
+SDO × FD ˜D
+SDO. Then, the fol-
+lowing mapping, denoted by ( ˜X, ˜y, ˜S) = M (x1;x2;...;xr;y;s1;s2;...;sr), with
+˜S = DArw(s1;...;sr),
+˜y = y,
+˜X ⪰ 0
+s.t.
+
+
+∑ui+ni
+j=ui ˜Xj j
+˜Xui,ui+1
+...
+˜Xui,ui+ni
+
+ =
+
+
+xi
+1
+xi
+2
+2...
+xin
+2
+
+
+, for all i = 1,...,r
+coupled with the inverse mapping denoted by (x1;x2;...;xr;y;s1;s2;...;sr) = M −1( ˜X, ˜y, ˜S) with
+(s1;...;sr) = DArw−1( ˜S),
+y = ˜y,
+(x1;...;xr) =
+
+
+∑ui+ni
+j=ui ˜Xj j
+2 ˜Xui,ui+1
+...
+2 ˜Xui,ui+ni
+
+,
+where ui = 1+∑i−1
+k=0 ni and n0 = 0, is an admissible mapping.
+Proof. The proof of this theorem is analogous to that of Theorem 3.1.
+□
+Remark 3.9. Note that a solution of (P ˜D
+SDO) is not necessarily block-diagonal.
+
+12
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+Using the conditions in Theorem 3.7, we can write
+˜S = DArw(s1,s2,...,sr) =
+
+
+Arw(s1)
+Arw(s2)
+...
+Arw(sr)
+
+,
+and,
+˜y = y.
+We can define matrix ˜X using our rank one mapping with vector ˜β. Vector ˜β consists of n
+elements partitioned into r sub-vectors each corresponds to the cones in the problem. Thus, we
+have
+˜X = ˜β ˜β T,
+where
+˜β = ((β 1)T,(β 2)T,...,(β r)T)T.
+One can calculate vector ˜β in the closed form as follows,
+β i =
+1
+�
+2(xi
+1 +δ i)
+�
+xi
+1 +δ i,xi
+2,...,xi
+ni
+�
+,
+where δ i =
+�
+(xi
+1)2 −||xi
+2:n||2. We can also have a separate rank-one mapping for each Lorentz
+cone as ˜X = ∑r
+i=1(β)i(β)T
+i , where
+(β j)i =
+
+
+
+1
+√
+2(xi
+1+δ i)
+�
+xi
+1 +δ i,xi
+2,...,xi
+ni
+�
+if j = i,
+0
+otherwise.
+In this case, ˜X is a block diagonal matrix with rank equal to r. If we choose (β j)i ∈ Rnj for
+j ∈ {1,...,r}/{i}, then ˜X = ∑r
+i=1(β)i(β)T
+i will be a positive semidefinite matrix, which is not
+necessarily block diagonal. Since all the input matrices are block diagonal, it is straightforward
+to check that the mapping will remain admissible. For any Lorentz cone i with xi
+1 > ∥xi
+2:n∥, the
+corresponding part of the matrix ˜X can have any rank from 1 to ni using higher rank mappings
+proposed in the previous section. For instance, if the solution x is in the interior of all Lorentz
+cones, i.e. xi
+1 > ∥xi
+2:n∥ for i = 1,...,r, then we can find ˜X = ∑r
+i=1∑nr
+k=1(β)k
+i (β)k
+i
+T whose rank is
+n×r. In this case, it is straightforward to produce mapping with arbitrary rank form 1 to n×r
+by using Algorithm 1 for filling diagonal blocks and fill other parts with arbitrary numbers.
+4. FROM SOCO TO SDO: STARTING FROM THE PRIMAL SIDE
+In this section, we analyze the case when we start to reformulate the standard primal SOCO
+as an equivalent SDO problem, which requires more complex reformulation and was left un-
+touched by Sim and Zhao [9]. The purpose is to investigate the SOCO-SDO relationship starting
+from the primal side and answer the following questions.
+(1) What happens if we force matrix X to be arrow-head?
+(2) Does this derivation results in similar mappings between the SOCO problems and their
+SDO counterparts?
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+13
+4.1. Derivation and Solution Mapping. Recall SOCO problems (P1
+SOCO) and (D1
+SOCO) and
+their corresponding feasible and optimal solution sets from Section 2. Recall that we have
+x ∈ L n if and only if X = Arw(x) ⪰ 0, thus starting from (P1
+SOCO) requires a different choice
+of ⃗C and ⃗Ai in order to properly represent the primal constraint of SOCO in an equivalent SDO
+reformulation. Thus, we define
+⃗C = Arw(1
+nc1, 1
+2c2,..., 1
+ncn),
+⃗Ai = Arw(1
+nai1, 1
+2ai2,..., 1
+2ain),
+i = 1,2,...,m.
+which leads to preserving aT
+(i)x = Tr(⃗AiX) = bi, i = 1,...,m, where X = Arw(x). Using the just
+introduced arrow-head representations of c, a(i), and x, we write the following SDO problem.
+min
+�
+Tr(⃗CX) : Tr(⃗AiX) = bi,
+i = 1,...,m,
+X ⪰ 0
+�
+.
+However, in this SDO problem X is a semidefinite matrix without arrow-head structure. Thus, in
+order to represent (P1
+SOCO) using this SDO model, we need to enforce the arrow-head structure
+on matrix X. In other words, we need to translate the arrow-head structure requirement into
+linear constraints. Thus, we introduce symmetric matrices ˇAhl for 2 ≤ h < l ≤ n with (ˇAhl)hl =
+(ˇAhl)lh = 1 for 2 ≤ h < l ≤ n, and all other entries are zero. Moreover, we introduce matrix ˆAk,
+with (ˆAk)11 = 1,(ˆAk)kk = −1 for k = 2,...,n, and rest of the entries are zero. Therefore, we write
+the following model
+min Tr(⃗CX)
+s.t. Tr(⃗AiX) = bi,
+i = 1,...,m
+Tr(ˇAhlX) = 0,
+h = 2,...,n−1, h < l ≤ n
+Tr(ˆAkX) = 0,
+k = 2,...,n
+X ⪰ 0,
+(PP
+SDO)
+which is an accurate SDO representation of (P1
+SOCO) as the arrow-head structure of the matrix
+X is enforced by the linear constraints. Let z∗
+PP
+SDO denote the optimal objective function value
+of (PP
+SDO), and
+FPP
+SDO = {X ∈ Sn :Tr(⃗AiX) = bi,i = 1,...,m,
+Tr(ˇAhlX) = 0,h = 2,...,n, h < l,
+Tr(ˆAkX) = 0,k = 2,...,n,
+X ⪰ 0},
+and
+PP
+SDO
+∗ = {X ∈ FPP
+SDO : Tr(⃗CX) = z∗
+PP
+SDO},
+
+14
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+be the feasible and optimal solution sets of problem (PP
+SDO), respectively. Next, we present the
+dual model of the SDO problem (PP
+SDO) as
+max bTv
+s.t.
+m
+∑
+i=1
+vi⃗Ai + ∑
+h̸=1,h<l
+whl ˇAhl +
+n
+∑
+k=2
+ukˆAk +S =⃗C,
+S ⪰ 0,
+(DP
+SDO)
+where whl denotes the dual variable corresponding to the matrix dedicated for setting entries
+(h,l) and (l,h) in X equal to zero, and uk corresponds to the linear constraints that are setting
+elements on the diagonal of each block equal to each other for that specific block.
+Similarly, let z∗
+DP
+SDO denote the optimal objective function value for the dual model (DP
+SDO), and
+FDP
+SDO =
+�
+(v,w,u,S) ∈ Rm ×R
+(n−1)(n−2)
+2
+×R(n−1) ×Sn :
+m
+∑
+i=1
+vi⃗Ai + ∑
+h̸=1,h<l
+whl ˇAhl +
+n
+∑
+k=2
+uk ˆAk +S =⃗C, S ⪰ 0
+�
+and
+DP
+SDO
+∗ = {(v,w,u,S) ∈ FDP
+SDO : bTv = z∗
+DP
+SDO},
+be the feasible and optimal solution sets of problem (DP
+SDO), respectively.
+Let’s analyze the structure of S. First, recall problem (D1
+SOCO), where we have
+s j = cj −
+m
+∑
+i=1
+yiai j,
+for all j = 1,...,n.
+Considering that, we seek to preserve the objective function value throughout the mapping, we
+have v = y. Therefore, using the non-zero structure of the matrices ˇAhl and ˆAk, we have
+S =
+
+
+1
+n(s1)−∑n
+k=2 uk
+1
+2(s2)
+...
+...
+1
+2(sn)
+1
+2(s2)
+1
+n(s1)+u2
+−w23
+...
+−w2n
+...
+−w23
+1
+n(s1)+u3
+...
+...
+...
+...
+...
+...
+−w(n−1)n
+1
+2(sn)
+−w2n
+...
+−w(n−1)n
+1
+n(s1)+un
+
+
+.
+(4.1)
+Furthermore, we have free variables w and u which can take values such that S is positive
+semidefinite. Now, let y = (v;w;u), and b = [b;0 (n−1)(n−2)
+2
+×1;0n−1×1]. Then, we can present the
+following theorem which presents a point to set admissible mapping, see Definition 2.4, with
+r = 1, based on (P1
+SOCO) and (D1
+SOCO), and their representations (PP
+SDO) and (DP
+SDO).
+Theorem 4.1. Consider the SOCO problem pairs (P1
+SOCO) and (D1
+SOCO) with (x,y,s) ∈ FP1
+SOCO ×
+FD1
+SOCO and SDO problem pairs (PP
+SDO) and (DP
+SDO) with (X,v,w,u,S) ∈ FPP
+SDO × FDP
+SDO.
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+15
+Then, the mapping (X,v,w,u,S) = M (x,y,s) defined as
+X = Arw(x),
+v = y,
+S =
+
+
+s1
+n −∑n
+k=2 uk
+s2
+2
+...
+...
+sn
+2
+s2
+2
+s1
+n +u2
+−w23
+...
+−w2n
+...
+−w23
+s1
+n +u3
+...
+...
+...
+...
+...
+...
+−w(n−1)n
+sn
+2
+−w2n
+...
+−w(n−1)n
+s1
+n +un
+
+
+,
+where w ∈ R
+(n−1)(n−2)
+2
+and u ∈ R(n−1) taking any values such that S ⪰ 0 is a point-to-set admis-
+sible mapping. The inverse mapping is denoted by (x,y,s) = M −1(X,v,w,u,S), with
+x = Arw−1(X),
+y = v,
+s =
+
+
+c1 −∑m
+i=1viai1
+...
+cn −∑m
+i=1viain
+
+.
+Proof. The proof of this theorem is presented in Appendix C.
+□
+Remark 4.2. The proposed mapping in Theorem 4.1 is not necessarily a point to point mapping
+regarding dual variables of (D1
+SOCO) and (DP
+SDO).
+Analogous to Corollaries 3.2-3.4, the proposed mapping in Theorem 4.1 preserves optimality
+and complementarity.
+4.2. Rank-one Mapping. Using the intuition from the dual side’s section, we seek to represent
+matrix S using a rank one matrix. Hence, we propose
+S = η(η)T,
+with
+n
+∑
+i=1
+(ηi)2 = s1,
+η1ηj = s j
+2 for all j = 2,...,n.
+We need to solve an n-variable-n-equation system. The solution of this system is
+η =
+1
+�
+2(s1 +δ ′)
+�
+s1 +δ ′,s2,...,sn
+�T,
+where δ ′ =
+�
+s1 −||s2:n||2. Using this vector, we can construct matrix S. Thus, we can compute
+uk for k = 2,...,n, and whl for h ̸= 1,h < l as follows,
+uk = (ηk)2 − 1
+ns1 =
+s2
+k
+2(s1 +δ ′) − 1
+ns1,
+k = 2,...,n,
+whl = ηhηl =
+shsl
+2(s1 +δ ′),
+h ̸= 1,h < l.
+
+16
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+Theorem 4.3. Consider the rank-one mapping
+S =
+�
+[0]n×n
+if s = (0,0,...,0),
+PMR1
+otherwise.
+(4.2)
+where
+PMR1 = η(η)T =
+
+
+s1+δ ′
+2
+s2
+2
+...
+sn
+2
+s2
+2
+s2
+2
+2[s1+δ ′]
+...
+s2sn
+2[s1+δ ′]
+...
+...
+...
+...
+sn
+2
+s2sn
+2[s1+δ ′]
+...
+s2n
+2[s1+δ ′]
+
+
+.
+Then, (4.2) together with (X,y) = (Arw(x),y) is a point-to-point admissible mapping.
+Proof. The proof is similar to that of Theorem 3.5.
+□
+4.3. Higher Rank Mapping. The derivation of full rank mapping is similar to the derivation
+discussed in Section 3.3, as we show that when a SOCO solution s ∈ int(L n), then we can use
+a full rank mapping, i.e.
+PMR =
+n
+∑
+i=1
+ηi(ηi)T.
+The proof of existence of a full rank mapping is similar to the proof presented in Appendix B.
+Theorem 4.4. There exist mappings PMR where rank(PMR(¯s)) = n for s ∈ int(L n).
+Proof. The proof of this theorem is similar to the proof presented in Appendix B. The difference
+is that we need to set π1 = s in the proof.
+□
+The reason why the procedure to show the existence of full rank mapping is similar for both
+directions is that although the mappings seem to be different, in fact both satisfy analogous
+conditions about the first row and column and the trace of the mapping matrix. From this point
+of view, they are very similar. In fact, the update vector in the procedure is meant to be satisfying
+those conditions. Thus, we can follow the procedure in Appendix B for both mappings.
+Similar to Theorem 1 of [9], we can develop the following mapping for the primal side.
+MR(s) =
+� 1
+4θ′
+1
+2sT
+2:n
+1
+2s2:n
+s1−∥s2:n∥
+2(n−1) I + s2:nsT
+2:n
+θ ′
+�
+,
+where θ′ = s1 +∥s2:n∥ +
+�
+(s1 +∥s2:n∥)2 −4∥s2:n∥2. Here, we can perform the same analysis
+we did in Section 3.3, too. Considering the case in which the solution is on the boundary of the
+second-order cone, i.e. s1 = ∥s2:n∥. Then, θ′ = 2s1, and
+MR1(s) =
+� 1
+2s1
+1
+2sT
+2:n
+1
+2s2:n
+s2:nsT
+2:n
+2s1
+�
+.
+We can see that this is exactly identical to the rank one mapping we presented in Theorem 4.3.
+We can also write it as
+MR1(s) = γ1(γ1)T
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+17
+where
+γ1 =
+1
+√
+θ′
+�1
+2θ′,s2,...,sn
+�T
+.
+To obtain MR(s) as the sum of rank-one matrices, we define
+γ j =
+�
+s1 −∥s2:n∥
+2(n−1)
+ej
+forj = 2,...,n,
+where ej is a unit vector with 1 in element j. By this setting, we have
+MR(s) =
+n
+∑
+j=1
+γ j(γ j)T.
+If the solution s of the SOCO dual problem is on the boundary of the cone, i.e. s1 = ∥s2:n∥, then
+we get
+γ1 =
+1
+√2s1
+(s1,s2,...,sn)T ,
+γ j = 0,
+for j = 2,...,n.
+This leads to the rank one mapping MR1(s). On the other hand, if the solution s ∈ int(L n),
+i.e. s1 > ∥s2:n∥, then γ j ̸= 0 for j = 2,...,n, and we obtain the full rank mapping MR(s). To
+construct an admissible rank-k mapping, here one cannot take a combination of first k vectors
+γ j as it violates the conditions given in Theorem 4.1, i.e. the sum of the diagonal elements will
+not be equal to s1. Recall the definition of N from Section 3.3. The correct choice of γ j to
+construct a rank-k mapping is
+γ j =
+�
+s1 −∥s2:n∥
+2(k −1)
+ej
+for j ∈ N , and γ j = 0 for j /∈ N .
+The resulting matrix MR(s) has rank k, and one can easily see that it satisfies the conditions of
+Theorem 4.1. Analogous to the dual side, we have the following theorem.
+Theorem 4.5. Let ρ(s) = max{rank(M (s)) : M is admissible map}. We have
+• ρ(s) = n if s ∈ int(L n).
+• ρ(s) = 1 if s ∈ ∂(L n).
+Proof. Proof of this theorem is similar to Theorem 1 of [9].
+□
+4.4. Generalization to Multiple SOCs. Similar to the discussion in Subsection 3.4, here we
+adopt the mapping presented in Theorem 3.8 to the primal side. Here we need to define the
+following notations. Let,
+˜C =
+
+
+⃗C1
+⃗C2
+...
+⃗Cr
+
+,
+where
+⃗Ck = Arw(1
+nck
+1, 1
+2ck
+2,..., 1
+2ck
+n)
+and
+˜Aj = DArw(⃗A1
+j,⃗A2
+j,...,⃗Ar
+j)
+for all j = 1,...,m,
+
+18
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+where
+⃗Ak
+j = Arw(1
+nak
+i1, 1
+2ak
+i2,..., 1
+2ak
+in)
+for all k = 1,...,r.
+Similar to the structure of ˜C and ˜Aj, matrix ˜X needs to have a block-diagonal structure. Thus,
+we need to enforce this structure on it. To this end, we define an operator. Let Cf(i) = j, be an
+operator which returns the corresponding cone j to input row i of the block-diagonal matrices.
+Let set I consists of all entries that need to be zero due to being either off-arrow within the
+blocks, or the off-block-diagonal structure.
+I =
+�
+(h,l) ∈ [1,
+r
+∑
+i=1
+ni]2
+�����
+�
+∑
+Cf(h)
+i=0 ni < l,
+∑
+Cf(h)−1
+i=0
+ni +1 < h < ∑
+Cf(h)
+i=0 ni,
+and
+h < l ≤ ∑
+Cf(h)
+i=0 ni.
+�
+,
+K =
+�
+k ∈ [1,
+r
+∑
+i=1
+ni]
+�����
+Cf(k)−1
+∑
+i=0
+ni +1 < k ≤
+Cf(k)
+∑
+i=0
+ni
+�
+,
+where n0 = 0. Note that here we introduce matrices ˜ˇAhl and ˜ˆAk which are generalizations of
+ˇAhl and ˆAk, for the multiple cone case, respectively. In detail, ˜ˇAhl enforces all entries off-block-
+diagonal and off-arrow within each block to be zero. Moreover, ˜ˆAk guarantees that in each
+block, diagonal entries are equal by setting entry (k,k) equal to −1 and entry (∑
+Cf(k)−1
+t=1
+nt +
+1,∑
+Cf(k)−1
+t=1
+nt +1) equal to +1. By this notation, we have
+min Tr(˜C ˜X)
+s.t. Tr(˜Aj ˜X) = bj,
+j = 1,...,m
+Tr(˜ˇAhl ˜X) = 0,
+(h,l) ∈ I
+Tr(˜ˆAk ˜X) = 0,
+k ∈ K
+˜X ⪰ 0.
+(P ˜P
+SDO)
+Then, we dualize and get
+max bTy
+s.t.
+m
+∑
+j=1
+yj ˜Aj + ∑
+h,l∈I
+vhl ˜ˇAhl + ∑
+k∈K
+zk˜ˆAk + ˜S = ˜C,
+˜S ⪰ 0.
+(D ˜P
+SDO)
+In similar fashion to other introduced models, we can define the sets of feasible and optimal
+solutions corresponding to models (P ˜P
+SDO) and (D ˜P
+SDO).
+Next, we can present the following theorem.
+Theorem 4.6. Consider the SOCO problem pairs (PSOCO) and (DSOCO) with
+(x1;x2;...;xr;y;s1;s2;...;sr) ∈ FPSOCO ×FDSOCO,
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+19
+and SDO problem pairs (P ˜P
+SDO) and (D ˜P
+SDO) with ( ˜X, ˜y, ˜S) ∈ FP ˜P
+SDO ×FD ˜P
+SDO. Then, the map-
+ping ( ˜X, ˜y,v,z, ˜S) = M (x,y,s) defined as
+˜X = DArw(X1,X2,...,Xr),
+˜y = y,
+˜S =
+
+
+S1
+S2
+...
+Sr
+
+ ⪰ 0 s.t.
+
+
+∑ui+ni
+j=ui ˜Sj j
+˜Sui,ui+1
+...
+˜Sui,ui+ni
+
+ =
+
+
+si
+1
+si
+2
+2...
+sin
+2
+
+
+with vectors v and z taking values such that ˜S ⪰ 0. Furthermore, the inverse mapping denoted
+by (x,y,s) = M −1( ˜X, ˜y,v,z, ˜S), with
+(x1;...;xr) with xi =
+
+
+˜xui,ui
+˜xui+1,ui
+...
+˜xui+ni,ui
+
+,
+y = ˜y,
+(s1;...;sr) with si =
+
+
+∑ui+ni
+j=ui ˜Sj j
+2 ˜Sui,ui+1
+...
+2 ˜Sui,ui+ni
+
+,
+where ui = 1+∑i−1
+k=0 ni and n0 = 0, is an admissible mapping.
+Proof. The proof of this theorem is analogous to that of Theorem 3.1.
+□
+We observe that although in the general case the mapping is similar to that of the dual side,
+but the details of the mapping are different. This mapping is different from the other one since it
+requires more work to enforce the arrow-head structure on matrix ˜X. An analogous analysis on
+different rank mappings can be done for the generalized multiple cone case, with the difference
+that here we have the arrow-head structure for matrix ˜X, and we can compute matrix ˜S with
+different ranks. We skip that similar analysis here for brevity.
+Now, that we have the generalized standard mapping for both sides, we can proceed with
+studying the mapping of the optimal partitions on both sides in the next section.
+5. MAPPING THE OPTIMAL PARTITION
+As shown in the previous sections, the proposed mappings represent a solution of SOCO
+depending on where it is located in the cone. For a solution on the boundary of the second-order
+cone, all admissible maps provide a rank-one positive semidefinite matrix, while a solution in
+the interior of the Lorentz cone can be mapped to semidefinite matrices with different ranks.
+This correspondence can be analyzed in order to see how the optimal partition of SOCO is
+mapped to that of the derived SDO counterparts. For mapping the optimal partition, maximally
+complementary solutions are of particular interest. First, we define a helpful notation, and then
+the following theorem discusses the preservation of maximal complementarity.
+Definition 5.1 (Proper Map). Mapping M is a proper map if M is admissible and rank(X) =
+ρ(x) for all x ∈ L n, or rank(S) = ρ(s) for all s ∈ L n
+Based on this definition, a rank-one mapping is not proper but map MR of [9] is proper. In
+the subsequent theorem, we show that a proper mapping preserves maximal complementarity
+using eigenvalues of the mapped solution. We used the mapping approach of Section 3 which
+
+20
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+starts from dual side. Let λ X
+i denotes the ith eigenvalue of a matrix X. Then, the eigenvalues
+of an arrow-head matrix X = Arw(x) are λ X
+1 = x1 − ∥x2:n∥, λ X
+2 = ··· = λ X
+n−1 = x1, and λ X
+n =
+x1 +∥x2:n∥, see e.g. [1].
+Theorem 5.2. For a maximally complementary solution (x;y;s) ∈ P∗
+SOCO×D∗
+SOCO, the mapped
+solution ( ˜X, ˜y, ˜S) = M (x;y;s) ∈ P ˜D∗
+SDO ×D ˜D∗
+SDO is maximally complementary if M is a proper
+map.
+Proof. Given a maximally complementary solution (x;y;s) ∈ P∗
+SOCO ×D∗
+SOCO, one can analyze
+the result of mapping based how the cones are partitioned. The following analysis holds for the
+dual side direction. In this direction, we have matrix ˜S as a block diagonal of arrow-head
+matrices. Since M is proper map, the eigenvalues of matrix ˜X can be partitioned based on
+Lorentz cones. If vector xi is on the non-zero boundary of Lorentz cone, the all corresponding
+eigenvalues are non-zero except the first one. If the vector xi is in the interior of Lorentz cone,
+the corresponding eigenvalues are positive. For the point of the second-order cone, all the
+corresponding eigenvalues are zero. Thus, we have
+if i ∈ ¯
+B =
+�
+xi
+1 > ∥xi
+2:ni∥
+→ full rank matrix Xi with λ Xi
+1 ,λ Xi
+2 ,...,λ Xi
+ni > 0,
+si = 0
+→ zero Si matrix with λ Si
+1 = ... = λ Si
+ni = 0,
+if i ∈
+¯
+N =
+�
+xi = 0
+→ zero matrix Xi with λ Xi
+1 = ... = λ Xi
+ni = 0,
+si
+1 > ∥si
+2:ni∥
+→ arrow-head matrix Si with λ Si
+1 ,λ Si
+2 ,...,λ Si
+ni > 0,
+if i ∈ ¯
+R =
+�
+xi
+1 = ∥xi
+2:ni∥ > 0
+→ rank-one matrix Xi with λ Xi
+1 > 0,λ Xi
+2 = ... = λ Xi
+ni = 0
+si
+1 = ∥si
+2:ni∥ > 0
+→ arrow-head Si matrix with λ Si
+1 = 0,λ Si
+2 ,...,λ Si
+ni > 0,
+if i ∈ ¯
+T1 =
+�
+xi = 0
+→ zero matrix Xi with λ Xi
+1 = ... = λ Xi
+ni = 0,
+si = 0
+→ zero matrix Si with λ Si
+1 = ... = λ Si
+ni = 0,
+if i ∈ ¯
+T2 =
+�
+xi
+1 = ∥xi
+2:ni∥ > 0
+→ rank-one matrix Xi with λ Xi
+1 > 0,λ Xi
+2 = ... = λ Xi
+ni = 0
+si = 0
+→ zero matrix Si with λ Si
+1 = ... = λ Si
+ni = 0,
+if i ∈ ¯
+T3 =
+�
+xi = 0
+→ zero matrix Xi with λ Xi
+1 = ... = λ Xi
+ni = 0,
+si
+1 = ∥si
+2:ni∥ > 0
+→ arrow-head matrix Si with λ Si
+1 = 0,λ Si
+2 ,...,λ Si
+ni > 0,
+Now, using that M is a proper, M maps a given maximally complementary optimal SOCO so-
+lution to ( ˜X, ˜y, ˜S) ∈ P ˜D∗
+SDO ×D ˜D∗
+SDO. Here, we show that this solution is maximally complemen-
+tary for the SDO counterpart problem. The proof goes by contradiction. Let us assume to the
+contrary that ( ˜X, ˜y, ˜S) is not a maximally complementary solution. Let ( ˆX, ˆy, ˆS) ∈ P ˜D∗
+SDO×D ˜D∗
+SDO
+be a maximally complementary solution. Since both ( ˜X, ˜y, ˜S) and ( ˆX, ˆy, ˆS) are optimal, then
+ˆS ˜X = 0. Accordingly, if λ ˜Si
+j > 0 then λ ˆSi
+j > 0. Now it is enough to show that there are no i, j
+such that λ ˜Si
+j = 0 and λ ˆSi
+j > 0. If there exist such index, by doing inverse mapping, the mapped
+solution in SOCO will have larger partition set either for
+¯
+N , ¯
+R, or
+¯
+T3 which contradicts with
+assumption that solution (x;y;s) being a maximally complementary solution of SOCO. With
+similar reasoning we can show that there are no i, j such that λ ˜Xi
+j
+= 0 and λ ˆXi
+j
+> 0. Thus, we
+can conclude that a proper map preserves maximal complementarity.
+□
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+21
+One can prove similar theorem as stated as follows for the mapping of Section 4 starting from
+primal side.
+Theorem 5.3. For a maximally complementary solution (x;y;s) ∈ P∗
+SOCO×D∗
+SOCO, the mapped
+solution ( ˜X, ˜y, ˜S) = M (x;y;s) ∈ P ˜P∗
+SDO ×D ˜P∗
+SDO is maximally complementary if M is a proper
+map.
+Proof. The proof is analogous to the proof of Theorem 5.2.
+□
+5.1. Mapping the Optimal Partition. Based on the analysis described in the proof of The-
+orem 5.2, one can develop the mapping for the optimal partitions. Based on the sign of the
+eigenvalues, if positive, their corresponding eigenvectors generate the subspaces B and N ,
+and if zero, to subspace T of the optimal partition of SDO. First we consider mapping starting
+from the dual side as discussed in Section 3.4 and Section 4.4. The summary of optimal parti-
+tion mapping on the dual side is presented in Table 1, where ei
+j represents the jth eigenvector
+corresponding to λ j for the semidefinite cone i.
+B
+N
+T
+if i ∈ ¯
+B
+{ei
+1,...,ei
+ni}
+/0
+/0
+if i ∈
+¯
+N
+/0
+{ei
+1,...,ei
+ni}
+/0
+if i ∈ ¯
+R
+{ei
+1}
+{ei
+2,...,ei
+ni}
+/0
+if i ∈ ¯T1
+/0
+/0
+{ei
+1,...,ei
+ni}
+if i ∈ ¯T2
+{ei
+1}
+/0
+{ei
+2,...,ei
+ni}
+if i ∈ ¯T3
+/0
+{ei
+2,...,eini}
+{ei
+1}
+TABLE 1. The Dual Side Optimal Partition Mapping
+A similar analysis can be conducted for the primal side derivation, where matrix X has an
+arrow-head structure and we represent matrix S using zero, rank-one or full rank matrices, based
+on the position of vector s in the second-order cone. The optimal partition mapping is presented
+in Table 2.
+
+22
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+B
+N
+T
+if i ∈ ¯
+B
+{ei
+1,...,ei
+ni}
+/0
+/0
+if i ∈
+¯
+N
+/0
+{ei
+1,...,ei
+ni}
+/0
+if i ∈ ¯
+R
+{ei
+2,...,ei
+ni}
+{ei
+1}
+/0
+if i ∈ ¯T1
+/0
+/0
+{ei
+1,...,ei
+ni}
+if i ∈ ¯T2 {ei
+2,...,ei
+ni}
+/0
+{ei
+1}
+if i ∈ ¯T3
+/0
+{ei
+1}
+{ei
+2,...,ei
+ni}
+TABLE 2. The Primal Side Optimal Partition Mapping
+We can see that the result of mapping the optimal partition for the two sides are different.
+We observe identical outcome for partitions ¯
+B,
+¯
+N ,
+¯
+T1, which was predictable as they include
+the simple case where at least one of the variables x and s is zero. However, notable difference
+arises for the ¯
+R, ¯T2, and ¯T3 partitions depending on whether the dual or the primal variable’s
+representation is forced to be arrow-head.
+Consider partition ¯
+R, where both x and s are on the non-zero boundary of the second-order
+cone. Adopting the dual (primal) side approach, variable s (x) will have arrow-head represen-
+tation, and the other has a rank one representation. We have seen in Example 2.3, this partition
+is where complementarity will not be preserved if we apply arrow-head representation on both
+primal and dual sides. While our derivations guarantee feasibility and duality, we can see that
+complementarity is also preserved as the arrow-head matrix has only one zero eigenvalue, while
+the rank-one matrix has only one none-zero. For partitions
+¯
+T2 and
+¯
+T3, the difference in tables
+simply comes from which variable is non-zero and its corresponding arrow-head representation
+eigenvalues.
+From the geometric point of view, we can see that for partitions ¯
+B and
+¯
+N , where the respec-
+tive solution is in the interior of the second-order cone, the mapped solution is in the interior of
+the semidefinite cone. For partition ¯
+R, we observe that the solution pair, which both are on the
+non-zero boundary of the second-order cone is mapped to a face of the semidefinite cone. For
+partition
+¯
+T1, the zero solution pair is mapped to the origin. Finally, for partitions
+¯
+T2, and
+¯
+T3,
+the solutions are on the boundary of the second-order cone in the primal and dual, respectively.
+We can see that the result of the mapping is a face of the semidefinite cone, for both cases,
+depending on if the primal or dual variable is on the boundary on the second-order cone.
+6. CONCLUSION
+In this paper, we study the relationship between a SOCO problem and its SDO representa-
+tion. Knowing about the fact that SOCO can be considered as a special case of SDO, we extend
+the literature by investigating both the primal and the dual side SDO representations of a SOCO
+problem. We demonstrate that using arrow-head matrix transformation on the primal or dual
+SOCO problem, does not result in an arrow-head matrix variable on its dual. In fact, nothing
+forces the dual variable to be arrow-head. We usually end up with dense matrices which can be
+represented by either rank-one or full rank mappings, based on the position of SOCO solutions
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+23
+in the second-order cone. We propose low-rank to full rank mappings which are admissible,
+meaning that they preserve feasibility and objective function value. First of all, these mappings
+are not unique. One can come up with different mappings that satisfies these conditions. In ad-
+dition, the dense structure of mapped solutions gives us an intuition about why solving SOCO
+as a SOCO problem is more efficient than solving as an SDO problem. In the SDO representa-
+tion, we have to deal with dense matrices when solving the SDO counterpart. Furthermore, we
+investigated the relationship between the optimal partitions of these problems. The optimal par-
+tition of SOCO is an index-based partition, while that of SDO is a subspace-based partition. We
+discussed how these partitions map to each other based on the eigenvalue analysis of mapped
+solutions.
+The theoretical study of the relationship between SOCO and its SDO counterparts remains
+with unsolved questions. One interesting question to look into is to study how degeneracy and
+singularity degree properties are affected throughout mappings.
+
+24
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+APPENDIX A. PROOF OF THEOREM 3.1
+Proof. To prove that the presented mapping in Theorem 3.1 is admissible, we need to show that
+it complies with the definition of admissible mapping, i.e. preservation of feasibility and the
+objective function value.
+Step 1. First, we show that if (y,s) ∈ FD1
+SOCO, then (y,S) = (y,Arw(s)) ∈ FDD
+SDO. Since S =
+Arw(s) and s ∈ L n, we know that S is positive semidefinite [1]. The only thing remains to
+prove is that ∑n
+i=1 yi⃗Ai +S = ⃗C. Since (y,s) is feasible, i.e., ATy+s = c, we have
+n
+∑
+i=1
+yi⃗Ai +S =
+
+
+∑n
+i=1yiai1 +s1
+∑n
+i=1yiai2 +s2
+...
+∑n
+i=1yiain +sn
+∑n
+i=1yiai2 +s2
+∑n
+i=1yiai1 +s1
+...
+...
+∑n
+i=1yiain +sn
+∑n
+i=1yiai1 +s1
+
+
+=
+
+
+c1
+c2
+...
+cn
+c2
+c1
+...
+...
+cn
+c1
+
+ = ⃗C.
+Step 2. We show that if x = (x1,...,xn) ∈ FP1
+SOCO, then matrix X as defined in Theorem 3.1,
+belongs to FPD
+SDO. By construction, X is positive semidefinite and we need to just show that
+Tr(⃗AiX) = bi. Based on the construction of X and feasibility of x, i.e., Ax = b, we have
+Tr(⃗AiX) = ai1(
+n
+∑
+i=1
+Xii)+2ai2X12 +...+2ainX1n
+= ai1x1 +ai2x2 +...+ainxn = bi.
+Step 3. It is obvious that bTy = bTy. Similar to step 2, we have
+Tr(⃗CX) = c1(
+n
+∑
+i=1
+Xii)+2c2X12 +...+2cnX1n
+= c1x1 +c2x2 +...+cnxn = cTx.
+Step 4. In this step, we show that if X ∈ FPD
+SDO, then x ∈ FP1
+SOCO. Suppose that X ∈ FPD
+SDO,
+then it is positive semidefinite. We need to show that x ∈ L n. To do so, we utilize that in a
+positive semidefinite matrix every principal submatrix, in particular every 2-by-2 submatrix is
+positive semidefinite [6]. Thus,
+|Xi j| ≤
+�
+XiiXj j
+for all i = 1,...,n and for all j = 1,...,n
+Thus, we have
+X2
+12 +X2
+13 +...+X2
+1n ≤ (X11X22)+(X11X33)+...+(X11Xnn)
+(A.1)
+≤ X11(X22 +X33 +...+Xnn),
+(A.2)
+�
+X2
+12 +X2
+13 +...+X2
+1n ≤
+�
+X11(X22 +X33 +...+Xnn)
+(A.3)
+≤ X11 +X22 +X33 +...+Xnn
+2
+,
+(A.4)
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+25
+where (A.4) is derived using the arithmetic-geometric mean inequality. The expression can be
+rewritten as
+n
+∑
+i=1
+Xii ≥ 2
+�
+X2
+12 +X2
+13 +...+X2
+1n
+=
+�
+(2X12)2 +(2X13)2 +...+(2X1n)2,
+or simply,
+n
+∑
+i=1
+Xii ≥
+�
+(2X12)2 +(2X13)2 +...+(2X1n)2,
+x1 ≥
+�
+(x2)2 +...+(xn)2 = ||x2:n||2,
+which shows that x ∈ L n. Now, we need to prove that Ax = b. Thus, we have
+Aix = ai1x1 +ai2x2 +...+ainxn
+for all i = 1,...,m,
+= ai1(
+n
+∑
+i=1
+Xii)+ai2(2X12)+...+ain(2X1n)
+= Tr(⃗AiX) = bi.
+Step 5. Finally, we need to show that if (y,S) ∈ FDD
+SDO, then (y,s) ∈ FD1
+SOCO. As (y,S) =
+(y,Arw(s)), according to [1], we know that S = Arw(s) ⪰ 0, implies s ∈ L n. Then, the only
+property that remains to prove is that ∑m
+i=1 yiAi + s = c. By feasibility of (y,S), we know that
+∑n
+i=1yi⃗Ai +S = ⃗C, i.e.,
+
+
+∑n
+i=1 yiai1 +s1
+∑n
+i=1 yiai2 +s2
+...
+∑n
+i=1 yiain +sn
+∑n
+i=1 yiai2 +s2
+∑n
+i=1 yiai1 +s1
+...
+...
+∑n
+i=1 yiain +sn
+∑n
+i=1 yiai1 +s1
+
+ =
+
+
+c1
+c2
+...
+cn
+c2
+c1
+...
+...
+cn
+c1
+
+,
+where each element of the matrix is a constraint in (D1
+SOCO) which means ∑m
+i=1 yiAi +s = c.
+Considering all of the previous steps, we conclude that presented mapping in Theorem 3.1 is
+an admissible mapping.
+□
+
+26
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+APPENDIX B. PROOF OF THEOREM 3.6
+Proof. A rank-n mapping of ¯x can be constructed by Algorithm 1.
+Algorithm 1 Full Rank Mapping
+π1 = ¯x,
+Choose sufficiently small ε
+for k = 1 : n−1 do
+τk ← πk
+2
+�
+τk
+k+1 = τk
+k+1 +ε
+if πk
+k+1 ≥ 0
+τk
+k+1 = τk
+k+1 −ε
+if πk
+k+1 < 0
+Calculate
+β k =
+1
+2
+�
+τk
+1+ ˆδ k
+2
+(τk
+1 + ˆδ k,τk
+2,...,τk
+n)T, where ˆδ k =
+�
+(τk
+1)2 −||τk
+2:n||2.
+πk+1 = πk −τk
+end for
+Calculate
+β n =
+1
+2
+�
+πn
+1+ ˆδ n
+2
+(πn
+1 + ˆδ n,πn
+2,...,πn
+n)T, where ˆδ n =
+�
+(πn
+1)2 −||πk
+2:n||2.
+If ∥πk
+2:n∥ < πk
+1 for all k, then we have
+0 < ||πk
+2:n||
+2
+≤ ||τk
+2:n|| < τk
+1 ≤ πk
+1
+2 < πk
+1,
+
+
+
+πk
+j
+2 ≤ τk
+j
+if π j ≥ 0
+πk
+j
+2 ≥ τk
+j
+if π j < 0
+,
+j = 2,...,n.
+(B.1)
+We need to show that in all loops ∥πk
+2:n∥ < πk
+1. For k = 1 this holds since ∥x2:n∥ < x1. Using
+induction, we need to prove ∥πk+1
+2:n ∥ < πk+1
+1
+assuming ∥πk
+2:n∥ < πk
+1. We have
+πk+1
+1
+−∥πk+1
+2:n ∥ = (πk
+1 −τk
+1)−∥(πk
+j −τk
+j)2:n∥ ≥ τk
+1 −∥τk
+2:n∥ > 0.
+Thus, πk
+1 > ∥πk
+2:n∥ for all k. The proposed MR is also admissible since
+n
+∑
+j=1
+MRj j =
+n
+∑
+k=1
+n
+∑
+j=1
+(β k
+j )2 = x1
+MR1j =
+n
+∑
+k=1
+β k
+j β k
+1 = x j
+2 .
+We need to prove that the generated β k vectors are linearly independent. We have
+β k =
+1
+2
+�
+τk
+1+ ˆδ k
+2
+(τk
+1 + ˆδ,τk
+2,...,τk
+n)T, where ˆδ k =
+�
+(τk
+1)2 −||τk
+2:n||2.
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+27
+Let us define the matrix
+B =
+
+
+τ1
+1 + ˆδ 1
+τ2
+1 + ˆδ 2
+...
+τn
+1 + ˆδ n
+τ1
+2
+τ2
+2
+...
+τn
+2
+...
+...
+...
+...
+τ1
+n
+τ2
+n
+...
+τn
+n
+
+
+We still need to prove that B has full rank. Now, w.l.o.g. we may assume that x ≥ 0. One can
+easily extend the following proof to general x. Then we have
+B =
+
+
+τ1
+1 + ˆδ 1
+τ2
+1 + ˆδ 2
+...
+τn
+1 + ˆδ n
+τ1
+2
+τ2
+2
+...
+τn
+2
+...
+...
+...
+...
+τ1
+n
+τ2
+n
+...
+τn
+n
+
+
+=
+
+
+π1
+1
+2 + ˆδ 1
+π2
+1
+2 + ˆδ 2
+...
+πn
+1 + ˆδ n
+π1
+2
+2 +ε
+π2
+2
+2
+...
+πn
+2
+π1
+3
+2
+π2
+3
+2 +ε
+...
+πn
+3
+...
+...
+...
+...
+π1n
+2
+π2n
+2
+...
+πn
+n
+
+
+=
+
+
+x1
+2 + ˆδ 1
+x1
+4 + ˆδ 2
+...
+x1
+(2)n + ˆδ n
+x2
+2 +ε
+x2
+4 − ε
+2
+...
+x1
+(2)n −
+ε
+(2)n−1
+x3
+2
+x3
+4 +ε
+...
+x1
+(2)n −
+ε
+(2)n−2
+...
+...
+...
+...
+xn
+2
+xn
+4
+...
+x1
+(2)n − ε
+(2)
+
+
+=
+
+
+x1
+2
+x1
+4
+...
+x1
+(2)n
+x2
+2
+x2
+4
+...
+x1
+(2)n
+x3
+2
+x3
+4
+...
+x1
+(2)n
+...
+...
+...
+...
+xn
+2
+xn
+4
+...
+x1
+(2)n
+
+
++
+
+
+ˆδ 1
+ˆδ 2
+...
+ˆδ n−1
+ˆδ n
+ε
+−ε
+2
+...
+−
+ε
+(2)n−2
+−
+ε
+(2)n−1
+0
+ε
+...
+−
+ε
+(2)n−3
+−
+ε
+(2)n−2
+...
+...
+...
+...
+...
+0
+0
+...
+ε
+− ε
+(2)
+
+
+,
+where
+
+
+ˆδ 1
+ˆδ 2
+...
+ˆδ n
+
+ =
+
+
+�
+(x1
+2 )2 −(x2
+2 +ε)2 −(x3
+2 )2 −···−(xn
+2 )2
+�
+(x1
+4 )2 −(x2
+4 − ε
+2)2 −(x3
+4 +ε)2 −···−(xn
+4 )2
+...
+�
+( x1
+(2)n)2 −( x1
+(2)n −
+ε
+(2)n−1)2 −(x3
+4 −
+ε
+(2)n−1)2 −···−( xn
+(2)n − ε
+2)2
+
+
+.
+
+28
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+By doing row eliminations on both matrices to sparsify the second matrix, we get
+
+
+x1
+2
+2x1
+4
+...
+n x1
+(2)n
+x2
+2
+2x2
+4
+...
+n x1
+(2)n
+x3
+2
+2x3
+4
+...
+n x1
+(2)n
+...
+...
+...
+...
+xn
+2
+2xn
+4
+...
+n x1
+(2)n
+
+
++
+
+
+0
+0
+...
+0
+¯δ
+ε
+0
+...
+0
+0
+0
+ε
+...
+0
+0
+...
+...
+...
+...
+...
+0
+0
+...
+ε
+0
+
+
+where
+¯δ =
+ˆδ 1
+(2)n−1 +
+ˆδ 2
+(2)n−2 +···+
+ˆδ n−1
+2
++ ˆδ n.
+As we can see ¯δ, as sum of strictly positive numbers, is strictly positive, and the matrix has
+full rank. By choosing small ε, we can show that B has full rank. The last piece is to find
+appropriate value for ε. We need to choose ε so that
+(x1
+2 )2 −(x2
+2 +ε)2 −(x3
+2 )2 −···−(xn
+2 )2 > 0
+(x1
+4 )2 −(x2
+4 − ε
+2)2 −(x3
+4 +ε)2 −···−(xn
+4 )2 > 0
+...
+( x1
+(2)n)2 −( x1
+(2)n −
+ε
+(2)n−1)2 −(x3
+4 −
+ε
+(2)n−1)2 −···−( xn
+(2)n − ε
+2)2 > 0.
+Let ρ = (x1)2 −∑n
+i=2(xi)2 > 0. Then, we have
+4ε2 +4x2ε −ρ < 0
+20ε2 +(8x3 −4x2)ε −ρ < 0
+...
+Since the second derivatives of all quadratic forms in these inequalities are positive, they are
+convex. Since coefficients of ε2 and ρ are strictly positive, they have two distinct roots. Thus,
+any epsilon in the intersection of all these intervals satisfies the required conditions. The in-
+tersection of them is non-empty since we have a valid choice ε = x1−∥x2:n∥
+2(n−1)
+as discussed in
+Section 3.3.
+□
+APPENDIX C. PROOF OF THEOREM 4.1
+Proof. Proof is similar to that of Theorem 3.1, showing that under the proposed setting (x,y,s)
+and (X,y,S) = M (x,y,s) satisfy the primal and dual constraints and preserves the objective
+function value. To this end, we need to take the following steps.
+Step 1. First, we show that if x ∈ FP1
+SOCO, then X = Arw(x) ∈ FPP
+SDO. First, we know that
+since x ∈ L n, then X = Arw(x) ⪰ 0. Next, we need to show that X satisfies the SDO primal
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+29
+constraint. Thus,
+Tr(⃗AiX) = 1
+nai1(
+n
+∑
+i=1
+Xii)+ 2
+2ai2X12 +...+ 2
+2ainX1n
+= ai1x1 +ai2x2 +...+ainxn
+= a(i)x = bi,
+for all i = 1,...,m.
+The second equality represents the product of ith row of matrix A and vector x. Hence, we can
+conclude that X satisfies the SDO primal constraint.
+Step 2. Next, we show that if (y,s) ∈ FD1
+SOCO, then (y,S) ∈ FDP
+SDO. By definition and using u
+and w in construction of S, we have S ⪰ 0. Next, we need to show that S satisfies the constraint
+of (DP
+SDO). Thus, since (y,s) is feasible for (D1
+SOCO), i.e. ∑m
+i=1 yiai j +sj = cj for all j = 1,...,n,
+we can write
+m
+∑
+i=1
+yi⃗Ai + ∑
+h̸=1,h<l
+whl ˜Ahl +
+n
+∑
+k=2
+uk ˆAk +S =
+
+
+1
+n ∑m
+i=1 yiai1
+1
+2 ∑m
+i=1 yiai2
+...
+1
+2 ∑m
+i=1 yiain
+1
+2 ∑m
+i=1 yiai2
+1
+n ∑m
+i=1 yiai1
+...
+...
+1
+2 ∑m
+i=1 yiain
+1
+n ∑m
+i=1 yiai1
+
+
++
+
+
+0
+0
+...
+...
+0
+0
+0
+w23
+...
+w2n
+...
+w23
+...
+...
+...
+...
+...
+...
+...
+w(n−1)n
+0
+w2n
+...
+w(n−1)n
+0
+
+
++
+
+
+∑n
+k=2 uk
+0
+...
+...
+...
+0
+0
+−u2
+0
+...
+...
+0
+...
+0
+...
+...
+...
+...
+...
+...
+−ui
+...
+...
+...
+...
+...
+...
+0
+0
+0
+...
+...
+0
+−un
+
+
++
+
+
+s1
+n −∑n
+k=2 uk
+s2
+2
+...
+...
+sn
+2
+s2
+2
+s1
+n +u2
+−w23
+...
+−w2n
+...
+−w23
+s1
+n +u3
+...
+...
+...
+...
+...
+...
+−w(n−1)n
+sn
+2
+−w2n
+...
+−w(n−1)n
+s1
+n +un
+
+
+=
+
+
+1
+nc1
+1
+2c2
+...
+1
+2cn
+1
+2c2
+1
+nc1
+...
+...
+1
+2cn
+1
+nc1
+
+ = ⃗C.
+
+30
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+Step 3. It is obvious that bTy = bTy. Similar to step 1, we have
+Tr(⃗CX) = 1
+nc1(
+n
+∑
+i=1
+Xii)+ 2
+2c2X12 +...+ 2
+2cnX1n
+= c1x1 +c2x2 +...+cnxn
+= cTx.
+Step 4. In this step, we need to show that if X ∈ FPP
+SDO, then x ∈ FP1
+SOCO. To this end, we
+know that as X = Arw(x) is positive semidefinite, then x ∈ L n. Next, we need to show that
+Ax = b. Thus, we have
+Aix = ai1x1 +ai2x2 +...+ainxn
+= 1
+nai1(nx1)+2(1
+2ai2)x2 +...+2(1
+2ain)xn
+= Tr(⃗AiX) = bi,
+for all i = 1,...,m.
+Step 5. Finally, we need to show that if (y,S) ∈ FDP
+SDO, then (y,s) ∈ FD1
+SOCO. Suppose that
+S ∈ FDP
+SDO, then it is positive semidefinite. We need to show that s ∈ L n. To do so, recall that
+in a positive semidefinite matrix every principal submatrix, in particular every 2-by-2 submatrix
+is positive semidefinite [6]. Thus,
+|Si j| ≤
+�
+SiiSj j
+for all i = 1,...,n and for all j = 1,...,n
+Thus, we have
+S2
+12 +S2
+13 +...+S2
+1n ≤ (S11S22)+(S11S33)+...+(S11Snn)
+(C.1)
+≤ S11(S22 +S33 +...+Snn),
+(C.2)
+�
+S2
+12 +S2
+13 +...+S2
+1n ≤
+�
+S11(S22 +S33 +...+Snn)
+(C.3)
+≤ S11 +S22 +S33 +...+Snn
+2
+,
+(C.4)
+where (C.4) is derived using the arithmetic-geometric mean inequality. The expression can be
+rewritten as
+n
+∑
+i=1
+Sii ≥ 2
+�
+S2
+12 +S2
+13 +...+S2
+1n
+=
+�
+(2S12)2 +(2S13)2 +...+(2S1n)2,
+or simply,
+s1 ≥
+�
+(s2)2 +...+(sn)2 = ||s2:n||2,
+which according to definition of (4.1) shows that s ∈ L n. Next, we need to show that vector
+s =
+
+
+c1 −∑m
+i=1yiai1
+...
+cn −∑m
+i=1yiain,
+
+
+satisfies the SOCO dual constraint. This is obvious since each entry of this vector is equal to
+the corresponding entry of vector s in the definition of (D1
+SOCO).
+□
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+31
+Acknowledgements
+This research is supported by the National Science Foundation (NSF) under Grant No. 2128527.
+REFERENCES
+[1] Farid Alizadeh and Donald Goldfarb. Second-order cone programming. Mathematical Programming,
+95(1):3–51, 2003.
+[2] Roberto Andreani, Gabriel Haeser, Leonardo M Mito, C H´ector Ram´ırez, and Thiago P Silveira. Global
+convergence of algorithms under constant rank conditions for nonlinear second-order cone programming.
+Journal of Optimization Theory and Applications, 195(1):42–78, 2022.
+[3] Aharon Ben-Tal and Arkadi Nemirovski. Lectures on Modern Convex Optimization: Analysis, Algorithms,
+and Engineering Applications. SIAM, 2001.
+[4] Etienne De Klerk. Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applica-
+tions, volume 65. Springer Science & Business Media, 2006.
+[5] Daniel Dueri, Jing Zhang, and Behc¸et Ac¸ikmese. Automated custom code generation for embedded, real-
+time second order cone programming. In Proceedings of the 19th IFAC World Congress. Ed. E. Boje, and X.
+Xia, volume 47, pages 1605–1612. Elsevier, 2014.
+[6] Roger A. Horn and Charles R. Johnson. Matrix Analysis. Cambridge University Press, 2012.
+[7] Qingwei Jin, Ye Tian, Zhibin Deng, Shu-Cherng Fang, and Wenxun Xing. Exact computable representation of
+some second-order cone constrained quadratic programming problems. Journal of the Operations Research
+Society of China, 1(1):107–134, 2013.
+[8] Ali Mohammad-Nezhad. Conic Optimization: Optimal Partition, Parametric, and Stability Analysis. PhD
+thesis, Lehigh University, 2019.
+[9] Chee-Khian Sim and Gongyun Zhao. A note on treating a second order cone program as a special case of a
+semidefinite program. Mathematical Programming, 102(3):609–613, 2005.
+[10] Henry Wolkowicz, Romesh Saigal, and Lieven Vandenberghe. Handbook of Semidefinite Programming: The-
+ory, Algorithms, and Applications, volume 27. Springer Science & Business Media, 2012.
+[11] Jinchuan Zhou, Jingyong Tang, and Jein-Shan Chen. Further relationship between second-order cone and
+positive semidefinite matrix cone. Optimization, 65(12):2115–2133, 2016.
+
diff --git a/JtFLT4oBgHgl3EQfKS8v/content/tmp_files/load_file.txt b/JtFLT4oBgHgl3EQfKS8v/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..4ae24d0c551f0f50bc28bfe51eb84639af931c29
--- /dev/null
+++ b/JtFLT4oBgHgl3EQfKS8v/content/tmp_files/load_file.txt
@@ -0,0 +1,1520 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf,len=1519
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='12007v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='OC]  27 Jan 2023  ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC  OPTIMIZATION PROBLEMS  POUYA SAMPOURMAHANI∗, MOHAMMADHOSSEIN MOHAMMADISIAHROUDI, TAM ´AS TERLAKY  Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA, USA, 18015  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Second-order conic optimization (SOCO) can be considered as a special case of semidefinite optimiza-  tion (SDO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In the literature it has been advised that a SOCO problem can be embedded in an SDO problem using  the arrow-head matrix transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' However, a primal-dual solution pair cannot be mapped simultaneously  using the arrow-head transformation as we might lose complementarity and duality in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' To address this  issue, we investigate the relationship between SOCO problems, and their SDO counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Through derivation of  standard semidefinite representations of SOCO problems, we introduce admissible mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We show that the  proposed mappings preserve both feasibility and optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Further, we discuss how the optimal partition of a  SOCO problem maps to the optimal partition of its SDO counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Second-order conic Optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Semidefinite Optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Semidefinite Representation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Mapping;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Optimal Partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' 90C25, 90C22, 90C99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' INTRODUCTION  In the hierarchy of convex optimization problems, second-order conic optimization (SOCO)  problems can be seen as a special case of semidefinite optimization (SDO) problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' SOCO  problems minimize a linear function over the intersection of an affine space with the Cartesian  product of second-order cones, also known as Lorentz cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' An SDO problem consists of  minimizing a linear objective function over the intersection of the cone of positive semidefinite  matrices with an affine space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' SDO encompasses other subclasses of conic optimization prob-  lems namely linear optimization (LO), and SOCO, in the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' This means that each one  can be represented as a special case of SDO [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  In this paper, we focus on the relationship of SOCO and SDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We investigate their relation-  ship in order to gain theoretical insight and realize how these problems get mapped to each  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Only a few papers [9, 11] were devoted to study this relationship from a theoretical point  of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Sim and Zhao [9], in particular, studied the relationship between a SOCO problem  and its counterpart SDO problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' They provided a mapping based on the direct correspon-  dence between the dual problems of SOCO and SDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Their SDO representation is defined on  the product of some cones of positive semidefinite matrices, which is a special case of standard  ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  E-mail address: pos220@lehigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='edu (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Sampourmahani), mom219@lehigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='edu (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Mohammadisiahroudi),  terlaky@lehigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='edu (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Terlaky).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Received xx, x, xxxx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Accepted xx, x, xxxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ©2023 Communications in Optimization Theory  1  2  SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY  SDO and needs further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In this paper, we extend their analysis by considering the actual  standard case which returns an SDO representation through a large positive semidefinite cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Furthermore, we propose a framework that allows full description of the point-to-set map  from SOCO to its SDO counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Then, we analyze how the optimal partition of a SOCO  problem is mapped to that of SDO, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' This is important in understanding the  relationship between these two problems as we are mapping between an index-based partition  and a subspace-based partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Throughout this paper, the following notation is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The Lorentz cone of dimension ni  is denoted by L ni, and Rn denotes the n-dimensional Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Superscripts are used  to represent cone-related information, and subscripts are used for matrix and vector entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  For a given matrix A, Ai j represents the (i, j)-th entry, while Ai denotes the i-th matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The  notation (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=') denotes the concatenation of the column vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The set of all p×q matrices  with real entries is denoted by Rp×q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' For a symmetric matrix X, X ⪰ 0 (X ≻ 0) means X is  positive semidefinite (positive definite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Furthermore, the trace operator is denoted by Tr(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  The remaining notations will be introduced at appropriate places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  This paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Section 2 reviews the preliminaries required for this pa-  per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Section 3 studies the relationship between SOCO and SDO relying on the correspondence  of dual problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Section 4 takes the other direction and proposes mappings focusing on corre-  spondence of primal problems as the starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Section 5 analyzes how the optimal partition  of SOCO maps to that of it’s SDO counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Section 6 concludes the paper, summarizing our  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' PRELIMINARIES  Let L ni denote the Lorentz cone of dimension ni, and L n = L n1 ×L n2 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='×L nr, where  n = ∑r  i=1 ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Then, the primal and dual SOCO problems are defined as follows  z∗  PSOCO := min (c1)Tx1 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+(cr)Txr  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' A1x1 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+Arxr = b,  xi ∈ L ni,  for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',r,  (PSOCO)  z∗  DSOCO := max bTy  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' (Ai)Ty+si = ci,  for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',r,  si ∈ L ni,  for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',r,  (DSOCO)  where ci ∈ Rni, Ai ∈ Rm×ni, b ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We define the feasible sets of the primal-dual problems as  follows,  FPSOCO = {(x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='.;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='xr) ∈ L n : A1x1 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+Arxr = b},  FDSOCO = {(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='.;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='sr) ∈ Rm ×L n : (Ai)Ty+si = ci for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',r},  and the sets of optimal solutions as  P∗  SOCO = {x∗ = (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='.;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='xr) ∈ FPSOCO : cTx∗ = z∗  PSOCO},  D∗  SOCO = {(y∗,s∗) = (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='.;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='sr) ∈ FDSOCO : bTy∗ = z∗  DSOCO},  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' An optimal solution of SOCO, if there exists any, is denoted by (x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='y∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS  3  Let Arw(·) denote the arrow-head (Lorentz) transformation [8, 10], with the structure of  Arw(xi) :=  � xi  1  (xi  2:ni)T  xi  2:ni  xi  1Ini−1  �  ,  where (xi  2:ni)T denotes the vector (xi  2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',xi  ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Then, the Jordan product is defined as  xi ◦si = Arw(xi)si = Arw(si)xi =  �  (xi)Tsi  xi  1si  2:ni +si  1xi  2:ni  �  ,  i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Any feasible solutions satisfying x◦s = 0 is called complementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Here, we have  x◦s := (x1 ◦s1,x2 ◦s2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',xr ◦sr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Feasible solution are complementary if and only if they are optimal with zero duality gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' An optimal solution (x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='y∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s∗) ∈ P∗  SOCO × D∗  SOCO is called maximally com-  plementary if x∗ ∈ ri(P∗  SOCO) and (y∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s∗) ∈ ri(D∗  SOCO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Further, (x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='y∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s∗) is called strictly  complementary if x∗ +s∗ ∈ int(L n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Next, we define the primal and dual SDO problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  z∗  PSDO := min Tr(CX)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Tr(AiX) = bi  for all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',m,  X ⪰ 0,  (PSDO)  z∗  PSDO := max bTy  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  m  ∑  i=1  yiAi +S = C,  S ⪰ 0,  (DSDO)  where X,S,C, and Ai for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',m are n×n symmetric matrices, and b,y ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We define  the feasible sets of SDO problems as  FPSDO = {X ∈ Sn : Tr(AiX) = bi,i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',m,X ⪰ 0},  FDSDO = {(y,S) ∈ Rm ×Sn :  m  ∑  i=1  yiAi +S = C,S ⪰ 0},  where Sn denotes the set of n ×n symmetric matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The sets of optimal solutions for a pair  of SDO problems are  P∗  SDO = {X∗ ∈ FPSDO : Tr(CX∗) = z∗  PSDO},  D∗  SDO = {(y∗,S∗) ∈ FDSDO : bTy∗ = z∗  DSDO}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  A feasible and an optimal solution of SDO are denoted as (X,y,S), and (X∗,y∗,S∗), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Any feasible solution (X,y,S) satisfying XS = 0 is called complementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Similar to SOCO, a  feasible solution is optimal and yields zero duality gap if and only if it is complementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' A primal-dual optimal solution (X∗,y∗,S∗) ∈ P∗  SDO × D∗  SDO is called maxi-  mally complementary if X∗ ∈ ri(P∗  SDO) and (y∗,S∗) ∈ ri(D∗  SDO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' A maximally complementary  optimal solution (X∗,y∗,S∗) is called strictly complementary if X∗ +S∗ ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  4  SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY  Next, we define the optimal partitions of SOCO and SDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The notion of the optimal partition  of LO can be extended to SOCO [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Even though a SOCO problem can be embedded in SDO,  the optimal partition in SOCO may be more nuanced when it is defined and analyzed directly  in the SOCO setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In SOCO, the index set {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',r} of the second-order cones is partitioned  into four subsets  ¯  B,  ¯  N , ¯  R, and  ¯  T , where  ¯  T is further partitioned to  ¯  T := ( ¯T1, ¯T2, ¯T3) as  follows,  ¯  B := { i | xi  1 > ||xi  2:ni||2, for some x ∈ P∗  SOCO},  ¯  N := { i | si  1 > ||si  2:ni||2, for some s ∈ D∗  SOCO},  ¯  R := { i | xi  1 = ||xi  2:ni||2 > 0,si  1 = ||si  2:ni||2 > 0, for some (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s) ∈ P∗  SOCO ×D∗  SOCO},  ¯  T1 := { i | xi = si = 0, for all (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s) ∈ P∗  SOCO ×D∗  SOCO},  ¯  T2 := { i | si = 0, for all (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s) ∈ D∗  SOCO, and xi  1 = ||xi  2:ni||2 > 0, for some x ∈ P∗  SOCO},  ¯  T3 := { i | xi = 0, for all x ∈ P∗  SOCO, and si  1 = ||si  2:ni||2 > 0, for some (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s) ∈ D∗  SOCO}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  It should be highlighted that, due to the convexity of the optimal set,  ¯  B,  ¯  N , ¯  R, and  ¯  T are  mutually disjoint and their union is the index set {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Therefore, it follows from the  complementarity condition that for all (˜x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' ˜y∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' ˜s∗) ∈ P∗  SOCO ×D∗  SOCO, ˜xi = 0 for all i ∈  ¯  N , and  ˜si = 0 for all i ∈ ¯  B [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  For SDO, let B := R(X∗) and N := R(S∗), where (X∗,y∗,S∗) is a maximally comple-  mentary optimal solution, meaning that we have R(X) ⊆ B and R(S) ⊆ N for all (X,y,S) ∈  P∗  SDO × D∗  SDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' By the complementarity condition, the subspaces B and N are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Moreover, let subspace T , be the orthogonal complement to B+N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The partition (B,N ,T )  of Rn is called the optimal partition of an SDO problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We can represent X∗ and S∗ using  a common eigenvector basis, Q∗, as X∗ = Q∗Λ(X∗)(Q∗)T, and S∗ = Q∗Λ(S∗)(Q∗)T, where  Λ(X∗) and Λ(S∗) corresponds to the diagonal matrices containing the eigenvalues of X∗ and S∗,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Thus, we have R(X∗) = R(Q∗Λ(X∗)), and R(S∗) = R(Q∗Λ(S∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In particular,  the columns of Q∗ corresponding to the positive eigenvalues of X∗ and S∗ can be chosen as an  orthonormal basis for B and N , respectively [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Current literature [1, 2, 3, 4, 5, 7, 9, 8, 10] suggests that a SOCO problem can be embedded  in an SDO problem using the arrow-head matrix transformation,  Arw(xi) :=  � xi  1  (xi  2:ni)T  xi  2:ni  xi  1Ini−1  �  ⪰ 0 ⇔ xi ∈ L ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1)  However, this transformation cannot be used to map both primal and dual solutions at the same  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Upon using the arrow-head representation of vectors xi and si simultaneously, we might  lose duality and complementarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The following example illustrates that we may lose comple-  mentarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Let (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s) be an optimal solution of SOCO and assume that there exist at least  one index i ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' For all i ∈ R, we can represent a solution as  xi = ζ i  \uf8eb  \uf8ed  1  xi  2:ni  ||xi  2:ni||2  \uf8f6  \uf8f8,  si = ξ i  \uf8eb  \uf8ed  1  si  2:ni  ||si  2:ni||2  \uf8f6  \uf8f8,  xi  2:ni  ||xi  2:ni||2  = −  si  2:ni  ||si  2:ni||2  ,  ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS  5  where ζ i = xi  1 ≥ 0, ξ i = si  1 ≥ 0, and for at least one (x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='y∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s∗) we have both ζ i,ξ i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Without  loss of generality, and for the sake of simplicity, assume that ζ i = ξ i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Moreover, let  uj =  xi  j  ||xi  2:ni||2  = −  si  j  ||si  2:ni||2  ,  j = 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',ni,  and u = (u2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='.;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='uni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Then, using the arrow-head matrix transformation, we have  Xi = Arw(xi) =  �1  uT  u  Ini−1  �  ,  Si = Arw(si) =  � 1  −uT  −u  Ini−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  While xi ◦si = 0, this transformation does not preserve complementarity as we have  XiSi =  �1  uT  u  Ini−1  �� 1  −uT  −u  Ini−1  �  =  �  0  [0]1×(n−1)  [0](n−1)×1  Ini−1 −uuT  �  ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='3 shows that the arrow-head matrix transformation is not sufficient to represent a  primal-dual pair of SOCO problems as an SDO problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Thus, it seems worth exploring the  actual relationship between an instance of SOCO and it’s SDO counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  To address this issue, Sim and Zhao [9] started from a SOCO dual problem and exploited the  arrowhead representation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1) of the dual SOCO problem, to obtain the SDO dual as follows,  max bTy  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  m  ∑  i=1  yiArw(aj  (i))+S j = Arw(cj) for all j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',r,  S j ⪰ 0 for all j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',r,  (DSZ)  where aj  (i) denotes ith row of the matrix A corresponding to Lorentz cone j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Observe that S j as a  linear combination of arrow-head matrices is an arrow-head matrix, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Using this dual model,  we get the SDO primal problem as  min  r  ∑  j=1  Tr(Arw(cj)X j)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  r  ∑  j=1  Tr(Arw(aj  (i))X j) = bi for all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',m,  X j ⪰ 0 for all j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  (PSZ)  They showed that X j = Arw(xj) is not a feasible solution for the SDO primal problem (PSZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  In fact, primal feasible solutions of (PSZ) are fully dense, and do not have an arrow-head  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' To fix this issue, they proposed the mapping  MR(xj) =  � 1  4θ j  1  2(xj  2:n)T  1  2xj  2:n  x j  1−∥x j  2:n∥  2(n−1) I + x j  2:n(x j  2:n)T  θ j  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='2)  where θ j = xj  1 +∥x j  2:n∥+  �  (xj  1 +∥xj  2:n∥)2 −4∥xj  2:n∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  In our study, a key concept is the notion of admissible map which is defined next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  6  SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' A mapping M is called admissible if it preserves feasibility and objective func-  tion value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  (x,y,s) ∈ FPSOCO ×FDSOCO ⇒ M (x,y,s) ∈ FPSDO ×FDSDO,  (X,y,S) ∈ FPSDO ×FDSDO ⇒ M −1(X,y,S) ∈ FPSOCO ×FDSOCO,  cTx = Tr(CX), bTy = bTy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  The mapping of Sim and Zhao [9] is admissible, and they proved that it maps a solution from  the boundary (interior) of the Lorentz cone to a solution on the boundary (interior) of the cone  of semidefinite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In this paper, we seek to extend their approach and explore mappings  that satisfy the definition of admissible mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Although the mapping of [9] is a point to  point map, the image of (x,y,s) might be a point or a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In addition, the SDO representation  of SOCO of [9], (PSZ) and (DSZ), is defined using the product of multiple cones of positive  semidefinite matrices, but we use a more general approach to get an SDO representation in  standard form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The major goal of this paper is clarifying more the relationship between SOCO  and the related SDO by developing different mappings and exploring the relationship between  the optimal partitions of these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Without loss of generality, in Sections 3 and 4, we first present the results in case of a single  second-order cone, and then we generalize the results to the multiple cone case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' To this end, we  consider the following primal and dual problems,  z∗  P1  SOCO = min  �  cTx : Ax = b, x ∈ L n�  ,  (P1  SOCO)  z∗  D1  SOCO = max  �  bTy : ATy+s = c, (y,s) ∈ Rm ×L n�  ,  (D1  SOCO)  with feasible sets FP1  SOCO = {x ∈ L n : Ax = b} and FD1  SOCO = {(y,s) ∈ Rm×L n : ATy+s = c},  and optimal solution sets P1  SOCO  ∗ = {x ∈ FP1  SOCO : cTx = z∗  P1  SOCO} and D1  SOCO  ∗ = {(y,s) ∈  FD1  SOCO : bTy = z∗  D1  SOCO}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' FROM SOCO TO SDO: STARTING FROM THE DUAL SIDE  One can derive the SDO counterpart of a SOCO problem starting with either the primal or  dual SOCO problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In this section, similar to [9], we initiate the derivation from the dual  side of SOCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Thus, as mentioned earlier, we preserve the arrow-head structure of the matrix S  corresponding to dual solution s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Derivation and Solution Mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We utilize the arrow-head transformation to the vec-  tor c and the rows of matrix A,  ⃗C = Arw(c),  ⃗Ai = Arw(a(i)),  i = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Since in the SOCO dual s = c−ATy, by applying the arrow-head structure to A and c, we have  that S = ⃗C −∑m  i=1 yi⃗Ai has the arrow-head structure, as it is a linear combination of arrow-head  matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Therefore, the SDO counterpart of the SOCO dual problem (D1  SOCO) is as follows,  z∗  DD  SDO := max  �  bTy :  m  ∑  i=1  yi⃗Ai +S = ⃗C, S ⪰ 0  �  ,  (DD  SDO)  ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS  7  which has the following dual,  z∗  PD  SDO := min  �  Tr(⃗CX) : Tr(⃗AiX) = bi,  i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',m,  X ⪰ 0  �  ,  (PD  SDO)  as it’s SDO primal problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' For the SDO problems (PD  SDO) and (DD  SDO), let  FDD  SDO = {(y,S) ∈ Rm ×Sn :  m  ∑  i=1  yi⃗Ai +S = ⃗C,S ⪰ 0},  FPD  SDO = {X ∈ Sn : Tr(⃗AiX) = bi,i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',m,X ⪰ 0}  represent the feasible sets, and  DD  SDO  ∗ = {(y,S) ∈ FDD  SDO : bTy = z∗  DD  SDO},  PD  SDO  ∗ = {X ∈ FPD  SDO : Tr(⃗CX) = z∗  PD  SDO},  represent the optimal solution sets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The following theorem provides a point to set  admissible mapping, see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='4 for r = 1, based on the (D1  SOCO) and (P1  SOCO), and their  representations (DD  SDO), (PD  SDO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Consider the SOCO problem pairs (P1  SOCO) and (D1  SOCO) with (x,y,s) ∈ FP1  SOCO ×  FD1  SOCO, and the SDO problem pairs (PD  SDO) and (DD  SDO) with (X,y,S) ∈ FPD  SDO × FDD  SDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Then, mapping (X,y,S) = M (x,y,s) with  S = Arw(s),  y = y,  X =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  X11  X12  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  X1n  X12  X22  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  X2n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  X1n  X2n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Xnn  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb ⪰ 0 with  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  ∑n  i=1Xii  X12  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  X1n  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  x1  x2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  xn  2  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb,  is a point-to-set admissible mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In addition, the inverse mapping denoted by (x,y,s) =  M −1(X,y,S), with  s = Arw−1(S),  y = y,  x =  �  ∑n  i=1 Xii,2X12,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',2X1n  �T ,  is a point-to-point admissible mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The proof of this theorem is presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  □  The following corollaries restate that the provided mapping preserves the objective function  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We have z∗  P1  SOCO = z∗  PD  SDO, and z∗  D1  SOCO = z∗  DD  SDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' A feasible solution (x,y,s) ∈ FP1  SOCO ×FD1  SOCO is optimal for a pair of SOCO  problems (P1  SOCO), and (D1  SOCO) with optimal value (z∗  P,z∗  D) if and only if the mapped solution  (X,y,S) is optimal for the SDO problems (PD  SDO), and (DD  SDO) with optimal value (z∗  P,z∗  D) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' A feasible solution (x,y,s) ∈ FP1  SOCO ×FD1  SOCO is optimal for a pair of SOCO  problems (P1  SOCO), and (D1  SOCO) with zero duality gap if and only if the mapped solution  (X,y,S) is optimal for the SDO problems (PD  SDO), and (DD  SDO) with zero duality gap, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',  x◦s = 0 ⇐⇒ Tr(XS) = 0  8  SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY  Note that these results are valid regardless of duality (strong / weak with gap), status of the  SOCO problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Furthermore, observe that one can propose different admissible mappings  that satisfies the conditions presented in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='2, we propose a rank-one  mapping, which is the simplest option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='3, we show that when x ∈ int(L n), full  rank mappings can also be obtained, which map a solution in the interior of SOCO to a solution  in the interior of the SDO cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Rank-one Mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In this section, we construct a rank-one matrix X for vector x that  satisfies the conditions in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Thus, we introduce the vector β ∈ Rn, and define the  rank-one matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  X = ββ T,  with  n  ∑  i=1  β 2  i = x1,  β1β j = xj  2 for all j = 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  We need to solve this n-variable-n-equation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The solution of this system is β = 0 if  x = 0, and if x ̸= 0 then  β =  1  �  2(x1 +δ)  (x1 +δ,x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',xn)T,  where δ =  �  (x1)2 −||x2:n||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' It is easy to see that if we are on the boundary of the second-  order cone, then δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Otherwise, we have δ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Using this vector, we can construct a  suitable matrix X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Consider the rank-one mapping with  X =  �  [0]n×n  if x = (0,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',0),  DMR1(x)  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1)  where  DMR1(x) = ββ T =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  x1+δ  2  x2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  xn  2  x2  2  x2  2  2[x1+δ]  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  x2xn  2[x1+δ]  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  xn  2  x2xn  2[x1+δ]  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  x2n  2[x1+δ]  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Then, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1) together with (y,S) = (y,Arw(s)) is a point-to-point admissible mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The proof is straightforward as it is enough to show that matrix DMR1(x) satisfies the  conditions in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Higher Rank Mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In this section, we show that when a SOCO solution is in the  interior of the cone, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' x ∈ int(L n), then we can use a full rank mapping, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  DMRn(x) =  n  ∑  i=1  β i(β i)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' There exist mappings DMR where rank(DMR(¯x)) = n for x ∈ int(L n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS  9  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The proof of this theorem is presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  □  One can easily modify Algorithm 1 (presented in proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='6) to map a solution  x ∈ int(L n) to a matrix DMRk(x) with rank 1 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' This means that the primal feasible set  of a SOCO can be mapped to different subsets of the SDO primal feasible region, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=', rank-one  mapping maps the primal feasible set of a SOCO to a one-dimensional face of the SDO primal  feasible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Recall that Theorem 1 of [9] proposes the mapping  MR(x) =  � 1  4θ  1  2xT  2:n  1  2x2:n  x1−∥x2:n∥  2(n−1) I + x2:nxT  2:n  θ  �  ,  where θ = x1 +∥x2:n∥+  �  (x1 +∥x2:n∥)2 −4∥x2:n∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' They showed that this map is admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Moreover, it has full rank when x1 > ∥x2:n∥, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' x ∈ int(L n), and it is a rank-one matrix when  x1 = ∥x2:n∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' This map also proves Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='6, while our proof follows a different approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  In our approach, we can generate different mappings and explore the feasible set of SDO by  changing parameter ε in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' However, to prove that the set of all maps with different  rank produced by our approach can build the whole feasible set of the SDO representation is an  ongoing research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Given the mapping of Sim and Zhao [9], consider the case in which the solution is on the  boundary of the second-order cone, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' x1 = ∥x2:n∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Then, θ = 2x1, and  MR1(x) =  � 1  2x1  1  2xT  2:n  1  2x2:n  x2:nxT  2:n  2x1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  We can see that this is exactly identical to the rank one mapping we presented in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  We can write it as  MR1(x) = ν1(ν1)T,  where  ν1 =  1  √  θ  �1  2θ,x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',xn  �T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  In order to construct MR(x) as the sum of rank-one matrices, we define  ν j =  �  x1 −∥x2:n∥  2(n−1)  ej  for j = 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n,  where ej is a unit vector with 1 in element j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' By this setting, we have  MR(x) =  n  ∑  j=1  ν j(ν j)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  If the solution x of the SOCO primal problem is on the boundary of the cone, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' x1 = ∥x2:n∥,  then we get  ν1 =  1  √2x1  (x1,x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',xn)T ,  ν j = 0,  for j = 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  10  SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY  This results in the rank one mapping MR1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' On the other hand, if the solution is in the interior  of the cone, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' x1 > ∥x2:n∥, then ν j ̸= 0 for j = 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n, and we can construct the full rank  mapping MR(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Note that one cannot take a combination of first k vectors ν j to construct a  rank-k mapping as it would not be admissible since it violates the conditions given in Theorem  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' sum of diagonals will not be equal to x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' To overcome this issue and construct a rank-k  mapping, let N ⊆ {2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n} with |N | = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' One can take the following definition of ν j,  ν j =  �  x1 −∥x2:n∥  2(k −1)  ej  for j ∈ N , and ν j = 0 for j /∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  The resulting matrix MRk(x) has rank k, and one can easily see that it satisfies the conditions  of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Considering k = n, this choice of ν j results in identical mapping to MR(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Next theorem shows that we can have mappings with different ranks when we are mapping a  solution from interior of the Lorentz cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Let ρ(x) = max{rank(M (x)) : M is admissible map}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We have  ρ(x) = n if x ∈ int(L n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ρ(x) = 1 if x ∈ ∂(L n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Proof of this theorem is similar to Theorem 1 of [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  □  In the next section, we generalize our result for the case there are multiple Lorentz cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Generalization to Multiple SOCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In this section, we extend our rank-one mapping to  the case with multiple second-order cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We can use similar conditions as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1 to  extend our mapping to the case of multiple second-order cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Although, instead of the “Arw”  operator, we need to introduce a new operator called “DArw” which constructs a block diagonal  matrix with arrow-head matrices of input vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' First, we transform the objective coefficient  vectors and coefficient matrices into proper block-diagonal structure as  ˜C = DArw(c1,c2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',cr) =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  ⃗C1  ⃗C2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ⃗Cr  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb,  where ⃗Ci is the arrow-head matrix corresponding to vector ci for all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Moreover, we  define a similar block diagonal matrix  ˜Aj = DArw(A1  j,A2  j,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',Ar  j) =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  ⃗A1  j  ⃗A2  j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ⃗Ar  j  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb  for all j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',m,  where Ai  j corresponds to the row j of matrix Ai for all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Now, the SDO representation  of the SOCO problem can be derived as (PSDO) and (DSDO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Compared to the SDO represen-  tation of [9], (PSZ) and (DSZ), our representation is a standard SDO over one cone of positive  ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS  11  semidefinite matrices of dimension n×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Given the introduced notations ˜C and ˜A, we have  z∗  P ˜D  SDO := min Tr( ˜C ˜X)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Tr( ˜Ai ˜X) = bi  for all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',m,  ˜X ⪰ 0,  (P ˜D  SDO)  and the dual problem is  z∗  D ˜D  SDO := max bTy  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  m  ∑  i=1  yi ˜Ai + ˜S = ˜C,  ˜S ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  (D ˜D  SDO)  Analogously, we can define the sets of feasible and optimal solutions for the problems (P ˜D  SDO)  and (D ˜D  SDO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  We can present the following theorem to specify admissible mappings for the general form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Consider the SOCO problem pairs (PSOCO) and (DSOCO) with  (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='.;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='xr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='.;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='sr) ∈ FPSOCO ×FDSOCO,  and SDO problem pairs (P ˜D  SDO) and (D ˜D  SDO) with ( ˜X, ˜y, ˜S) ∈ FP ˜D  SDO × FD ˜D  SDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Then, the fol-  lowing mapping, denoted by ( ˜X, ˜y, ˜S) = M (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='.;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='xr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='.;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='sr), with  ˜S = DArw(s1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='.;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='sr),  ˜y = y,  ˜X ⪰ 0  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  ∑ui+ni  j=ui ˜Xj j  ˜Xui,ui+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ˜Xui,ui+ni  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  xi  1  xi  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  xin  2  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  , for all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',r  coupled with the inverse mapping denoted by (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='.;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='xr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='.;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='sr) = M −1( ˜X, ˜y, ˜S) with  (s1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='.;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='sr) = DArw−1( ˜S),  y = ˜y,  (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='.;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='xr) =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  ∑ui+ni  j=ui ˜Xj j  2 ˜Xui,ui+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  2 ˜Xui,ui+ni  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb,  where ui = 1+∑i−1  k=0 ni and n0 = 0, is an admissible mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The proof of this theorem is analogous to that of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Note that a solution of (P ˜D  SDO) is not necessarily block-diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  12  SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY  Using the conditions in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='7, we can write  ˜S = DArw(s1,s2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',sr) =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  Arw(s1)  Arw(s2)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Arw(sr)  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb,  and,  ˜y = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  We can define matrix ˜X using our rank one mapping with vector ˜β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Vector ˜β consists of n  elements partitioned into r sub-vectors each corresponds to the cones in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Thus, we  have  ˜X = ˜β ˜β T,  where  ˜β = ((β 1)T,(β 2)T,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',(β r)T)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  One can calculate vector ˜β in the closed form as follows,  β i =  1  �  2(xi  1 +δ i)  �  xi  1 +δ i,xi  2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',xi  ni  �  ,  where δ i =  �  (xi  1)2 −||xi  2:n||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We can also have a separate rank-one mapping for each Lorentz  cone as ˜X = ∑r  i=1(β)i(β)T  i , where  (β j)i =  \uf8f1  \uf8f2  \uf8f3  1  √  2(xi  1+δ i)  �  xi  1 +δ i,xi  2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',xi  ni  �  if j = i,  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  In this case, ˜X is a block diagonal matrix with rank equal to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' If we choose (β j)i ∈ Rnj for  j ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',r}/{i}, then ˜X = ∑r  i=1(β)i(β)T  i will be a positive semidefinite matrix, which is not  necessarily block diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Since all the input matrices are block diagonal, it is straightforward  to check that the mapping will remain admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' For any Lorentz cone i with xi  1 > ∥xi  2:n∥, the  corresponding part of the matrix ˜X can have any rank from 1 to ni using higher rank mappings  proposed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' For instance, if the solution x is in the interior of all Lorentz  cones, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' xi  1 > ∥xi  2:n∥ for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',r, then we can find ˜X = ∑r  i=1∑nr  k=1(β)k  i (β)k  i  T whose rank is  n×r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In this case, it is straightforward to produce mapping with arbitrary rank form 1 to n×r  by using Algorithm 1 for filling diagonal blocks and fill other parts with arbitrary numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' FROM SOCO TO SDO: STARTING FROM THE PRIMAL SIDE  In this section, we analyze the case when we start to reformulate the standard primal SOCO  as an equivalent SDO problem, which requires more complex reformulation and was left un-  touched by Sim and Zhao [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The purpose is to investigate the SOCO-SDO relationship starting  from the primal side and answer the following questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  (1) What happens if we force matrix X to be arrow-head?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  (2) Does this derivation results in similar mappings between the SOCO problems and their  SDO counterparts?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS  13  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Derivation and Solution Mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Recall SOCO problems (P1  SOCO) and (D1  SOCO) and  their corresponding feasible and optimal solution sets from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Recall that we have  x ∈ L n if and only if X = Arw(x) ⪰ 0, thus starting from (P1  SOCO) requires a different choice  of ⃗C and ⃗Ai in order to properly represent the primal constraint of SOCO in an equivalent SDO  reformulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Thus, we define  ⃗C = Arw(1  nc1, 1  2c2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=', 1  ncn),  ⃗Ai = Arw(1  nai1, 1  2ai2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=', 1  2ain),  i = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  which leads to preserving aT  (i)x = Tr(⃗AiX) = bi, i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',m, where X = Arw(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Using the just  introduced arrow-head representations of c, a(i), and x, we write the following SDO problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  min  �  Tr(⃗CX) : Tr(⃗AiX) = bi,  i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',m,  X ⪰ 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  However, in this SDO problem X is a semidefinite matrix without arrow-head structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Thus, in  order to represent (P1  SOCO) using this SDO model, we need to enforce the arrow-head structure  on matrix X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In other words, we need to translate the arrow-head structure requirement into  linear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Thus, we introduce symmetric matrices ˇAhl for 2 ≤ h < l ≤ n with (ˇAhl)hl =  (ˇAhl)lh = 1 for 2 ≤ h < l ≤ n, and all other entries are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Moreover, we introduce matrix ˆAk,  with (ˆAk)11 = 1,(ˆAk)kk = −1 for k = 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n, and rest of the entries are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Therefore, we write  the following model  min Tr(⃗CX)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Tr(⃗AiX) = bi,  i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',m  Tr(ˇAhlX) = 0,  h = 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n−1, h < l ≤ n  Tr(ˆAkX) = 0,  k = 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n  X ⪰ 0,  (PP  SDO)  which is an accurate SDO representation of (P1  SOCO) as the arrow-head structure of the matrix  X is enforced by the linear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Let z∗  PP  SDO denote the optimal objective function value  of (PP  SDO), and  FPP  SDO = {X ∈ Sn :Tr(⃗AiX) = bi,i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',m,  Tr(ˇAhlX) = 0,h = 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n, h < l,  Tr(ˆAkX) = 0,k = 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n,  X ⪰ 0},  and  PP  SDO  ∗ = {X ∈ FPP  SDO : Tr(⃗CX) = z∗  PP  SDO},  14  SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY  be the feasible and optimal solution sets of problem (PP  SDO), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Next, we present the  dual model of the SDO problem (PP  SDO) as  max bTv  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  m  ∑  i=1  vi⃗Ai + ∑  h̸=1,h<l  whl ˇAhl +  n  ∑  k=2  ukˆAk +S =⃗C,  S ⪰ 0,  (DP  SDO)  where whl denotes the dual variable corresponding to the matrix dedicated for setting entries  (h,l) and (l,h) in X equal to zero, and uk corresponds to the linear constraints that are setting  elements on the diagonal of each block equal to each other for that specific block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Similarly, let z∗  DP  SDO denote the optimal objective function value for the dual model (DP  SDO), and  FDP  SDO =  �  (v,w,u,S) ∈ Rm ×R  (n−1)(n−2)  2  ×R(n−1) ×Sn :  m  ∑  i=1  vi⃗Ai + ∑  h̸=1,h<l  whl ˇAhl +  n  ∑  k=2  uk ˆAk +S =⃗C, S ⪰ 0  �  and  DP  SDO  ∗ = {(v,w,u,S) ∈ FDP  SDO : bTv = z∗  DP  SDO},  be the feasible and optimal solution sets of problem (DP  SDO), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Let’s analyze the structure of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' First, recall problem (D1  SOCO), where we have  s j = cj −  m  ∑  i=1  yiai j,  for all j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Considering that, we seek to preserve the objective function value throughout the mapping, we  have v = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Therefore, using the non-zero structure of the matrices ˇAhl and ˆAk, we have  S =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  1  n(s1)−∑n  k=2 uk  1  2(s2)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  1  2(sn)  1  2(s2)  1  n(s1)+u2  −w23  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  −w2n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  −w23  1  n(s1)+u3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  −w(n−1)n  1  2(sn)  −w2n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  −w(n−1)n  1  n(s1)+un  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1)  Furthermore, we have free variables w and u which can take values such that S is positive  semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Now, let y = (v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='u), and b = [b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='0 (n−1)(n−2)  2  ×1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='0n−1×1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Then, we can present the  following theorem which presents a point to set admissible mapping, see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='4, with  r = 1, based on (P1  SOCO) and (D1  SOCO), and their representations (PP  SDO) and (DP  SDO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Consider the SOCO problem pairs (P1  SOCO) and (D1  SOCO) with (x,y,s) ∈ FP1  SOCO ×  FD1  SOCO and SDO problem pairs (PP  SDO) and (DP  SDO) with (X,v,w,u,S) ∈ FPP  SDO × FDP  SDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS  15  Then, the mapping (X,v,w,u,S) = M (x,y,s) defined as  X = Arw(x),  v = y,  S =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  s1  n −∑n  k=2 uk  s2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  sn  2  s2  2  s1  n +u2  −w23  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  −w2n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  −w23  s1  n +u3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  −w(n−1)n  sn  2  −w2n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  −w(n−1)n  s1  n +un  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  ,  where w ∈ R  (n−1)(n−2)  2  and u ∈ R(n−1) taking any values such that S ⪰ 0 is a point-to-set admis-  sible mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The inverse mapping is denoted by (x,y,s) = M −1(X,v,w,u,S), with  x = Arw−1(X),  y = v,  s =  \uf8ee  \uf8ef\uf8f0  c1 −∑m  i=1viai1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  cn −∑m  i=1viain  \uf8f9  \uf8fa\uf8fb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The proof of this theorem is presented in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The proposed mapping in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1 is not necessarily a point to point mapping  regarding dual variables of (D1  SOCO) and (DP  SDO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Analogous to Corollaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='2-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='4, the proposed mapping in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1 preserves optimality  and complementarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Rank-one Mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Using the intuition from the dual side’s section, we seek to represent  matrix S using a rank one matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Hence, we propose  S = η(η)T,  with  n  ∑  i=1  (ηi)2 = s1,  η1ηj = s j  2 for all j = 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  We need to solve an n-variable-n-equation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The solution of this system is  η =  1  �  2(s1 +δ ′)  �  s1 +δ ′,s2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',sn  �T,  where δ ′ =  �  s1 −||s2:n||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Using this vector, we can construct matrix S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Thus, we can compute  uk for k = 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n, and whl for h ̸= 1,h < l as follows,  uk = (ηk)2 − 1  ns1 =  s2  k  2(s1 +δ ′) − 1  ns1,  k = 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n,  whl = ηhηl =  shsl  2(s1 +δ ′),  h ̸= 1,h < l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  16  SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Consider the rank-one mapping  S =  �  [0]n×n  if s = (0,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',0),  PMR1  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='2)  where  PMR1 = η(η)T =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  s1+δ ′  2  s2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  sn  2  s2  2  s2  2  2[s1+δ ′]  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  s2sn  2[s1+δ ′]  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  sn  2  s2sn  2[s1+δ ′]  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  s2n  2[s1+δ ′]  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Then, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='2) together with (X,y) = (Arw(x),y) is a point-to-point admissible mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The proof is similar to that of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Higher Rank Mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The derivation of full rank mapping is similar to the derivation  discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='3, as we show that when a SOCO solution s ∈ int(L n), then we can use  a full rank mapping, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  PMR =  n  ∑  i=1  ηi(ηi)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  The proof of existence of a full rank mapping is similar to the proof presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' There exist mappings PMR where rank(PMR(¯s)) = n for s ∈ int(L n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The proof of this theorem is similar to the proof presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The difference  is that we need to set π1 = s in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  □  The reason why the procedure to show the existence of full rank mapping is similar for both  directions is that although the mappings seem to be different, in fact both satisfy analogous  conditions about the first row and column and the trace of the mapping matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' From this point  of view, they are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In fact, the update vector in the procedure is meant to be satisfying  those conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Thus, we can follow the procedure in Appendix B for both mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Similar to Theorem 1 of [9], we can develop the following mapping for the primal side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  MR(s) =  � 1  4θ′  1  2sT  2:n  1  2s2:n  s1−∥s2:n∥  2(n−1) I + s2:nsT  2:n  θ ′  �  ,  where θ′ = s1 +∥s2:n∥ +  �  (s1 +∥s2:n∥)2 −4∥s2:n∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Here, we can perform the same analysis  we did in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='3, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Considering the case in which the solution is on the boundary of the  second-order cone, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' s1 = ∥s2:n∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Then, θ′ = 2s1, and  MR1(s) =  � 1  2s1  1  2sT  2:n  1  2s2:n  s2:nsT  2:n  2s1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  We can see that this is exactly identical to the rank one mapping we presented in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  We can also write it as  MR1(s) = γ1(γ1)T  ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS  17  where  γ1 =  1  √  θ′  �1  2θ′,s2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',sn  �T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  To obtain MR(s) as the sum of rank-one matrices, we define  γ j =  �  s1 −∥s2:n∥  2(n−1)  ej  forj = 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n,  where ej is a unit vector with 1 in element j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' By this setting, we have  MR(s) =  n  ∑  j=1  γ j(γ j)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  If the solution s of the SOCO dual problem is on the boundary of the cone, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' s1 = ∥s2:n∥, then  we get  γ1 =  1  √2s1  (s1,s2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',sn)T ,  γ j = 0,  for j = 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  This leads to the rank one mapping MR1(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' On the other hand, if the solution s ∈ int(L n),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' s1 > ∥s2:n∥, then γ j ̸= 0 for j = 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n, and we obtain the full rank mapping MR(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' To  construct an admissible rank-k mapping, here one cannot take a combination of first k vectors  γ j as it violates the conditions given in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' the sum of the diagonal elements will  not be equal to s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Recall the definition of N from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The correct choice of γ j to  construct a rank-k mapping is  γ j =  �  s1 −∥s2:n∥  2(k −1)  ej  for j ∈ N , and γ j = 0 for j /∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  The resulting matrix MR(s) has rank k, and one can easily see that it satisfies the conditions of  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Analogous to the dual side, we have the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Let ρ(s) = max{rank(M (s)) : M is admissible map}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We have  ρ(s) = n if s ∈ int(L n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ρ(s) = 1 if s ∈ ∂(L n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Proof of this theorem is similar to Theorem 1 of [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Generalization to Multiple SOCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Similar to the discussion in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='4, here we  adopt the mapping presented in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='8 to the primal side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Here we need to define the  following notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Let,  ˜C =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  ⃗C1  ⃗C2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ⃗Cr  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb,  where  ⃗Ck = Arw(1  nck  1, 1  2ck  2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=', 1  2ck  n)  and  ˜Aj = DArw(⃗A1  j,⃗A2  j,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',⃗Ar  j)  for all j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',m,  18  SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY  where  ⃗Ak  j = Arw(1  nak  i1, 1  2ak  i2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=', 1  2ak  in)  for all k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Similar to the structure of ˜C and ˜Aj, matrix ˜X needs to have a block-diagonal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Thus,  we need to enforce this structure on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' To this end, we define an operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Let Cf(i) = j, be an  operator which returns the corresponding cone j to input row i of the block-diagonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Let set I consists of all entries that need to be zero due to being either off-arrow within the  blocks, or the off-block-diagonal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  I =  �  (h,l) ∈ [1,  r  ∑  i=1  ni]2  �����  �  ∑  Cf(h)  i=0 ni < l,  ∑  Cf(h)−1  i=0  ni +1 < h < ∑  Cf(h)  i=0 ni,  and  h < l ≤ ∑  Cf(h)  i=0 ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  �  ,  K =  �  k ∈ [1,  r  ∑  i=1  ni]  �����  Cf(k)−1  ∑  i=0  ni +1 < k ≤  Cf(k)  ∑  i=0  ni  �  ,  where n0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Note that here we introduce matrices ˜ˇAhl and ˜ˆAk which are generalizations of  ˇAhl and ˆAk, for the multiple cone case, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In detail, ˜ˇAhl enforces all entries off-block-  diagonal and off-arrow within each block to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Moreover, ˜ˆAk guarantees that in each  block, diagonal entries are equal by setting entry (k,k) equal to −1 and entry (∑  Cf(k)−1  t=1  nt +  1,∑  Cf(k)−1  t=1  nt +1) equal to +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' By this notation, we have  min Tr(˜C ˜X)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Tr(˜Aj ˜X) = bj,  j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',m  Tr(˜ˇAhl ˜X) = 0,  (h,l) ∈ I  Tr(˜ˆAk ˜X) = 0,  k ∈ K  ˜X ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  (P ˜P  SDO)  Then, we dualize and get  max bTy  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  m  ∑  j=1  yj ˜Aj + ∑  h,l∈I  vhl ˜ˇAhl + ∑  k∈K  zk˜ˆAk + ˜S = ˜C,  ˜S ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  (D ˜P  SDO)  In similar fashion to other introduced models, we can define the sets of feasible and optimal  solutions corresponding to models (P ˜P  SDO) and (D ˜P  SDO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Next, we can present the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Consider the SOCO problem pairs (PSOCO) and (DSOCO) with  (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='.;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='xr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='.;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='sr) ∈ FPSOCO ×FDSOCO,  ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS  19  and SDO problem pairs (P ˜P  SDO) and (D ˜P  SDO) with ( ˜X, ˜y, ˜S) ∈ FP ˜P  SDO ×FD ˜P  SDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Then, the map-  ping ( ˜X, ˜y,v,z, ˜S) = M (x,y,s) defined as  ˜X = DArw(X1,X2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',Xr),  ˜y = y,  ˜S =  \uf8ee  \uf8ef\uf8f0  S1  S2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Sr  \uf8f9  \uf8fa\uf8fb ⪰ 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  ∑ui+ni  j=ui ˜Sj j  ˜Sui,ui+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ˜Sui,ui+ni  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  si  1  si  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  sin  2  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  with vectors v and z taking values such that ˜S ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Furthermore, the inverse mapping denoted  by (x,y,s) = M −1( ˜X, ˜y,v,z, ˜S), with  (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='.;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='xr) with xi =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  ˜xui,ui  ˜xui+1,ui  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ˜xui+ni,ui  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb,  y = ˜y,  (s1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='.;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='sr) with si =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  ∑ui+ni  j=ui ˜Sj j  2 ˜Sui,ui+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  2 ˜Sui,ui+ni  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb,  where ui = 1+∑i−1  k=0 ni and n0 = 0, is an admissible mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The proof of this theorem is analogous to that of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  □  We observe that although in the general case the mapping is similar to that of the dual side,  but the details of the mapping are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' This mapping is different from the other one since it  requires more work to enforce the arrow-head structure on matrix ˜X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' An analogous analysis on  different rank mappings can be done for the generalized multiple cone case, with the difference  that here we have the arrow-head structure for matrix ˜X, and we can compute matrix ˜S with  different ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We skip that similar analysis here for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Now, that we have the generalized standard mapping for both sides, we can proceed with  studying the mapping of the optimal partitions on both sides in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' MAPPING THE OPTIMAL PARTITION  As shown in the previous sections, the proposed mappings represent a solution of SOCO  depending on where it is located in the cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' For a solution on the boundary of the second-order  cone, all admissible maps provide a rank-one positive semidefinite matrix, while a solution in  the interior of the Lorentz cone can be mapped to semidefinite matrices with different ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  This correspondence can be analyzed in order to see how the optimal partition of SOCO is  mapped to that of the derived SDO counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' For mapping the optimal partition, maximally  complementary solutions are of particular interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' First, we define a helpful notation, and then  the following theorem discusses the preservation of maximal complementarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1 (Proper Map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Mapping M is a proper map if M is admissible and rank(X) =  ρ(x) for all x ∈ L n, or rank(S) = ρ(s) for all s ∈ L n  Based on this definition, a rank-one mapping is not proper but map MR of [9] is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In  the subsequent theorem, we show that a proper mapping preserves maximal complementarity  using eigenvalues of the mapped solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We used the mapping approach of Section 3 which  20  SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY  starts from dual side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Let λ X  i denotes the ith eigenvalue of a matrix X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Then, the eigenvalues  of an arrow-head matrix X = Arw(x) are λ X  1 = x1 − ∥x2:n∥, λ X  2 = ··· = λ X  n−1 = x1, and λ X  n =  x1 +∥x2:n∥, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' For a maximally complementary solution (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s) ∈ P∗  SOCO×D∗  SOCO, the mapped  solution ( ˜X, ˜y, ˜S) = M (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s) ∈ P ˜D∗  SDO ×D ˜D∗  SDO is maximally complementary if M is a proper  map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Given a maximally complementary solution (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s) ∈ P∗  SOCO ×D∗  SOCO, one can analyze  the result of mapping based how the cones are partitioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The following analysis holds for the  dual side direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In this direction, we have matrix ˜S as a block diagonal of arrow-head  matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Since M is proper map, the eigenvalues of matrix ˜X can be partitioned based on  Lorentz cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' If vector xi is on the non-zero boundary of Lorentz cone, the all corresponding  eigenvalues are non-zero except the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' If the vector xi is in the interior of Lorentz cone,  the corresponding eigenvalues are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' For the point of the second-order cone, all the  corresponding eigenvalues are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Thus, we have  if i ∈ ¯  B =  �  xi  1 > ∥xi  2:ni∥  → full rank matrix Xi with λ Xi  1 ,λ Xi  2 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',λ Xi  ni > 0,  si = 0  → zero Si matrix with λ Si  1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' = λ Si  ni = 0,  if i ∈  ¯  N =  �  xi = 0  → zero matrix Xi with λ Xi  1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' = λ Xi  ni = 0,  si  1 > ∥si  2:ni∥  → arrow-head matrix Si with λ Si  1 ,λ Si  2 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',λ Si  ni > 0,  if i ∈ ¯  R =  �  xi  1 = ∥xi  2:ni∥ > 0  → rank-one matrix Xi with λ Xi  1 > 0,λ Xi  2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' = λ Xi  ni = 0  si  1 = ∥si  2:ni∥ > 0  → arrow-head Si matrix with λ Si  1 = 0,λ Si  2 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',λ Si  ni > 0,  if i ∈ ¯  T1 =  �  xi = 0  → zero matrix Xi with λ Xi  1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' = λ Xi  ni = 0,  si = 0  → zero matrix Si with λ Si  1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' = λ Si  ni = 0,  if i ∈ ¯  T2 =  �  xi  1 = ∥xi  2:ni∥ > 0  → rank-one matrix Xi with λ Xi  1 > 0,λ Xi  2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' = λ Xi  ni = 0  si = 0  → zero matrix Si with λ Si  1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' = λ Si  ni = 0,  if i ∈ ¯  T3 =  �  xi = 0  → zero matrix Xi with λ Xi  1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' = λ Xi  ni = 0,  si  1 = ∥si  2:ni∥ > 0  → arrow-head matrix Si with λ Si  1 = 0,λ Si  2 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',λ Si  ni > 0,  Now, using that M is a proper, M maps a given maximally complementary optimal SOCO so-  lution to ( ˜X, ˜y, ˜S) ∈ P ˜D∗  SDO ×D ˜D∗  SDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Here, we show that this solution is maximally complemen-  tary for the SDO counterpart problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The proof goes by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Let us assume to the  contrary that ( ˜X, ˜y, ˜S) is not a maximally complementary solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Let ( ˆX, ˆy, ˆS) ∈ P ˜D∗  SDO×D ˜D∗  SDO  be a maximally complementary solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Since both ( ˜X, ˜y, ˜S) and ( ˆX, ˆy, ˆS) are optimal, then  ˆS ˜X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Accordingly, if λ ˜Si  j > 0 then λ ˆSi  j > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Now it is enough to show that there are no i, j  such that λ ˜Si  j = 0 and λ ˆSi  j > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' If there exist such index, by doing inverse mapping, the mapped  solution in SOCO will have larger partition set either for  ¯  N , ¯  R, or  ¯  T3 which contradicts with  assumption that solution (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s) being a maximally complementary solution of SOCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' With  similar reasoning we can show that there are no i, j such that λ ˜Xi  j  = 0 and λ ˆXi  j  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Thus, we  can conclude that a proper map preserves maximal complementarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  □  ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS  21  One can prove similar theorem as stated as follows for the mapping of Section 4 starting from  primal side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' For a maximally complementary solution (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s) ∈ P∗  SOCO×D∗  SOCO, the mapped  solution ( ˜X, ˜y, ˜S) = M (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='s) ∈ P ˜P∗  SDO ×D ˜P∗  SDO is maximally complementary if M is a proper  map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The proof is analogous to the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Mapping the Optimal Partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Based on the analysis described in the proof of The-  orem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='2, one can develop the mapping for the optimal partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Based on the sign of the  eigenvalues, if positive, their corresponding eigenvectors generate the subspaces B and N ,  and if zero, to subspace T of the optimal partition of SDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' First we consider mapping starting  from the dual side as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='4 and Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The summary of optimal parti-  tion mapping on the dual side is presented in Table 1, where ei  j represents the jth eigenvector  corresponding to λ j for the semidefinite cone i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  B  N  T  if i ∈ ¯  B  {ei  1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',ei  ni}  /0  /0  if i ∈  ¯  N  /0  {ei  1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',ei  ni}  /0  if i ∈ ¯  R  {ei  1}  {ei  2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',ei  ni}  /0  if i ∈ ¯T1  /0  /0  {ei  1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',ei  ni}  if i ∈ ¯T2  {ei  1}  /0  {ei  2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',ei  ni}  if i ∈ ¯T3  /0  {ei  2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',eini}  {ei  1}  TABLE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The Dual Side Optimal Partition Mapping  A similar analysis can be conducted for the primal side derivation, where matrix X has an  arrow-head structure and we represent matrix S using zero, rank-one or full rank matrices, based  on the position of vector s in the second-order cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The optimal partition mapping is presented  in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  22  SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY  B  N  T  if i ∈ ¯  B  {ei  1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',ei  ni}  /0  /0  if i ∈  ¯  N  /0  {ei  1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',ei  ni}  /0  if i ∈ ¯  R  {ei  2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',ei  ni}  {ei  1}  /0  if i ∈ ¯T1  /0  /0  {ei  1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',ei  ni}  if i ∈ ¯T2 {ei  2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',ei  ni}  /0  {ei  1}  if i ∈ ¯T3  /0  {ei  1}  {ei  2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',ei  ni}  TABLE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The Primal Side Optimal Partition Mapping  We can see that the result of mapping the optimal partition for the two sides are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  We observe identical outcome for partitions ¯  B,  ¯  N ,  ¯  T1, which was predictable as they include  the simple case where at least one of the variables x and s is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' However, notable difference  arises for the ¯  R, ¯T2, and ¯T3 partitions depending on whether the dual or the primal variable’s  representation is forced to be arrow-head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Consider partition ¯  R, where both x and s are on the non-zero boundary of the second-order  cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Adopting the dual (primal) side approach, variable s (x) will have arrow-head represen-  tation, and the other has a rank one representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We have seen in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='3, this partition  is where complementarity will not be preserved if we apply arrow-head representation on both  primal and dual sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' While our derivations guarantee feasibility and duality, we can see that  complementarity is also preserved as the arrow-head matrix has only one zero eigenvalue, while  the rank-one matrix has only one none-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' For partitions  ¯  T2 and  ¯  T3, the difference in tables  simply comes from which variable is non-zero and its corresponding arrow-head representation  eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  From the geometric point of view, we can see that for partitions ¯  B and  ¯  N , where the respec-  tive solution is in the interior of the second-order cone, the mapped solution is in the interior of  the semidefinite cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' For partition ¯  R, we observe that the solution pair, which both are on the  non-zero boundary of the second-order cone is mapped to a face of the semidefinite cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' For  partition  ¯  T1, the zero solution pair is mapped to the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Finally, for partitions  ¯  T2, and  ¯  T3,  the solutions are on the boundary of the second-order cone in the primal and dual, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  We can see that the result of the mapping is a face of the semidefinite cone, for both cases,  depending on if the primal or dual variable is on the boundary on the second-order cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' CONCLUSION  In this paper, we study the relationship between a SOCO problem and its SDO representa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Knowing about the fact that SOCO can be considered as a special case of SDO, we extend  the literature by investigating both the primal and the dual side SDO representations of a SOCO  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We demonstrate that using arrow-head matrix transformation on the primal or dual  SOCO problem, does not result in an arrow-head matrix variable on its dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In fact, nothing  forces the dual variable to be arrow-head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We usually end up with dense matrices which can be  represented by either rank-one or full rank mappings, based on the position of SOCO solutions  ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS  23  in the second-order cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We propose low-rank to full rank mappings which are admissible,  meaning that they preserve feasibility and objective function value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' First of all, these mappings  are not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' One can come up with different mappings that satisfies these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In ad-  dition, the dense structure of mapped solutions gives us an intuition about why solving SOCO  as a SOCO problem is more efficient than solving as an SDO problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In the SDO representa-  tion, we have to deal with dense matrices when solving the SDO counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Furthermore, we  investigated the relationship between the optimal partitions of these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The optimal par-  tition of SOCO is an index-based partition, while that of SDO is a subspace-based partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We  discussed how these partitions map to each other based on the eigenvalue analysis of mapped  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  The theoretical study of the relationship between SOCO and its SDO counterparts remains  with unsolved questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' One interesting question to look into is to study how degeneracy and  singularity degree properties are affected throughout mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  24  SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY  APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' PROOF OF THEOREM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' To prove that the presented mapping in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1 is admissible, we need to show that  it complies with the definition of admissible mapping, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' preservation of feasibility and the  objective function value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' First, we show that if (y,s) ∈ FD1  SOCO, then (y,S) = (y,Arw(s)) ∈ FDD  SDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Since S =  Arw(s) and s ∈ L n, we know that S is positive semidefinite [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The only thing remains to  prove is that ∑n  i=1 yi⃗Ai +S = ⃗C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Since (y,s) is feasible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=', ATy+s = c, we have  n  ∑  i=1  yi⃗Ai +S =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  ∑n  i=1yiai1 +s1  ∑n  i=1yiai2 +s2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ∑n  i=1yiain +sn  ∑n  i=1yiai2 +s2  ∑n  i=1yiai1 +s1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ∑n  i=1yiain +sn  ∑n  i=1yiai1 +s1  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb  =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  c1  c2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  cn  c2  c1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  cn  c1  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb = ⃗C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We show that if x = (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',xn) ∈ FP1  SOCO, then matrix X as defined in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1,  belongs to FPD  SDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' By construction, X is positive semidefinite and we need to just show that  Tr(⃗AiX) = bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Based on the construction of X and feasibility of x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=', Ax = b, we have  Tr(⃗AiX) = ai1(  n  ∑  i=1  Xii)+2ai2X12 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+2ainX1n  = ai1x1 +ai2x2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+ainxn = bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' It is obvious that bTy = bTy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Similar to step 2, we have  Tr(⃗CX) = c1(  n  ∑  i=1  Xii)+2c2X12 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+2cnX1n  = c1x1 +c2x2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+cnxn = cTx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In this step, we show that if X ∈ FPD  SDO, then x ∈ FP1  SOCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Suppose that X ∈ FPD  SDO,  then it is positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We need to show that x ∈ L n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' To do so, we utilize that in a  positive semidefinite matrix every principal submatrix, in particular every 2-by-2 submatrix is  positive semidefinite [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Thus,  |Xi j| ≤  �  XiiXj j  for all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n and for all j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n  Thus, we have  X2  12 +X2  13 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+X2  1n ≤ (X11X22)+(X11X33)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+(X11Xnn)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1)  ≤ X11(X22 +X33 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+Xnn),  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='2)  �  X2  12 +X2  13 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+X2  1n ≤  �  X11(X22 +X33 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+Xnn)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='3)  ≤ X11 +X22 +X33 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+Xnn  2  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='4)  ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS  25  where (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='4) is derived using the arithmetic-geometric mean inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The expression can be  rewritten as  n  ∑  i=1  Xii ≥ 2  �  X2  12 +X2  13 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+X2  1n  =  �  (2X12)2 +(2X13)2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+(2X1n)2,  or simply,  n  ∑  i=1  Xii ≥  �  (2X12)2 +(2X13)2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+(2X1n)2,  x1 ≥  �  (x2)2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+(xn)2 = ||x2:n||2,  which shows that x ∈ L n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Now, we need to prove that Ax = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Thus, we have  Aix = ai1x1 +ai2x2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+ainxn  for all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',m,  = ai1(  n  ∑  i=1  Xii)+ai2(2X12)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+ain(2X1n)  = Tr(⃗AiX) = bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Finally, we need to show that if (y,S) ∈ FDD  SDO, then (y,s) ∈ FD1  SOCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' As (y,S) =  (y,Arw(s)), according to [1], we know that S = Arw(s) ⪰ 0, implies s ∈ L n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Then, the only  property that remains to prove is that ∑m  i=1 yiAi + s = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' By feasibility of (y,S), we know that  ∑n  i=1yi⃗Ai +S = ⃗C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  ∑n  i=1 yiai1 +s1  ∑n  i=1 yiai2 +s2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ∑n  i=1 yiain +sn  ∑n  i=1 yiai2 +s2  ∑n  i=1 yiai1 +s1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ∑n  i=1 yiain +sn  ∑n  i=1 yiai1 +s1  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  c1  c2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  cn  c2  c1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  cn  c1  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb,  where each element of the matrix is a constraint in (D1  SOCO) which means ∑m  i=1 yiAi +s = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Considering all of the previous steps, we conclude that presented mapping in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1 is  an admissible mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  □  26  SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY  APPENDIX B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' PROOF OF THEOREM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='6  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' A rank-n mapping of ¯x can be constructed by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Algorithm 1 Full Rank Mapping  π1 = ¯x,  Choose sufficiently small ε  for k = 1 : n−1 do  τk ← πk  2  �  τk  k+1 = τk  k+1 +ε  if πk  k+1 ≥ 0  τk  k+1 = τk  k+1 −ε  if πk  k+1 < 0  Calculate  β k =  1  2  �  τk  1+ ˆδ k  2  (τk  1 + ˆδ k,τk  2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',τk  n)T, where ˆδ k =  �  (τk  1)2 −||τk  2:n||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  πk+1 = πk −τk  end for  Calculate  β n =  1  2  �  πn  1+ ˆδ n  2  (πn  1 + ˆδ n,πn  2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',πn  n)T, where ˆδ n =  �  (πn  1)2 −||πk  2:n||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  If ∥πk  2:n∥ < πk  1 for all k, then we have  0 < ||πk  2:n||  2  ≤ ||τk  2:n|| < τk  1 ≤ πk  1  2 < πk  1,  \uf8f1  \uf8f2  \uf8f3  πk  j  2 ≤ τk  j  if π j ≥ 0  πk  j  2 ≥ τk  j  if π j < 0  ,  j = 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1)  We need to show that in all loops ∥πk  2:n∥ < πk  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' For k = 1 this holds since ∥x2:n∥ < x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Using  induction, we need to prove ∥πk+1  2:n ∥ < πk+1  1  assuming ∥πk  2:n∥ < πk  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We have  πk+1  1  −∥πk+1  2:n ∥ = (πk  1 −τk  1)−∥(πk  j −τk  j)2:n∥ ≥ τk  1 −∥τk  2:n∥ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Thus, πk  1 > ∥πk  2:n∥ for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The proposed MR is also admissible since  n  ∑  j=1  MRj j =  n  ∑  k=1  n  ∑  j=1  (β k  j )2 = x1  MR1j =  n  ∑  k=1  β k  j β k  1 = x j  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  We need to prove that the generated β k vectors are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We have  β k =  1  2  �  τk  1+ ˆδ k  2  (τk  1 + ˆδ,τk  2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',τk  n)T, where ˆδ k =  �  (τk  1)2 −||τk  2:n||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS  27  Let us define the matrix  B =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  τ1  1 + ˆδ 1  τ2  1 + ˆδ 2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  τn  1 + ˆδ n  τ1  2  τ2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  τn  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  τ1  n  τ2  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  τn  n  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb  We still need to prove that B has full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Now, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' we may assume that x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' One can  easily extend the following proof to general x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Then we have  B =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  τ1  1 + ˆδ 1  τ2  1 + ˆδ 2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  τn  1 + ˆδ n  τ1  2  τ2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  τn  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  τ1  n  τ2  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  τn  n  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb  =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  π1  1  2 + ˆδ 1  π2  1  2 + ˆδ 2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  πn  1 + ˆδ n  π1  2  2 +ε  π2  2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  πn  2  π1  3  2  π2  3  2 +ε  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  πn  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  π1n  2  π2n  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  πn  n  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  x1  2 + ˆδ 1  x1  4 + ˆδ 2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  x1  (2)n + ˆδ n  x2  2 +ε  x2  4 − ε  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  x1  (2)n −  ε  (2)n−1  x3  2  x3  4 +ε  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  x1  (2)n −  ε  (2)n−2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  xn  2  xn  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  x1  (2)n − ε  (2)  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  x1  2  x1  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  x1  (2)n  x2  2  x2  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  x1  (2)n  x3  2  x3  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  x1  (2)n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  xn  2  xn  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  x1  (2)n  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  +  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  ˆδ 1  ˆδ 2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ˆδ n−1  ˆδ n  ε  −ε  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  −  ε  (2)n−2  −  ε  (2)n−1  0  ε  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  −  ε  (2)n−3  −  ε  (2)n−2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ε  − ε  (2)  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  ,  where  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  ˆδ 1  ˆδ 2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ˆδ n  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  �  (x1  2 )2 −(x2  2 +ε)2 −(x3  2 )2 −···−(xn  2 )2  �  (x1  4 )2 −(x2  4 − ε  2)2 −(x3  4 +ε)2 −···−(xn  4 )2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  �  ( x1  (2)n)2 −( x1  (2)n −  ε  (2)n−1)2 −(x3  4 −  ε  (2)n−1)2 −···−( xn  (2)n − ε  2)2  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  28  SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY  By doing row eliminations on both matrices to sparsify the second matrix, we get  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  x1  2  2x1  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  n x1  (2)n  x2  2  2x2  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  n x1  (2)n  x3  2  2x3  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  n x1  (2)n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  xn  2  2xn  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  n x1  (2)n  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  +  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  0  ¯δ  ε  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  0  0  0  ε  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ε  0  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  where  ¯δ =  ˆδ 1  (2)n−1 +  ˆδ 2  (2)n−2 +···+  ˆδ n−1  2  + ˆδ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  As we can see ¯δ, as sum of strictly positive numbers, is strictly positive, and the matrix has  full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' By choosing small ε, we can show that B has full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The last piece is to find  appropriate value for ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We need to choose ε so that  (x1  2 )2 −(x2  2 +ε)2 −(x3  2 )2 −···−(xn  2 )2 > 0  (x1  4 )2 −(x2  4 − ε  2)2 −(x3  4 +ε)2 −···−(xn  4 )2 > 0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  ( x1  (2)n)2 −( x1  (2)n −  ε  (2)n−1)2 −(x3  4 −  ε  (2)n−1)2 −···−( xn  (2)n − ε  2)2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Let ρ = (x1)2 −∑n  i=2(xi)2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Then, we have  4ε2 +4x2ε −ρ < 0  20ε2 +(8x3 −4x2)ε −ρ < 0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Since the second derivatives of all quadratic forms in these inequalities are positive, they are  convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Since coefficients of ε2 and ρ are strictly positive, they have two distinct roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Thus,  any epsilon in the intersection of all these intervals satisfies the required conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The in-  tersection of them is non-empty since we have a valid choice ε = x1−∥x2:n∥  2(n−1)  as discussed in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  □  APPENDIX C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' PROOF OF THEOREM 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Proof is similar to that of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1, showing that under the proposed setting (x,y,s)  and (X,y,S) = M (x,y,s) satisfy the primal and dual constraints and preserves the objective  function value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' To this end, we need to take the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' First, we show that if x ∈ FP1  SOCO, then X = Arw(x) ∈ FPP  SDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' First, we know that  since x ∈ L n, then X = Arw(x) ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Next, we need to show that X satisfies the SDO primal  ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS  29  constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Thus,  Tr(⃗AiX) = 1  nai1(  n  ∑  i=1  Xii)+ 2  2ai2X12 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+ 2  2ainX1n  = ai1x1 +ai2x2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+ainxn  = a(i)x = bi,  for all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  The second equality represents the product of ith row of matrix A and vector x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Hence, we can  conclude that X satisfies the SDO primal constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Next, we show that if (y,s) ∈ FD1  SOCO, then (y,S) ∈ FDP  SDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' By definition and using u  and w in construction of S, we have S ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Next, we need to show that S satisfies the constraint  of (DP  SDO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Thus, since (y,s) is feasible for (D1  SOCO), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' ∑m  i=1 yiai j +sj = cj for all j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n,  we can write  m  ∑  i=1  yi⃗Ai + ∑  h̸=1,h<l  whl ˜Ahl +  n  ∑  k=2  uk ˆAk +S =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  1  n ∑m  i=1 yiai1  1  2 ∑m  i=1 yiai2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  1  2 ∑m  i=1 yiain  1  2 ∑m  i=1 yiai2  1  n ∑m  i=1 yiai1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  1  2 ∑m  i=1 yiain  1  n ∑m  i=1 yiai1  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb  +  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  0  0  0  w23  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  w2n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  w23  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  w(n−1)n  0  w2n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  w(n−1)n  0  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  +  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  ∑n  k=2 uk  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
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+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  0  0  −u2  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
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+page_content='  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
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+page_content='  −ui  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
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+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
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+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  0  −un  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  +  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  s1  n −∑n  k=2 uk  s2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  sn  2  s2  2  s1  n +u2  −w23  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  −w2n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  −w23  s1  n +u3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  −w(n−1)n  sn  2  −w2n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  −w(n−1)n  s1  n +un  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  1  nc1  1  2c2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  1  2cn  1  2c2  1  nc1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  1  2cn  1  nc1  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb = ⃗C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  30  SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' It is obvious that bTy = bTy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Similar to step 1, we have  Tr(⃗CX) = 1  nc1(  n  ∑  i=1  Xii)+ 2  2c2X12 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+ 2  2cnX1n  = c1x1 +c2x2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+cnxn  = cTx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In this step, we need to show that if X ∈ FPP  SDO, then x ∈ FP1  SOCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' To this end, we  know that as X = Arw(x) is positive semidefinite, then x ∈ L n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Next, we need to show that  Ax = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Thus, we have  Aix = ai1x1 +ai2x2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+ainxn  = 1  nai1(nx1)+2(1  2ai2)x2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+2(1  2ain)xn  = Tr(⃗AiX) = bi,  for all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Finally, we need to show that if (y,S) ∈ FDP  SDO, then (y,s) ∈ FD1  SOCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Suppose that  S ∈ FDP  SDO, then it is positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' We need to show that s ∈ L n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' To do so, recall that  in a positive semidefinite matrix every principal submatrix, in particular every 2-by-2 submatrix  is positive semidefinite [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Thus,  |Si j| ≤  �  SiiSj j  for all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n and for all j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=',n  Thus, we have  S2  12 +S2  13 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+S2  1n ≤ (S11S22)+(S11S33)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+(S11Snn)  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1)  ≤ S11(S22 +S33 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+Snn),  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='2)  �  S2  12 +S2  13 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+S2  1n ≤  �  S11(S22 +S33 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+Snn)  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='3)  ≤ S11 +S22 +S33 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+Snn  2  ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='4)  where (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='4) is derived using the arithmetic-geometric mean inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' The expression can be  rewritten as  n  ∑  i=1  Sii ≥ 2  �  S2  12 +S2  13 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+S2  1n  =  �  (2S12)2 +(2S13)2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+(2S1n)2,  or simply,  s1 ≥  �  (s2)2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='+(sn)2 = ||s2:n||2,  which according to definition of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='1) shows that s ∈ L n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Next, we need to show that vector  s =  \uf8ee  \uf8ef\uf8f0  c1 −∑m  i=1yiai1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  cn −∑m  i=1yiain,  \uf8f9  \uf8fa\uf8fb  satisfies the SOCO dual constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' This is obvious since each entry of this vector is equal to  the corresponding entry of vector s in the definition of (D1  SOCO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  □  ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS  31  Acknowledgements  This research is supported by the National Science Foundation (NSF) under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' 2128527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  REFERENCES  [1] Farid Alizadeh and Donald Goldfarb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Second-order cone programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Mathematical Programming,  95(1):3–51, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  [2] Roberto Andreani, Gabriel Haeser, Leonardo M Mito, C H´ector Ram´ırez, and Thiago P Silveira.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Global  convergence of algorithms under constant rank conditions for nonlinear second-order cone programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Journal of Optimization Theory and Applications, 195(1):42–78, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  [3] Aharon Ben-Tal and Arkadi Nemirovski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Lectures on Modern Convex Optimization: Analysis, Algorithms,  and Engineering Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' SIAM, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  [4] Etienne De Klerk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applica-  tions, volume 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Springer Science & Business Media, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  [5] Daniel Dueri, Jing Zhang, and Behc¸et Ac¸ikmese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Automated custom code generation for embedded, real-  time second order cone programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' In Proceedings of the 19th IFAC World Congress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Boje, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  Xia, volume 47, pages 1605–1612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Elsevier, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  [6] Roger A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Horn and Charles R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Johnson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Matrix Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Cambridge University Press, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  [7] Qingwei Jin, Ye Tian, Zhibin Deng, Shu-Cherng Fang, and Wenxun Xing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Exact computable representation of  some second-order cone constrained quadratic programming problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Journal of the Operations Research  Society of China, 1(1):107–134, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  [8] Ali Mohammad-Nezhad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Conic Optimization: Optimal Partition, Parametric, and Stability Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' PhD  thesis, Lehigh University, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  [9] Chee-Khian Sim and Gongyun Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' A note on treating a second order cone program as a special case of a  semidefinite program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Mathematical Programming, 102(3):609–613, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  [10] Henry Wolkowicz, Romesh Saigal, and Lieven Vandenberghe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Handbook of Semidefinite Programming: The-  ory, Algorithms, and Applications, volume 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Springer Science & Business Media, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content='  [11] Jinchuan Zhou, Jingyong Tang, and Jein-Shan Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Further relationship between second-order cone and  positive semidefinite matrix cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
+page_content=' Optimization, 65(12):2115–2133, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFLT4oBgHgl3EQfKS8v/content/2301.12007v1.pdf'}
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+Platoon Leader Selection, User Association and
+Resource Allocation on a C-V2X based highway: A
+Reinforcement Learning Approach
+Mohammad Farzanullah and Tho Le-Ngoc
+Department of Electrical & Computer Engineering, McGill University, Montr´eal, QC, Canada
+Abstract—We consider the problem of dynamic platoon leader
+selection, user association, channel assignment, and power allo-
+cation on a cellular vehicle-to-everything (C-V2X) based high-
+way, where multiple vehicle-to-vehicle (V2V) and vehicle-to-
+infrastructure (V2I) links share the frequency resources. There
+are multiple roadside units (RSUs) on a highway, and vehicles can
+form platoons, which has been identified as an advanced use case
+to increase road efficiency. The traditional optimization methods,
+requiring global channel information at a central controller, are
+not viable for high-mobility vehicular networks. To deal with this
+challenge, we propose a distributed multi-agent reinforcement
+learning (MARL) for resource allocation (RA). Each platoon
+leader, acting as an agent, can collaborate with other agents for
+joint sub-band selection and power allocation for its V2V links,
+and joint user association and power control for its V2I links.
+Moreover, each platoon can dynamically select the vehicle most
+suitable to be the platoon leader. We aim to maximize the V2V
+and V2I packet delivery probability in the desired latency using
+the deep Q-learning algorithm. Simulation results indicate that
+our proposed MARL outperforms the centralized hill-climbing
+algorithm, and platoon leader selection helps to improve both
+V2V and V2I performance.
+Index Terms—New radio, cellular vehicle-to-everything, rein-
+forcement learning, resource allocation
+I. INTRODUCTION
+Cellular vehicle-to-everything (C-V2X) is a vehicular stan-
+dard that enables communication between vehicles and other
+entities on the road, such as pedestrians and infrastructure, to
+increase road safety and efficiency [1]. The C-V2X system
+consists of communication between different entities, such
+as vehicle-to-vehicle (V2V), vehicle-to-infrastructure (V2I),
+vehicle-to-network (V2N), and vehicle-to-pedestrian (V2P)
+communications. The C-V2X is envisioned to support high
+throughput, ultra-reliable, and low latency communications.
+3GPP has provided the 5G new radio (NR) standards
+for vehicular communications concerning architecture in TS
+23.287 [2], advanced use cases in TS 22.886 [3], and eval-
+uation methodology in TR 37.885 [4]. One component of
+infrastructure will be the roadside units (RSUs). RSU is a
+stationary wireless C-V2X device that can exchange messages
+with vehicles and other C-V2X entities. It uses the PC5
+side-link interface to communicate with the vehicles and
+transmit information about road signs and traffic lights [3].
+It can also receive information from the vehicles to make
+a dynamic map of the surroundings and share it with other
+vehicles/pedestrians. Furthermore, we consider the use-case of
+platooning, where multiple vehicles form a train-like structure
+and travel closely together in a line. The platoon leader (PL)
+organizes communications between vehicles. Vehicle platoon-
+ing has been identified as an advanced use case in [3] and has
+gained significant interest since it reduces fuel consumption
+and traffic congestion. The RSU and platoon will need to
+exchange a maximum of 1200 bytes in 500 ms for real-
+time traffic updates and 600 bytes in 10 ms for conditional
+automated driving [3]. We grouped these two requirements
+for an aggregate of 624 bytes in 10 ms. Moreover, the PL
+and members need to exchange 50-1200 bytes in 10 ms for
+cooperative driving and up to 2000 bytes in 10 ms for collision
+avoidance [3]. We aggregate these service requirements and
+keep the exchange of 1200-2800 bytes in 10 ms for our
+simulations. An intelligent resource allocation (RA) design is
+necessary for these stringent requirements.
+The 5G new radio (NR) C-V2X supports two RA modes
+for sidelink PC5 communications: mode 1, the under-coverage
+mode, and mode 2, the out-of-coverage mode [5]. In mode 1,
+the gNB allocates the communication resources to vehicles.
+Meanwhile, in mode 2, the vehicles autonomously select
+the resources. For mode 2, the current RA technique used
+in standards is the sensing-based Semi-Persistent Scheduling
+(SPS) algorithm, which periodically selects random resources.
+However, the probability of resource selection collision can be
+high, and many works have considered either improving the
+SPS algorithm or alternate techniques to increase reliability.
+[6] proposes a novel sensing-based SPS algorithm in an urban
+scenario to reduce the collision probability. The authors in
+[7] considered a highway scenario and suggested a stochastic
+reservation scheme for aperiodic traffic. For the platooning
+use case, [8] shows that the SPS algorithm can not achieve
+the required performance.
+Due to the fast channel variations in vehicular networks,
+centralized optimization schemes that require global channel
+state information (CSI) will no longer be feasible. The high
+CSI overhead and the corresponding increase in latency make
+such methods impractical. To deal with this issue, distributed
+RA algorithms have been suggested in the literature, e.g., [9],
+[10]. Furthermore, traditional optimization techniques have
+limitations, requiring complete information about the environ-
+ment and needing to be retrained for rapidly varying envi-
+ronments [11]. Recently, distributed multi-agent reinforcement
+learning (MARL) has been proposed as an alternative approach
+to resolving such issues. The authors in [12] used Deep Q
+Networks (DQN) for joint channel assignment and power
+allocation to maximize the V2V delivery probability and V2N
+arXiv:2301.03145v1  [cs.MA]  9 Jan 2023
+
+sum-rate in an urban setting, where the V2V links share the
+time-frequency resources with V2N links. Inspired by these
+results, [13] used double DQN for a platoon-based scenario
+for the same objectives. [14] uses the actor-critic method for
+mode selection, subchannel selection, and power control in an
+urban platoon scenario to increase the transmission probability
+of cooperative awareness messages.
+This paper considers a highway C-V2X system consisting
+of multiple platoons. We consider the periodic payload deliv-
+ery from PLs to RSUs, termed V2I links. Furthermore, we
+consider the periodic transmission of messages from PLs to
+members, termed V2V links. We assume a limited spectrum
+is available for the V2V and V2I transmission, and they share
+the frequency resources for efficient spectrum usage. Given
+this system, this paper formulates a dynamic PL selection,
+user association, channel assignment, and power level control
+to maximize the packet delivery probability for both V2V and
+V2I links. Reliability is defined as the successful transmission
+of the packet within a time constraint T [12], [13].
+We utilize MARL in a distributed manner. The RL works
+on a trial-and-error strategy, and each agent slowly improves
+the action taken based on the feedback from the vehicular
+environment. We use the Deep Q-learning algorithm, which
+DeepMind developed for Atari video games [15]. Deep Q-
+Learning has been used for joint channel assignment and
+power allocation in C-V2X systems [12], [13]. However,
+we also use Deep Q-learning for user association and PL
+selection. As per our knowledge, dynamic PL selection has
+not been investigated in the literature. In our work, there
+are multiple collaborative agents for PL selection, V2V joint
+channel assignment and power allocation, and V2I joint user
+association and power allocation. The objective is to increase
+reliability for both V2V and V2I links. Simulation results
+indicate that the proposed MARL algorithm can outperform
+other benchmarks, such as the hill-climbing algorithm, which
+requires global CSI at the central controller. Moreover, the
+dynamic PL selection offers a gain in V2V and V2I reliability.
+II. SYSTEM MODEL AND PROBLEM FORMULATION
+Road Side Unit (RSU)
+V2V link
+V2I link
+Platoon Leader
+Platoon Member
+Fig. 1.
+Illustrative C-V2X based highway, where multiple platoon leaders
+are transmitting to RSUs using the V2I links, and each platoon leader is
+transmitting to its platoon members using the V2V links.
+As illustrated in Fig. 1, we consider a highway-based C-
+V2X System, outlined in [4]. The highway consists of 3
+lanes on both sides. The roadside units (RSUs) are placed in
+the middle of the highway, with 100 m between them. RSU
+is a stationary communicating device capable of exchanging
+messages with other V2X devices. We consider there are K
+RSUs on the highway. Furthermore, we consider there are
+M platoons, with O vehicles in each platoon. The PL is
+required to share the real-time data with the RSUs so that
+the RSUs can form a dynamic map of the surrounding traffic.
+We refer to the message exchange from PLs to RSUs as
+V2I communication. Moreover, the PLs need to periodically
+transmit the cooperative awareness messages and the traffic
+data received from RSUs to the platoon members. In this
+paper, we refer to the communication from the PL to platoon
+members as V2V communications. Each platoon is denoted as
+m, and the platoon leader is denoted as m′. In our simulations,
+the PL selection is dynamic, and all vehicles in a platoon are
+candidates for becoming the PL. The vehicles in a platoon are
+in the same lane, each with a single PL at a given time. It is
+considered that the vehicles are separated by a fixed distance
+of d meters and are traveling with a velocity of v m/s. It is
+considered that the vehicles and RSUs use a single antenna to
+transmit/receive the signal.
+We consider that a fixed and limited number of sub-bands
+are available for both V2V and V2I links, denoted as N.
+Each sub-band has a bandwidth of W. The PL needs to
+transmit a payload of size BV 2I to the RSUs, and a payload
+of BV 2V to the platoon members, within a time constraint
+of T. We assume that both V2V and V2I links use the
+same spectrum for efficient spectrum utilization. However, all
+the V2V and V2I links can interfere, making an intelligent
+design for interference management necessary. The set of
+RSUs, platoons, and sub-bands are denoted as K, M, and N,
+respectively. Meanwhile, the set of members in each platoon
+is denoted as Vm, m = 1, . . . , O. In the paper, Lab refers to
+the large-scale fading power from transmitter a to receiver b.
+The small-scale fading power from a to b in the sub-band n
+is given by gab[n]. The ρab[n] is used as an indicator function
+set to 1 if the link ab reuses the sub-band n and 0 otherwise.
+The V2I links consist of communication from the PL to the
+RSU. Each PL m′ is required to transmit a fixed payload of
+BV 2I to the RSUs within a time constraint of T. We assume
+each RSU has a sub-band preassigned to it. Each RSU k will
+experience interference from other V2V and V2I links using
+the same sub-band n. Thus, the typical signal-to-interference
+(SINR) for a V2I link m′k can be written as:
+SINRm′k[n] = Pm′kLm′kgm′k[n]
+Ik + σ2
+k
+(1)
+Here, the interference at the RSU receiver is denoted by Ik
+Ik =
+�
+k∈K
+�
+a̸=m
+ρa′k[n]Pa′kLa′kga′k[n]+
+M
+�
+m=1
+�
+o∈Vm
+ρm′o[n]Pm′oLm′ogm′o[n]
+(2)
+In (1), the σ2
+k is the noise power at the RSU. For simplicity,
+we assume that the noise power at all RSUs is equal.
+
+Given the SINR, the achievable rate for the V2I link from
+m can be written as:
+Rm = W log2(1 + SINRm′k[n])
+(3)
+Meanwhile, the V2V link consists of communications be-
+tween the PL and the members. Each PL m′ is required to
+transmit a fixed payload of size BV 2V to each of its members o
+in the time constraint T. The platoon member o will experience
+interference from the other V2V and V2I links using the same
+sub-band n. The SINR for a platoon member o in platoon m
+can be written as:
+SINRmo[n] = Pm′oLm′ogm′o[n]
+Io + σ2o
+(4)
+Here, the interference at member o is denoted by Io
+Io =
+�
+k∈K
+�
+m∈M
+ρm′k[n]Pm′kLm′o,kgm′o,k[n]+
+M
+�
+a=1,a̸=m
+ρa′x[n]Pa′xLa′,moga′,mo[n]
+(5)
+where x denotes the platoon members in platoon a.
+Given the SINR, the achievable rate for the platoon member
+o in platoon m can be written as:
+Rmo =
+W
+O − 1 log2(1 + SINRmo[n])
+(6)
+where we divide the bandwidth equally among the platoon
+members.
+A. Problem Formulation
+We consider a multi-objective optimization problem, where
+we simultaneously maximize the payload delivery proba-
+bility for the V2V and V2I links. For the V2I link, the
+objective for each PL is to transmit the payload BV 2I
+to the RSUs within a time limit of T. This is given by
+P(∆T
+�T
+t=1
+�N
+n=1 ρmk[n, t] ≥ BV 2I), ∀m ∈ M, where
+∆T is the channel coherence time. Meanwhile, for the V2V
+links, the objective is to maximize the delivery of pay-
+load BV 2V
+within a time limit of T. This is given by
+P(∆T
+�T
+t=1
+�N
+n=1 ρmo[n, t] ≥ BV 2V ), ∀m ∈ M, o ∈ Vm.
+Due to the spectrum sharing between the V2V and V2I
+links, we need to optimize two competing objectives of simul-
+taneously maximizing the V2V and the V2I payload delivery
+probability. To achieve this, we use MARL for multiple
+objectives:
+1) Platoon Leader selection: For each platoon m, the PL
+will be selected dynamically and periodically. The pla-
+toon will decide which vehicle is the most suitable for
+being the leader so that both objectives can be met.
+2) Joint User Association and power allocation for V2I
+links: Each PL m′ will need to decide which RSU k
+it needs to be served by, along with its transmit power
+level Pm′k
+3) Joint channel assignment and power allocation for V2V
+links: Each PL m′ needs to decide the channel n and
+transmit power Pm′o to transmit to its platoon members.
+III. OPTIMAL ACTION SELECTION USING DEEP
+REINFORCEMENT LEARNING
+A. Reinforcement Learning and Deep Q Learning
+Reinforcement Learning (RL) is a discipline of Machine
+Learning (ML) where an agent can make a sequence of
+decisions by interacting with an environment. Based on the
+reward received by taking action, the agent learns to become
+intelligent. The agent aims to take actions that maximize
+the long-term cumulative reward. Markov Decision Process
+(MDP) is used to model an RL problem. According to the
+Markov property, the current state captures all relevant infor-
+mation from history. At each time-step t, the agent observes
+the environment through the state st and takes an action at.
+The agent receives a reward rt and transitions into a new
+state st+1. In RL, the goal of the agents is to maximize the
+cumulative reward Gt it receives in the long run, given by
+Gt = �∞
+k=0 γkrk+t where γ ∈ [0, 1] represents the discount
+factor which reduces the present value of the future rewards.
+The action-value function Qπ(s, a) is defined as the ex-
+pected return by taking an action a in state s by following
+a policy π: Qπ(s, a) = Eπ[Gt|St = s, At = a] where the
+expectation is taken over all possible transitions following the
+distribution π. The goal of RL is to find the optimal policy
+π∗ that maximizes the Q-function over all the policies.
+Q-learning is an off-policy RL algorithm that learns the
+value of an action a in a state s. It repetitively updates the
+action-value function for each state-action pair (s, a) until
+they converge to the optimal action-value function Q∗(s, a).
+The update equation is given by: Q(st, at) ← Q(st, at) +
+α[rt+1 + γ maxa′ Q(st+1, a′) − Q(st, at)] where α represents
+the learning rate. If the Q-function is estimated accurately,
+the optimal policy π∗ at a given state s would be to select
+the action a∗ that yields the highest value. The Deep Q-
+Learning [16] uses a Deep Neural Network (DNN) as a
+function approximator to learn the Q-function. The state space
+is input to the DNN, and it learns to predict the Q-value for
+each output action. The state-action space is explored with a
+soft policy such as ϵ-greedy, where the agent takes random
+action at a given state st at time t with a probability of
+ϵ. Otherwise, greedy action a∗ = arg maxa∈A Q∗(st, a) is
+selected. The tuple < st, at, rt, st+1 > is stored in the replay
+memory at each time instance. At each time-step, a mini-
+batch is sampled from the replay memory to update the DNN
+parameters θ, and the gradient descent algorithms are used to
+update the parameters θ.
+B. Multi-Agent Reinforcement Learning for optimization
+In this section, we formulate the multi-agent RL algorithm
+for optimization. There will be three different types of agents,
+all based within the vehicles in the platoon. The first agent
+type will be the PL selection, which will dynamically decide
+the PL every 100 ms. The second type of agent will be the
+
+V2V platoon, which needs to determine the joint channel
+assignment and power allocation. Meanwhile, the third type
+of agent will be the V2I agent, which will need to optimize
+joint user association and power allocation. All the agents will
+interact with the environment and learn to take optimal actions
+by trial and error. Furthermore, we use a common reward for
+all the agents to ensure collaboration. Moreover, each agent
+has a separate DQN and only uses its own experience to train
+the DNN.
+We develop two phases for the MARL problem: training
+and testing. During the training phase, each agent can access
+the common reward to train the DQN. Meanwhile, during the
+testing phase, each agent uses the trained DQN to select the
+action optimally.
+1) State and Action Space for Platoon Leader Selection:
+For the platoon leader selection agent, the state space Zpl(t)
+consist of the measurements at time-step t. The state space
+consists of the following measurements: i) The large-scale fad-
+ing information between all members within a platoon m, i.e.,
+{Lab}(a,b)∈Vm; ii) The large-scale fading information between
+all vehicles in platoon m to all RSUs, i.e., {Lok}k∈K,o∈Vm.
+Meanwhile, the action space consists of the PL selection. The
+action is updated every 100 ms.
+2) State and Action Space for V2V agent:
+The state
+space of the V2V agent, denoted by Zv2v(t), consists of
+the measurements from the last time-step t and consists of
+the following groups: i) Direct channel measurements from
+the PL m′ to the members, i.e., {Lm′ogm′o[n]}o∈Vm ii)
+The interfering channels from other PLs sharing the same
+sub-band with the V2V agent m, which occupies the sub-
+band n, i.e., {ρa′x[n]Pa′xLa′,moga′mo}a∈M,a̸=m,x∈Va iii) The
+interfering channels from the V2I links to the RSU, i.e.,
+{ρm′k[n]Pm′kLm′o,kgm′ok}o∈Vm,k∈K iv) The remaining pay-
+load and time limitation after the current time-step.
+Meanwhile, the action space consists of the combination
+of sub-band selection and power allocation. The sub-band
+consists of N disjoint sub-bands, and the power levels are
+broken down into multiple discrete levels in the range [0, Pd],
+where Pd denotes the maximum power.
+3) State and Action Space for V2I agent: The state space
+of the V2I agent, denoted by Zv2i(t), for the PL m′, consists
+of the measurements from the last time-step t and consists of
+the following groups: i) Direct channel measurements from
+the PL m′ to all the RSUs, i.e., {Lm′kgm′k}k∈K ii) The
+remaining payload and time remaining after current time-step.
+iii) The training iteration number e and the agent’s probability
+of random action selection ϵ.
+The action space consists of the RSU selected and the power
+level. We assume that each RSU uses a fixed sub-band. There
+are K RSUs to select from and transmit at a power divided
+into multiple discrete levels in the range [0, Pd].
+4) Reward function design: We use a common reward
+for all the agents in our proposed MARL design to ensure
+collaboration. We have a multi-objective problem, which is
+to maximize the payload delivery probability for the PL to
+RSU V2I links, and maximize the payload delivery probability
+for the PL to platoon members V2V links, within the time
+constraint T. The V2V and V2I agents need to select actions
+to minimize interference between each other. To achieve this
+purpose, we define the reward at time-step t, denoted as rt,
+as:
+rt = wc
+M
+�
+m=1
+�
+o∈Vm
+Umo(t) + wd
+M
+�
+m=1
+Vm(t)
+(7)
+where Umo(t) is the contribution towards the reward of the
+V2V link m′o and Vm(t) is the contribution of the V2I link
+from PL m′ to RSU. Furthermore, wc, wd ∈ [0, 1] are weights
+to balance the two objectives.
+Umo(t) is the achievable rate of the PL to platoon member
+link mo, defined as:
+Umo(t) =
+�
+Rmo(t),
+if Bmo(t) ≥ 0,
+U,
+otherwise.
+(8)
+where Bmo(t) is the remaining payload for the V2V link mo at
+time-step t. Furthermore, if the payload intended for link mo
+has been delivered, the agent is given an award U, which needs
+to be greater than the maximum rate achievable, to indicate
+to the agent the successful transmission of the payload. U is
+a hyperparameter that needs to be adjusted empirically [12].
+Similarly, Vm(t) is the achievable rate of the PL to RSU
+link, defined as:
+Vm(t) =
+�
+Rm(t),
+if Bm(t) ≥ 0,
+V,
+otherwise.
+(9)
+where Bm(t) is the payload that PL m′ needs to transmit to
+the RSUs. V is a hyperparameter, which needs to be greater
+than the maximum achievable rate of the V2I link.
+C. Training Algorithm and Testing Strategy
+We devise the problem as an episodic setting, where each
+episode corresponds to the time limit T for the V2V and V2I
+links to complete their transmission. Each episode consists
+of multiple time-steps t. The vehicle location and large-scale
+fading are updated every 100 ms [4]. Meanwhile, the small-
+scale fading is updated at each time-step t, changing the state
+space for the V2V and V2I agents and prompting the agents to
+adjust their actions. Each agent stops its transmission once its
+payload has been delivered. The training is centralized, where
+each agent has access to the common reward rt. Deep Q-
+Learning is used to train the agent. The algorithm is outlined
+in Algorithm 1.
+During the testing phase, each agent observes the state. The
+state is input to the trained DQN, which is used to select the
+optimal action. The testing is implemented in a distributed
+manner, where each agent takes action based on their local
+state observation only.
+IV. ILLUSTRATIVE RESULTS
+This section presents the simulation results to illustrate the
+performance of our algorithm. We consider a highway setting
+as described in TR 37.885 [4], with the carrier frequency
+of 6 GHz. The technical report provides all details, such
+
+Algorithm 1 Training Algorithm
+1: Initiate the environment and generate the V2I and V2V links
+2: Initiate the DQN with random parameters θ
+3: for each episode i do
+4:
+if i%20 = 0 then
+5:
+Update the vehicle locations and large-scale fading
+6:
+for each PL selection agent n1 do
+7:
+Observe the state st1 for PL selection and take action
+at1
+8:
+end for
+9:
+end if
+10:
+for each time-step t do
+11:
+Update small-scale channel fading
+12:
+for each V2V agent n2 do
+13:
+Observe the state st2
+14:
+end for
+15:
+for each V2I agent n3 do
+16:
+Observe the state st3
+17:
+end for
+18:
+All agents take actions simultaneously according to the
+ϵ−greedy policy and receive the common reward rt
+19:
+for each agent {n1, n2, n3} do
+20:
+Observe the next state st+1
+21:
+Store et = [st, at, rt, st+1] in the replay memory
+22:
+end for
+23:
+end for
+24:
+for each agent {n1, n2, n3} do
+25:
+Uniformly sample mini-batch data D from replay memory
+26:
+Train the deep Q-networks using the mini-batch data.
+27:
+end for
+28: end for
+as evaluation scenarios, vehicle drop and mobility modeling,
+RSU deployment, and channel models for V2V and V2I links.
+The small-scale fading is modeled as Rayleigh fading. We
+consider a highway with a length of 1 km, with 3 lanes for
+traffic on both sides. The RSUs are placed in the middle of
+the highway, with a distance of 100 m between them. We use
+option A in UE drop options in Section 6.1.2 of TR 37.885
+[4]. The type 3 vehicles (bus/tracks) are used, with a length of
+13 m and 2 m distance between each in a platoon. All vehicles
+travel with a velocity of 140 km/h. Each V2V platoon consists
+of 3 vehicles. Moreover, the antenna on vehicles is placed in
+the middle of each vehicle. As per TS 22.185 [3], the V2V
+and V2I links need to complete their transmission in 10 ms.
+However, we set T as 5 ms, assuming the other 5 ms will
+be used for communication in other directions, i.e., platoon
+member to platoon leader. The main simulation parameters
+are listed in Table I.
+The DQN was implemented in Python using the Tensorflow
+package. The DNN for all 3 types of agents consisted of 3
+hidden layers. The DNN of PL selection agents had 71, 35,
+and 17 neurons, the DNN of V2V agents had 100, 50, and
+24 neurons; and the DNN of V2I agents had 166, 83, and
+40 neurons in their hidden layers, respectively. The rectified
+linear unit (ReLU) was us as the activation function for all
+3 types of agents. RMSProp was used for optimization for
+all agents, and learning rates of 0.0001, 0.0001, and 0.001
+were used for PL selection agents, V2V agents, and V2I
+agents, respectively. The wc and wd in (7) were set as 0.3
+and 0.7, respectively. Meanwhile, the hyperparameters U in
+TABLE I
+SIMULATION PARAMETERS
+Carrier frequency
+6 GHz
+Bandwidth of each sub-band
+1 MHz
+Number of sub-bands N
+2
+Number of RSUs K
+11
+Number of platoons M
+[4,6]
+Number of vehicles in each platoon O
+3
+Vehicle velocity v
+140 km/h
+Tx power for V2V links
+[23, 15, 5, -100] dBm
+Tx power for V2I links
+[23, -100] dBm
+Vehicle Antenna gain
+3 dBi
+Vehicle receiver noise figure
+9 dB
+Noise PSD
+-169 dBm/Hz
+Time constraint T
+5 ms
+Platoon Leader update interval
+100 ms
+V2V payload BV 2V
+[1200,.....,2800] bytes
+V2I payload BV 2I
+624 bytes
+(8) and V in (9) were selected as 25 and 15, respectively. The
+training phase consisted of 2000 episodes, and the testing was
+performed for 100 episodes. The ϵ-greedy policy was used
+during the training, and the value of ϵ was reduced linearly
+from 1 to 0.02 for 1600 episodes. The training was performed
+setting BV 2V as 2400 bytes and was varied between 1200-
+2800 bytes during testing. Meanwhile, BV 2I was set to 624
+bytes during the training and testing phases.
+We developed three benchmarks for comparison: 1) Hill-
+climbing algorithm [17]: Hill-climbing algorithm is a local
+search optimization method, guaranteed to reach a local opti-
+mum. It is a centralized iterative algorithm, which starts with a
+random solution, and then iteratively keeps improving it until
+it reaches an optimum. The algorithm is used as an upper
+benchmark in our paper. 2) Greedy Algorithm: Each agent
+uses the best link available to transmit at maximum power. 3)
+RL algorithm without PL selection: We run the RL algorithm,
+fixing the leading vehicle in each platoon as the platoon leader.
+This is to show the effectiveness of PL selection agents.
+0
+200
+400
+600
+800
+1000
+1200
+1400
+1600
+1800
+2000
+Training episodes
+2
+3
+4
+5
+6
+7
+8
+Cummulative reward per episode
+Fig. 2. Cumulative reward per episode for M = 4
+Fig. 2 shows the cumulative reward for each episode for
+the case of 4 platoons. The reward increases during training,
+indicating that the agents can collaborate.
+
+6
+7
+8
+9
+10
+11
+12
+13
+14
+V2V payload size BV2V ( 
+ 200 bytes)
+0.7
+0.75
+0.8
+0.85
+0.9
+0.95
+1
+Average V2V payload success rate
+Proposed RL algorithm
+Hill climbing algorithm
+RL algorithm without PL selection
+Greedy Algorithm
+4 platoons
+6 platoons
+(a) Average V2V payload delivery probability
+6
+7
+8
+9
+10
+11
+12
+13
+14
+V2V payload size BV2V ( 
+ 200 bytes)
+0.8
+0.82
+0.84
+0.86
+0.88
+0.9
+0.92
+0.94
+0.96
+0.98
+1
+Average V2I payload success rate
+Proposed RL algorithm
+Hill climbing algorithm
+RL algorithm without PL selection
+Greedy Algorithm
+4 platoons
+6 platoons
+(b) Average V2I payload delivery probability
+Fig. 3. V2V and V2I performance for M = 4 and M = 6.
+Fig. 3 shows the reliability of V2V and V2I links as we
+increase BV 2V for the cases of 4 and 6 platoons. Fig. 3a
+shows that the V2V performance decreases as we increase the
+packet size. This is because a larger payload size requires
+a longer time to transmit. For 4 platoons, we achieve a
+reliability of 1 for up to 2200 bytes, outperforming all the other
+benchmarks. Fig. 3b shows that for 4 platoons, the payload
+success rate for V2I links is 0.9975 for all cases. However,
+the hill-climbing and greedy algorithm performance decrease,
+indicating more significant interference as we increase V2V
+payload size. When we increase the number of platoons to 6,
+the performance gap between the proposed algorithm and the
+hill-climbing algorithm increases, which shows the superiority
+of our algorithm for a higher number of agents. Furthermore,
+it can be seen that dynamic platoon leader selection improves
+performance for both V2V and V2I links.
+V. CONCLUSION
+In this work, we proposed a distributed multi-agent rein-
+forcement learning algorithm to optimize the performance of
+the V2V and the V2I links. The V2V and V2I links used
+the same spectrum, making an intelligent resource allocation
+design necessary to manage interference. Each platoon leader
+had an agent for joint channel assignment and power allocation
+for V2V links, and another agent for joint user association
+and power allocation for V2I links. Further, another agent was
+able to select the platoon leader, to maximize the reliability
+of both V2V and V2I links. Based on RL, the agents could
+collaborate to take optimal actions. The proposed approach
+is decentralized, and the agents were able to make decisions
+based on their local state observations only. Simulation results
+indicate that the proposed algorithm could perform well for
+variable V2V packet size and different numbers of platoons,
+outperforming the centralized hill-climbing algorithm. More-
+over, the PL selection improved reliability for both V2V and
+V2I links.
+REFERENCES
+[1] H. Abou-Zeid, F. Pervez, A. Adinoyi, M. Aljlayl, and H. Yanikomeroglu,
+“Cellular V2X transmission for connected and autonomous vehicles
+standardization, applications, and enabling technologies,” IEEE Con-
+sumer Electronics Magazine, vol. 8, no. 6, pp. 91–98, 2019.
+[2] “3rd Generation Partnership Project; Technical Specification Group
+Services and System Aspects; Architecture enhancements for 5G System
+(5GS) to support Vehicle-to-Everything (V2X) services (Release 17) ,”
+TS 23.287 V17.4.0, Sep. 2022.
+[3] “3rd Generation Partnership Project; Technical Specification Group
+Services and System Aspects; Study on enhancement of 3GPP Support
+for 5G V2X Services (Release 16) ,” TR 22.886 V16.2.0, Dec. 2018.
+[4] “3rd Generation Partnership Project; Technical Specification Group Ra-
+dio Access Network; Study on evaluation methodology of new Vehicle-
+to-Everything (V2X) use cases for LTE and NR; (Release 15) ,” TR
+37.885 V15.3.0, Jun. 2019.
+[5] K. Sehla, T. M. T. Nguyen, G. Pujolle, and P. B. Velloso, “Resource
+Allocation Modes in C-V2X: From LTE-V2X to 5G-V2X,” IEEE
+Internet of Things Journal, vol. 9, no. 11, pp. 8291–8314, 2022.
+[6] S. Yi, G. Sun, and X. Wang, “Enhanced resource allocation for 5G V2X
+in congested smart intersection,” in 2020 IEEE 92nd Veh. Tech. Conf.
+[7] Y. Yoon and H. Kim, “A stochastic reservation scheme for aperiodic
+traffic in NR V2X communication,” in 2021 IEEE Wireless Commun.
+Networking Conf. (WCNC).
+[8] M. Segata, P. Arvani, and R. L. Cigno, “A critical assessment of C-V2X
+resource allocation scheme for platooning applications,” in ”2021 IEEE
+Wireless On-demand Network Systems and Services Conf. (WONS)”.
+[9] A. Masmoudi, K. Mnif, and F. Zarai, “A survey on radio resource
+allocation for V2X communication,” Wireless Communications and
+Mobile Computing, vol. 2019.
+[10] M. Allouch, S. Kallel, A. Soua, O. Shagdar, and S. Tohme, “Survey on
+radio resource allocation in long-term evolution-vehicle,” Concurrency
+and Computation: Practice and Experience, vol. 34, no. 7, 2022.
+[11] A. Alwarafy, M. Abdallah, B. S. Ciftler, A. Al-Fuqaha, and M. Hamdi,
+“Deep reinforcement learning for radio resource allocation and manage-
+ment in next generation heterogeneous wireless networks: A survey,”
+arXiv preprint arXiv:2106.00574, 2021.
+[12] L. Liang, H. Ye, and G. Y. Li, “Spectrum sharing in vehicular networks
+based on multi-agent reinforcement learning,” IEEE J. Sel. Areas Com-
+mun., vol. 37, no. 10, pp. 2282–2292, 2019.
+[13] H. V. Vu, M. Farzanullah, Z. Liu, D. H. Nguyen, R. Morawski,
+and T. Le-Ngoc, “Multi-Agent Reinforcement Learning for Channel
+Assignment and Power Allocation in Platoon-Based C-V2X Systems,”
+in 2022 IEEE 95th Veh. Tech. Conf.
+[14] M. Parvini et al., “AoI-aware resource allocation for platoon-based C-
+V2X networks via multi-agent multi-task reinforcement learning,” arXiv
+preprint arXiv:2105.04196, 2021.
+[15] Mnih et al., “Human-level control through deep reinforcement learning,”
+nature, vol. 518, no. 7540, pp. 529–533, 2015.
+[16] C. J. C. H. Watkins, “Learning from delayed rewards,” Ph.D. disserta-
+tion, King’s College, Cambridge United Kingdom, May 1989.
+[17] S. J. Russell, Artificial intelligence a modern approach.
+Pearson
+Education, Inc., 2010.
+
diff --git a/LtE1T4oBgHgl3EQfZAR1/content/tmp_files/load_file.txt b/LtE1T4oBgHgl3EQfZAR1/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..dddb48243a8ef6abd0c692ebf28913646a56e866
--- /dev/null
+++ b/LtE1T4oBgHgl3EQfZAR1/content/tmp_files/load_file.txt
@@ -0,0 +1,481 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf,len=480
+page_content='Platoon Leader Selection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' User Association and  Resource Allocation on a C-V2X based highway: A  Reinforcement Learning Approach  Mohammad Farzanullah and Tho Le-Ngoc  Department of Electrical & Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' McGill University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Montr´eal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' QC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Canada  Abstract—We consider the problem of dynamic platoon leader  selection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' user association,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' channel assignment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' and power allo-  cation on a cellular vehicle-to-everything (C-V2X) based high-  way,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' where multiple vehicle-to-vehicle (V2V) and vehicle-to-  infrastructure (V2I) links share the frequency resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' There  are multiple roadside units (RSUs) on a highway, and vehicles can  form platoons, which has been identified as an advanced use case  to increase road efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The traditional optimization methods,  requiring global channel information at a central controller, are  not viable for high-mobility vehicular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' To deal with this  challenge, we propose a distributed multi-agent reinforcement  learning (MARL) for resource allocation (RA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Each platoon  leader, acting as an agent, can collaborate with other agents for  joint sub-band selection and power allocation for its V2V links,  and joint user association and power control for its V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Moreover, each platoon can dynamically select the vehicle most  suitable to be the platoon leader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We aim to maximize the V2V  and V2I packet delivery probability in the desired latency using  the deep Q-learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Simulation results indicate that  our proposed MARL outperforms the centralized hill-climbing  algorithm, and platoon leader selection helps to improve both  V2V and V2I performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Index Terms—New radio, cellular vehicle-to-everything, rein-  forcement learning, resource allocation  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' INTRODUCTION  Cellular vehicle-to-everything (C-V2X) is a vehicular stan-  dard that enables communication between vehicles and other  entities on the road, such as pedestrians and infrastructure, to  increase road safety and efficiency [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The C-V2X system  consists of communication between different entities, such  as vehicle-to-vehicle (V2V), vehicle-to-infrastructure (V2I),  vehicle-to-network (V2N), and vehicle-to-pedestrian (V2P)  communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The C-V2X is envisioned to support high  throughput, ultra-reliable, and low latency communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  3GPP has provided the 5G new radio (NR) standards  for vehicular communications concerning architecture in TS  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='287 [2], advanced use cases in TS 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='886 [3], and eval-  uation methodology in TR 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='885 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' One component of  infrastructure will be the roadside units (RSUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' RSU is a  stationary wireless C-V2X device that can exchange messages  with vehicles and other C-V2X entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' It uses the PC5  side-link interface to communicate with the vehicles and  transmit information about road signs and traffic lights [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  It can also receive information from the vehicles to make  a dynamic map of the surroundings and share it with other  vehicles/pedestrians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Furthermore, we consider the use-case of  platooning, where multiple vehicles form a train-like structure  and travel closely together in a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The platoon leader (PL)  organizes communications between vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Vehicle platoon-  ing has been identified as an advanced use case in [3] and has  gained significant interest since it reduces fuel consumption  and traffic congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The RSU and platoon will need to  exchange a maximum of 1200 bytes in 500 ms for real-  time traffic updates and 600 bytes in 10 ms for conditional  automated driving [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We grouped these two requirements  for an aggregate of 624 bytes in 10 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Moreover, the PL  and members need to exchange 50-1200 bytes in 10 ms for  cooperative driving and up to 2000 bytes in 10 ms for collision  avoidance [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We aggregate these service requirements and  keep the exchange of 1200-2800 bytes in 10 ms for our  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' An intelligent resource allocation (RA) design is  necessary for these stringent requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  The 5G new radio (NR) C-V2X supports two RA modes  for sidelink PC5 communications: mode 1, the under-coverage  mode, and mode 2, the out-of-coverage mode [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' In mode 1,  the gNB allocates the communication resources to vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Meanwhile, in mode 2, the vehicles autonomously select  the resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' For mode 2, the current RA technique used  in standards is the sensing-based Semi-Persistent Scheduling  (SPS) algorithm, which periodically selects random resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  However, the probability of resource selection collision can be  high, and many works have considered either improving the  SPS algorithm or alternate techniques to increase reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  [6] proposes a novel sensing-based SPS algorithm in an urban  scenario to reduce the collision probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The authors in  [7] considered a highway scenario and suggested a stochastic  reservation scheme for aperiodic traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' For the platooning  use case, [8] shows that the SPS algorithm can not achieve  the required performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Due to the fast channel variations in vehicular networks,  centralized optimization schemes that require global channel  state information (CSI) will no longer be feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The high  CSI overhead and the corresponding increase in latency make  such methods impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' To deal with this issue, distributed  RA algorithms have been suggested in the literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=', [9],  [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Furthermore, traditional optimization techniques have  limitations, requiring complete information about the environ-  ment and needing to be retrained for rapidly varying envi-  ronments [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Recently, distributed multi-agent reinforcement  learning (MARL) has been proposed as an alternative approach  to resolving such issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The authors in [12] used Deep Q  Networks (DQN) for joint channel assignment and power  allocation to maximize the V2V delivery probability and V2N  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='03145v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='MA]  9 Jan 2023  sum-rate in an urban setting, where the V2V links share the  time-frequency resources with V2N links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Inspired by these  results, [13] used double DQN for a platoon-based scenario  for the same objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' [14] uses the actor-critic method for  mode selection, subchannel selection, and power control in an  urban platoon scenario to increase the transmission probability  of cooperative awareness messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  This paper considers a highway C-V2X system consisting  of multiple platoons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We consider the periodic payload deliv-  ery from PLs to RSUs, termed V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Furthermore, we  consider the periodic transmission of messages from PLs to  members, termed V2V links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We assume a limited spectrum  is available for the V2V and V2I transmission, and they share  the frequency resources for efficient spectrum usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Given  this system, this paper formulates a dynamic PL selection,  user association, channel assignment, and power level control  to maximize the packet delivery probability for both V2V and  V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Reliability is defined as the successful transmission  of the packet within a time constraint T [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  We utilize MARL in a distributed manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The RL works  on a trial-and-error strategy, and each agent slowly improves  the action taken based on the feedback from the vehicular  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We use the Deep Q-learning algorithm, which  DeepMind developed for Atari video games [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Deep Q-  Learning has been used for joint channel assignment and  power allocation in C-V2X systems [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' However,  we also use Deep Q-learning for user association and PL  selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' As per our knowledge, dynamic PL selection has  not been investigated in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' In our work, there  are multiple collaborative agents for PL selection, V2V joint  channel assignment and power allocation, and V2I joint user  association and power allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The objective is to increase  reliability for both V2V and V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Simulation results  indicate that the proposed MARL algorithm can outperform  other benchmarks, such as the hill-climbing algorithm, which  requires global CSI at the central controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Moreover, the  dynamic PL selection offers a gain in V2V and V2I reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' SYSTEM MODEL AND PROBLEM FORMULATION  Road Side Unit (RSU)  V2V link  V2I link  Platoon Leader  Platoon Member  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Illustrative C-V2X based highway, where multiple platoon leaders  are transmitting to RSUs using the V2I links, and each platoon leader is  transmitting to its platoon members using the V2V links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 1, we consider a highway-based C-  V2X System, outlined in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The highway consists of 3  lanes on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The roadside units (RSUs) are placed in  the middle of the highway, with 100 m between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' RSU  is a stationary communicating device capable of exchanging  messages with other V2X devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We consider there are K  RSUs on the highway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Furthermore, we consider there are  M platoons, with O vehicles in each platoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The PL is  required to share the real-time data with the RSUs so that  the RSUs can form a dynamic map of the surrounding traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  We refer to the message exchange from PLs to RSUs as  V2I communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Moreover, the PLs need to periodically  transmit the cooperative awareness messages and the traffic  data received from RSUs to the platoon members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' In this  paper, we refer to the communication from the PL to platoon  members as V2V communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Each platoon is denoted as  m, and the platoon leader is denoted as m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' In our simulations,  the PL selection is dynamic, and all vehicles in a platoon are  candidates for becoming the PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The vehicles in a platoon are  in the same lane, each with a single PL at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' It is  considered that the vehicles are separated by a fixed distance  of d meters and are traveling with a velocity of v m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' It is  considered that the vehicles and RSUs use a single antenna to  transmit/receive the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  We consider that a fixed and limited number of sub-bands  are available for both V2V and V2I links, denoted as N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Each sub-band has a bandwidth of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The PL needs to  transmit a payload of size BV 2I to the RSUs, and a payload  of BV 2V to the platoon members, within a time constraint  of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We assume that both V2V and V2I links use the  same spectrum for efficient spectrum utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' However, all  the V2V and V2I links can interfere, making an intelligent  design for interference management necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The set of  RSUs, platoons, and sub-bands are denoted as K, M, and N,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Meanwhile, the set of members in each platoon  is denoted as Vm, m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' , O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' In the paper, Lab refers to  the large-scale fading power from transmitter a to receiver b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  The small-scale fading power from a to b in the sub-band n  is given by gab[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The ρab[n] is used as an indicator function  set to 1 if the link ab reuses the sub-band n and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  The V2I links consist of communication from the PL to the  RSU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Each PL m′ is required to transmit a fixed payload of  BV 2I to the RSUs within a time constraint of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We assume  each RSU has a sub-band preassigned to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Each RSU k will  experience interference from other V2V and V2I links using  the same sub-band n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Thus, the typical signal-to-interference  (SINR) for a V2I link m′k can be written as:  SINRm′k[n] = Pm′kLm′kgm′k[n]  Ik + σ2  k  (1)  Here, the interference at the RSU receiver is denoted by Ik  Ik =  �  k∈K  �  a̸=m  ρa′k[n]Pa′kLa′kga′k[n]+  M  �  m=1  �  o∈Vm  ρm′o[n]Pm′oLm′ogm′o[n]  (2)  In (1), the σ2  k is the noise power at the RSU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' For simplicity,  we assume that the noise power at all RSUs is equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Given the SINR, the achievable rate for the V2I link from  m can be written as:  Rm = W log2(1 + SINRm′k[n])  (3)  Meanwhile, the V2V link consists of communications be-  tween the PL and the members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Each PL m′ is required to  transmit a fixed payload of size BV 2V to each of its members o  in the time constraint T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The platoon member o will experience  interference from the other V2V and V2I links using the same  sub-band n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The SINR for a platoon member o in platoon m  can be written as:  SINRmo[n] = Pm′oLm′ogm′o[n]  Io + σ2o  (4)  Here, the interference at member o is denoted by Io  Io =  �  k∈K  �  m∈M  ρm′k[n]Pm′kLm′o,kgm′o,k[n]+  M  �  a=1,a̸=m  ρa′x[n]Pa′xLa′,moga′,mo[n]  (5)  where x denotes the platoon members in platoon a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Given the SINR, the achievable rate for the platoon member  o in platoon m can be written as:  Rmo =  W  O − 1 log2(1 + SINRmo[n])  (6)  where we divide the bandwidth equally among the platoon  members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Problem Formulation  We consider a multi-objective optimization problem, where  we simultaneously maximize the payload delivery proba-  bility for the V2V and V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' For the V2I link, the  objective for each PL is to transmit the payload BV 2I  to the RSUs within a time limit of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' This is given by  P(∆T  �T  t=1  �N  n=1 ρmk[n, t] ≥ BV 2I), ∀m ∈ M, where  ∆T is the channel coherence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Meanwhile, for the V2V  links, the objective is to maximize the delivery of pay-  load BV 2V  within a time limit of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' This is given by  P(∆T  �T  t=1  �N  n=1 ρmo[n, t] ≥ BV 2V ), ∀m ∈ M, o ∈ Vm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Due to the spectrum sharing between the V2V and V2I  links, we need to optimize two competing objectives of simul-  taneously maximizing the V2V and the V2I payload delivery  probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' To achieve this, we use MARL for multiple  objectives:  1) Platoon Leader selection: For each platoon m, the PL  will be selected dynamically and periodically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The pla-  toon will decide which vehicle is the most suitable for  being the leader so that both objectives can be met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  2) Joint User Association and power allocation for V2I  links: Each PL m′ will need to decide which RSU k  it needs to be served by, along with its transmit power  level Pm′k  3) Joint channel assignment and power allocation for V2V  links: Each PL m′ needs to decide the channel n and  transmit power Pm′o to transmit to its platoon members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' OPTIMAL ACTION SELECTION USING DEEP  REINFORCEMENT LEARNING  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Reinforcement Learning and Deep Q Learning  Reinforcement Learning (RL) is a discipline of Machine  Learning (ML) where an agent can make a sequence of  decisions by interacting with an environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Based on the  reward received by taking action, the agent learns to become  intelligent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The agent aims to take actions that maximize  the long-term cumulative reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Markov Decision Process  (MDP) is used to model an RL problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' According to the  Markov property, the current state captures all relevant infor-  mation from history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' At each time-step t, the agent observes  the environment through the state st and takes an action at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  The agent receives a reward rt and transitions into a new  state st+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' In RL, the goal of the agents is to maximize the  cumulative reward Gt it receives in the long run, given by  Gt = �∞  k=0 γkrk+t where γ ∈ [0, 1] represents the discount  factor which reduces the present value of the future rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  The action-value function Qπ(s, a) is defined as the ex-  pected return by taking an action a in state s by following  a policy π: Qπ(s, a) = Eπ[Gt|St = s, At = a] where the  expectation is taken over all possible transitions following the  distribution π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The goal of RL is to find the optimal policy  π∗ that maximizes the Q-function over all the policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Q-learning is an off-policy RL algorithm that learns the  value of an action a in a state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' It repetitively updates the  action-value function for each state-action pair (s, a) until  they converge to the optimal action-value function Q∗(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  The update equation is given by: Q(st, at) ← Q(st, at) +  α[rt+1 + γ maxa′ Q(st+1, a′) − Q(st, at)] where α represents  the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' If the Q-function is estimated accurately,  the optimal policy π∗ at a given state s would be to select  the action a∗ that yields the highest value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The Deep Q-  Learning [16] uses a Deep Neural Network (DNN) as a  function approximator to learn the Q-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The state space  is input to the DNN, and it learns to predict the Q-value for  each output action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The state-action space is explored with a  soft policy such as ϵ-greedy, where the agent takes random  action at a given state st at time t with a probability of  ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Otherwise, greedy action a∗ = arg maxa∈A Q∗(st, a) is  selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The tuple < st, at, rt, st+1 > is stored in the replay  memory at each time instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' At each time-step, a mini-  batch is sampled from the replay memory to update the DNN  parameters θ, and the gradient descent algorithms are used to  update the parameters θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Multi-Agent Reinforcement Learning for optimization  In this section, we formulate the multi-agent RL algorithm  for optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' There will be three different types of agents,  all based within the vehicles in the platoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The first agent  type will be the PL selection, which will dynamically decide  the PL every 100 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The second type of agent will be the  V2V platoon, which needs to determine the joint channel  assignment and power allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Meanwhile, the third type  of agent will be the V2I agent, which will need to optimize  joint user association and power allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' All the agents will  interact with the environment and learn to take optimal actions  by trial and error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Furthermore, we use a common reward for  all the agents to ensure collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Moreover, each agent  has a separate DQN and only uses its own experience to train  the DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  We develop two phases for the MARL problem: training  and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' During the training phase, each agent can access  the common reward to train the DQN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Meanwhile, during the  testing phase, each agent uses the trained DQN to select the  action optimally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  1) State and Action Space for Platoon Leader Selection:  For the platoon leader selection agent, the state space Zpl(t)  consist of the measurements at time-step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The state space  consists of the following measurements: i) The large-scale fad-  ing information between all members within a platoon m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=',  {Lab}(a,b)∈Vm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' ii) The large-scale fading information between  all vehicles in platoon m to all RSUs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=', {Lok}k∈K,o∈Vm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Meanwhile, the action space consists of the PL selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The  action is updated every 100 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  2) State and Action Space for V2V agent:  The state  space of the V2V agent, denoted by Zv2v(t), consists of  the measurements from the last time-step t and consists of  the following groups: i) Direct channel measurements from  the PL m′ to the members, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=', {Lm′ogm′o[n]}o∈Vm ii)  The interfering channels from other PLs sharing the same  sub-band with the V2V agent m, which occupies the sub-  band n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=', {ρa′x[n]Pa′xLa′,moga′mo}a∈M,a̸=m,x∈Va iii) The  interfering channels from the V2I links to the RSU, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=',  {ρm′k[n]Pm′kLm′o,kgm′ok}o∈Vm,k∈K iv) The remaining pay-  load and time limitation after the current time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Meanwhile, the action space consists of the combination  of sub-band selection and power allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The sub-band  consists of N disjoint sub-bands, and the power levels are  broken down into multiple discrete levels in the range [0, Pd],  where Pd denotes the maximum power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  3) State and Action Space for V2I agent: The state space  of the V2I agent, denoted by Zv2i(t), for the PL m′, consists  of the measurements from the last time-step t and consists of  the following groups: i) Direct channel measurements from  the PL m′ to all the RSUs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=', {Lm′kgm′k}k∈K ii) The  remaining payload and time remaining after current time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  iii) The training iteration number e and the agent’s probability  of random action selection ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  The action space consists of the RSU selected and the power  level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We assume that each RSU uses a fixed sub-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' There  are K RSUs to select from and transmit at a power divided  into multiple discrete levels in the range [0, Pd].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  4) Reward function design: We use a common reward  for all the agents in our proposed MARL design to ensure  collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We have a multi-objective problem, which is  to maximize the payload delivery probability for the PL to  RSU V2I links, and maximize the payload delivery probability  for the PL to platoon members V2V links, within the time  constraint T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The V2V and V2I agents need to select actions  to minimize interference between each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' To achieve this  purpose, we define the reward at time-step t, denoted as rt,  as:  rt = wc  M  �  m=1  �  o∈Vm  Umo(t) + wd  M  �  m=1  Vm(t)  (7)  where Umo(t) is the contribution towards the reward of the  V2V link m′o and Vm(t) is the contribution of the V2I link  from PL m′ to RSU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Furthermore, wc, wd ∈ [0, 1] are weights  to balance the two objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Umo(t) is the achievable rate of the PL to platoon member  link mo, defined as:  Umo(t) =  �  Rmo(t),  if Bmo(t) ≥ 0,  U,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  (8)  where Bmo(t) is the remaining payload for the V2V link mo at  time-step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Furthermore, if the payload intended for link mo  has been delivered, the agent is given an award U, which needs  to be greater than the maximum rate achievable, to indicate  to the agent the successful transmission of the payload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' U is  a hyperparameter that needs to be adjusted empirically [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Similarly, Vm(t) is the achievable rate of the PL to RSU  link, defined as:  Vm(t) =  �  Rm(t),  if Bm(t) ≥ 0,  V,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  (9)  where Bm(t) is the payload that PL m′ needs to transmit to  the RSUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' V is a hyperparameter, which needs to be greater  than the maximum achievable rate of the V2I link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Training Algorithm and Testing Strategy  We devise the problem as an episodic setting, where each  episode corresponds to the time limit T for the V2V and V2I  links to complete their transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Each episode consists  of multiple time-steps t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The vehicle location and large-scale  fading are updated every 100 ms [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Meanwhile, the small-  scale fading is updated at each time-step t, changing the state  space for the V2V and V2I agents and prompting the agents to  adjust their actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Each agent stops its transmission once its  payload has been delivered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The training is centralized, where  each agent has access to the common reward rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Deep Q-  Learning is used to train the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The algorithm is outlined  in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  During the testing phase, each agent observes the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The  state is input to the trained DQN, which is used to select the  optimal action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The testing is implemented in a distributed  manner, where each agent takes action based on their local  state observation only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' ILLUSTRATIVE RESULTS  This section presents the simulation results to illustrate the  performance of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We consider a highway setting  as described in TR 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='885 [4], with the carrier frequency  of 6 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The technical report provides all details,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' such  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='Algorithm 1 Training Algorithm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='1: Initiate the environment and generate the V2I and V2V links  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='2: Initiate the DQN with random parameters θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='3: for each episode i do  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='4:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='if i%20 = 0 then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='5:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='Update the vehicle locations and large-scale fading  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='6:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='for each PL selection agent n1 do  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='7:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='Observe the state st1 for PL selection and take action  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='at1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='8:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='end for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='9:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='end if  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='10:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='for each time-step t do  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='11:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='Update small-scale channel fading  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='12:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='for each V2V agent n2 do  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='13:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='Observe the state st2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='14:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='end for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='15:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='for each V2I agent n3 do  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='16:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='Observe the state st3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='17:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='end for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='18:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='All agents take actions simultaneously according to the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='ϵ−greedy policy and receive the common reward rt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='19:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='for each agent {n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' n3} do  20:  Observe the next state st+1  21:  Store et = [st,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' at,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' rt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' st+1] in the replay memory  22:  end for  23:  end for  24:  for each agent {n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' n3} do  25:  Uniformly sample mini-batch data D from replay memory  26:  Train the deep Q-networks using the mini-batch data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  27:  end for  28: end for  as evaluation scenarios, vehicle drop and mobility modeling,  RSU deployment, and channel models for V2V and V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  The small-scale fading is modeled as Rayleigh fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We  consider a highway with a length of 1 km, with 3 lanes for  traffic on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The RSUs are placed in the middle of  the highway, with a distance of 100 m between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We use  option A in UE drop options in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='2 of TR 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='885  [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The type 3 vehicles (bus/tracks) are used, with a length of  13 m and 2 m distance between each in a platoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' All vehicles  travel with a velocity of 140 km/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Each V2V platoon consists  of 3 vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Moreover, the antenna on vehicles is placed in  the middle of each vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' As per TS 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='185 [3], the V2V  and V2I links need to complete their transmission in 10 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  However, we set T as 5 ms, assuming the other 5 ms will  be used for communication in other directions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=', platoon  member to platoon leader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The main simulation parameters  are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  The DQN was implemented in Python using the Tensorflow  package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The DNN for all 3 types of agents consisted of 3  hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The DNN of PL selection agents had 71, 35,  and 17 neurons, the DNN of V2V agents had 100, 50, and  24 neurons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' and the DNN of V2I agents had 166, 83, and  40 neurons in their hidden layers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The rectified  linear unit (ReLU) was us as the activation function for all  3 types of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' RMSProp was used for optimization for  all agents, and learning rates of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='0001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='0001, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='001  were used for PL selection agents, V2V agents, and V2I  agents, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The wc and wd in (7) were set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='3  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Meanwhile,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' the hyperparameters U in  TABLE I  SIMULATION PARAMETERS  Carrier frequency  6 GHz  Bandwidth of each sub-band  1 MHz  Number of sub-bands N  2  Number of RSUs K  11  Number of platoons M  [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='6]  Number of vehicles in each platoon O  3  Vehicle velocity v  140 km/h  Tx power for V2V links  [23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' -100] dBm  Tx power for V2I links  [23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' -100] dBm  Vehicle Antenna gain  3 dBi  Vehicle receiver noise figure  9 dB  Noise PSD  169 dBm/Hz  Time constraint T  5 ms  Platoon Leader update interval  100 ms  V2V payload BV 2V  [1200,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=',2800] bytes  V2I payload BV 2I  624 bytes  (8) and V in (9) were selected as 25 and 15, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The  training phase consisted of 2000 episodes, and the testing was  performed for 100 episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The ϵ-greedy policy was used  during the training, and the value of ϵ was reduced linearly  from 1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='02 for 1600 episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The training was performed  setting BV 2V as 2400 bytes and was varied between 1200-  2800 bytes during testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Meanwhile, BV 2I was set to 624  bytes during the training and testing phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  We developed three benchmarks for comparison: 1) Hill-  climbing algorithm [17]: Hill-climbing algorithm is a local  search optimization method, guaranteed to reach a local opti-  mum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' It is a centralized iterative algorithm, which starts with a  random solution, and then iteratively keeps improving it until  it reaches an optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The algorithm is used as an upper  benchmark in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 2) Greedy Algorithm: Each agent  uses the best link available to transmit at maximum power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 3)  RL algorithm without PL selection: We run the RL algorithm,  fixing the leading vehicle in each platoon as the platoon leader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  This is to show the effectiveness of PL selection agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  0  200  400  600  800  1000  1200  1400  1600  1800  2000  Training episodes  2  3  4  5  6  7  8  Cummulative reward per episode  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Cumulative reward per episode for M = 4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 2 shows the cumulative reward for each episode for  the case of 4 platoons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The reward increases during training,  indicating that the agents can collaborate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  6  7  8  9  10  11  12  13  14  V2V payload size BV2V (  200 bytes)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='95  1  Average V2V payload success rate  Proposed RL algorithm  Hill climbing algorithm  RL algorithm without PL selection  Greedy Algorithm  4 platoons  6 platoons  (a) Average V2V payload delivery probability  6  7  8  9  10  11  12  13  14  V2V payload size BV2V (  200 bytes)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='86  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='98  1  Average V2I payload success rate  Proposed RL algorithm  Hill climbing algorithm  RL algorithm without PL selection  Greedy Algorithm  4 platoons  6 platoons  (b) Average V2I payload delivery probability  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' V2V and V2I performance for M = 4 and M = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 3 shows the reliability of V2V and V2I links as we  increase BV 2V for the cases of 4 and 6 platoons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 3a  shows that the V2V performance decreases as we increase the  packet size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' This is because a larger payload size requires  a longer time to transmit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' For 4 platoons, we achieve a  reliability of 1 for up to 2200 bytes, outperforming all the other  benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 3b shows that for 4 platoons, the payload  success rate for V2I links is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='9975 for all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' However,  the hill-climbing and greedy algorithm performance decrease,  indicating more significant interference as we increase V2V  payload size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' When we increase the number of platoons to 6,  the performance gap between the proposed algorithm and the  hill-climbing algorithm increases, which shows the superiority  of our algorithm for a higher number of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Furthermore,  it can be seen that dynamic platoon leader selection improves  performance for both V2V and V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' CONCLUSION  In this work, we proposed a distributed multi-agent rein-  forcement learning algorithm to optimize the performance of  the V2V and the V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The V2V and V2I links used  the same spectrum, making an intelligent resource allocation  design necessary to manage interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Each platoon leader  had an agent for joint channel assignment and power allocation  for V2V links, and another agent for joint user association  and power allocation for V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Further, another agent was  able to select the platoon leader, to maximize the reliability  of both V2V and V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Based on RL, the agents could  collaborate to take optimal actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The proposed approach  is decentralized, and the agents were able to make decisions  based on their local state observations only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Simulation results  indicate that the proposed algorithm could perform well for  variable V2V packet size and different numbers of platoons,  outperforming the centralized hill-climbing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' More-  over, the PL selection improved reliability for both V2V and  V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
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+Draft version February 1, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+Planetesimal Initial Mass Functions following Diffusion Regulated Gravitational Collapse
+Konstantin Gerbig
+1 and Rixin Li (李日新)
+2
+1Department of Astronomy, Yale University, 52 Hillhouse Ave, New Haven, CT 06511, USA
+2Center for Astrophysics and Planetary Science, Department of Astronomy, Cornell University, Ithaca, NY 14853, USA
+ABSTRACT
+The initial mass function (IMF) of planetesimals is of key importance for understanding the
+initial stages of planet formation, yet theoretical predictions so far have been insufficient in
+explaining the variety of IMFs found in simulations. Here, we connect diffusion-tidal-shear limited
+planetesimal formation within the framework of a Toomre-like instability in the particle mid-plane of
+a protoplanetary disk to an analytic prediction for the planetesimal IMF. The shape of the IMF is
+set by the stability parameter Qp, which in turn depends on the particle Stokes number, the Toomre
+Q value of the gas, the local dust concentration and the local diffusivity. We compare our prediction
+to high-resolution numerical simulations of the streaming instability and planetesimal formation via
+gravitational collapse. We find that our IMF prediction agrees with numerical results, and is consistent
+with both the ‘planetesimals are born big’ paradigm and the power law description commonly found
+in simulations.
+1. INTRODUCTION
+Planetesimals
+are
+the
+initial
+building
+blocks
+of
+planets and their Initial Mass Function (IMF) is
+consequently of great interest.
+The planetesimal
+formation
+process
+in
+protoplanetary
+disks,
+which
+ultimately dictates the planetesimal IMF, is connected
+to
+an
+ensemble
+of
+challenges,
+one
+of
+the
+most
+prominent of which is the so-called meter barrier (see
+the review by Chiang & Youdin 2010).
+Particles
+exceeding sizes of order meters are expected to be
+limited
+in
+their
+capacity
+to
+grow
+via
+continued
+coagulation due to both rapid radial drift inwards and
+increased relative velocities that result in preferentially
+destructive collisions (Birnstiel et al. 2012).
+One
+well-received solution to this growth barrier is to
+rapidly form planetesimals via gravitational collapse of
+over-dense particle clouds. Initial ideas focused on the
+gravitational instability of the entire particle mid-plane
+(Safronov 1969; Goldreich & Ward 1973).
+However,
+Kelvin-Helmholtz stirring prevents razor-thin particle
+settling and thus renders such a global gravitational
+instability challenging (Weidenschilling 1980; Sekiya
+1998; Youdin & Shu 2002; Johansen et al. 2006a). On
+the other hand, local patches have been shown to
+Corresponding author: Konstantin Gerbig
+konstantin.gerbig@yale.edu
+collapse if local particle concentrations are sufficiently
+high (Johansen et al. 2006b). In particular, instabilities
+energized by the relative dust-gas streaming velocity
+(Youdin & Goodman 2005; Squire & Hopkins 2018)
+can concentrate particles to densities sufficient for
+gravitational collapse to trigger and planetesimals to
+form (e.g., Johansen et al. 2015; Gerbig et al. 2020). In
+the past, the IMF of planetesimals has been obtained
+by performing numerical simulations of this setup and
+then counting produced planetesimals, a process that
+resulted in power-laws of various kinds (Simon et al.
+2016; Sch¨afer et al. 2017; Li et al. 2019).
+Recently, the formation of planetesimals has been
+connected to the diffusion of particles as well as their
+stability to stellar tidal forces (Klahr & Schreiber 2016;
+Gerbig et al. 2020; Klahr & Schreiber 2020, 2021).
+This framework envisions particles to be subject to a
+diffusive flux away from the concentration maximum,
+much how pressurized gas clouds resist collapse in star
+formation. In addition, any particle cloud on the verge
+of gravitational collapse must be stable to stellar tidal
+gravity. These effects limit planetesimal formation on
+small scales and large scales respectively, thus together
+prescribe a characteristic scale on which planetesimal
+formation is expected to occur.
+This in turn, can be
+translated to a characteristic planetesimal mass, which
+Klahr & Schreiber (2020) hypothesized to be the center
+of a Gaussian-shaped IMF, in agreement with IMFs of
+arXiv:2301.13297v1  [astro-ph.EP]  30 Jan 2023
+
+ID2
+Gerbig and Li
+primordial asteroids (Delbo et al. 2019), yet seemingly
+implying a mismatch to numerically obtained IMFs.
+In this paper,
+we connect these two paradigms
+by deriving planetesimal IMFs within the framework
+of diffusion-tidal-shear limited planetesimal formation.
+Thereby, we argue that the probability density function
+of a given scale to collapse scales with the scale’s
+Toomre-like growth rate.
+We also directly test our
+prediction by conducting numerical simulations using
+proven setups in ATHENA (Stone et al. 2008), that
+produce the streaming instability (Youdin & Goodman
+2005) and planetesimal formation. In the process, we
+for the first time conduct diffusion measurements in
+large-scale stratified streaming instability simulations,
+as well as develop a method for obtaining local particle
+concentrations that are appropriate for characterizing
+the onset of planetesimal formation.
+The paper is structured as follows.
+In Sect. 2
+we review the Toomre-like instability for planetesimal
+formation.
+In Sect. 3, we derive planetesimal IMFs,
+which we compare to numerical simulations in Sect. 4.
+We discuss our results, namely applicability, caveats and
+implications, in Sect. 5.
+2. TOOMRE-LIKE INSTABILITY
+2.1. Stability parameter
+The stability of self-gravitating particles subject to
+a diffusive flux induced by coupling to turbulent gas
+velocities has been studied in numerous occasions
+(Youdin 2011; Gerbig et al. 2020; Klahr & Schreiber
+2020, 2021), and can be assessed using a Toomre-like
+value Qp, such as
+Qp ≡
+�
+δr
+St
+1
+Z
+csΩ
+πGΣg
+=
+�
+δr
+St
+Q
+Z
+(1)
+where Z = Σp/Σg is the (local) dust concentration,
+Q = csΩ/(πGΣg) is the standard Toomre value (Toomre
+1964), St = tsΩ is the dimensionless stopping time,
+and δr
+= Dp,r/(csH) is the dimensionless (radial)
+diffusion coefficient for particles.
+Ω is the orbital
+frequency, cs the sound-speed of the gas, H = cs/Ω the
+disk pressure scale-height, Σp and Σg particle and gas
+surface densities respectively, and G is the gravitational
+constant.
+In the Epstein regime, the stopping time
+relates to particle size a, and dust material density ρ•
+via ts = ρ•a/(
+√
+2πΣg). We consider dust and pebbles
+where ts is such that St remains below unity (e.g.,
+Birnstiel et al. 2012).
+This corresponds to well or at
+least marginally coupled particles.
+If Qp < 1, the system is unstable and expected to
+collapse and form planetesimals.
+We review the
+corresponding
+instability
+analysis
+in
+the
+following
+section.
+2.2. Dynamical equations and dispersion relation
+Following (Klahr & Schreiber 2021), we start with the
+a set of dynamical equations for dust particles in the
+shearing sheet approximation and under the assumption
+of a razor-thin disk. We adopt the coordinates r, φ, z
+for radial, azimuthal and vertical directions respectively,
+and consider a patch at distance R from a solar-mass
+star. The linearized set of equation is given by
+1
+Σp
+∂Σ′
+p
+∂t + ∂v′
+r
+∂r = 0,
+(2)
+∂v′
+r
+∂r − 2Ωv′
+φ = − 1
+Σp
+Dp,r
+ts
+∂Σ′
+p
+∂r − ∂Φ′
+∂r − v′
+r
+ts
+,
+(3)
+∂v′
+φ
+∂t + Ωv′
+r
+2
+= 0,
+(4)
+where the prime denotes perturbed quantities.
+They
+describe mass continuity, and conservation of radial and
+azimuthal momenta.
+Notably, we ignore an explicit
+azimuthal drag term as in e.g., Youdin (2011), an
+assumption that is justified at dust-to-gas ratios of
+order unity.
+Instead, coupling to the gas is assumed
+to be wholly described by gas pressure counteracting
+radial contraction and the diffusive flux in the radial
+momentum equation, which can be understood as an
+effective pressure flux induced by turbulent gas motions.
+As a consequence, there is also no mass diffusion term
+in the continuity equation.
+We continue by introducing axisymmetric WKB waves
+scaling with Σ′
+p ∝ exp(−i(kr − ωt)). Φ′ = −2πGΣ′/|k|
+is the potential for a perturbed disk assuming ρp(k, z) =
+(kΣp/2) exp(−|k|z). The resulting dispersion relation is
+given by (Eq. (B22) in Klahr & Schreiber 2021),
+ω2
+0 = δr
+Stc2
+sk2 − 2πΣpG|k| + Ω2,
+(5)
+and can be expressed in terms of the stability parameter
+Qp
+ω2
+0
+Ω2 = δr
+St(kH)2 − 2
+Qp
+�
+δr
+St|k|H + 1.
+(6)
+Here, ω2
+0 is defined as ω2
+0 = ω(ω − i/ts), and represents
+the complex frequency without the drag term in Eq. (3).
+Note that the herein used solution to the Poisson
+equation does not take into account a softening term
+caused by the particle layer thickness (see Eq. (12) in
+Youdin 2011).
+
+Planetesimal IMFs under diffusion regulated collapse
+3
+10−2
+10−1
+100
+101
+k[kfgm]
+−6
+−4
+−2
+0
+2
+ω2
+0[Ω−2]
+Qp = 0.4
+Qp = 0.5
+Qp = 0.6
+Qp = 0.8
+10−1
+100
+Qp
+10−3
+10−2
+10−1
+100
+λ[H]
+λfgm
+lc
+2π
+�
+δ/StH
+−1.5
+−1.0
+−0.5
+0.0
+0.5
+1.0
+log(γΩ)
+Figure 1. Dispersion relation ω2
+0 in Eq. (5) (left panel) and growth rates γ in Eq. (11) (right panel) for different values of
+Qp. Parameters are δ = 10−5 and St = 0.4. The gray shaded region in the left panel indicates stability against axisymmetric
+perturbations. As Qp decreases, more modes become unstable, and the fastest growing mode (dashed lines in right panel) shifts
+to smaller scales. For reference, we plot the scale 2π
+�
+δ/StH (dotted line right panel).
+10−4
+10−2
+100
+mp[MCeres]
+0.0000
+0.0025
+0.0050
+0.0075
+0.0100
+0.0125
+0.0150
+0.0175
+p(mp)
+Qp = 0.15
+Qp = 0.4
+Qp = 0.8
+10−4
+10−2
+100
+mp[MCeres]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+P(mp)
+δr = δz = 10−6
+δr = δz = 10−5
+10−4
+10−2
+100
+mp[MCeres]
+10−2
+10−1
+100
+N(> mp)
+Figure 2. Probability density functions (left panel), cumulative probability density functions (center panel) and initial mass
+functions (right panel) for the growth rates in Fig. 2 where St = 0.4. Diffusion is isotropic and set to δ = 10−6 (solid lines)
+and δ = 10−5 (dashed lines). Different colors correspond to different values of Qp, all of which chosen to be unstable. We use
+Eq. (16) for converting unstable scale to seed mass, which assumes collapse at Hill density, and requires knowledge of the aspect
+ratio which we set to h = 0.05. We assume the system to produce N = 1000 planetesimals total to calculate the normalized
+IMF N(< mp).
+2.3. Fastest growing mode and growth rates.
+The
+fastest
+growing
+mode
+is
+found
+by
+solving
+∂ω2
+0/∂k = 0, which yields
+kfgm =
+�
+St
+δr
+1
+QpH .
+(7)
+The complex frequency of the fastest growing mode is
+simply
+ω2
+0,fgm
+Ω2
+= 1 − 1
+Q2p
+,
+(8)
+which highlights that only for Qp < 1 exist k for which
+ω2
+0 < 0 and the instability can grow, thus justifying the
+definition of Qp as a stability parameter. We can also
+calculate largest and smallest unstable scales λ = 2π/k
+by solving ω2
+0 = 0, resulting in
+λmin/max
+λfgm
+= 1
+Qp
+�
+1
+Qp
+±
+�
+1
+Q2p
+− 1
+�
+,
+(9)
+with λfgm = 2π/kfgm. The range of unstable scales is
+λmax − λmin = 2λfgm/Qp
+�
+1/Q2p − 1, which increases
+for decreasing Qp.
+Note, that the fastest growing mode relates to the
+critical cloud radius lc in Klahr & Schreiber (2020) and
+Gerbig et al. (2020) via
+λfgm = 2π
+�
+δr
+StQpH = 6πQplc.
+(10)
+
+4
+Gerbig and Li
+The growth rate γ(k) can be found solving γ = iω
+or equivalently γ(γ + 1/ts) = −ω2
+0 (Klahr & Schreiber
+2021):
+γ(k)
+Ω
+= − 1
+2St +
+�
+1
+4St2 − ω2
+0(k)
+Ω2
+.
+(11)
+Hence, the fastest growing mode kfgm grows with
+γ(k = kfgm)
+Ω
+= − 1
+2St +
+�
+1
+4St2 + 1
+Q2p
+− 1
+(12)
+Dispersion relation and growth rates are shown for a set
+of unstable Qp in Fig. 1. In the right panel, we also
+plot λfgm, lc as well as the scale 2π
+�
+δr/StH which is of
+order the radial extent of small-scale particle structures
+(Gerbig et al. 2020).
+3. PLANETESIMAL INITIAL MASS FUNCTIONS
+3.1. Planetesimal masses at Hill density
+In order for a region in the particle disk to be
+stable against stellar tidal gravity, its mass must be be
+contained within its Hill-radius.
+In other words, the
+region’s density must be at least Hill density
+ρH = 9Ω2
+4πG.
+(13)
+Assuming a given scale k = 2π/λ is unstable under the
+previously discussed dispersion relation, at Hill density
+ρH, the mass available to the produced planetesimal1
+can be estimated with
+mp = π
+4
+� λ
+2π
+�2
+Σp(ρp = ρH)
+(14)
+We assumed that the seed mass has access to a region of
+size λ/2π. The surface density if the mid-plane has Hill-
+density can be estimated with Σp =
+√
+2πHpρp, where
+the particle scale height relates to the vertical diffusion
+coefficient via (Youdin & Lithwick 2007)
+Hp =
+�
+δz
+StH.
+(15)
+Together, the mass associated with an unstable scale λ
+is of order,
+mp = 9
+64
+�
+2
+π3 h3
+�
+δz
+St
+� λ
+H
+�2
+M⊙,
+(16)
+1 The mass mp is only equal to the initial planetesimal mass if
+the entire particle cloud collapses to material density of the
+planetesimal, and thus should be understood as an approximate
+mass scale.
+As such, in Klahr & Schreiber (2020, 2021), this
+mass scale is called ‘equivalent mass’, and associated with some
+conversion efficiency that quantifies the fraction that ultimately
+ends up in the formed planetesimal.
+and equivalently, the unstable scale λ that is expected
+to produce a planetesimal of mass of order mp is
+λ
+H =
+�
+64
+9
+�
+π3
+2
+1
+h3
+�
+St
+δz
+mp
+M⊙
+�1/2
+(17)
+Here h = H/R is the disk aspect ratio, typically of order
+0.03 < h < 0.1. The mass associated with the fastest
+growing mode is given by
+mp,fgm = 9
+8
+�π
+2
+�
+δz
+St
+δr
+Sth3Q2
+pM⊙
+(18)
+which scales as
+mp,fgm ≈ 0.37MCeres ·
+� δz
+10−5
+� 1
+2 � δr
+10−5
+�
+·
+�0.1
+St
+� 3
+2 � h
+0.05
+�3 �Qp
+1
+�2
+(19)
+Under isotropic diffusion, this equals the characteristic
+mass in Klahr & Schreiber (2020) if Qp ≈ 0.28.
+Note, that the Toomre instability a priori does not
+require Hill-density to operate. Indeed, as pointed out
+by Klahr & Schreiber (2021), if the vertical density
+structure is not set by vertical diffusion, but instead
+stringently follows ρp(k, z) = (kΣp/2) exp(−|k|z), then
+the fastest growing linear instability would be achieved
+at a mid-plane density of ρp(k = kfgm, z = 0) = 2ρH/9.
+By presupposing Hill density, our approach detaches
+from this assumption of mode-dependent stratification,
+and in the process excludes Toomre unstable clouds that
+fail to withstand tidal gravity and are thus of little
+physical importance for the IMF.
+3.2. IMF from Toomre growth rate
+Given this context of Toomre-like instability and
+Hill density, we proceed by providing predictions for
+the IMF. Our ansatz is to take the mode-depended
+instability growth rates as a probability density function
+p(λ) for unstable scales, and then convert unstable
+modes to planetesimal masses via the recipes discussed
+in Sect. 3.1. This yields a probability density function
+p(mp) such that the probability of a seed mass within
+[mp, mp + dmp] is p(mp)dmp. This first order approach
+agrees with intuition in that fastest growing modes
+should most preferentially collapse,
+slowly growing
+modes only sometimes, stable modes never.
+We write the probability density function p(λ) for
+scale λ to collapse as
+p(λ)dλ ∝ max[ˆγ−1γ(λ), 0]
+(20)
+
+Planetesimal IMFs under diffusion regulated collapse
+5
+ˆγ−1
+is
+a
+normalization
+constant
+given
+by
+ˆγ =
+� λmax
+λmin γ(λ)dλ. Since λ ∝ m1/2
+p
+via Eq. (17), we can
+map p(λ)dλ onto the probability density function for
+seed planetesimal masses p(mp)dmp.
+The
+probability
+density
+function
+p(mp),
+the
+cumulative probability function P(m′
+p ≤ mp), and the
+resulting IMF are shown in Fig. 2, for St = 0.4 and
+different values of δ and Qp.
+The smaller Qp, the
+flatter the IMF, as more scales become unstable. The
+steepest IMF is achieved for Qp → 1.
+Indeed, for
+Qp = 1, the probability density function becomes a
+Dirac delta function, i.e., p(mp) = δ(mp,H − mp).
+Increasing diffusivity shifts the IMF to larger masses,
+provided Qp remains constant, which would require a
+corresponding decrease in Q/Z.
+3.3. Comparison to statistical approaches
+Our ansatz is distinct from past means of deriving
+planetesimal IMFs. Cuzzi et al. (2008, 2010); Hartlep
+& Cuzzi (2020) approach the problem statistically,
+and consider turbulent clustering of particles.
+In
+particular, the assumed scale invariance of the turbulent
+spectrum implies the statistically appearance of regions
+of highly enhanced particle density.
+The argument is
+that only sufficiently dense clumps can withstand ram
+pressure disruption and thus contract to planetesimals
+in a process called primary accretion, thus limiting
+the formation of planetesimals on the low-mass end.
+Our mechanism likewise prohibits the formation of
+arbitrarily small planetesimals, yet the physical intuition
+differs in that (1) the system is, in fact, gravitationally
+unstable under the Toomre-like instability discussed in
+Sect. 2 resulting in (2) ram pressure being negligible
+compared to diffusion (also see Klahr & Schreiber
+(2020)), and (3), that the smallest planetesimals are
+those resulting from the smallest scale that can be
+gravitationally unstable under the dispersion relation in
+Eq. (5).
+Another statistical ansatz was taken by Hopkins &
+Christiansen
+(2013),
+where
+turbulent
+density
+fluctuations can render local regions of the (gas) disk
+unstable to gravitational collapse.
+The initial mass
+function of collapsing clumps therein depends on the
+critical density for self-gravitating clumps and as well
+as the properties of the ambient turbulence.
+In our
+ansatz, the gas remains stable throughout. Moreover,
+we take all clumps to collapse exactly at Hill density in
+Eq. (13). Different masses are the result of collapse of
+differently sized regions, i.e. on different scales λ.
+4. NUMERICAL TESTS
+We perform high-resolution numerical tests using
+ATHENA (Stone et al. 2008; Bai & Stone 2010a) to
+−0.1
+0.0
+0.1
+y/H
+t = 60.00 Ω−1
+−0.1
+0.0
+0.1
+x/H
+−0.05
+0.00
+0.05
+z/H
+−2.5
+−2.0
+−1.5
+−1.0
+−0.5
+0.0
+log(Σp/Σg)
+−2
+0
+2
+log(max(ρp)y/ρ0)
+Figure 3. Map of the particle surface density at after 60
+Ω−1, which is when self-gravity is turned on. Top panel is
+vertically integrated, wheares bottom panel is azimuthally
+integrated. The white dashed lines mark the border of the
+physical simulation domain and the ghost cells. Since, one
+of the two filaments at 60 Ω−1 is located right at the radial
+simulation boundary, we chose to include the ghost cells in
+this figure, which results in that filament being depicted
+twice.
+directly test our predictions for the IMF. The procedure
+is similar to that in e.g. Li et al. (2019); Gerbig
+et al. (2020), in that we let a small patch around
+a protoplanetary disk mid-plane evolve into some
+turbulent state.
+We then turn on self-gravity with
+different values for Qp by varying the self-gravity
+parameter ˆG in code units. The measurement of Qp,
+as well as the calculation of the predicted IMF requires
+measurement of radial and vertical diffusion, prior to
+turning on-self gravity. Since the streaming instability
+will concentrate particles into filaments, we must also
+determine the local particle concentration.
+4.1. Numerical Setup
+We employ ATHENA (Stone et al. 2008) to solve the
+hydrodynamic equations on an Eulerian grid including
+Lagrangian super-particles (Bai & Stone 2010a). Our
+numerical setup is similar to simulations of streaming
+instability regulated planetesimal formation such as
+Johansen et al. (2007); Simon et al. (2016); Li et al.
+(2019); Gerbig et al. (2020), in that we use the local
+
+6
+Gerbig and Li
+−0.1
+0.0
+0.1
+x/H
+−0.1
+0.0
+0.1
+y/H
+Q = 16, t = 60 Ω−1
+−0.1
+0.0
+0.1
+x/H
+Q = 8, t = 60 Ω−1
+−0.1
+0.0
+0.1
+x/H
+Q = 4, t = 60 Ω−1
+−3.5
+−3.0
+−2.5
+−2.0
+−1.5
+−1.0
+−0.5
+0.0
+log(Σp/Σg)
+101
+103
+NColumns
+all columns
+Q = 4
+Q = 8
+Q = 16
+Z
+Figure 4. Maps of the columns that, at 60 Ω−1, contain at least one cell at or above Hill-density for Q = 16, Q = 8, and Q = 4
+(top panels, from left to right), and corresponding column density histograms (bottom panel). The bottom panel also indicates
+the chosen values for Z in dashed lines, i.e. Z = 0.137, Z = 0.100, and Z = 0.0075 for Q = 16, Q = 8, and Q = 4 respectively.
+The full surface density map of this snapshot is shown in Fig. 3.
+shearing box approximation (Goldreich & Lynden-Bell
+1965) with coordinates (x, y, z).
+We consider a non-
+magnetized gas with an isothermal equation of state.
+Gas is initialized in hydrostatic equilibrium. Particles
+are forced into a super-Keplerian rotation by an external
+pressure gradient, which we parameterize using
+Π = ηvK
+cs
+,
+(21)
+where
+η
+=
+−(1/2)h2d ln ρ/(d ln r).
+This
+is
+mathematically equivalent to real disks where particles
+orbit Keplerian, and gas experiences sub-Keplerian
+forcing (see Bai & Stone 2010a for details on the
+pressure gradient implementation in ATHENA). Note,
+that Π relates to the pressure gradient parameter used
+in Schreiber & Klahr (2018); Gerbig et al. (2020)
+simply via Π = β/2.
+Particles are initialized with a
+narrower Gaussian, however at an scale height of ηr/2,
+which is the characteristic scale of Kelvin-Helmholtz
+instability in protoplanetary disks (Gerbig et al. 2020).
+As
+such,
+particles
+are
+not
+expected
+to
+undergo
+significant settling or lofting due to vertical stellar
+gravity or Kelvin-Helmholtz stirring respectively.
+The pressure gradient and the resulting relative
+velocity between particle and gas flow energize the linear
+streaming instability (Youdin & Goodman 2005; Squire
+& Hopkins 2018), which then saturates non-linearly and
+in the process concentrates particles into high density
+regions (Johansen & Youdin 2007). The degree of which
+the streaming instability operates, and as a consequence
+the strength of particle diffusion and concentration prior
+to gravitational collapse and planetesimal formation, is
+largely set by two quantities. First, the Stokes number
+St, and second, the ratio of metallicity and pressure
+gradient Z0/Π (Sekiya & Onishi 2018).
+The former
+quantifies the coupling of particles to the gas flow, in
+particular the gas turbulence. The latter traces the ratio
+of dust abundance relative to gas and dust layer scale-
+height, and thus maps onto the mid-plane dust-to-gas
+ratio. Note, that the metallicity Z0 is the global (as in
+simulation domain averaged) particle concentration and
+thus not necessarily equal to the local enhancement Z we
+introduced in Eq. (1). We elaborate on this important
+difference in Sect. 4.2.
+In this work, we choose St = 0.4, Z0 = 0.02 and
+Π = 0.05. This setup is specifically designed to provide
+favorable conditions for the streaming instability and,
+provided Qp < 1, planetesimal formation, in order to
+produce a large number of planetesimals which allows
+for a more statistically robust determination of the
+numerical IMF. Our simulations use a computational
+
+Planetesimal IMFs under diffusion regulated collapse
+7
+domain of 0.2H × 0.2H × 0.15H, with a resolution of
+2560/H (i.e., ∆x ≈ 3.9 · 10−4H) and Npar = 226 ≈
+6.71·107 particles. The vertical extent is slightly reduced
+from a cube to mitigate computational costs and is
+still tall enough because the particle layer remains thin
+all the time.
+Moreover, in the vertical direction, we
+adopt outflow boundary conditions that are known to
+reduce boundary artifacts, especially in shorter boxes
+(Li et al. 2018). In the radial and azimuthal directions,
+the standard shearing-periodic boundary conditions are
+imposed.
+Following the precedent set by many past works of
+streaming instability, we express the results of our scale-
+free simulations in the dimensionless unit system of
+dynamical timescale Ω−1, H, and ρ0. The simulation
+is run for t = 60Ω−1 without self-gravity. This allows
+for sufficient amount of time in the non-linear phase of
+the streaming instability to measure diffusion.
+Figure.
+3 shows the vertically
+(top
+panel)
+and
+azimuthally
+(bottom
+panel)
+integrated
+particle
+densities at 60Ω−1, i.e.
+just before self-gravity is
+turned on.
+The streaming instability in a stratified
+disk collects particles into two azimuthally elongated
+filaments,
+which are enhanced in particle density
+relative to the prescribed average of Z0 = 0.02.
+Following the snapshot at 60 Ω−1 depicted in Fig. 3
+we turn on self-gravity,
+the strength of which is
+parameterized by the self-gravity parameter
+˜G
+≡
+4πGρ0/Ω2, which relates to Q via Q = 4/(
+√
+2π ˜G). Self-
+gravity is required for the concept of Hill-density to be
+meaningful. Specifically, the Hill-density depends on the
+self-gravity parameter via
+ρH
+ρ0
+= 9
+˜G
+= 9
+�π
+8 Q
+(22)
+We conduct three self-gravity runs with Q = 16, Q = 8
+and Q = 4 which correspond to Hill densities of ρH/ρ0 ∼
+90, ρH/ρ0 ∼ 45 and ρH/ρ0 ∼ 23 respectively. Note that,
+we can also now associate a Hill-radius rH with a given
+mass m, i.e.,
+rH = R
+� m
+M⊙
+� 1
+3
+=
+�4πρ0m
+˜G
+� 1
+3
+=
+�4π
+9 ρHm
+� 1
+3
+. (23)
+4.2. Local particle concentration
+The stability parameter Qp depends on the local
+particle concentration Z = Σp/Σg. This is importantly
+not equal to the initial, global particle concentration
+Z0, which in our case is set to Z0 = 0.02. The reason
+for this lies within predominanetly radial concentration
+of particle surface density within two filaments as
+evident in Fig. 3, where the typical particle column
+density exceeds Z = 0.02 by between one and two
+orders of magnitude.
+This challenge in applying the
+diffusion-limited collapse criterion to proven shearing
+box simulation simulations of the streaming instability
+was also recognized in Gerbig et al. (2020), where
+the radial extent of streaming instability filaments was
+related to a radial enhancement. While this argument
+was sufficient to demonstrate the general applicability
+of a diffusion-limited collapse criterion to simulations,
+it lacks the precision to reliably predict the Qp as it
+by construction cannot account for additional azimuthal
+enhancement within filaments.
+Because of this, in this work, we choose a different
+pathway and utilize the requirement of Hill-density
+for planetesimal formation to measure the average
+particle concentration of the cells that are expected to
+participate in planetesimal formation. More precisely, in
+order for a given vertical column to be counted towards
+the metallicity, it must contain at least one cell at or
+above Hill density.
+Figure. 4 shows maps of the columns that satisfy this
+requirement for all three Q-values. Note, we are here
+depicting the same snapshot at 60Ω−1 as in Fig. 3,
+i.e.
+before self-gravity has impacted the simulation.
+We are merely evaluating which columns satisfy our
+requirement for a given Hill-density.
+A comparison
+to Fig. 3 confirms that this scheme constrains the
+metallicity measurement to the over-dense filaments
+only. In the bottom panel of Fig. 4, we show histograms
+of the surface density of both the entire simulation
+domain in grey, as well only those columns that contain
+at least one cell above Hill density. The dashed lines
+indicate the average particle concentration for the three
+runs given, and correspond to Z = 0.137, Z = 0.100,
+and Z = 0.075 for Q = 16, Q = 8, and Q = 4
+respectively.
+4.3. Radial diffusion
+We measure radial particle diffusion via (Youdin &
+Lithwick 2007; Johansen & Youdin 2007; Schreiber &
+Klahr 2018; Baehr et al. 2022)
+δr
+csH = Dp,r = 1
+2
+∂⟨|xi(t) − xi(t0)|2⟩i
+∂t
+,
+(24)
+The idea is,
+that as the turbulent state evolves
+particles, the underlying diffusion will widen the particle
+distribution, and as such allow for a calculation of
+a value for particle diffusion.
+If the particles were
+normally distributed, this would correspond to the
+time evolution of the distribution’s variance.
+Note,
+that since we are investigating the streaming instability
+in stratified disks, this method cannot be used to
+
+8
+Gerbig and Li
+−0.8
+−0.6
+−0.4
+−0.2
+0.0
+∆x/H
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+Nbin/Npar
+t0 = 18Ω−1
+∆t =6Ω−1
+∆t =15Ω−1
+∆t =24Ω−1
+∆t =42Ω−1
+−0.3
+−0.2
+−0.1
+0.0
+∆x/H
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+Nbin/Npar
+t0 = 50.25Ω−1
+∆t =3.75Ω−1
+∆t =5.0Ω−1
+∆t =7.5Ω−1
+∆t =9.75Ω−1
+Figure 5.
+Evolution of an initial particle distribution
+at different times ∆t for t0
+= 18Ω−1 (top panel) and
+t0 = 50.25Ω−1 (bottom panel).
+Due to the two over-
+dense filaments the particle distribution develops bi-modality
+— an effect not seen in unstratified streaming instability
+simulations (e.g., Johansen & Youdin 2007)
+determine vertical diffusivity which we will measure in
+the subsequent section.
+Figure. 5 shows the evolution of the initial particle
+distribution from 18Ω−1 to 60Ω−1, when we turn on
+self-gravity.
+Our particle distribution depicts much
+stronger non-Gaussianity than Johansen & Youdin
+(2007) Fig. 17,
+since our simulation is stratified.
+This leads the streaming instability to produce as
+azimuthally-extended filaments, where particles drift
+slower due to the enhanced local dust-to-gas ratio. This
+can lead to multiple peaks in the particle distribution
+— in our specific case there are two peaks as there are
+two dominant filaments.
+If we take all particles into account when evaluating
+Eq. (24), we will determine a global value for the
+radial diffusivity.
+However, the strength of diffusion
+is expected to vary locally and correlate with particle
+density (see e.g., Schreiber & Klahr 2018).
+As such,
+we sort all particles by local density and measure
+the evolution of the distribution up to 60Ω−1 for
+each density bin, and then calculate the corresponding
+diffusion coefficients via Eq. (24).
+Fig. 6 shows that
+denser regions are less diffusive than less dense regions.
+More specifically, up to a dust-to-gas ratio of order 101,
+the diffusion is approximately constant at δr = 1.3·10−4.
+This is a value consistent with that found in Li & Youdin
+(2021).
+For more dense regions,
+the diffusion decreases
+significantly down to δ ∼ 10−6 for ρp/ρ0 > 4 · 102. This
+turbophoretic behavior of the particles (Caporaloni
+et al. 1975; Belan et al. 2014) makes it challenging to
+pinpoint an exact value of δ that is appropriate for the
+determination
+of
+Qp
+and
+the
+subsequent
+IMF.
+Additionally,
+calculation
+of
+δ
+requires
+temporal
+averaging. In Fig. 6 we show multiple averaging times
+in order to highlight the trend of longer averaging
+times yielding larger diffusivity.
+In this work, we choose diffusivities by evaluating
+δ(ρp) at the particle density weighted average
+˜ρp(Q) =
+�
+i,ρp,i>ρH(Q) Niρp,i
+�
+i,ρp,i>ρH(Q)
+.
+(25)
+Note that we only count those particles in Hill-stable
+regions.
+The resulting densities are indicated in the
+right panel of Fig. 6. We evaluate the radial diffusion
+coefficient δr (see Eq. 24) with a second-order one-side
+derivative at t = 60/Ω (by supplying edge order=2
+to numpy.gradient).
+This results in diffusivities of
+δr = 2.4 · 10−6, δr = 2.0 · 10−6, and δr = 2.6 · 10−6
+for Q = 16, Q = 8, and Q = 4 respectively.
+We
+acknowledge
+that
+due
+to
+the
+steepness
+of
+δx(ρH/ρ0) at high dust-to-gas ratios, as well as the
+dependence on averaging time ∆tave, there remains a
+relatively large ambiguity about the most appropriate
+values for δr.
+Indeed, we find any diffusivity within
+1 · 10−6 ≲ δx ≲ 1 · 10−5 justifiable, and note that this
+full range is consistent with diffusivities obtained by
+Schreiber & Klahr (2018) at similar dust-to-gas ratios
+(although with smaller particles).
+4.4. Vertical diffusion
+While vertical diffusion is not required to calculate
+the stability parameter Qp which arises from a purely
+2D
+consideration,
+it
+is
+required
+to
+calculate
+planetesimal masses by setting the surface density
+necessary to achieve tidal-stability.
+We first measure
+the particle scale-height in each vertical column of grid
+cells by calculating the standard deviation of the
+
+Planetesimal IMFs under diffusion regulated collapse
+9
+52
+54
+56
+58
+60
+t [Ω−1]
+0.0000
+0.0005
+0.0010
+0.0015
+0.0020
+⟨|xi(t) − xi(t0 = 50Ω−1)|2⟩i [H2]
+10−1
+100
+101
+102
+103
+ρp/ρ0
+10−6
+10−5
+10−4
+δx
+t = 60Ω−1
+∆tavg = 1Ω−1
+∆tavg = 2Ω−1
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+Number of particles
+×106
+ρH(Q = 16)/ρ0
+ρH(Q = 8)/ρ0
+ρH(Q = 4)/ρ0
+˜ρp(Q)/ρ0
+Figure 6. Radial diffusion depends on particle density. Left panel: Evolution of the variance of the particle distribution for
+different densities. Colors correspond to densities. Right panel: diffusion vs particle density. Colors correspond to density
+and map onto the left panel. Different line styles indicate different averaging times when calculating the gradient in Eq. (24).
+We overlay a histogram of the density distribution of the particles, where vertical solid lines indicate Hill density for chosen
+Q-values. Vertical dashed-dotted lines correspond to the weighted average density ˜ρp(Q), which pinpoint the diffusivities to
+δr = 2.4 · 10−6, δr = 2.0 · 10−6, and δr = 2.6 · 10−6 for Q = 16, Q = 8, and Q = 4 respectively
+−0.1
+0.0
+0.1
+x/H
+−0.1
+0.0
+0.1
+y/H
+−0.1
+0.0
+0.1
+x/H
+−0.1
+0.0
+0.1
+y/H
+10−3
+10−2
+10−1
+Σp/Σg
+10−6
+10−5
+δz
+Q = 16
+Q = 8
+Q = 4
+0.002
+0.004
+0.006
+0.008
+0.010
+Hp/H
+−6.5
+−6.0
+−5.5
+−5.0
+−4.5
+log(δz)
+Figure 7. Map of particle scale height Hp(left panel) and vertical diffusion coefficient δz(center panel) at 60 Ω−1. The right
+panel shows the mean vertical diffusion ℏδz vs the local particle concentration (compare to top panel of Fig. 3). The shaded
+region characterizes ℏδz ± σδ, where σδ is the standard deviation of the vertical diffusivities at a given Σp/Σ. The vertical
+lines indicate the the determined values for Z (see Sect. 4.2, Fig. 4), which set the chosen values for vertical diffusivities to
+δz = 2.3 · 10−6, δz = 3.1 · 10−6, and δz = 3.3 · 10−6 for Q = 16, Q = 8, and Q = 4 respectively.
+vertical particle positions, and then convert to vertical
+diffusivity δz using Eq. (15). Similar procedures were
+also done by e.g., Bai & Stone (2010b); Yang et al.
+(2018); Li & Youdin (2021) when analyzing the vertical
+diffusivity within a stratified particle layer subject to
+the streaming instability. The resulting maps are seen
+in the left and center panel of Fig. 7. The right panel
+of Fig. 7 depicts the mean vertical diffusivity plotted vs
+local particle concentration, where the gray shaded
+region indicates one standard deviation from the mean.
+While there is significant spread of diffusivity for a
+given particle concentration, the vertical diffusivity
+behaves similar to the radial diffusivity in that it
+decreases in more concentrated regions. While in the
+plateau region, the diffusion anisotropy is of order
+δr/δz ∼ 10 which is consistent with e.g., Fig. 6 in Li &
+
+10
+Gerbig and Li
+Table 1. Simulation parameters for the three self-gravity
+runs.
+For all simulations, St = 0.4, Z0 = 0.02, Π =
+0.05, and self-gravity is turned on after 60Ω−1. Calculated
+masses assume h = 0.05 and collapse at Hill-density. Our
+simulations have a radial and azimuthal domain size of 0.2H,
+and the smallest resolved scale is ∆x = 3.9 × 10−4H. The
+first two rows are prescribed values. Rows three, four and
+five correspond to quantities measured at 60Ω−1 as described
+in Sects. 4.2, 4.3 and 4.4. Subsequent rows are properties
+calculated within the diffusion-limited collapse framework
+outlined in Sects. 2 and 3.
+Q
+16
+8
+4
+ρH/ρ0
+90.2
+45.1
+22.6
+Z
+0.137
+0.100
+0.075
+δr
+2.42 · 10−6
+2.04 · 10−6
+2.60 · 10−6
+δz
+2.35 · 10−6
+3.07 · 10−6
+3.28 · 10−6
+Qp
+0.287
+0.180
+0.136
+λmin[H]
+2.26 · 10−3
+1.29 · 10−3
+1.09 · 10−3
+λfgm[H]
+4.42 · 10−3
+2.56 · 10−3
+2.18 · 10−3
+λmax[H]
+1.05 · 10−1
+1.56 · 10−1
+2.34 · 10−1
+mp,min[MCeres]
+1.16 · 10−4
+4.32 · 10−5
+3.22 · 10−5
+mp,fgm[MCeres]
+4.44 · 10−4
+1.70 · 10−4
+1.27 · 10−4
+mp,max[MCeres]
+0.253
+0.630
+1.47
+Youdin (2021), in denser regions the diffusion is of
+order isotropic.
+Indeed, we evaluate δz(Σp/Σ) at the
+particle concentrations identified in Sect. 4.2, which
+yields δz
+=
+2.3 · 10−6,
+δz
+=
+3.1 · 10−6,
+and
+δz = 3.3 · 10−6 for Q = 16, Q = 8, and Q = 4
+respectively.
+This
+relatively
+low,
+and
+isotropic
+diffusion
+is
+consistent with high particle concentrations damping
+diffusion from gas, regardless of direction (Ida et al.
+2021).
+4.5. Planetesimals and mass scalings
+Table 1 summarizes the simulation parameters, the
+measured
+quantities
+δr,
+δz
+and
+local
+particle
+concentration Z, as well as the resulting predicted
+properties, namely Qp as well as characteristic size and
+mass scales. All three simulations have Qp < 1 and are
+thus expected to be gravitationally unstable.
+This is
+unsurprising as the simulation parameters were chosen
+in
+line
+with
+past
+setups
+proven
+to
+produce
+planetesimals (e.g., Li et al. 2019).
+Our smallest
+resolved scale is ∆x = 3.9 · 10−4H, which is about an
+order
+of
+magnitude
+smaller
+than
+λmin
+for
+our
+simulations.
+We thus expect to capture the small
+scales
+of
+diffusion
+regulated
+collapse
+in
+these
+simulations.
+On the other hand, the domain size is
+0.2H, which is of order λmax.
+Moreover, the largest
+scale is further limited by the maximum extent of
+over-dense regions, which radially and vertically is of
+order ηr
+=
+ΠH (Gerbig et al. 2020),
+which for
+Π = 0.05 is just 5 · 10−2H, and thus almost an order of
+magnitude less than λmax. Hence, we do not expect to
+the simulation to collapse on the largest unstable scales
+and produce planetesimals on the very high mass end.
+Fig. 8 shows snapshots of all three runs at 62Ω−1
+which is 2.0Ω−1 after self-gravity is turned on.
+In
+addition
+to
+the
+practical
+benefit
+of
+reducing
+computational costs,
+we chose this short time for
+self-gravity to act, in order to capture a relatively
+pristine mass distribution that is devoid of mergers.
+We used the clump finder algorithm PLAN (Li 2019; Li
+et al. 2019) to identify gravitationally bound clumps2,
+which are produced in all three runs. Indeed, all runs
+collapse rapidly enough for diffusion measurements to
+be unfeasible post 60 Ω−1.
+Thus, we cannot confirm
+whether
+or
+not
+diffusion
+increases
+in
+this
+gravo-turbulent state as it does in Klahr & Schreiber
+(2021).
+Note, that the requirement of Hill-density imposes a
+physical mass unit onto the analytic prediciton, and only
+implicitly depends on distance to the star (see Klahr &
+Schreiber 2020, for a discussion on the radial dependence
+of the diffusion-limited collapse criterion). On the other
+hand, the numerical results, i.e. the densities of bound
+clumps in Fig. 8 are fully scale-free in gas surface density
+ρ0. However, the two can be connected using the self-
+gravity parameter. More specifically, the mass unit of
+the simulation M0 can be expressed as
+M0 = ρ0H3 = h3 ˜GM⊙
+4π =
+1
+4
+√
+2π3
+h3
+Q M⊙
+(26)
+= 5.21 · 102
+� Q
+16
+�−1 � h
+0.05
+�3
+MCeres,
+(27)
+which introduces the same cubic dependence on aspect
+ratio as the analytic predictions in e.g., Eq. (16). The
+aspect ratio relates to the pressure gradient parameter
+via
+h =
+�
+−2η d ln ρ
+d ln r
+�1/2
+= −2Πd ln ρ
+d ln r .
+(28)
+2 Throughout
+this
+paper,
+we
+use
+‘planetesimals’
+and
+‘gravitationally
+bound
+clumps’
+relatively
+interchangeably.
+The latter is more precise, as our simulations do not resolve the
+scale of planetesimals. Likewise, the Toomre-like paradigm only
+predicts the mass of a Hill-stable region subject to gravitational
+collapse.
+The actual planetesimal mass requires knowledge of
+contraction efficiency and subsequent accretion of additional
+pebbles.
+
+Planetesimal IMFs under diffusion regulated collapse
+11
+−0.10
+−0.05
+0.00
+0.05
+0.10
+x [H]
+−0.10
+−0.05
+0.00
+0.05
+0.10
+y [H]
+Q = 16, Qp = 0.29
+−0.10
+−0.05
+0.00
+0.05
+0.10
+x [H]
+Q = 8, Qp = 0.18
+−0.10
+−0.05
+0.00
+0.05
+0.10
+x [H]
+Q = 4, Qp = 0.14
+−2.0
+−1.5
+−1.0
+−0.5
+0.0
+0.5
+1.0
+log(Σp/Σg)
+−2.0
+−1.5
+−1.0
+−0.5
+0.0
+0.5
+1.0
+log(Σp/Σg)
+−2.0
+−1.5
+−1.0
+−0.5
+0.0
+0.5
+1.0
+log(Σp/Σg)
+Figure 8. Snapshots of the three self-gravity simulations at 62Ω−1, which is 2Ω−1 after self-gravity is turned on. All three
+simulations formed gravitationally bound clumps indiciated by red circles.
+Throughout this work, we choose h = 0.05, which
+together with our simulation parameter of Π = 0.05
+implicitly assumes d ln ρ/d ln r = −1/2. These choices
+are relatively generic for disk models (compare to e.g.,
+Dullemond et al. 2007; Gerbig et al. 2019; Gerbig &
+Laughlin 2022).
+The smallest mass above which a planetesimal is well
+resolved is the mass associated with a Hill-radius that
+equals the cell-size, i.e., mrH=∆x.
+This limitation is
+caused by the accuracy of the self-gravity solver set
+by the finite grid scale (Simon et al. 2016).
+PLAN
+automatically discards clumps, if any, less massive than
+mrH=∆x.
+Using Eq. (23) and the mass scaling in
+Eq. (26), this mass limit is independent of Q, i.e.
+mrH=∆x = 9
+4π ρH∆x3 = 9
+4π
+�ρH
+ρ0
+� �∆x
+H
+�3
+M0
+= 81
+64h3
+�∆x
+H
+�3
+M⊙ = 1.97 · 10−5MCeres
+(29)
+for ∆x = 3.9 · 10−4H. As shown in Table 1, mrH=∆x
+is smaller than mp,min, and λmin is larger than 2∆x
+for all Q, suggesting we sufficiently resolve the smallest
+expected planetesimals.
+4.6. IMFs and K-S test
+Figure 9 shows the IMFs obtained from the snapshot
+shown in Fig. 8 in solid lines.
+We compare these
+numerically obtained IMFs, to the predicted IMF from
+the Toomre-like instability paradigm, evaluated for the
+properties shown in Tab. 1.
+Hereby, we draw N
+planetesimals from the theoretical PDF, where N is
+the number of bound clumps found in the respective
+simulation. We conduct this random draw 1000 times
+in order to obtain an average theoretical IMF (dashed
+curves), together with one and three σ intervals (dark
+and light shaded regions respectively). In addition, we
+draw a dotted vertical line, corresponding to the mass
+associated with the fastest growing mode of the IMF
+prediction.
+In order to assess the quality of the analytical
+predictions,
+we
+perform
+two
+sample
+Kolmogorov-Smirnov (K-S) tests, which quantify the
+likelihood p of two samples (in our case the numerical
+and
+analytically
+obtained
+planetesimal
+masses)
+originating from the same underlying distribution. The
+null hypothesis, that is the two samples indeed coming
+from the same distribution, can be rejected if p < 0.05.
+For the growth-rate based PDF, the K-S test yields
+p-values of 0.501, 0.395, and 0.0629 for Q = 16, Q = 8,
+and Q = 4 respectively.
+The numerical IMFs are
+therefore consistent with the analytical IMFs obtained
+from the Toomre-like analysis.
+5. DISCUSSION
+We developed a framework of analytically obtaining
+planetesimal IMFs, based on the Toomre-like instability
+in the particle layer in protoplanetary disks. In Sect. 4
+we presented a simulation of the streaming instability,
+measured diffusivity and particle concentration for three
+self-gravity runs, and then compared resulting Qp-values
+and corresponding analytical IMFs to the numerically
+obtained IMFs. We find that the IMFs are consistent
+with each other, as seen Fig. 9.
+Specifically, our
+prediction can explain a number of key properties of the
+planetesimal IMF, which we will outline in the following.
+Our IMF prediction explains the somewhat counter-
+intuitive property of the least massive clumps being
+present in the simulation with the most mass, i.e.
+Q = 4. Since this simulation has the smallest Qp with
+Qp = 0.14, the fastest growing mode shifts to smaller
+scales (see Fig. 1) and thus the most likely planetesimal
+becomes less massive.
+
+12
+Gerbig and Li
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+mp[MCeres]
+10−2
+10−1
+100
+N(> mp)
+p = 5.01e-01
+Q = 16, Z = 0.137
+δz = 2.3e-06, δr = 2.4e-06
+Qp = 0.29, N = 90
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+mp[MCeres]
+p = 3.95e-01
+Q = 8, Z = 0.1
+δz = 3.1e-06, δr = 2.0e-06
+Qp = 0.18, N = 102
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+mp[MCeres]
+p = 6.29e-02
+Q = 4, Z = 0.075
+δz = 3.3e-06, δr = 2.6e-06
+Qp = 0.14, N = 84
+Figure 9. Initial mass functions N(> mp)for the three self-gravity runs. The numerically obtained IMFs are shown in solid
+lines and are based on the snapshots seen in Fig. 8. The IMF resulting from the growth rate-based PDF are shown in dashed
+lines. The dark and light shaded regions indicate one and three standard deviations from the distribution of IMFs that originates
+from only drawing N ∼ 100 planetesimals from the distribution. The vertical dotted lines indicate the masses associated with
+the fastest growing mode mp,fgm. Lastly, the p-value from the K-S-Test is shown in the top right corner of each panel.
+Next, we observe that the IMF becomes flatter if Q
+and therefore Qp decreases, which is due to the increase
+of the range of unstable scales.
+Notably, all three
+numerical IMFs fall short of the analytical prediction
+on the high mass end. As alluded to in Sect. 4.5, this
+is expected as these planetesimals would have to form
+from scales that exceed the maximum scale of particle
+over-densities ηr (see Gerbig et al. 2020). Indeed, the
+idea that the most massive planetesimal is not set by
+the largest Toomre-scale λ but by ηr, is consistent
+with our simulations where the most massive clump
+has ∼ 0.1MCeres in all three simulations, and also with
+Fig. 10 in Simon et al. (2016), where the most massive
+planetesimal shortly after self-gravity is turned on is
+relatively independent of ˜G (or equivalently Q).
+Lastly, the fact that λfgm is much closer to λmin
+when Qp is small, provides an explanation for why only
+high-resolution simulations can sufficiently capture the
+turnover in the mass frequency distribution at the low-
+mass end of planetesimal IMFs (Li et al. 2019).
+5.1. Caveats
+In this section, we outline a number of caveats that are
+to be kept in mind when relating the herein presented
+theory and analysis to real disks.
+First, unstable scales λ were converted to planetesimal
+masses by assuming the contraction of a circular
+sheet of radius λ/(2π) at Hill-density.
+While such a
+conversion agrees with timescale arguments presented
+in e.g. Gerbig et al. (2020); Klahr & Schreiber (2020),
+it certainly is an order of magnitude calculation.
+In
+particular, as pointed out in Polak & Klahr (2022), a
+sphere at Hill-density is not fully tidally stable.
+The
+Hill-sphere is derived by considering the tidal forces on
+either end of the Roche lobe in the restricted three-
+body potential. This has the Hill-sphere extend beyond
+the tidally-stable Roche lobe, and implies Hill-density is
+insufficient for tidal stability by an order of unity factor.
+Moreover, it is not the case that all unstable scales are
+at Hill-density exactly. In fact, as evidenced by Fig. 6,
+many particles are located within regions which exceed
+Hill-density by up to an order of magnitude.
+Additionally,
+our
+model
+assumed
+planetesimal
+formation to be perfectly efficient.
+Fig. 8 however
+demonstrates that there are still particles in the
+filaments that are not bound to any clump. Moreover,
+previous
+works
+showed
+that
+higher
+resolution
+simulations may extend the mass distribution to lower
+masses (Li et al. 2019). Another possibility that is not
+considered in our model is a single clump forming
+binary planetesimals upon contraction to material
+density (Nesvorn´y et al. 2019, 2021).
+Further, our
+model does not take into account the possibility of
+“clumps
+within
+clumps”
+which
+would
+result
+in
+double-counting of planetesimals (compare to Hopkins
+&
+Christiansen
+2013,
+who
+consider
+this
+when
+investigating the statistical instability of the gas disk).
+We note,
+that while multiplicity and inefficient
+contraction would shift the IMF towards smaller masses,
+collapse beyond Hill-density and double counting would
+shift the IMF towards larger masses. We expect that
+this allows the numerically obtained IMFs to remain
+consistent with our analytical predictions.
+Other simplifying assumptions include gas density
+being considered constant.
+For stable, yet small Q
+values, the gas develops a mid-plane cusp with a non-
+
+Planetesimal IMFs under diffusion regulated collapse
+13
+neglible vertical density gradient (see e.g., Armitage
+2015, for a review). We also assumed a mono-disperse
+dust population with a constant Stokes number. This
+choice is likely reasonable to first order as in real disks,
+most solid mass is contained in the largest grains (e.g.,
+Birnstiel et al. 2011; McNally et al. 2021), and the
+largest grains dominate dust-gas interactions in the mid-
+plane and participate most vigorously in planetesimal
+formation (Yang & Zhu 2021).
+Still, a multi-species
+dust fluid can alter the turbulent behavior of the system
+itself (see e.g., Krapp et al. 2019, who pointed out
+the damping effect multi-species dust can have on the
+streaming instability).
+Therefore, the assumption of
+single-species dust is to be kept in mind when applying
+our results to real disks.
+Lastly, we acknowledge that our numerical setup does
+not perfectly replicate the assumptions the Toomre-like
+instability is based on. More specifically, the Toomre
+analysis,
+assumes a thin,
+two-dimensional disk of
+constant density which is then linearly perturbed in form
+of axisymmetric waves.
+This picture evidently differs
+from the turbulent state, i.e.
+non-linear streaming
+instability in vertically-stratified disks, our simulations
+are in when self-gravity is turned on.
+We attempted
+to mitigate this discrepancy by evaluating diffusivities
+and concentration locally, rather than globally, which,
+as discussed in Sect. 4.6, yielded consistent IMFs.
+5.2. The shape of the initial mass function in real disks
+The diffusion-tidal-shear collapse paradigm suggests
+that the shape of planetesimal IMF depends on the
+gravitational,
+Toomre-like
+instability
+and
+on
+the
+stability parameter Qp
+only.
+While independent
+information on diffusivities δr and δz, as well as Stokes
+number St, and disk aspect ratio h are required to
+match the IMF to the appropriate mass range, they do
+not affect the steepness of the IMF (for a fixed Qp). As
+such, as a disk region collects more massive particles, it
+experiences an increase in average grain size, cools or
+becomes less turbulent, its Qp-value decreases, thus
+flattening the planetesimal IMF. Within this context,
+the role of the streaming instability (or other processes
+affecting particle dynamics) for the planetesimal IMF
+in this work, is to set the initial condition at the onset
+of gravitational collapse. The key question within our
+framework therefore is to determine what value for Qp
+takes on in real disks.
+Our simulations, much like many in previous works
+(e.g., Simon et al. 2016; Sch¨afer et al. 2017; Li et al. 2019;
+Gerbig et al. 2020) show collapse as soon as self-gravity
+is turned on. The turbulent state before self-gravity is
+turned on is therefore not physical, and thus the Qp-
+values and IMFs we calculate not necessarily reflective
+of real disks.
+Indeed, as discussed in Gerbig et al.
+(2020), if a system softly evolves towards instability —
+for example by collecting more particle mass — its Qp-
+value likewise softly evolve from Qp > 1 to Qp < 1.
+If Qp ∼ 1 is sufficient for the system to fully collapse
+and form planetesimals as assumed in Klahr & Schreiber
+(2020, 2021), then the resulting IMF is expected to be
+rather steep (Polak & Klahr 2022). On the other hand,
+the system may be unable to fully contract at Qp ∼ 1
+if it is below Hill-density.
+To a similar effect, as the
+contraction time, given by (Gerbig et al. 2020),
+tc =
+Ω
+4πGρpSt =
+�π
+8
+�
+δz
+δr
+Qp
+St Ω−1,
+(30)
+only falls below the dynamical timescale Ω−1 once Qp
+is substantially less than unity, it is plausible that the
+system can collect relevant amounts of mass during
+contraction.
+Both situations would lead to smaller
+values for Qp and thus flatter IMFs like the ones in our
+simulations in Fig. 9.
+One observational constraint may be provided by
+the Asteroids size distribution,
+which,
+at least in
+part, constitutes pristine planetesimals from the Solar
+System’s formation era (see Klahr et al. 2022, for a
+review). Taking, the collisional evolution into account,
+it is possible to reconstruct model of the primordial size
+distribution of Asteroids (see e.g., Delbo et al. 2019).
+Within the here presented framework, the “knee” at
+∼ 100 km (e.g., Gladman et al. 2001) typically found
+in the resulting mass functions can be interpreted as
+an indication of a marginally unstable origin system
+with Qp close to unity — that is assuming the asteroids
+in question indeed formed via gravitational collapse of
+locally over-dense regions, which very well may not have
+been the case.
+Cold Classical Kuiper Belt Objects
+(KBOs) are believed to be even more primordial with
+little to no collisional evolution (Morbidelli & Nesvorn´y
+2020).
+Kavelaars et al. (2021) found a exponentially
+tapered power law as the best fit for the Cold Classical
+KBO mass distribution, and connected the lack of
+large planetesimals (> 400 km in size) to streaming
+instability regulated planetesimal formation. Our work
+is consistent with this interpretation, as Qp imposes a
+limit on the maximum planetesimal mass.
+One often invoked pathway of forming planetesimals
+and circumventing the meter barrier, that is worth
+discussing in the context of our work, is the existence
+of long-lived pressure bumps associated with disk sub-
+structures. While such structures are robustly confirmed
+observationally (e.g., Andrews et al. 2018), their role
+in producing the first generation of planetesimals is
+
+14
+Gerbig and Li
+elusive,
+as already existing planets seem to most
+promisingly explain the sub-structures in the first
+place (Teague et al. 2021).
+Either way, a pressure
+bump is characterized by very small pressure gradients,
+which diminishes the relative velocity between dust
+and gas.
+If Π = 0, then particles can trivially settle
+razor-thin and go gravitationally unstable (as seen in
+e.g., Abod et al. 2019).
+In this state, both vertical
+and radial diffusivities are expected to trend towards
+zero, implying Qp
+→
+0 as well, which our work
+associates with the planetesimal IMF shifting towards
+very small masses. Unless hierarchical mergers between
+small planetesimal immediately after formation is very
+efficient, such a bottom-heavy mass function seems
+unlikely. We therefore suspect that either the majority
+of planetesimal formation does not occur in Π = 0
+regions; or, that other mechanisms, possibly gravito-
+turbulence (see e.g., Riols et al. 2017, for the gas disk),
+provide a lower bound on diffusivities, thus also setting
+a lower limit on Qp.
+To conclude, our analysis that connects the PDF of
+formed
+planetesimals
+to
+the
+growth
+rates
+of
+the
+Toomre-like instability of a particle layer subject to a
+diffusive flux, unifies both the flat, power-law shaped,
+IMFs previously obtained numerically (e.g., Simon
+et al. 2016; Li et al. 2019), and the ‘Asteroids are born
+big’ (Morbidelli et al. 2009) paradigm that arises when
+investigating marginally unstable systems (Klahr &
+Schreiber 2020).
+The analytically obtained IMFs are
+consistent with our numerical setups. Further work is
+required
+to
+test
+the
+predictions
+for
+different
+numerically setups, and in particular, to assess how the
+value of Qp at the on-set of planetesimal formation,
+and thus the steepness of the resulting IMF, depend on
+disk properties and radii.
+Such an analysis informs
+initial conditions for planet formation models (e.g.,
+Emsenhuber
+et
+al.
+2021),
+and
+conversely,
+if
+the
+formation
+of
+an
+observed
+exoplanetary
+population
+presupposes a specific planetesimal IMF(e.g., Batygin
+& Morbidelli 2023, for rocky Super-Earths), provides
+constraints
+on
+the
+disk
+state
+during
+the
+era
+of
+planetesimal formation.
+6. ACKNOWLEDGEMENTS
+This work and KG and RL benefited from the
+2022 Exoplanet Summer Program in the Other Worlds
+Laboratory (OWL) at the University of California,
+Santa Cruz, a program funded by the Heising-Simons
+Foundation. The authors thank Greg Laughlin, Hubert
+Klahr, Andrew Youdin, Ruth Murray-Clay and Malena
+Rice for insightful comments and discussions.
+Software:
+Athena (Stone et al. 2008), PLAN (Li
+2019), NumPy (Harris et al. 2020), Matplotlib (Hunter
+2007), CMasher (van der Velden 2020)
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diff --git a/LtFQT4oBgHgl3EQfUjZg/content/tmp_files/load_file.txt b/LtFQT4oBgHgl3EQfUjZg/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..933f42ea8cbe7aa617130010bbdbf7758bcd4135
--- /dev/null
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@@ -0,0 +1,1285 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf,len=1284
+page_content='Draft version February 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2023  Typeset using LATEX twocolumn style in AASTeX631  Planetesimal Initial Mass Functions following Diffusion Regulated Gravitational Collapse  Konstantin Gerbig  1 and Rixin Li (李日新)  2  1Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Yale University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 52 Hillhouse Ave,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' New Haven,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' CT 06511,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' USA  2Center for Astrophysics and Planetary Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Cornell University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Ithaca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' NY 14853,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' USA  ABSTRACT  The initial mass function (IMF) of planetesimals is of key importance for understanding the  initial stages of planet formation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' yet theoretical predictions so far have been insufficient in  explaining the variety of IMFs found in simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Here, we connect diffusion-tidal-shear limited  planetesimal formation within the framework of a Toomre-like instability in the particle mid-plane of  a protoplanetary disk to an analytic prediction for the planetesimal IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The shape of the IMF is  set by the stability parameter Qp, which in turn depends on the particle Stokes number, the Toomre  Q value of the gas, the local dust concentration and the local diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' We compare our prediction  to high-resolution numerical simulations of the streaming instability and planetesimal formation via  gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' We find that our IMF prediction agrees with numerical results, and is consistent  with both the ‘planetesimals are born big’ paradigm and the power law description commonly found  in simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' INTRODUCTION  Planetesimals  are  the  initial  building  blocks  of  planets and their Initial Mass Function (IMF) is  consequently of great interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  The planetesimal  formation  process  in  protoplanetary  disks,  which  ultimately dictates the planetesimal IMF, is connected  to  an  ensemble  of  challenges,  one  of  the  most  prominent of which is the so-called meter barrier (see  the review by Chiang & Youdin 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Particles  exceeding sizes of order meters are expected to be  limited  in  their  capacity  to  grow  via  continued  coagulation due to both rapid radial drift inwards and  increased relative velocities that result in preferentially  destructive collisions (Birnstiel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  One  well-received solution to this growth barrier is to  rapidly form planetesimals via gravitational collapse of  over-dense particle clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Initial ideas focused on the  gravitational instability of the entire particle mid-plane  (Safronov 1969;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Goldreich & Ward 1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  However,  Kelvin-Helmholtz stirring prevents razor-thin particle  settling and thus renders such a global gravitational  instability challenging (Weidenschilling 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Sekiya  1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Youdin & Shu 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Johansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2006a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' On  the other hand, local patches have been shown to  Corresponding author: Konstantin Gerbig  konstantin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='gerbig@yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='edu  collapse if local particle concentrations are sufficiently  high (Johansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2006b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' In particular, instabilities  energized by the relative dust-gas streaming velocity  (Youdin & Goodman 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Squire & Hopkins 2018)  can concentrate particles to densities sufficient for  gravitational collapse to trigger and planetesimals to  form (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=', Johansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Gerbig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' In  the past, the IMF of planetesimals has been obtained  by performing numerical simulations of this setup and  then counting produced planetesimals, a process that  resulted in power-laws of various kinds (Simon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Sch¨afer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Recently, the formation of planetesimals has been  connected to the diffusion of particles as well as their  stability to stellar tidal forces (Klahr & Schreiber 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Gerbig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Klahr & Schreiber 2020, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  This framework envisions particles to be subject to a  diffusive flux away from the concentration maximum,  much how pressurized gas clouds resist collapse in star  formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' In addition, any particle cloud on the verge  of gravitational collapse must be stable to stellar tidal  gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' These effects limit planetesimal formation on  small scales and large scales respectively, thus together  prescribe a characteristic scale on which planetesimal  formation is expected to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  This in turn, can be  translated to a characteristic planetesimal mass, which  Klahr & Schreiber (2020) hypothesized to be the center  of a Gaussian-shaped IMF, in agreement with IMFs of  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='13297v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='EP]  30 Jan 2023  ID2  Gerbig and Li  primordial asteroids (Delbo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2019), yet seemingly  implying a mismatch to numerically obtained IMFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  In this paper,  we connect these two paradigms  by deriving planetesimal IMFs within the framework  of diffusion-tidal-shear limited planetesimal formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Thereby, we argue that the probability density function  of a given scale to collapse scales with the scale’s  Toomre-like growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  We also directly test our  prediction by conducting numerical simulations using  proven setups in ATHENA (Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2008), that  produce the streaming instability (Youdin & Goodman  2005) and planetesimal formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' In the process, we  for the first time conduct diffusion measurements in  large-scale stratified streaming instability simulations,  as well as develop a method for obtaining local particle  concentrations that are appropriate for characterizing  the onset of planetesimal formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2  we review the Toomre-like instability for planetesimal  formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 3, we derive planetesimal IMFs,  which we compare to numerical simulations in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  We discuss our results, namely applicability, caveats and  implications, in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' TOOMRE-LIKE INSTABILITY  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Stability parameter  The stability of self-gravitating particles subject to  a diffusive flux induced by coupling to turbulent gas  velocities has been studied in numerous occasions  (Youdin 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Gerbig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Klahr & Schreiber  2020, 2021), and can be assessed using a Toomre-like  value Qp, such as  Qp ≡  �  δr  St  1  Z  csΩ  πGΣg  =  �  δr  St  Q  Z  (1)  where Z = Σp/Σg is the (local) dust concentration,  Q = csΩ/(πGΣg) is the standard Toomre value (Toomre  1964), St = tsΩ is the dimensionless stopping time,  and δr  = Dp,r/(csH) is the dimensionless (radial)  diffusion coefficient for particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Ω is the orbital  frequency, cs the sound-speed of the gas, H = cs/Ω the  disk pressure scale-height, Σp and Σg particle and gas  surface densities respectively, and G is the gravitational  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  In the Epstein regime, the stopping time  relates to particle size a, and dust material density ρ•  via ts = ρ•a/(  √  2πΣg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' We consider dust and pebbles  where ts is such that St remains below unity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=',  Birnstiel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  This corresponds to well or at  least marginally coupled particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  If Qp < 1, the system is unstable and expected to  collapse and form planetesimals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  We review the  corresponding  instability  analysis  in  the  following  section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Dynamical equations and dispersion relation  Following (Klahr & Schreiber 2021), we start with the  a set of dynamical equations for dust particles in the  shearing sheet approximation and under the assumption  of a razor-thin disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' We adopt the coordinates r, φ, z  for radial, azimuthal and vertical directions respectively,  and consider a patch at distance R from a solar-mass  star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The linearized set of equation is given by  1  Σp  ∂Σ′  p  ∂t + ∂v′  r  ∂r = 0,  (2)  ∂v′  r  ∂r − 2Ωv′  φ = − 1  Σp  Dp,r  ts  ∂Σ′  p  ∂r − ∂Φ′  ∂r − v′  r  ts  ,  (3)  ∂v′  φ  ∂t + Ωv′  r  2  = 0,  (4)  where the prime denotes perturbed quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  They  describe mass continuity, and conservation of radial and  azimuthal momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Notably, we ignore an explicit  azimuthal drag term as in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=', Youdin (2011), an  assumption that is justified at dust-to-gas ratios of  order unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Instead, coupling to the gas is assumed  to be wholly described by gas pressure counteracting  radial contraction and the diffusive flux in the radial  momentum equation, which can be understood as an  effective pressure flux induced by turbulent gas motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  As a consequence, there is also no mass diffusion term  in the continuity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  We continue by introducing axisymmetric WKB waves  scaling with Σ′  p ∝ exp(−i(kr − ωt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Φ′ = −2πGΣ′/|k|  is the potential for a perturbed disk assuming ρp(k, z) =  (kΣp/2) exp(−|k|z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The resulting dispersion relation is  given by (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (B22) in Klahr & Schreiber 2021),  ω2  0 = δr  Stc2  sk2 − 2πΣpG|k| + Ω2,  (5)  and can be expressed in terms of the stability parameter  Qp  ω2  0  Ω2 = δr  St(kH)2 − 2  Qp  �  δr  St|k|H + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  (6)  Here, ω2  0 is defined as ω2  0 = ω(ω − i/ts), and represents  the complex frequency without the drag term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Note that the herein used solution to the Poisson  equation does not take into account a softening term  caused by the particle layer thickness (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (12) in  Youdin 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Planetesimal IMFs under diffusion regulated collapse  3  10−2  10−1  100  101  k[kfgm]  −6  −4  −2  0  2  ω2  0[Ω−2]  Qp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='4  Qp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  Qp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='6  Qp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='8  10−1  100  Qp  10−3  10−2  10−1  100  λ[H]  λfgm  lc  2π  �  δ/StH  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  log(γΩ)  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Dispersion relation ω2  0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (5) (left panel) and growth rates γ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (11) (right panel) for different values of  Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Parameters are δ = 10−5 and St = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The gray shaded region in the left panel indicates stability against axisymmetric  perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' As Qp decreases, more modes become unstable, and the fastest growing mode (dashed lines in right panel) shifts  to smaller scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' For reference, we plot the scale 2π  �  δ/StH (dotted line right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  10−4  10−2  100  mp[MCeres]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0075  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0125  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0150  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0175  p(mp)  Qp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='15  Qp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='4  Qp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='8  10−4  10−2  100  mp[MCeres]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  P(mp)  δr = δz = 10−6  δr = δz = 10−5  10−4  10−2  100  mp[MCeres]  10−2  10−1  100  N(> mp)  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Probability density functions (left panel), cumulative probability density functions (center panel) and initial mass  functions (right panel) for the growth rates in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2 where St = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Diffusion is isotropic and set to δ = 10−6 (solid lines)  and δ = 10−5 (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Different colors correspond to different values of Qp, all of which chosen to be unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' We use  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (16) for converting unstable scale to seed mass, which assumes collapse at Hill density, and requires knowledge of the aspect  ratio which we set to h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' We assume the system to produce N = 1000 planetesimals total to calculate the normalized  IMF N(< mp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Fastest growing mode and growth rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  The  fastest  growing  mode  is  found  by  solving  ∂ω2  0/∂k = 0, which yields  kfgm =  �  St  δr  1  QpH .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  (7)  The complex frequency of the fastest growing mode is  simply  ω2  0,fgm  Ω2  = 1 − 1  Q2p  ,  (8)  which highlights that only for Qp < 1 exist k for which  ω2  0 < 0 and the instability can grow, thus justifying the  definition of Qp as a stability parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' We can also  calculate largest and smallest unstable scales λ = 2π/k  by solving ω2  0 = 0, resulting in  λmin/max  λfgm  = 1  Qp  �  1  Qp  ±  �  1  Q2p  − 1  �  ,  (9)  with λfgm = 2π/kfgm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The range of unstable scales is  λmax − λmin = 2λfgm/Qp  �  1/Q2p − 1, which increases  for decreasing Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Note, that the fastest growing mode relates to the  critical cloud radius lc in Klahr & Schreiber (2020) and  Gerbig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (2020) via  λfgm = 2π  �  δr  StQpH = 6πQplc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  (10)  4  Gerbig and Li  The growth rate γ(k) can be found solving γ = iω  or equivalently γ(γ + 1/ts) = −ω2  0 (Klahr & Schreiber  2021):  γ(k)  Ω  = − 1  2St +  �  1  4St2 − ω2  0(k)  Ω2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  (11)  Hence, the fastest growing mode kfgm grows with  γ(k = kfgm)  Ω  = − 1  2St +  �  1  4St2 + 1  Q2p  − 1  (12)  Dispersion relation and growth rates are shown for a set  of unstable Qp in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' In the right panel, we also  plot λfgm, lc as well as the scale 2π  �  δr/StH which is of  order the radial extent of small-scale particle structures  (Gerbig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' PLANETESIMAL INITIAL MASS FUNCTIONS  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Planetesimal masses at Hill density  In order for a region in the particle disk to be  stable against stellar tidal gravity, its mass must be be  contained within its Hill-radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  In other words, the  region’s density must be at least Hill density  ρH = 9Ω2  4πG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  (13)  Assuming a given scale k = 2π/λ is unstable under the  previously discussed dispersion relation, at Hill density  ρH, the mass available to the produced planetesimal1  can be estimated with  mp = π  4  � λ  2π  �2  Σp(ρp = ρH)  (14)  We assumed that the seed mass has access to a region of  size λ/2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The surface density if the mid-plane has Hill-  density can be estimated with Σp =  √  2πHpρp, where  the particle scale height relates to the vertical diffusion  coefficient via (Youdin & Lithwick 2007)  Hp =  �  δz  StH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  (15)  Together, the mass associated with an unstable scale λ  is of order,  mp = 9  64  �  2  π3 h3  �  δz  St  � λ  H  �2  M⊙,  (16)  1 The mass mp is only equal to the initial planetesimal mass if  the entire particle cloud collapses to material density of the  planetesimal, and thus should be understood as an approximate  mass scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  As such, in Klahr & Schreiber (2020, 2021), this  mass scale is called ‘equivalent mass’, and associated with some  conversion efficiency that quantifies the fraction that ultimately  ends up in the formed planetesimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  and equivalently, the unstable scale λ that is expected  to produce a planetesimal of mass of order mp is  λ  H =  �  64  9  �  π3  2  1  h3  �  St  δz  mp  M⊙  �1/2  (17)  Here h = H/R is the disk aspect ratio, typically of order  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='03 < h < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The mass associated with the fastest  growing mode is given by  mp,fgm = 9  8  �π  2  �  δz  St  δr  Sth3Q2  pM⊙  (18)  which scales as  mp,fgm ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='37MCeres ·  � δz  10−5  � 1  2 � δr  10−5  � �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  St  � 3  2 � h  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05  �3 �Qp  1  �2  (19)  Under isotropic diffusion, this equals the characteristic  mass in Klahr & Schreiber (2020) if Qp ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Note, that the Toomre instability a priori does not  require Hill-density to operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Indeed, as pointed out  by Klahr & Schreiber (2021), if the vertical density  structure is not set by vertical diffusion, but instead  stringently follows ρp(k, z) = (kΣp/2) exp(−|k|z), then  the fastest growing linear instability would be achieved  at a mid-plane density of ρp(k = kfgm, z = 0) = 2ρH/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  By presupposing Hill density, our approach detaches  from this assumption of mode-dependent stratification,  and in the process excludes Toomre unstable clouds that  fail to withstand tidal gravity and are thus of little  physical importance for the IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' IMF from Toomre growth rate  Given this context of Toomre-like instability and  Hill density, we proceed by providing predictions for  the IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Our ansatz is to take the mode-depended  instability growth rates as a probability density function  p(λ) for unstable scales, and then convert unstable  modes to planetesimal masses via the recipes discussed  in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' This yields a probability density function  p(mp) such that the probability of a seed mass within  [mp, mp + dmp] is p(mp)dmp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' This first order approach  agrees with intuition in that fastest growing modes  should most preferentially collapse,  slowly growing  modes only sometimes, stable modes never.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  We write the probability density function p(λ) for  scale λ to collapse as  p(λ)dλ ∝ max[ˆγ−1γ(λ), 0]  (20)  Planetesimal IMFs under diffusion regulated collapse  5  ˆγ−1  is  a  normalization  constant  given  by  ˆγ =  � λmax  λmin γ(λ)dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Since λ ∝ m1/2  p  via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (17), we can  map p(λ)dλ onto the probability density function for  seed planetesimal masses p(mp)dmp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  The  probability  density  function  p(mp),  the  cumulative probability function P(m′  p ≤ mp), and the  resulting IMF are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2, for St = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='4 and  different values of δ and Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  The smaller Qp, the  flatter the IMF, as more scales become unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The  steepest IMF is achieved for Qp → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Indeed, for  Qp = 1, the probability density function becomes a  Dirac delta function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=', p(mp) = δ(mp,H − mp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Increasing diffusivity shifts the IMF to larger masses,  provided Qp remains constant, which would require a  corresponding decrease in Q/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Comparison to statistical approaches  Our ansatz is distinct from past means of deriving  planetesimal IMFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Cuzzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (2008, 2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Hartlep  & Cuzzi (2020) approach the problem statistically,  and consider turbulent clustering of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  In  particular, the assumed scale invariance of the turbulent  spectrum implies the statistically appearance of regions  of highly enhanced particle density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  The argument is  that only sufficiently dense clumps can withstand ram  pressure disruption and thus contract to planetesimals  in a process called primary accretion, thus limiting  the formation of planetesimals on the low-mass end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Our mechanism likewise prohibits the formation of  arbitrarily small planetesimals, yet the physical intuition  differs in that (1) the system is, in fact, gravitationally  unstable under the Toomre-like instability discussed in  Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2 resulting in (2) ram pressure being negligible  compared to diffusion (also see Klahr & Schreiber  (2020)), and (3), that the smallest planetesimals are  those resulting from the smallest scale that can be  gravitationally unstable under the dispersion relation in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Another statistical ansatz was taken by Hopkins &  Christiansen  (2013),  where  turbulent  density  fluctuations can render local regions of the (gas) disk  unstable to gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  The initial mass  function of collapsing clumps therein depends on the  critical density for self-gravitating clumps and as well  as the properties of the ambient turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  In our  ansatz, the gas remains stable throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Moreover,  we take all clumps to collapse exactly at Hill density in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Different masses are the result of collapse of  differently sized regions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' on different scales λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' NUMERICAL TESTS  We perform high-resolution numerical tests using  ATHENA (Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Bai & Stone 2010a) to  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  y/H  t = 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='00 Ω−1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  x/H  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05  z/H  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  log(Σp/Σg)  −2  0  2  log(max(ρp)y/ρ0)  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Map of the particle surface density at after 60  Ω−1, which is when self-gravity is turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Top panel is  vertically integrated, wheares bottom panel is azimuthally  integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The white dashed lines mark the border of the  physical simulation domain and the ghost cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Since, one  of the two filaments at 60 Ω−1 is located right at the radial  simulation boundary, we chose to include the ghost cells in  this figure, which results in that filament being depicted  twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  directly test our predictions for the IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The procedure  is similar to that in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Gerbig  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (2020), in that we let a small patch around  a protoplanetary disk mid-plane evolve into some  turbulent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  We then turn on self-gravity with  different values for Qp by varying the self-gravity  parameter ˆG in code units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The measurement of Qp,  as well as the calculation of the predicted IMF requires  measurement of radial and vertical diffusion, prior to  turning on-self gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Since the streaming instability  will concentrate particles into filaments, we must also  determine the local particle concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Numerical Setup  We employ ATHENA (Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2008) to solve the  hydrodynamic equations on an Eulerian grid including  Lagrangian super-particles (Bai & Stone 2010a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Our  numerical setup is similar to simulations of streaming  instability regulated planetesimal formation such as  Johansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Simon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Gerbig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (2020), in that we use the local  6  Gerbig and Li  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  x/H  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  y/H  Q = 16, t = 60 Ω−1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  x/H  Q = 8, t = 60 Ω−1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  x/H  Q = 4, t = 60 Ω−1  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  log(Σp/Σg)  101  103  NColumns  all columns  Q = 4  Q = 8  Q = 16  Z  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Maps of the columns that, at 60 Ω−1, contain at least one cell at or above Hill-density for Q = 16, Q = 8, and Q = 4  (top panels, from left to right), and corresponding column density histograms (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The bottom panel also indicates  the chosen values for Z in dashed lines, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='137, Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='100, and Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0075 for Q = 16, Q = 8, and Q = 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  The full surface density map of this snapshot is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  shearing box approximation (Goldreich & Lynden-Bell  1965) with coordinates (x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  We consider a non-  magnetized gas with an isothermal equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Gas is initialized in hydrostatic equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Particles  are forced into a super-Keplerian rotation by an external  pressure gradient, which we parameterize using  Π = ηvK  cs  ,  (21)  where  η  =  −(1/2)h2d ln ρ/(d ln r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  This  is  mathematically equivalent to real disks where particles  orbit Keplerian, and gas experiences sub-Keplerian  forcing (see Bai & Stone 2010a for details on the  pressure gradient implementation in ATHENA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Note,  that Π relates to the pressure gradient parameter used  in Schreiber & Klahr (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Gerbig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (2020)  simply via Π = β/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Particles are initialized with a  narrower Gaussian, however at an scale height of ηr/2,  which is the characteristic scale of Kelvin-Helmholtz  instability in protoplanetary disks (Gerbig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  As  such,  particles  are  not  expected  to  undergo  significant settling or lofting due to vertical stellar  gravity or Kelvin-Helmholtz stirring respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  The pressure gradient and the resulting relative  velocity between particle and gas flow energize the linear  streaming instability (Youdin & Goodman 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Squire  & Hopkins 2018), which then saturates non-linearly and  in the process concentrates particles into high density  regions (Johansen & Youdin 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The degree of which  the streaming instability operates, and as a consequence  the strength of particle diffusion and concentration prior  to gravitational collapse and planetesimal formation, is  largely set by two quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' First, the Stokes number  St, and second, the ratio of metallicity and pressure  gradient Z0/Π (Sekiya & Onishi 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  The former  quantifies the coupling of particles to the gas flow, in  particular the gas turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The latter traces the ratio  of dust abundance relative to gas and dust layer scale-  height, and thus maps onto the mid-plane dust-to-gas  ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Note, that the metallicity Z0 is the global (as in  simulation domain averaged) particle concentration and  thus not necessarily equal to the local enhancement Z we  introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' We elaborate on this important  difference in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  In this work, we choose St = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='4, Z0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='02 and  Π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' This setup is specifically designed to provide  favorable conditions for the streaming instability and,  provided Qp < 1, planetesimal formation, in order to  produce a large number of planetesimals which allows  for a more statistically robust determination of the  numerical IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Our simulations use a computational  Planetesimal IMFs under diffusion regulated collapse  7  domain of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='2H × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='2H × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='15H, with a resolution of  2560/H (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=', ∆x ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='9 · 10−4H) and Npar = 226 ≈  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='71·107 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The vertical extent is slightly reduced  from a cube to mitigate computational costs and is  still tall enough because the particle layer remains thin  all the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Moreover, in the vertical direction, we  adopt outflow boundary conditions that are known to  reduce boundary artifacts, especially in shorter boxes  (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' In the radial and azimuthal directions,  the standard shearing-periodic boundary conditions are  imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Following the precedent set by many past works of  streaming instability, we express the results of our scale-  free simulations in the dimensionless unit system of  dynamical timescale Ω−1, H, and ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The simulation  is run for t = 60Ω−1 without self-gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' This allows  for sufficient amount of time in the non-linear phase of  the streaming instability to measure diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  3 shows the vertically  (top  panel)  and  azimuthally  (bottom  panel)  integrated  particle  densities at 60Ω−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  just before self-gravity is  turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  The streaming instability in a stratified  disk collects particles into two azimuthally elongated  filaments,  which are enhanced in particle density  relative to the prescribed average of Z0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Following the snapshot at 60 Ω−1 depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 3  we turn on self-gravity,  the strength of which is  parameterized by the self-gravity parameter  ˜G  ≡  4πGρ0/Ω2, which relates to Q via Q = 4/(  √  2π ˜G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Self-  gravity is required for the concept of Hill-density to be  meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Specifically, the Hill-density depends on the  self-gravity parameter via  ρH  ρ0  = 9  ˜G  = 9  �π  8 Q  (22)  We conduct three self-gravity runs with Q = 16, Q = 8  and Q = 4 which correspond to Hill densities of ρH/ρ0 ∼  90, ρH/ρ0 ∼ 45 and ρH/ρ0 ∼ 23 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Note that,  we can also now associate a Hill-radius rH with a given  mass m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=',  rH = R  � m  M⊙  � 1  3  =  �4πρ0m  ˜G  � 1  3  =  �4π  9 ρHm  � 1  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (23)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Local particle concentration  The stability parameter Qp depends on the local  particle concentration Z = Σp/Σg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' This is importantly  not equal to the initial, global particle concentration  Z0, which in our case is set to Z0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The reason  for this lies within predominanetly radial concentration  of particle surface density within two filaments as  evident in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 3, where the typical particle column  density exceeds Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='02 by between one and two  orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  This challenge in applying the  diffusion-limited collapse criterion to proven shearing  box simulation simulations of the streaming instability  was also recognized in Gerbig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (2020), where  the radial extent of streaming instability filaments was  related to a radial enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' While this argument  was sufficient to demonstrate the general applicability  of a diffusion-limited collapse criterion to simulations,  it lacks the precision to reliably predict the Qp as it  by construction cannot account for additional azimuthal  enhancement within filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Because of this, in this work, we choose a different  pathway and utilize the requirement of Hill-density  for planetesimal formation to measure the average  particle concentration of the cells that are expected to  participate in planetesimal formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' More precisely, in  order for a given vertical column to be counted towards  the metallicity, it must contain at least one cell at or  above Hill density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 4 shows maps of the columns that satisfy this  requirement for all three Q-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Note, we are here  depicting the same snapshot at 60Ω−1 as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 3,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  before self-gravity has impacted the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  We are merely evaluating which columns satisfy our  requirement for a given Hill-density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  A comparison  to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 3 confirms that this scheme constrains the  metallicity measurement to the over-dense filaments  only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' In the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 4, we show histograms  of the surface density of both the entire simulation  domain in grey, as well only those columns that contain  at least one cell above Hill density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The dashed lines  indicate the average particle concentration for the three  runs given, and correspond to Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='137, Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='100,  and Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='075 for Q = 16, Q = 8, and Q = 4  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Radial diffusion  We measure radial particle diffusion via (Youdin &  Lithwick 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Johansen & Youdin 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Schreiber &  Klahr 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Baehr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2022)  δr  csH = Dp,r = 1  2  ∂⟨|xi(t) − xi(t0)|2⟩i  ∂t  ,  (24)  The idea is,  that as the turbulent state evolves  particles, the underlying diffusion will widen the particle  distribution, and as such allow for a calculation of  a value for particle diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  If the particles were  normally distributed, this would correspond to the  time evolution of the distribution’s variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Note,  that since we are investigating the streaming instability  in stratified disks, this method cannot be used to  8  Gerbig and Li  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='8  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='6  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  ∆x/H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05  Nbin/Npar  t0 = 18Ω−1  ∆t =6Ω−1  ∆t =15Ω−1  ∆t =24Ω−1  ∆t =42Ω−1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='3  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  ∆x/H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='07  Nbin/Npar  t0 = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='25Ω−1  ∆t =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='75Ω−1  ∆t =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0Ω−1  ∆t =7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5Ω−1  ∆t =9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='75Ω−1  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Evolution of an initial particle distribution  at different times ∆t for t0  = 18Ω−1 (top panel) and  t0 = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='25Ω−1 (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Due to the two over-  dense filaments the particle distribution develops bi-modality  — an effect not seen in unstratified streaming instability  simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=', Johansen & Youdin 2007)  determine vertical diffusivity which we will measure in  the subsequent section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 5 shows the evolution of the initial particle  distribution from 18Ω−1 to 60Ω−1, when we turn on  self-gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Our particle distribution depicts much  stronger non-Gaussianity than Johansen & Youdin  (2007) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 17,  since our simulation is stratified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  This leads the streaming instability to produce as  azimuthally-extended filaments, where particles drift  slower due to the enhanced local dust-to-gas ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' This  can lead to multiple peaks in the particle distribution  — in our specific case there are two peaks as there are  two dominant filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  If we take all particles into account when evaluating  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (24), we will determine a global value for the  radial diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  However, the strength of diffusion  is expected to vary locally and correlate with particle  density (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=', Schreiber & Klahr 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  As such,  we sort all particles by local density and measure  the evolution of the distribution up to 60Ω−1 for  each density bin, and then calculate the corresponding  diffusion coefficients via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 6 shows that  denser regions are less diffusive than less dense regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  More specifically, up to a dust-to-gas ratio of order 101,  the diffusion is approximately constant at δr = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='3·10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  This is a value consistent with that found in Li & Youdin  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  For more dense regions,  the diffusion decreases  significantly down to δ ∼ 10−6 for ρp/ρ0 > 4 · 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' This  turbophoretic behavior of the particles (Caporaloni  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Belan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2014) makes it challenging to  pinpoint an exact value of δ that is appropriate for the  determination  of  Qp  and  the  subsequent  IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Additionally,  calculation  of  δ  requires  temporal  averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 6 we show multiple averaging times  in order to highlight the trend of longer averaging  times yielding larger diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  In this work, we choose diffusivities by evaluating  δ(ρp) at the particle density weighted average  ˜ρp(Q) =  �  i,ρp,i>ρH(Q) Niρp,i  �  i,ρp,i>ρH(Q)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  (25)  Note that we only count those particles in Hill-stable  regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  The resulting densities are indicated in the  right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' We evaluate the radial diffusion  coefficient δr (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 24) with a second-order one-side  derivative at t = 60/Ω (by supplying edge order=2  to numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='gradient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  This results in diffusivities of  δr = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='4 · 10−6, δr = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0 · 10−6, and δr = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='6 · 10−6  for Q = 16, Q = 8, and Q = 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  We  acknowledge  that  due  to  the  steepness  of  δx(ρH/ρ0) at high dust-to-gas ratios, as well as the  dependence on averaging time ∆tave, there remains a  relatively large ambiguity about the most appropriate  values for δr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Indeed, we find any diffusivity within  1 · 10−6 ≲ δx ≲ 1 · 10−5 justifiable, and note that this  full range is consistent with diffusivities obtained by  Schreiber & Klahr (2018) at similar dust-to-gas ratios  (although with smaller particles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Vertical diffusion  While vertical diffusion is not required to calculate  the stability parameter Qp which arises from a purely  2D  consideration,  it  is  required  to  calculate  planetesimal masses by setting the surface density  necessary to achieve tidal-stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  We first measure  the particle scale-height in each vertical column of grid  cells by calculating the standard deviation of the  Planetesimal IMFs under diffusion regulated collapse  9  52  54  56  58  60  t [Ω−1]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0020  ⟨|xi(t) − xi(t0 = 50Ω−1)|2⟩i [H2]  10−1  100  101  102  103  ρp/ρ0  10−6  10−5  10−4  δx  t = 60Ω−1  ∆tavg = 1Ω−1  ∆tavg = 2Ω−1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  Number of particles  ×106  ρH(Q = 16)/ρ0  ρH(Q = 8)/ρ0  ρH(Q = 4)/ρ0  ˜ρp(Q)/ρ0  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Radial diffusion depends on particle density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Left panel: Evolution of the variance of the particle distribution for  different densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Colors correspond to densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Right panel: diffusion vs particle density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Colors correspond to density  and map onto the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Different line styles indicate different averaging times when calculating the gradient in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  We overlay a histogram of the density distribution of the particles, where vertical solid lines indicate Hill density for chosen  Q-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Vertical dashed-dotted lines correspond to the weighted average density ˜ρp(Q), which pinpoint the diffusivities to  δr = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='4 · 10−6, δr = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0 · 10−6, and δr = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='6 · 10−6 for Q = 16, Q = 8, and Q = 4 respectively  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  x/H  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  y/H  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  x/H  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  y/H  10−3  10−2  10−1  Σp/Σg  10−6  10−5  δz  Q = 16  Q = 8  Q = 4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='010  Hp/H  −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  log(δz)  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Map of particle scale height Hp(left panel) and vertical diffusion coefficient δz(center panel) at 60 Ω−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The right  panel shows the mean vertical diffusion ℏδz vs the local particle concentration (compare to top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The shaded  region characterizes ℏδz ± σδ, where σδ is the standard deviation of the vertical diffusivities at a given Σp/Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The vertical  lines indicate the the determined values for Z (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='2, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 4), which set the chosen values for vertical diffusivities to  δz = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='3 · 10−6, δz = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1 · 10−6, and δz = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='3 · 10−6 for Q = 16, Q = 8, and Q = 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  vertical particle positions, and then convert to vertical  diffusivity δz using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Similar procedures were  also done by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=', Bai & Stone (2010b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Li & Youdin (2021) when analyzing the vertical  diffusivity within a stratified particle layer subject to  the streaming instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The resulting maps are seen  in the left and center panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The right panel  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 7 depicts the mean vertical diffusivity plotted vs  local particle concentration, where the gray shaded  region indicates one standard deviation from the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  While there is significant spread of diffusivity for a  given particle concentration, the vertical diffusivity  behaves similar to the radial diffusivity in that it  decreases in more concentrated regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' While in the  plateau region, the diffusion anisotropy is of order  δr/δz ∼ 10 which is consistent with e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 6 in Li &  10  Gerbig and Li  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Simulation parameters for the three self-gravity  runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  For all simulations, St = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='4, Z0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='02, Π =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05, and self-gravity is turned on after 60Ω−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Calculated  masses assume h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05 and collapse at Hill-density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Our  simulations have a radial and azimuthal domain size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='2H,  and the smallest resolved scale is ∆x = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='9 × 10−4H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The  first two rows are prescribed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Rows three, four and  five correspond to quantities measured at 60Ω−1 as described  in Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Subsequent rows are properties  calculated within the diffusion-limited collapse framework  outlined in Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Q  16  8  4  ρH/ρ0  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='2  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='6  Z  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='137  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='075  δr  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='42 · 10−6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='04 · 10−6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='60 · 10−6  δz  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='35 · 10−6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='07 · 10−6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='28 · 10−6  Qp  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='287  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='180  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='136  λmin[H]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='26 · 10−3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='29 · 10−3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='09 · 10−3  λfgm[H]  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='42 · 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='56 · 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='18 · 10−3  λmax[H]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05 · 10−1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='56 · 10−1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='34 · 10−1  mp,min[MCeres]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='16 · 10−4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='32 · 10−5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='22 · 10−5  mp,fgm[MCeres]  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='44 · 10−4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='70 · 10−4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='27 · 10−4  mp,max[MCeres]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='253  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='630  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='47  Youdin (2021), in denser regions the diffusion is of  order isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Indeed, we evaluate δz(Σp/Σ) at the  particle concentrations identified in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='2, which  yields δz  =  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='3 · 10−6,  δz  =  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1 · 10−6,  and  δz = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='3 · 10−6 for Q = 16, Q = 8, and Q = 4  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  This  relatively  low,  and  isotropic  diffusion  is  consistent with high particle concentrations damping  diffusion from gas, regardless of direction (Ida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Planetesimals and mass scalings  Table 1 summarizes the simulation parameters, the  measured  quantities  δr,  δz  and  local  particle  concentration Z, as well as the resulting predicted  properties, namely Qp as well as characteristic size and  mass scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' All three simulations have Qp < 1 and are  thus expected to be gravitationally unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  This is  unsurprising as the simulation parameters were chosen  in  line  with  past  setups  proven  to  produce  planetesimals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=', Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Our smallest  resolved scale is ∆x = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='9 · 10−4H, which is about an  order  of  magnitude  smaller  than  λmin  for  our  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  We thus expect to capture the small  scales  of  diffusion  regulated  collapse  in  these  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  On the other hand, the domain size is  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='2H, which is of order λmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Moreover, the largest  scale is further limited by the maximum extent of  over-dense regions, which radially and vertically is of  order ηr  =  ΠH (Gerbig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2020),  which for  Π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05 is just 5 · 10−2H, and thus almost an order of  magnitude less than λmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Hence, we do not expect to  the simulation to collapse on the largest unstable scales  and produce planetesimals on the very high mass end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 8 shows snapshots of all three runs at 62Ω−1  which is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0Ω−1 after self-gravity is turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  In  addition  to  the  practical  benefit  of  reducing  computational costs,  we chose this short time for  self-gravity to act, in order to capture a relatively  pristine mass distribution that is devoid of mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  We used the clump finder algorithm PLAN (Li 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Li  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2019) to identify gravitationally bound clumps2,  which are produced in all three runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Indeed, all runs  collapse rapidly enough for diffusion measurements to  be unfeasible post 60 Ω−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Thus, we cannot confirm  whether  or  not  diffusion  increases  in  this  gravo-turbulent state as it does in Klahr & Schreiber  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Note, that the requirement of Hill-density imposes a  physical mass unit onto the analytic prediciton, and only  implicitly depends on distance to the star (see Klahr &  Schreiber 2020, for a discussion on the radial dependence  of the diffusion-limited collapse criterion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' On the other  hand, the numerical results, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' the densities of bound  clumps in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 8 are fully scale-free in gas surface density  ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' However, the two can be connected using the self-  gravity parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' More specifically, the mass unit of  the simulation M0 can be expressed as  M0 = ρ0H3 = h3 ˜GM⊙  4π =  1  4  √  2π3  h3  Q M⊙  (26)  = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='21 · 102  � Q  16  �−1 � h  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05  �3  MCeres,  (27)  which introduces the same cubic dependence on aspect  ratio as the analytic predictions in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The  aspect ratio relates to the pressure gradient parameter  via  h =  �  −2η d ln ρ  d ln r  �1/2  = −2Πd ln ρ  d ln r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  (28)  2 Throughout  this  paper,  we  use  ‘planetesimals’  and  ‘gravitationally  bound  clumps’  relatively  interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  The latter is more precise, as our simulations do not resolve the  scale of planetesimals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Likewise, the Toomre-like paradigm only  predicts the mass of a Hill-stable region subject to gravitational  collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  The actual planetesimal mass requires knowledge of  contraction efficiency and subsequent accretion of additional  pebbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Planetesimal IMFs under diffusion regulated collapse  11  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='10  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='10  x [H]  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='10  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='10  y [H]  Q = 16, Qp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='29  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='10  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='10  x [H]  Q = 8, Qp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='18  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='10  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='10  x [H]  Q = 4, Qp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='14  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  log(Σp/Σg)  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  log(Σp/Σg)  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0  log(Σp/Σg)  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Snapshots of the three self-gravity simulations at 62Ω−1, which is 2Ω−1 after self-gravity is turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' All three  simulations formed gravitationally bound clumps indiciated by red circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Throughout this work, we choose h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05, which  together with our simulation parameter of Π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05  implicitly assumes d ln ρ/d ln r = −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' These choices  are relatively generic for disk models (compare to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=',  Dullemond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Gerbig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Gerbig &  Laughlin 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  The smallest mass above which a planetesimal is well  resolved is the mass associated with a Hill-radius that  equals the cell-size, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=', mrH=∆x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  This limitation is  caused by the accuracy of the self-gravity solver set  by the finite grid scale (Simon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  PLAN  automatically discards clumps, if any, less massive than  mrH=∆x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (23) and the mass scaling in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (26), this mass limit is independent of Q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  mrH=∆x = 9  4π ρH∆x3 = 9  4π  �ρH  ρ0  � �∆x  H  �3  M0  = 81  64h3  �∆x  H  �3  M⊙ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='97 · 10−5MCeres  (29)  for ∆x = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='9 · 10−4H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' As shown in Table 1, mrH=∆x  is smaller than mp,min, and λmin is larger than 2∆x  for all Q, suggesting we sufficiently resolve the smallest  expected planetesimals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' IMFs and K-S test  Figure 9 shows the IMFs obtained from the snapshot  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 8 in solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  We compare these  numerically obtained IMFs, to the predicted IMF from  the Toomre-like instability paradigm, evaluated for the  properties shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Hereby, we draw N  planetesimals from the theoretical PDF, where N is  the number of bound clumps found in the respective  simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' We conduct this random draw 1000 times  in order to obtain an average theoretical IMF (dashed  curves), together with one and three σ intervals (dark  and light shaded regions respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' In addition, we  draw a dotted vertical line, corresponding to the mass  associated with the fastest growing mode of the IMF  prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  In order to assess the quality of the analytical  predictions,  we  perform  two  sample  Kolmogorov-Smirnov (K-S) tests, which quantify the  likelihood p of two samples (in our case the numerical  and  analytically  obtained  planetesimal  masses)  originating from the same underlying distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The  null hypothesis, that is the two samples indeed coming  from the same distribution, can be rejected if p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  For the growth-rate based PDF, the K-S test yields  p-values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='501, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='395, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0629 for Q = 16, Q = 8,  and Q = 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  The numerical IMFs are  therefore consistent with the analytical IMFs obtained  from the Toomre-like analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' DISCUSSION  We developed a framework of analytically obtaining  planetesimal IMFs, based on the Toomre-like instability  in the particle layer in protoplanetary disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 4  we presented a simulation of the streaming instability,  measured diffusivity and particle concentration for three  self-gravity runs, and then compared resulting Qp-values  and corresponding analytical IMFs to the numerically  obtained IMFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' We find that the IMFs are consistent  with each other, as seen Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Specifically, our  prediction can explain a number of key properties of the  planetesimal IMF, which we will outline in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Our IMF prediction explains the somewhat counter-  intuitive property of the least massive clumps being  present in the simulation with the most mass, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Q = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Since this simulation has the smallest Qp with  Qp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='14, the fastest growing mode shifts to smaller  scales (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 1) and thus the most likely planetesimal  becomes less massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  12  Gerbig and Li  10−5  10−4  10−3  10−2  10−1  100  mp[MCeres]  10−2  10−1  100  N(> mp)  p = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='01e-01  Q = 16, Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='137  δz = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='3e-06, δr = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='4e-06  Qp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='29, N = 90  10−5  10−4  10−3  10−2  10−1  100  mp[MCeres]  p = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='95e-01  Q = 8, Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1  δz = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1e-06, δr = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='0e-06  Qp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='18, N = 102  10−5  10−4  10−3  10−2  10−1  100  mp[MCeres]  p = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='29e-02  Q = 4, Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='075  δz = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='3e-06, δr = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='6e-06  Qp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='14, N = 84  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Initial mass functions N(> mp)for the three self-gravity runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The numerically obtained IMFs are shown in solid  lines and are based on the snapshots seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The IMF resulting from the growth rate-based PDF are shown in dashed  lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The dark and light shaded regions indicate one and three standard deviations from the distribution of IMFs that originates  from only drawing N ∼ 100 planetesimals from the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The vertical dotted lines indicate the masses associated with  the fastest growing mode mp,fgm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Lastly, the p-value from the K-S-Test is shown in the top right corner of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Next, we observe that the IMF becomes flatter if Q  and therefore Qp decreases, which is due to the increase  of the range of unstable scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Notably, all three  numerical IMFs fall short of the analytical prediction  on the high mass end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' As alluded to in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='5, this  is expected as these planetesimals would have to form  from scales that exceed the maximum scale of particle  over-densities ηr (see Gerbig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Indeed, the  idea that the most massive planetesimal is not set by  the largest Toomre-scale λ but by ηr, is consistent  with our simulations where the most massive clump  has ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1MCeres in all three simulations, and also with  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 10 in Simon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (2016), where the most massive  planetesimal shortly after self-gravity is turned on is  relatively independent of ˜G (or equivalently Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Lastly, the fact that λfgm is much closer to λmin  when Qp is small, provides an explanation for why only  high-resolution simulations can sufficiently capture the  turnover in the mass frequency distribution at the low-  mass end of planetesimal IMFs (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Caveats  In this section, we outline a number of caveats that are  to be kept in mind when relating the herein presented  theory and analysis to real disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  First, unstable scales λ were converted to planetesimal  masses by assuming the contraction of a circular  sheet of radius λ/(2π) at Hill-density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  While such a  conversion agrees with timescale arguments presented  in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Gerbig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Klahr & Schreiber (2020),  it certainly is an order of magnitude calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  In  particular, as pointed out in Polak & Klahr (2022), a  sphere at Hill-density is not fully tidally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  The  Hill-sphere is derived by considering the tidal forces on  either end of the Roche lobe in the restricted three-  body potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' This has the Hill-sphere extend beyond  the tidally-stable Roche lobe, and implies Hill-density is  insufficient for tidal stability by an order of unity factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Moreover, it is not the case that all unstable scales are  at Hill-density exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' In fact, as evidenced by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 6,  many particles are located within regions which exceed  Hill-density by up to an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Additionally,  our  model  assumed  planetesimal  formation to be perfectly efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 8 however  demonstrates that there are still particles in the  filaments that are not bound to any clump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Moreover,  previous  works  showed  that  higher  resolution  simulations may extend the mass distribution to lower  masses (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Another possibility that is not  considered in our model is a single clump forming  binary planetesimals upon contraction to material  density (Nesvorn´y et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2019, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Further, our  model does not take into account the possibility of  “clumps  within  clumps”  which  would  result  in  double-counting of planetesimals (compare to Hopkins  &  Christiansen  2013,  who  consider  this  when  investigating the statistical instability of the gas disk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  We note,  that while multiplicity and inefficient  contraction would shift the IMF towards smaller masses,  collapse beyond Hill-density and double counting would  shift the IMF towards larger masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' We expect that  this allows the numerically obtained IMFs to remain  consistent with our analytical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Other simplifying assumptions include gas density  being considered constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  For stable, yet small Q  values, the gas develops a mid-plane cusp with a non-  Planetesimal IMFs under diffusion regulated collapse  13  neglible vertical density gradient (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=', Armitage  2015, for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' We also assumed a mono-disperse  dust population with a constant Stokes number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' This  choice is likely reasonable to first order as in real disks,  most solid mass is contained in the largest grains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=',  Birnstiel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' McNally et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2021), and the  largest grains dominate dust-gas interactions in the mid-  plane and participate most vigorously in planetesimal  formation (Yang & Zhu 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Still, a multi-species  dust fluid can alter the turbulent behavior of the system  itself (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=', Krapp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2019, who pointed out  the damping effect multi-species dust can have on the  streaming instability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Therefore, the assumption of  single-species dust is to be kept in mind when applying  our results to real disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Lastly, we acknowledge that our numerical setup does  not perfectly replicate the assumptions the Toomre-like  instability is based on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' More specifically, the Toomre  analysis,  assumes a thin,  two-dimensional disk of  constant density which is then linearly perturbed in form  of axisymmetric waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  This picture evidently differs  from the turbulent state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  non-linear streaming  instability in vertically-stratified disks, our simulations  are in when self-gravity is turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  We attempted  to mitigate this discrepancy by evaluating diffusivities  and concentration locally, rather than globally, which,  as discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='6, yielded consistent IMFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The shape of the initial mass function in real disks  The diffusion-tidal-shear collapse paradigm suggests  that the shape of planetesimal IMF depends on the  gravitational,  Toomre-like  instability  and  on  the  stability parameter Qp  only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  While independent  information on diffusivities δr and δz, as well as Stokes  number St, and disk aspect ratio h are required to  match the IMF to the appropriate mass range, they do  not affect the steepness of the IMF (for a fixed Qp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' As  such, as a disk region collects more massive particles, it  experiences an increase in average grain size, cools or  becomes less turbulent, its Qp-value decreases, thus  flattening the planetesimal IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Within this context,  the role of the streaming instability (or other processes  affecting particle dynamics) for the planetesimal IMF  in this work, is to set the initial condition at the onset  of gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The key question within our  framework therefore is to determine what value for Qp  takes on in real disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Our simulations, much like many in previous works  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=', Simon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Sch¨afer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Gerbig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2020) show collapse as soon as self-gravity  is turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The turbulent state before self-gravity is  turned on is therefore not physical, and thus the Qp-  values and IMFs we calculate not necessarily reflective  of real disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Indeed, as discussed in Gerbig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  (2020), if a system softly evolves towards instability —  for example by collecting more particle mass — its Qp-  value likewise softly evolve from Qp > 1 to Qp < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  If Qp ∼ 1 is sufficient for the system to fully collapse  and form planetesimals as assumed in Klahr & Schreiber  (2020, 2021), then the resulting IMF is expected to be  rather steep (Polak & Klahr 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' On the other hand,  the system may be unable to fully contract at Qp ∼ 1  if it is below Hill-density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  To a similar effect, as the  contraction time, given by (Gerbig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2020),  tc =  Ω  4πGρpSt =  �π  8  �  δz  δr  Qp  St Ω−1,  (30)  only falls below the dynamical timescale Ω−1 once Qp  is substantially less than unity, it is plausible that the  system can collect relevant amounts of mass during  contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Both situations would lead to smaller  values for Qp and thus flatter IMFs like the ones in our  simulations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  One observational constraint may be provided by  the Asteroids size distribution,  which,  at least in  part, constitutes pristine planetesimals from the Solar  System’s formation era (see Klahr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2022, for a  review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Taking, the collisional evolution into account,  it is possible to reconstruct model of the primordial size  distribution of Asteroids (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=', Delbo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Within the here presented framework, the “knee” at  ∼ 100 km (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=', Gladman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2001) typically found  in the resulting mass functions can be interpreted as  an indication of a marginally unstable origin system  with Qp close to unity — that is assuming the asteroids  in question indeed formed via gravitational collapse of  locally over-dense regions, which very well may not have  been the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Cold Classical Kuiper Belt Objects  (KBOs) are believed to be even more primordial with  little to no collisional evolution (Morbidelli & Nesvorn´y  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Kavelaars et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' (2021) found a exponentially  tapered power law as the best fit for the Cold Classical  KBO mass distribution, and connected the lack of  large planetesimals (> 400 km in size) to streaming  instability regulated planetesimal formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Our work  is consistent with this interpretation, as Qp imposes a  limit on the maximum planetesimal mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  One often invoked pathway of forming planetesimals  and circumventing the meter barrier, that is worth  discussing in the context of our work, is the existence  of long-lived pressure bumps associated with disk sub-  structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' While such structures are robustly confirmed  observationally (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=', Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2018), their role  in producing the first generation of planetesimals is  14  Gerbig and Li  elusive,  as already existing planets seem to most  promisingly explain the sub-structures in the first  place (Teague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Either way, a pressure  bump is characterized by very small pressure gradients,  which diminishes the relative velocity between dust  and gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  If Π = 0, then particles can trivially settle  razor-thin and go gravitationally unstable (as seen in  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=', Abod et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  In this state, both vertical  and radial diffusivities are expected to trend towards  zero, implying Qp  →  0 as well, which our work  associates with the planetesimal IMF shifting towards  very small masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Unless hierarchical mergers between  small planetesimal immediately after formation is very  efficient, such a bottom-heavy mass function seems  unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' We therefore suspect that either the majority  of planetesimal formation does not occur in Π = 0  regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' or, that other mechanisms, possibly gravito-  turbulence (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=', Riols et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2017, for the gas disk),  provide a lower bound on diffusivities, thus also setting  a lower limit on Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  To conclude, our analysis that connects the PDF of  formed  planetesimals  to  the  growth  rates  of  the  Toomre-like instability of a particle layer subject to a  diffusive flux, unifies both the flat, power-law shaped,  IMFs previously obtained numerically (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=', Simon  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2019), and the ‘Asteroids are born  big’ (Morbidelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' 2009) paradigm that arises when  investigating marginally unstable systems (Klahr &  Schreiber 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  The analytically obtained IMFs are  consistent with our numerical setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' Further work is  required  to  test  the  predictions  for  different  numerically setups, and in particular, to assess how the  value of Qp at the on-set of planetesimal formation,  and thus the steepness of the resulting IMF, depend on  disk properties and radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Such an analysis informs  initial conditions for planet formation models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=',  Emsenhuber  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  2021),  and  conversely,  if  the  formation  of  an  observed  exoplanetary  population  presupposes a specific planetesimal IMF(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=', Batygin  & Morbidelli 2023, for rocky Super-Earths), provides  constraints  on  the  disk  state  during  the  era  of  planetesimal formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' ACKNOWLEDGEMENTS  This work and KG and RL benefited from the  2022 Exoplanet Summer Program in the Other Worlds  Laboratory (OWL) at the University of California,  Santa Cruz, a program funded by the Heising-Simons  Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content=' The authors thank Greg Laughlin, Hubert  Klahr, Andrew Youdin, Ruth Murray-Clay and Malena  Rice for insightful comments and discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='  Software:  Athena (Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
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+page_content='3847/1538-4357/aae7d4  Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
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+page_content='1093/mnras/stab2959  Youdin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
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+page_content='1088/0004-637X/731/2/99  Youdin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
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+page_content=' 2005, The Astrophysical  Journal, 620, 459, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1086/426895  Youdin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
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+page_content=' 2007, Icarus, 192, 588,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
+page_content='icarus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
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+page_content='012  Youdin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFQT4oBgHgl3EQfUjZg/content/2301.13297v1.pdf'}
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+“A Handbook of Integer Sequences” Fifty Years Later
+N. J. A. Sloane,
+The OEIS Foundation Inc.,
+11 So. Adelaide Ave., Highland Park, NJ 08904, USA
+Email: njasloane@gmail.com
+January 8, 2023
+Abstract
+Until 1973 there was no database of integer sequences. Someone coming across
+the sequence 1,2,4,9,21,51,127,... would have had no way of discovering that
+it had been studied since 1870 (today these are called the Motzkin numbers,
+and form entry A001006 in the database). Everything changed in 1973 with the
+publication of A Handbook of Integer Sequences, which listed 2372 entries. This
+report describes the fifty-year evolution of the database from the Handbook to its
+present form as The On-Line Encyclopedia of Integer Sequences (or OEIS), which
+contains 360,000 entries, receives a million visits a day, and has been cited 10,000
+times, often with a comment saying “discovered thanks to the OEIS”.
+1
+Introduction
+Number sequences arise in all branches of science: for example, 1,1,2,4,9,20,48,115,...
+gives the number of rooted trees with n nodes (A000081,1 see also Fig. 1), and in daily
+life: how many pieces can you cut a pancake into with n knife-cuts? (The pieces need
+not all be the same size.) That one is easy: 1,2,4,7,11,16,..., n(n+1)/2+1 (A000124).
+But what is the answer for cutting up an (ideal) bagel, or torus? That is a lot harder:
+with a sharp knife you might get a few terms, perhaps 1,2,6,13,..., but probably not
+enough to guess the formula, which is n(n2 + 3n + 8)/6 for n > 0. For that you would
+need to to consult the database: go to https://oeis.org and enter “cutting bagel”,
+or go directly to A003600.
+My fascination with these sequences began in 1964 when I was a graduate student
+at Cornell University in Ithaca, NY, studying neural networks. I had encountered a
+sequence of numbers, 1,8,78,944,13800,..., and I badly needed a formula for the n-th
+term, in order to determine the rate of growth of the terms (this would indicate how
+long the activity in this very simple neural network would persist). I will say more
+about this sequence in Section 2.1.
+I noticed that although several books in the Cornell library contained sequences
+somewhat similar to mine, as far as I could tell this particular sequence was not men-
+tioned. I expected to have to analyze many related sequences, so in order to keep track
+of the sequences in these books, I started recording them on 3” × 5” file cards.
+1Six-digit numbers prefixed by A refer to entries in the current version of the Handbook, The On-Line
+Encyclopedia of Integer Sequences [11].
+1
+arXiv:2301.03149v1  [math.NT]  9 Jan 2023
+
+Figure 1: Left: one of 48 unlabeled rooted trees with 7 nodes (the root node is at the
+bottom); center: four cuts of a pancake can produce 11 pieces; right: three cuts of a
+bagel can produce 13 pieces.
+The collection grew rapidly as I searched though more books, and once the word got
+out, people started sending me sequences. Richard Guy was an enthusiastic supporter
+right from the start.
+In 1973 I formalized the collection as A Handbook of Integer
+Sequences, which was published by Academic Press (Fig. 2). It contained 2372 entries.
+Figure 2: Front cover of the Handbook. The embossed figures show side views of the two
+ways of folding a strip of three (blank) stamps, and the five ways of folding a strip of
+four stamps. The full sequence begins 1,1,2,5,14,38,120,353,1148,3527,..., A001011.
+No formula is known.
+Once the book appeared, the flood of correspondence increased, and it took twenty
+years to prepare the next version.
+Simon Plouffe helped a great deal, and in 1995
+Academic Press published our sequel, The Encyclopedia of Integer Sequences, with 5487
+entries. From this point on the collection grew even more rapidly. I waited a year,
+until it had doubled in size, and then put it on the Internet, calling it The On-Line
+Encyclopedia of Integer Sequences.
+In the rest of this article I will first say more about the evolution of the database:
+the Handbook (§2.1), the 1995 Encyclopedia (§2.2), the On-Line Encyclopedia (§2.3),
+and the OEIS Foundation (§2.4). The next sections describe the database itself: what
+2
+
+AHANDBOOKOF
+INTEGER SEQUENCES
+N.JLA.SLOANEsequences are—or are not—included (§3.1), how the database is used (§3.2), the layout
+of a typical entry (§3.3), the arrangement of the entries (§3.4), and a Fact Sheet (§3.5).
+The final sections describe some especially interesting sequences: Recam´an’s sequence
+(§4.1), Iteration of number-theoretic functions (§4.2), Gijswijt’s sequence (§4.3), Lex-
+icographically Earliest Sequences (§4.4), The Stepping Stones problem (§4.5), Stained
+glass windows (§4.6), and other sequences I would have liked to include (§4.7).
+Several open questions are mentioned to which I would very much like to know the
+answers.
+Notation.
+a(n) will denote the n-th term of the sequence being discussed, and
+A001006(n) (for example) would denote the n-th term of A001006. σ(n) is the sum of
+the divisors of n (A000203).
+2
+Evolution of the database
+2.1
+The Handbook of Integer Sequences
+Once the collection had grown to a few hundred entries, I entered them on punched
+cards,2 which made it easier to check and sort them. The Handbook was type-set directly
+from the punched cards. There were a few errors in the book, but almost all of them
+were caused by errors in the original publications. Accuracy was a primary concern in
+that book, as it is today in the OEIS.
+The book was an instant success. It was, I believe, the world’s first dictionary of
+integer sequences (and my original title said Dictionary rather than Handbook). Many
+people said “What a great idea”, and wondered why no one had done it before. Martin
+Gardner recommended it in the Scientific American of July 1974. Lynn A. Steen, writing
+in the American Mathematical Monthly said “Incomparable, eccentric, yet very useful.
+Contains thousands of ‘well-defined and interesting’ infinite integer sequences together
+with references for each ... If you ever wondered what comes after 1,2,4,8,17,35,71,...,
+this is the place to look it up”’.
+Harvey J. Hindin, writing from New York City, exuberantly concluded a letter to
+me by saying: “There’s the Old Testament, the New Testament, and the Handbook of
+Integer Sequences.”
+I never did find the sequence that started it all in the literature, but I learned P´olya’s
+theory of counting, and with John Riordan’s help found the answer, which appears in
+[13] and A000435.
+2These were never called “punch cards” (sic). To anyone who worked with them in the 1960s, “punch
+cards” sounds like “grill cheese” (sic) for “grilled cheese”, or “barb wire” (sic) for “barbed wire”, both
+of which I have recently seen in print.
+3
+
+2.2
+The Encyclopedia of Integer Sequences
+Following the publication of the Handbook, a large amount of correspondence ensued,
+with suggestions for further sequences and updates to the entries. By the early 1990’s
+over a cubic meter of new material had accumulated.
+A Canadian mathematician,
+Simon Plouffe, offered to help in preparing a revised edition of the book, and in 1995
+The Encyclopedia of Integer Sequence, by me and Simon Plouffe, was published by
+Academic Press. It contained 5487 sequences, occupying 587 pages. By now punched
+cards were obsolete, and the entries were stored on magnetic tape.
+2.3
+The On-Line Encyclopedia of Integer Sequences
+Again, once the book appeared, many further sequences and updates were submitted
+from people all over the world. I waited a year, until the size of the collection had
+doubled, to 10000 entries, and then in 1996 I launched The On-Line Encyclopedia of
+Integer Sequences (now usually called simply the OEIS) on the Internet. From 1996
+until October 26, 2009, it was part of my homepage on the AT&T Labs website.
+Incidentally, in 2004 the database was mentioned by the Internet website slashdot
+(“News for Nerds. Stuff that Matters”), and this brought so much traffic to my Bell
+Labs homepage that it briefly crashed the whole Bell Labs website. My boss was quite
+proud of this, since it was a rare accomplishment for the Mathematics and Statistics
+Research Center.
+2.4
+The OEIS Foundation
+In 2009, in order to ensure the long-term future of the database, I set up a non-profit
+foundation, The OEIS Foundation Inc., a 501(c)(3) Public Charity, whose purpose is
+to own, maintain and raise funds to support The On-Line Encyclopedia of Integer Se-
+quences or OEIS.
+On October 26, 2009, I transferred the intellectual property of The On-Line Ency-
+clopedia of Integer Sequences to the Foundation. A new OEIS with multiple editors was
+launched on November 11, 2010.
+Since then it has been possible for anyone in the world to propose a new sequence
+or an update to an existing sequence. To do this, users must first register, and then
+submissions are reviewed by the editors before they become a permanent part of the
+OEIS. Technically the OEIS is now a “moderated wiki”.
+I started writing this article on November 11, 2022, noting that this marked twelve
+years of successful operation of the online OEIS, and also that the database is in its
+59th year of existence.
+4
+
+3
+The database today
+3.1
+What sequences are included?
+From the very beginning the goal of the database has been to include all “interest-
+ing” sequences of integers.
+This is a vague definition, but some further examples
+will make it clearer.
+The database includes a huge number of familiar and unfa-
+miliar sequences from mathematics (the prime numbers 2,3,5,7,11,13,..., A000040;
+60,168,360,504,660,1092,..., the orders of noncyclic simple groups, A001034), com-
+puter science (0,1,3,5,8,11,14,..., Number of comparisons needed for merge sort,
+A001855), physics (see “self-avoiding walks on lattices”, Ising model, etc., e.g. A002921),
+chemistry (the enumeration of chemical compounds was one of the motivations behind
+P´olya’s theory of counting, see e.g. A000602), and not least, from puzzles and I.Q.
+tests (1,8,11,69,99,96,111,..., the “strobogrammatic” numbers, guess!, or see A000787;
+4,14,23,34,42,50,59,..., the numbered stops on the New York City A train subway,
+A011554. That entry has links to a map and the train schedule).
+Sequences that have arisen in the course of someone’s work—especially if published—
+have always been welcomed. On the other hand, sequences that have been proposed
+simply because they were missing from the database are less likely to be accepted.
+There are a few hard and fast rules. The sequence must be well-defined and the
+terms must not be time-dependent—if the next term is only known to be either 14 or
+15, for instance, then the sequence must end with the last term that is known for certain.
+The sequence may not have any missing terms or gaps. In the case of Mersenne primes,
+for instance (A000043) it is common for later primes to be known before all intermediate
+numbers having been tested. The later primes get mentioned in comments, but they
+are not as part of the main sequence until their position has been confirmed.
+Very short sequences and sequences that are subsequences of many other sequences
+are not accepted. A sequence for which the only known terms are 2,3,5,7 would not
+be accepted since it is matched by a large number of existing sequences. The definition
+may not involve an arbitrary but large parameter (primes ending in 1 are fine, A030430,
+but not primes ending in 2023).3
+Most entries give an ordered list of integers. But triangles of numbers are included
+by reading them row-by-row: Pascal’s triangle becomes 1, 1,1, 1,2,1, 1,3,3,1,...,
+A007318.
+Doubly-infinite square arrays are included by reading them by antidiago-
+nals: the standard multiplication table for positive integers becomes 1, 2,2, 3,4,3,
+4,6,6,4,..., A003991.
+Sequences of fractions are included as a linked pair giving the numerators and denom-
+inators separately (the Bernoulli numbers are A027641/A027642). Important individual
+real numbers are included by giving their decimal or continued fraction expansions (for
+π see A000796 and A001203). A relatively small number of sequences of nonintegral
+3The OEIS Wiki, which has a great deal of useful information about the database, has a section
+listing additional examples of “what not to submit”.
+5
+
+real numbers are included by rounding them to the nearest integer, or by taking floors
+or ceilings (the imaginary parts of the zeros of Riemann’s zeta function give A002410).
+Two less obvious sources for sequences are binomial coefficient identities and number-
+theoretic inequalities. The values of either side of the identity
+n
+∑
+k=0
+(2n
+k )
+2
+= 1
+2(4n
+2n) − 1
+2(2n
+n )
+2
+[7, (3.68)] give A036910. From the inequality σ(n) < n√n for n > 2, [10, Sect. III.1.1.b],
+we get the integer sequence ⌊n√n⌋ − σ(n), A055682. The point is that if you want to
+know if this inequality is known, you look up the difference sequence, and find A055682
+and a reference to the proof. Many more sequences of these two types should be added
+to the database.
+3.2
+How the database is used
+The main applications of the database are in identifying sequences or in finding out the
+current status of a known sequence. Barry Cipra has called it a mathematical analogue
+of a “fingerprint file”. You encounter a number sequence, and wish to know if anyone
+has ever come across it before.
+If your sequence is in the database, the reply will give a definition, the first 50 or
+so terms, and, when available, formulas, references, computer code for producing the
+sequence, links to any relevant web sites, and so on.
+Figures 3 and 4 show what happens if you submit 1,2,5,14,42,132,429, the first few
+Catalan numbers, one of the most famous sequences of all.
+Figure 3: The result of submitting 1,2,5,14,42,132,429 to the database. This figure
+shows the banner at the top of the reply. There are 26 matches, ranked in order of
+importance, the top match being the one we want, the Catalan numbers. A shortened
+version of the top match is shown in the next figure.
+6
+
+TheOEisissupportedbythemanygenerousdonorstotheOEisFoundation
+013627
+THE ON-LINE ENCYCLOPEDIA
+20
+OF INTEGER SEOUENCES
+?
+10221121
+founded in 1964 by N. J. A. Sloane
+1.2.5.14,42.132,429
+Search
+Hints
+(GreetingsfromTheOn-LineEncyclopediaofIntegerSequences!)
+Search:seq:1.2.5.14.42.132.429
+Displaying1-10of26resultsfound.
+page 1 2 3
+Sort:relevanceIreferencesInumberImodifiedIcreated
+Format:longIshortIdataA000108
+Catalan numbers: C(n) = binomial(2n,n)/(n + 1) = (2n)!/(n!(n + 1)!).
+(Formerly M1459 N0577)
+DATA
+1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ...
+COMMENTS
+These were formerly sometimes called Segner numbers.
+A very large number of combinatorial interpretations are known -
+see references, esp. R. P. Stanley, Catalan Numbers, Camb., 2015.
+This is probably the longest entry in the OEIS, and rightly so.
+The solution to Schr¨oder’s first problem: number of ways to insert n pairs
+of parentheses in a word of n + 1 letters. E.g., for n = 2 there are 2 ways:
+((ab)c) or (a(bc)); for n=3 there are 5 ways: ((ab)(cd)), (((ab)c)d), ...
+...
+REFERENCES
+The large number of references and links demonstrates the ubiquity
+of the Catalan numbers.
+R. Alter, Some remarks and results on Catalan numbers, pp. 109-132
+in Proc. Louisiana Conf. Combinatorics, Graph Theory and
+Computer Science. Vol. 2, edited R. C. Mullin et al., 1971.
+M. Bona, ed., Handbook of Enumerative Combinatorics, CRC Press, 2015
+L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 53.
+J. H. Conway & R. K. Guy, The Book of Numbers, Springer, 1995, 96-106.
+...
+LINKS
+Robert G. Wilson v, Table of n, a(n) for n = 0..1000
+...
+F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Disc. Math.,
+Vol. 204, No. 1-3 (1999), 73-112.
+...
+W. G. Brown, Historical Note on a Recurrent Combinatorial Problem,
+Amer. Math. Monthly,. 72, No. 9 (1965), 973-977.
+...
+FORMULA
+Recurrence: a(n) = 2 ∗ (2 ∗ n − 1) ∗ a(n − 1)/(n + 1) with a(0) = 1.
+...
+MAPLE
+A000108 := n-¿binomial(2*n, n)/(n+1);
+...
+MATHEMATICA
+A000108[n ] := (2 n)!/n!/(n+1)!
+...
+PARI
+a(n)=binomial(2*n, n)/(n+1)
+...
+KEYWORD
+core,nonn,easy,nice
+AUTHOR
+N. J. A. Sloane
+Figure 4: The entry for the Catalan number A000108. The full entry has over 750 lines,
+which have been edited here to show samples of the different fields.
+I could have chosen a simpler example, like the Fibonacci numbers, but I have a
+particular reason for choosing the Catalan numbers. When the OEIS was new, people
+would sometimes say to me that they had a sequence they were trying to understand,
+and would I show them how to use the database.
+At least twice when I used the
+Catalan sequence as an illustration, they said, why, that is my sequence, how on earth
+did you know? It was no mind-reading trick, the Catalan numbers are certainly the
+most common sequence that people don’t know about. This entry is the longest—and
+7
+
+one of the most important—in the whole database.
+If we do not find your sequence in the database, we will send you a message inviting
+you to submit it (if you consider it is of general interest), so that the next person who
+comes across it will be helped, and your name will go on record as the person who
+submitted it.
+The second main use of the database is to find out the latest information about a
+particular sequence.
+Of course we cannot hope to keep all 360000 entries up-to-date. But when a new
+paper is published that mentions the OEIS, Google will tell us, and we then add links to
+that paper from any sequence that it mentions. People have told us that this is one of
+the main ways they use the OEIS. After all, even a specialist in (say) permutation groups
+cannot keep track of all the papers published worldwide in that area. And if a paper in
+a physics journal happens to mention a number-theoretic sequence, for example, that is
+unlikely to be noticed by mathematicians.
+There are also many other ways in which the database has proved useful.
+For example, it is an excellent source of problems to work on. The database is con-
+stantly being updated. Every day we get thirty to fifty submissions of new sequences,
+and an equal number of comments on existing entries (new formulas, references, ad-
+ditional terms, etc.).
+The new sequences are often sent in by non-mathematicians,
+and are a great source of problems. You can see the current submissions at https:
+//oeis.org/draft. Often enough you will see a sequence that is so interesting you
+want to drop everything and work on it. And remember that we are always in need of
+more volunteer editors. In fact anyone who has registered with the OEIS can suggest
+edits, you do not even need to be an official editor. We have been the source of many
+international collaborations.
+There is also an educational side: several people have told us that they were led into
+mathematics through working as an editor. Here is a typical story.
+Subject: Reminiscence from a young mathematician
+I wanted to relay a bit of nostalgia and my heartfelt thanks. Back in the late 1990s, I was a high
+school student in Oregon. While I was interested in mathematics, I had no significant mathematically
+creative outlet until I discovered the OEIS in the course of trying to invent some puzzles for myself. I
+remember becoming a quite active contributor through the early 2000s, and eventually at one point, an
+editor. My experience with the OEIS, and the eventual intervention of one of my high school teachers,
+catalyzed my interest in studying mathematics, which I eventually did at [...] College. I went on to a
+Ph.D. in algebraic geometry at the University of [...], and am currently at [...].
+I wanted to thank you for seriously engaging with an 18-year old kid, even though I likely submitted
+my fair share of mathematically immature sequences. I doubt I would have become a mathematician
+without the OEIS!
+A less-obvious use of the database is to quickly tell you how hard a problem is. I use
+it myself in this way all the time. Is the sequence “Catalan” or “Collatz”? If a sequence
+comes up in your own work, or when reviewing someone else’s work, it is useful to know
+right away if this is a well-understood sequence, like the Catalan numbers, or if it is one
+of the notoriously intractable problems like the Collatz or 3x + 1 problem (A006577).
+8
+
+Finally, the OEIS is a welcome escape when you feel the world is falling apart. Take
+a look at Scott Shannon’s drawings of stained glass windows in A331452; or Jonathan
+Wild’s delicate illustrations of the ways to draw four circles (in A250001); or ´Eric An-
+gelini’s “1995” puzzle (A131744) or any of his “lexicographically earliest sequences”
+(A121053, A307720, and many more); or find better solutions to the Stepping Stones
+Problem (§4.5,A337663). You can find brand new problems at any hour of the day or
+night by looking at the stack of recent submissions: but beware, you may see a problem
+there that will keep you awake for days. Or search in the database for phrases like “It
+appears that ...”, or “Conjecture: ...”, or “It would be nice to know more!”
+3.3
+Layout of a typical entry
+This is a good place to mention some of the features of an OEIS entry. Most of the
+fields (see Figs. 3 and 4) are self-explanatory. At the top it tells you how many matches
+were found to your query (26 in the example). These are ranked in order of importance.
+The DATA section shows the start of the sequence, usually enough terms to fill a few
+lines on the screen (typically 300 to 500 decimal digits). Often one wants more terms
+than are shown, and the first link in the entry will point to a plain text file with perhaps
+10000 or 20000 terms. That file will have a name like b001006.txt, and is called the
+“b-file” for the sequence. Some entries also have much larger tables, giving a million or
+more terms.
+If you click the “graph” button near the top of the reply, you will be shown two plots
+of the sequence, and if you click the “listen” button, you could listen to the sequence
+played on an instrument of your choice. The default instrument is the grand piano, and
+the terms of the sequence would be mapped to the 80 keys by reducing the numbers
+mod 80 and adding 1.
+I conclude this section with a philosophical comment.
+When you are seriously trying to analyze a sequence, and are prepared to spend
+any amount of time needed (searching for a formula or recurrence, for instance), you
+need all the help you can get, which is why we provide the b-files and other data files,
+and why we give computer programs in so many languages. This is also the reason we
+give as many references and links as possible for a sequence. Even if the reference is to
+an ancient or obscure journal, or one that has been accused as being “predatory”, we
+still give the reference, especially for sequences that are not well-understood. The same
+thing holds for formulas, comments, and cross-references to other sequences. When you
+are desperate, you will accept help from anywhere. And do not forget “Superseeker”!
+3.4
+Arrangement of the entries
+The entries in the database are (virtually) arranged in two different ways, the first
+essentially chronological, the second lexicographic.
+9
+
+The first is by their absolute identification number, or A-number.4 Once the collec-
+tion reached a few hundred entries, I sorted them into lexicographic order and numbered
+them A1, A2, A3, .... A1 gives the number of symmetry groups of order n, A2 is the
+famous Kolakoski sequence, and so on. This numbering is still used today, only A1 has
+become A000001, A2 is A000002, ..., and as each new submission comes in it gets a
+number from the stack. Current sequences are being issued numbers around A360000.
+Rejected A-numbers are recycled, so there are no gaps in the order. We reached 100000
+entries in 2004, and 250000 in 2015. The present growth rate is about 12000 new entries
+each year.
+The second arrangement is a kind of lexicographic ordering.
+First I describe an
+idealized, theoretical, lexicographic order. Sequences of nonnegative numbers can be
+arranged in lexicographic (or dictionary) order.
+For example, sequences beginning
+1,2,4,... come before 1,2,5,..., 1,2,4,3,..., 1,3,..., etc., but after 1,2,3,....
+Also
+1,2,4,... comes after the two-term sequence 1,2 (because blanks precede numbers).
+More formally, we compare the two sequences term-by-term, and in the first position
+where they differ whichever is smaller (or blank) is the lexicographically earlier sequence.
+For sequences with negative terms, we ignore the signs and sort according to the
+absolute values.
+Here is the actual ordering used in the OEIS. The sequences are arranged (virtually)
+into a version of lexicographic order, according to the following rules. First, delete all
+minus signs. Then find the first term that is greater than 1, and discard all the terms
+before it. What’s left determines its position in the lexicographic order. For example,
+to place −1,0,1,1,2,1,17,3,2,1,... in the ordering, we would ignore the terms before
+the underlined 2, and consider the sequence as beginning 2,1,17,3,2,1,....
+Sequences that contain only 0s, 1s and −1s are sorted into lexicographic order by
+absolute value and appear at the beginning of the ordering. The first sequence in the
+database is therefore the zero sequence A000004.
+In this way every sequence has a unique position in the ordering. The sequences
+have been sorted in this way since the 1960s. For the first ten years the punched card
+entries were physically sorted into this order.
+When you look at an OEIS entry, A005132 say (the subject of Section 4.1), towards
+the bottom you will see two lines like5
+Sequence in context: A277558 A350578 A335299 * A064388 A064387 A064389
+Adjacent sequences: A005129 A005130 A005131 * A005133 A005134 A005135
+which tell you the three entries immediately before and after that entry in the lexico-
+graphic ordering, and the three entries before and after it in the A-numbering. The
+asterisks represent the sequence you are looking at. The first group can be useful if you
+are uncertain about a term in your sequence, the second in case you want to look at
+other sequences submitted around that time.
+Today the sequences are actually stored internally in an SQLite database. However,
+4The sequences in the 1973 and 1995 books were numbered N0001, ..., and M0001, ..., respectively.
+5If you don’t see these, click on the A-number at the top of the entry.
+10
+
+the punched card format has been so useful that when you view a sequence, as in Fig. 4,
+it is still presented to you in something very like the old punched card format.
+3.5
+Summary: “A Handbook of Integer Sequences” today
+– Now The On-Line Encyclopedia of Integer Sequences or OEIS: https://oeis.org
+– Accurate information about 360000 sequences.
+– Definition, formulas, references, links, programs. View as list, table, graph, music!
+– Traffic: 1 million hits/day.
+– 30 new entries, 50 updates every day.
+– Often called one of best math sites on the Web. Fingerprint file for mathematics.
+– Street creds: 10000 citations.
+– A moderated Wiki, owned by OEIS Foundation, a 501(c)(3) public charity.
+– Uses: to see if your sequence is new, to find references, formulas, programs.
+– Catalan or Collatz? (Very easy or very hard?)
+– Source of fascinating research problems;6 low-hanging fruit from recent submis-
+sions.
+– Accessible (free, friendly).
+– Fun (1,2,4,6,3,9,12,8,10,5,15,...?). Interesting, educational. Escape.
+– Addictive (better than video games).
+– Has led many people into mathematics.
+– One of the most successful international collaborations, a modest contribution
+towards world peace.
+– Need editors.
+4
+Some favorite sequences
+I’m sometimes asked what my favorite sequence is. This is a difficult question. I’m
+tempted to reply by saying: If you were the keeper of the only zoo in the world, how
+would you answer that question? (Because that is roughly the situation I’m in.) Would
+you pick one of the exotic animals, a giraffe, a kangaroo, or a blue whale? Or one of the
+essential animals, like a horse, a cow, or a duck? If the question came from a visiting
+alien, of course, there is only one possible answer: a human being.
+6Look for “Conjecture”, “It appears that”, “It would be nice to”, ...
+11
+
+For sequences, the essential ones are the primes, the powers of 2, the Catalan num-
+bers, or (especially if the question came from an alien with no fingers or toes), the
+counting sequence 0, 1, 2, 3, 4, ... (A001477).
+But here I’ll mention a few that are fairly exotic. The Recam´an and Gijswijt se-
+quences have simple recursive definitions, yet are astonishingly hard to understand.
+4.1
+Recam´an’s sequence (A005132)
+This remarkable sequence has resisted analysis for over 30 years, even though we have
+computed an astronomical number of terms. It was contributed to the database by
+Bernardo Recam´an Santos in 1991.
+The definition is deceptively simple. The first term is 0. We now add or subtract 1,
+then we add or subtract 2, then add or subtract 3, and so on. The rule is that we always
+first try to subtract, but we can only subtract if that leaves a nonnegative number that
+is not yet in the sequence. Otherwise we must add.
+Here is how the sequence starts. We have the initial 0. We can’t subtract 1, because
+that would give a negative number, so we add 1 to 0. So the second term is 1. We can’t
+subtract 2 from 1, so we add it, getting the third term 1+2 = 3. Again we can’t subtract
+3, for that would give 0, which has already appeared, so we add 3, getting the fourth
+tern 3 + 3 = 6.
+Now we must add or subtract 4, and this time we can subtract, because 6−4 = 2, and
+2 is nonnegative and a number that hasn’t yet appeared. So at this point the sequence
+is 0,1,3,6,2. Then it continues 7(= 2 + 5), 13(= 7 + 6),20(= 13 + 7),12(= 20 − 8), and so
+on. The first 16 terms are
+0,1,3,6,2,7,13,20,12,21,11,22,10,23,9,24,...
+When adding rather than subtracting, repeated terms are permitted (42 is repeated
+at the 24th term).
+Edmund Harriss has found an elegant was to draw the sequence as a spiral on the
+number line. Start at 0, and when we subtract n, draw a semicircle of diameter n to
+the left from the last point, or to the right if we are adding n. Draw the semicircles
+alternately below and above the horizontal axis so as to produce a smooth spiral.
+The main question about this sequence is: Does every positive number appear?
+What makes this sequence so interesting is that certain numbers (for reasons we do not
+understand) are extremely reluctant to appear. 4 does not appear until 131 steps, and
+19 takes 95734 steps.
+A group of us at AT&T Bell Labs worked on this sequence in 2001, and developed a
+way to greatly speed up the computation. Allan Wilks used it to compute the first 1015
+terms, and found that 2406 (which had been missing for a long time) finally appeared
+at step 394178473633984.
+At this point the smallest missing number was 852655 = 5⋅31⋅5501. Benjamin Chaffin
+has continued this work, and in 2018 reached 10230 terms. However, 852655 was still
+12
+
+Figure 5: Harriss’s drawing of the first 64 terms of Recam´an’s sequence. (The tiny
+initial semicircle, at the extreme left, is below the axis. It has diameter 1 and joins the
+points 0 and 1. It continues as a semicircle of diameter 2, above the axis, joining the
+points 1 and 3.)
+missing, and there has been no progress since then.
+Thirty years ago I thought that every number would eventually appear.
+Now I
+am not so sure. My current belief is that there are two possibilities: either there are
+infinitely many numbers that never appear, and 852655 just happens to be the smallest
+of them, but has no other special property. A similar phenomenon appears to occur
+when iterating various number-theoretic functions—see the next section.
+Or, every
+number will eventually appear (just as presumably every one of Shakespeare’s plays will
+eventually appear in the expansion of π in base 60), although we may never be able
+to extend the sequence far enough to hit 852655. For the latest information about this
+sequence (or any other sequence mentioned in this article), consult the OEIS.
+Open question: Does 852655 appear in A005132?
+4.2
+Iteration of number-theoretic functions
+Many mysterious sequences arise from the iteration of number-theoretic functions. A
+classic problem concerns the iteration of the function f(n) = σ(n) − n, the sum of the
+“aliquot parts” of n (see Guy [8, §B6]). For an initial value of n, what happens to the
+trajectory n,f(n),f(f(n)),...? All n < 276 terminate by entering a cycle (such n are
+called “perfect”, “amicable”, or “sociable” numbers), or reaching a prime, then 1, then
+0. But it appears likely that n = 276 and perhaps infinitely many even numbers, will
+never terminate. The trajectory of 276 is A008892, and the b-file gives the first 2140
+13
+
+terms, term 2140 being a 213-digit number. The continuation of the trajectory has
+stalled at term 2051, where a 202-digit number is waiting to be factored. A098007 gives
+the number of distinct terms in the trajectory of n, or −1 if the trajectory is unbounded.
+The value of A098007(276) is unknown.
+If indeed 276 does go to infinity, it is natural to ask, how did 276 know it was destined
+to be the first immortal number under the map f? The answer may be that there are
+infinitely many immortal numbers, and 276 just happens to be the first. It got lucky,
+that’s all! Just as 852655 got lucky in Recam´an’s problem.
+A similar question, also discussed by Guy [8, §B41], which has received much less
+attention, concerns the map g(n) = (σ(n) + φ(n))/2, where φ(n) is the Euler totient
+function A000010. The trajectory may end at 1, a prime, or a fraction, or it may increase
+monotonically to infinity. Sequence A292108 gives the number of steps in the trajectory,
+or −1 if the trajectory is infinite. All numbers n < 270 have finite trajectories, but it
+appears that 270 goes increases forever. The trajectory of 270 is A291789. Andrew
+Booker has given a heuristic argument showing that almost all numbers go to infinity.
+What makes 270 the first immortal number under g? Again I suspect it just got lucky!
+Open questions: Does the trajectory of 276 under f increase forever? What about
+the trajectory of 270 under g?
+4.3
+Gijswijt’s sequence (A090822)
+For this sequence it will be helpful to remember that chemists do not write H − H − O,
+they write H2O, they do not write AlAlAlSOOOOSOOOO, they write Al3(SO4)2.
+We will apply a similar compression to sequences of numbers, except that we indicate
+repetition by superscripts rather than by subscripts.
+For this problem, when we look at a sequence of numbers, we want to write it in the
+form XY Y ...Y , or XY k, where X and Y are themselves sequences of numbers, X can
+be missing, and the exponent k is as large as possible.
+For example, we can write 1,2,2,2,2 as XY k, where X = 1, Y = 2, and k = 4. The
+highest k we can achieve for a sequence is called its “curling number”. So 1,2,2,2,2
+has curling number 4. Think of an animal with its head looking to the left, with a very
+curly tail. X represents the head and body of the animal, and Y k represents the curls
+in its tail.
+Consider the sequence 3,2,4,4,2,4,4,2,4,4. We could take X = 3,2,4,4,2,4,4,2 and
+Y = 4, getting XY 2, with k = 2, or we could take X = 3, Y = 2,4,4, getting XY 3, with
+k = 3, which is larger. So this sequence has curling number 3.
+Remember that X may be missing. So the sequence with a single term 99, say, can
+be written as Y 1 where Y is the number 99, and it has curling number 1. (The notion
+of curling number is independent of the base in which the numbers are written.)
+We are now ready to define Dion Gijswijt’s absolutely brilliant sequence, which he
+sent to the OEIS in 2004.
+14
+
+The rule for finding the next term is simple: it is the curling number of the sequence
+so far. And you start with 1. That’s the sequence!
+So let’s construct it. We start with 1, and the curling number of 1 is 1. So now we
+have 1,1. This has curling number 2, so now we have 1,1,2. At each step we recompute
+the curling number, and make that the next term.
+Here are the first few generations.
+1
+1 1
+1 1 2
+1 1 2 1
+1 1 2 1 1
+1 1 2 1 1 2
+1 1 2 1 1 2 2 (we took Y = 1 1 2)
+1 1 2 1 1 2 2 2
+1 1 2 1 1 2 2 2 3
+and we have found the first 3, at the 9th term. After a while, a 4 appears at term 220.
+But Gijswijt was unable to find a 5, and left that question open when he submitted
+the sequence. Some Bell Labs colleagues computed many millions of terms, but no 5
+appeared.
+Finally, over the course of a long weekend, Fokko van der Bult (a fellow student of
+Gijswijt’s in Amsterdam) and I independently showed that there is a 5. In fact there
+are infinitely many 5’s, but the first one does not appear until about term 101023. The
+universe would be cold long before any computer search would find it.
+In the paper we wrote about the sequence [4], we also conjectured that the first time
+a number N > 4 appears is at about term
+2 ↑ (2 ↑ (3 ↑ (4 ↑ (5 ↑ ... ↑ (N − 1))))),
+where the up-arrows (↑) indicate exponentiation. This is a tower of exponents of height
+N − 1.
+A very recent manuscript by a student of Gijswijt’s, Levi van de Pol, still under
+review, has extended our work, and may have proved the above conjecture.
+I cannot resist adding a further comment about curling numbers, which if true shows
+that the Gijswijt sequence is in a sense universal.
+The Curling Number Conjecture asserts that if any finite starting sequence is ex-
+tended by the rule that the next term is the curling number of the sequence so far, then
+eventually the curling number will be 1.
+If true, this implies that if the starting sequence contains no 1s, then the sequence
+eventually becomes Gijswijt’s sequence [5, Th. 23]. In fact I conjecture that this is true
+for any starting sequence.
+Open question: Is the Curling Number Conjecture true?
+15
+
+4.4
+Lexicographically Earliest Sequences
+Although there is no space to discuss them in detail, let me just mention that there
+are many fascinating and difficult sequences in the OEIS whose definition has the form
+“Lexicographically Earliest Sequence of distinct positive numbers with the property that
+...”, where now we are using lexicographic in its pure sense, as defined in Section 3.4.
+A favorite example is the EKG (or ECG) sequence A064413, whose definition is the
+lexicographically earlier infinite sequence of distinct positive numbers with the property
+that each term after the first has a nontrivial common factor with the previous term [9].
+Other L.E.S. examples are the Yellowstone permutation A098550 [2], the Enots Wolley
+sequence A336957 (the name suggests the definition), and the Binary Two-Up sequence
+A354169 [6].
+Open question: Show that the terms of the Enots Wolley sequence are precisely 1,
+2, and all numbers with at least two distinct prime factors.
+4.5
+The Stepping Stones Problem (A337663)
+This lovely problem was invented in 2020 by two undergraduates, Thomas Ladouceur
+and Jeremy Rebenstock. You have an infinite chessboard, and a handful of brown stones,
+which are worth one point each. You also have an infinite number of white stones, of
+values 2, 3, 4,..., one of each value. Suppose you have n brown stones. You start by
+placing them anywhere on the board. Now you place the white stones, trying to place
+as many as you can. The rules are that you can only place a white stone labeled k on
+a square if the values of the stones on the eight squares around it add up to k. And
+you must place the white stones in order, first 2, then 3, and so on. You stop when
+you cannot place the next higher-numbered white stone. The goal is to maximize the
+highest value that you place. Call this a(n).
+9
+5
+10
+11
+4
+1
+12
+8
+3
+2
+16∗
+6
+1
+15
+13
+7
+14
+Figure 6: A solution to the Stepping Stones problem for two starting stones. The high
+point a(2) = 16 here is indicated by an asterisk, as it is in the next three tables.
+Say we start with n = 2 brown stones. There are infinitely many squares where they
+can be placed, but it turns out that the best thing is to place them so they are separated
+diagonally by a single blank square, as in Fig. 6. Now we start trying to place the white
+16
+
+stones. The 2 stone has to go between the two brown (or 1) stones, and then the 3 goes
+on a square adjacent to the 1 and the 2. There is now a choice for where the 4 goes, but
+the choice shown in Fig. 6 is the best. (After we have placed the 4, the neighbors of the
+3 no longer add up to 3, but that is OK. It is only when we place the 3 that its neighbors
+must add to 3.) Continuing in this way, we eventually reach 16. There is nowhere to
+place the 17, so we stop. Ladouceur and Rebenstock showed, using a computer and
+considering all possible arrangements, that 16 is the highest value that can be attained
+with two starting stones. So a(2) = 16.
+This is clearly a hard problem, since the number of possibilities grows rapidly with
+the number of brown stones. Only six terms of this sequence are known: a(1) through
+a(6) are 1,16,28,38,49,60. A solution for n = 4 found by Arnauld Chevallier is shown in
+Fig. 7. There are lower bounds for larger values of n which may turn out to be optimal.
+For n = 7,...,10 the current best constructions give 71,80,90,99. See A337663 for the
+latest information.
+35
+18
+36
+23
+21
+32
+17
+1
+14
+9
+12
+20
+34
+16
+15
+5
+4
+8
+26
+27
+31
+10
+1
+3
+19
+25
+1
+28
+11
+2
+6
+33
+29
+24
+13
+22
+1
+7
+37
+30
+38∗
+Figure 7: A solution to the Stepping Stones problem for four starting stones.
+We don’t know how fast a(n) grows. There have been a series of upper and lower
+bounds, initiated by Robert Gerbicz and Andrew Howroyd. The simple linear construc-
+tion shown in Fig. 8 shows that a(n) ≥ 6(n − 1) for n ≥ 3.
+1
+2
+3
+4
+5
+6
+7
+8
+9
+1
+1
+1
+10
+18∗
+17
+16
+15
+14
+13
+12
+11
+Figure 8: Every additional 1 on the middle row increases the number of white stones
+by 6, showing that a(n) ≥ 6(n − 1) for n ≥ 3.
+17
+
+By combining the constructions of Figs. 6 and 8, Menno Verhoeven obtained a(n) ≥
+6n + 3 for n ≥ 3 (Table 9).
+25
+24
+1
+26
+23
+27
+22
+28
+21
+1
+29
+20
+30
+19
+31
+9
+5
+10
+11
+18
+1
+32
+4
+1
+17
+33∗
+12
+8
+3
+2
+16
+6
+1
+15
+13
+7
+14
+Figure 9: Combining the constructions of of Figs. 6 and 8 gives a(n) ≥ 6n + 3 for n ≥ 3.
+The case n = 5 is shown. For other values of n, adjust the height of the “chimney” on
+the right.
+The best lower bound for large n is due to Robert Gerbicz, who has shown by a
+remarkable extension of the construction in Figs. 8 and 9 that limn→∞ a(n)/n > 6. (A
+preliminary version of his bound gives a(n) > 6.0128n−5621 for all n, although the exact
+values of the constants have not been confirmed.) In his construction the “chimney” on
+the right of Fig. 9 gets expanded into a whole trellis.
+One might think that with a sufficiently clever arrangement, perhaps extending the
+construction in Fig. 8 so that the path wraps around itself in a spiral, one could achieve
+large numbers with only a few starting stones. But a simple counting argument due
+to Robert Gerbicz shows this is impossible. The current best upper bound is due to
+Jonathan F. Waldmann, who has shown that a(n) < 79n + C for some constant C. See
+A337663 for the latest information, including proofs of of the results mentioned here.
+Open question: Improve the upper and lower bounds on a(n).
+4.6
+Stained glass windows
+In 1998 Poonen and Rubinstein [12] famously determined the numbers of vertices and
+cells in the planar graph formed from a regular n-gon by joining every pair of vertices
+by a chord. The answers are in A006561 and A007678. Lars Blomberg, Scott Shannon,
+and I have studied versions of this question when the regular n-gon is replaced by other
+18
+
+Figure 10: A 4×2 grid of squares with every pair of boundary points joined by a chord.
+The graph has 213 vertices and 296 cells.
+The cells are color-coded to distinguish
+triangles (red), quadrilaterals (yellow), and pentagons (blue).
+polygons, for instance by a square in which n equally-spaced points are placed along each
+side and each pair of boundary points is joined by a chord. We also studied rectangles,
+triangles, etc. In most cases we were unable to find formulas for the numbers of vertices
+or cells, but we collected a lot of data, and the graphs, when colored, often resemble
+stained glass windows (see [3] and the illustrations in A331452 and other sequences
+cross-referenced there).7 So we consoled ourselves with the motto: if we can’t solve it,
+make art!
+The most promising case to analyze seemed to be the n × 2 grid (although we did
+not succeed even there).
+Open question: How many vertices and cells are there in the graph for the n×2 grid,
+7There is no fee for downloading images in the OEIS, but if you use any of them, please credit the
+source!
+19
+
+as illustrated for n = 4 in Fig. 10? Sequences A331763 and A331766 give the first 100
+terms, yet even with all that data we have not found a formula.
+The case of an n × n grid seems even harder.
+Figure 11 shows the 6 × 6 graph.
+Sequences A331449 and A255011 give the numbers of vertices and cells for n ≤ 42.
+A334699 enumerates the cells by number of sides.
+Figure 11: A 6×6 grid with every pair of boundary points joined by a chord. There are
+4825 vertices and 6264 cells.
+In the summer of 2022 Scott Shannon and I considered several other families of planar
+graphs. I cannot resist showing one of Shannon’s graphs, a 16 × 16 grid, illustrating the
+16th term of A355798 (Fig. 12). There are 61408 cells. Although Shannon has calculated
+40 terms of this sequence, again no formula is known.
+20
+
+Figure 12: Scott Shannon’s “Magic Carpet” graph, illustrating A355798(16).
+4.7
+If I had more space
+If I had had more space I would also have discussed some very interesting sequences
+arising from:
+– Dissecting a square to get a regular n-gon (A110312).
+– Gerrymandering (A341578, A348453, and many others).
+– In how many ways can circles overlap? (A250001).
+– The Inventory sequence A342585.
+– Kaprekar’s junction numbers (A006064, [1]).
+– The kissing number problem (A001116, A257479).
+– The neural network problem that started it all (A000435).
+– Squares in the plane (A051602).
+And, maybe, meta-sequences such as A051070 (a(n) is the nth term of An) and
+A107357 (the nth term is 1 + the nth term of An).
+21
+
+A final comment: there are many videos on the Internet of talks I have given about
+sequences. There are over twenty videos that Brady Haran and I have made that have
+appeared on the Youtube Numberphile channel (and have been viewed over eight million
+times). See for example “Terrific Toothpick Patterns”.
+5
+Acknowledgments
+I would like to thank some good friends who have helped me and the OEIS over the
+years: David L. Applegate, William Cheswick, Russ Cox, Susanna S. Cuyler, Harvey
+P. Dale, Ronald L. Graham, Richard K. Guy, Marc LeBrun, John Riordan, and Doron
+Zeilberger.
+There are many active volunteer editors, and it is impossible to thank them all.
+But I would like to give particular thanks to J¨org Arndt, Michael S. Branicky, Michael
+De Vlieger, Amiram Eldar, Charles R. Greathouse IV, Maximilian F. Hasler, Alois P.
+Heinz, Andrew Howroyd, Sean A. Irvine, Michel Marcus, Richard J. Mathar, Peter
+Munn, Hugo Pfoertner, Kevin Ryde, Jon E. Schoenfield, R´emy Sigrist, and Chaiwah
+Wu.
+I also thank the members of the Board of Trustees of the OEIS Foundation, past
+and present, for all their help, both to me personally and to the OEIS.
+Figure credits: Figure 1(c): Clifford A. Pickover. Figure 5: Edmund Harriss. Fig-
+ures 6, 7, 8, and 9 are based on communications from Thomas Ladouceur and Jeremy
+Rebenstock, Arnauld Chevallier, Skylark Xentha Murphy-Davies, and Menno Verho-
+even, respectively. Figures 10 and 11: Lars Blomberg and Scott Shannon. Figure 12:
+Scott R. Shannon. Other figures: the author.
+References
+[1] M. A. Alekseyev and N. J. A. Sloane, On Kaprekar’s Junction Numbers, J. Com-
+binat. and Number Theory, 2023, in press.
+[2] D. L. Applegate, H. Havermann, B. Selcoe, V. Shevelev, N. J. A. Sloane, and
+R. Zumkeller, The Yellowstone permutation, J. Integer Seqs., 18 (2015), #15.6.7.
+[3] L. Blomberg, S. R. Shannon, and N. J. A. Sloane, Graphical enumeration and
+stained glass windows, 1: Rectangular grids, Integers, Ron Graham Memorial Vol-
+ume 21A (2021), #A5.
+[4] F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane, and Allan
+Wilks, A slow-growing sequence defined by an unusual recurrence, J. Integer Seqs.,
+10 (2007), #07.1.2.
+[5] B. Chaffin, J. P. Linderman, N. J. A. Sloane and A. R. Wilks, On curling numbers
+of integer sequences, J. Integer Seqs., 16 (2013), #13.4.3.
+22
+
+[6] M. De Vlieger, T. Scheuerle, R. Sigrist, N. J. A. Sloane, and W. Trump, The binary
+Two-Up sequence, arXiv:2209.04108, Sep. 11 2022.
+[7] H. W. Gould, Combinatorial Identities, Morgantown, WV, 1972.
+[8] R. K. Guy, Unsolved Problems in Number Theory, 3rd. ed., Springer, 2010.
+[9] J. C. Lagarias, E. M. Rains, and N. J. A. Sloane, The EKG sequence, Experimental
+Math., 11 (2002), 437–446.
+[10] D. S. Mitrinovi´c, J. S´andor and B. Crstici, Handbook of Number Theory, Kluwer,
+Dordrecht, 1996.
+[11] The OEIS Foundation Inc. (2023), The On-Line Encyclopedia of Integer Sequences,
+https://oeis.org.
+[12] B. Poonen and M. Rubinstein, The number of intersection points made by the
+diagonals of a regular polygon, SIAM J. Discrete Mathematics, 11.1 (1998) 135–
+156.
+[13] J. Riordan and N. J. A. Sloane, Enumeration of rooted trees by total height, J.
+Austral. Math. Soc., 10 (1969), 278–282.
+2020 Mathematics Subject Classification: 05-00, 11-00, 11Bxx, 48-00, 68-00
+23
+
diff --git a/MtE1T4oBgHgl3EQfZQTY/content/tmp_files/load_file.txt b/MtE1T4oBgHgl3EQfZQTY/content/tmp_files/load_file.txt
new file mode 100644
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--- /dev/null
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@@ -0,0 +1,824 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf,len=823
+page_content='“A Handbook of Integer Sequences” Fifty Years Later  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Sloane,  The OEIS Foundation Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=',  11 So.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Adelaide Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', Highland Park, NJ 08904, USA  Email: njasloane@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='com  January 8, 2023  Abstract  Until 1973 there was no database of integer sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Someone coming across  the sequence 1,2,4,9,21,51,127,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' would have had no way of discovering that  it had been studied since 1870 (today these are called the Motzkin numbers,  and form entry A001006 in the database).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Everything changed in 1973 with the  publication of A Handbook of Integer Sequences, which listed 2372 entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' This  report describes the fifty-year evolution of the database from the Handbook to its  present form as The On-Line Encyclopedia of Integer Sequences (or OEIS), which  contains 360,000 entries, receives a million visits a day, and has been cited 10,000  times, often with a comment saying “discovered thanks to the OEIS”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  1  Introduction  Number sequences arise in all branches of science: for example, 1,1,2,4,9,20,48,115,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  gives the number of rooted trees with n nodes (A000081,1 see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 1), and in daily  life: how many pieces can you cut a pancake into with n knife-cuts?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' (The pieces need  not all be the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=') That one is easy: 1,2,4,7,11,16,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', n(n+1)/2+1 (A000124).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  But what is the answer for cutting up an (ideal) bagel, or torus?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' That is a lot harder:  with a sharp knife you might get a few terms, perhaps 1,2,6,13,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', but probably not  enough to guess the formula, which is n(n2 + 3n + 8)/6 for n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' For that you would  need to to consult the database: go to https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='org and enter “cutting bagel”,  or go directly to A003600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  My fascination with these sequences began in 1964 when I was a graduate student  at Cornell University in Ithaca, NY, studying neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' I had encountered a  sequence of numbers, 1,8,78,944,13800,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', and I badly needed a formula for the n-th  term, in order to determine the rate of growth of the terms (this would indicate how  long the activity in this very simple neural network would persist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' I will say more  about this sequence in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  I noticed that although several books in the Cornell library contained sequences  somewhat similar to mine, as far as I could tell this particular sequence was not men-  tioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' I expected to have to analyze many related sequences, so in order to keep track  of the sequences in these books, I started recording them on 3” × 5” file cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  1Six-digit numbers prefixed by A refer to entries in the current version of the Handbook, The On-Line  Encyclopedia of Integer Sequences [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='03149v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='NT]  9 Jan 2023  Figure 1: Left: one of 48 unlabeled rooted trees with 7 nodes (the root node is at the  bottom);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' center: four cuts of a pancake can produce 11 pieces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' right: three cuts of a  bagel can produce 13 pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  The collection grew rapidly as I searched though more books, and once the word got  out, people started sending me sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Richard Guy was an enthusiastic supporter  right from the start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  In 1973 I formalized the collection as A Handbook of Integer  Sequences, which was published by Academic Press (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' It contained 2372 entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Figure 2: Front cover of the Handbook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The embossed figures show side views of the two  ways of folding a strip of three (blank) stamps, and the five ways of folding a strip of  four stamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The full sequence begins 1,1,2,5,14,38,120,353,1148,3527,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', A001011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  No formula is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Once the book appeared, the flood of correspondence increased, and it took twenty  years to prepare the next version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Simon Plouffe helped a great deal, and in 1995  Academic Press published our sequel, The Encyclopedia of Integer Sequences, with 5487  entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' From this point on the collection grew even more rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' I waited a year,  until it had doubled in size, and then put it on the Internet, calling it The On-Line  Encyclopedia of Integer Sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  In the rest of this article I will first say more about the evolution of the database:  the Handbook (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='1), the 1995 Encyclopedia (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='2), the On-Line Encyclopedia (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='3),  and the OEIS Foundation (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The next sections describe the database itself: what  2  AHANDBOOKOF  INTEGER SEQUENCES  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='JLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='SLOANEsequences are—or are not—included (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='1), how the database is used (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='2), the layout  of a typical entry (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='3), the arrangement of the entries (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='4), and a Fact Sheet (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  The final sections describe some especially interesting sequences: Recam´an’s sequence  (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='1), Iteration of number-theoretic functions (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='2), Gijswijt’s sequence (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='3), Lex-  icographically Earliest Sequences (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='4), The Stepping Stones problem (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='5), Stained  glass windows (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='6), and other sequences I would have liked to include (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Several open questions are mentioned to which I would very much like to know the  answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  a(n) will denote the n-th term of the sequence being discussed, and  A001006(n) (for example) would denote the n-th term of A001006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' σ(n) is the sum of  the divisors of n (A000203).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  2  Evolution of the database  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='1  The Handbook of Integer Sequences  Once the collection had grown to a few hundred entries, I entered them on punched  cards,2 which made it easier to check and sort them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The Handbook was type-set directly  from the punched cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' There were a few errors in the book, but almost all of them  were caused by errors in the original publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Accuracy was a primary concern in  that book, as it is today in the OEIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  The book was an instant success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' It was, I believe, the world’s first dictionary of  integer sequences (and my original title said Dictionary rather than Handbook).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Many  people said “What a great idea”, and wondered why no one had done it before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Martin  Gardner recommended it in the Scientific American of July 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Lynn A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Steen, writing  in the American Mathematical Monthly said “Incomparable, eccentric, yet very useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Contains thousands of ‘well-defined and interesting’ infinite integer sequences together  with references for each .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' If you ever wondered what comes after 1,2,4,8,17,35,71,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=',  this is the place to look it up”’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Harvey J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Hindin, writing from New York City, exuberantly concluded a letter to  me by saying: “There’s the Old Testament, the New Testament, and the Handbook of  Integer Sequences.”  I never did find the sequence that started it all in the literature, but I learned P´olya’s  theory of counting, and with John Riordan’s help found the answer, which appears in  [13] and A000435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  2These were never called “punch cards” (sic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' To anyone who worked with them in the 1960s, “punch  cards” sounds like “grill cheese” (sic) for “grilled cheese”, or “barb wire” (sic) for “barbed wire”, both  of which I have recently seen in print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='2  The Encyclopedia of Integer Sequences  Following the publication of the Handbook, a large amount of correspondence ensued,  with suggestions for further sequences and updates to the entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' By the early 1990’s  over a cubic meter of new material had accumulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  A Canadian mathematician,  Simon Plouffe, offered to help in preparing a revised edition of the book, and in 1995  The Encyclopedia of Integer Sequence, by me and Simon Plouffe, was published by  Academic Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' It contained 5487 sequences, occupying 587 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' By now punched  cards were obsolete, and the entries were stored on magnetic tape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='3  The On-Line Encyclopedia of Integer Sequences  Again, once the book appeared, many further sequences and updates were submitted  from people all over the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' I waited a year, until the size of the collection had  doubled, to 10000 entries, and then in 1996 I launched The On-Line Encyclopedia of  Integer Sequences (now usually called simply the OEIS) on the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' From 1996  until October 26, 2009, it was part of my homepage on the AT&T Labs website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Incidentally, in 2004 the database was mentioned by the Internet website slashdot  (“News for Nerds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Stuff that Matters”), and this brought so much traffic to my Bell  Labs homepage that it briefly crashed the whole Bell Labs website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' My boss was quite  proud of this, since it was a rare accomplishment for the Mathematics and Statistics  Research Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='4  The OEIS Foundation  In 2009, in order to ensure the long-term future of the database, I set up a non-profit  foundation, The OEIS Foundation Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', a 501(c)(3) Public Charity, whose purpose is  to own, maintain and raise funds to support The On-Line Encyclopedia of Integer Se-  quences or OEIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  On October 26, 2009, I transferred the intellectual property of The On-Line Ency-  clopedia of Integer Sequences to the Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' A new OEIS with multiple editors was  launched on November 11, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Since then it has been possible for anyone in the world to propose a new sequence  or an update to an existing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' To do this, users must first register, and then  submissions are reviewed by the editors before they become a permanent part of the  OEIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Technically the OEIS is now a “moderated wiki”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  I started writing this article on November 11, 2022, noting that this marked twelve  years of successful operation of the online OEIS, and also that the database is in its  59th year of existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  4  3  The database today  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='1  What sequences are included?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  From the very beginning the goal of the database has been to include all “interest-  ing” sequences of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  This is a vague definition, but some further examples  will make it clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  The database includes a huge number of familiar and unfa-  miliar sequences from mathematics (the prime numbers 2,3,5,7,11,13,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', A000040;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  60,168,360,504,660,1092,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', the orders of noncyclic simple groups, A001034), com-  puter science (0,1,3,5,8,11,14,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', Number of comparisons needed for merge sort,  A001855), physics (see “self-avoiding walks on lattices”, Ising model, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' A002921),  chemistry (the enumeration of chemical compounds was one of the motivations behind  P´olya’s theory of counting, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' A000602), and not least, from puzzles and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  tests (1,8,11,69,99,96,111,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', the “strobogrammatic” numbers, guess!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', or see A000787;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  4,14,23,34,42,50,59,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', the numbered stops on the New York City A train subway,  A011554.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' That entry has links to a map and the train schedule).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Sequences that have arisen in the course of someone’s work—especially if published—  have always been welcomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' On the other hand, sequences that have been proposed  simply because they were missing from the database are less likely to be accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  There are a few hard and fast rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The sequence must be well-defined and the  terms must not be time-dependent—if the next term is only known to be either 14 or  15, for instance, then the sequence must end with the last term that is known for certain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  The sequence may not have any missing terms or gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' In the case of Mersenne primes,  for instance (A000043) it is common for later primes to be known before all intermediate  numbers having been tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The later primes get mentioned in comments, but they  are not as part of the main sequence until their position has been confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Very short sequences and sequences that are subsequences of many other sequences  are not accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' A sequence for which the only known terms are 2,3,5,7 would not  be accepted since it is matched by a large number of existing sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The definition  may not involve an arbitrary but large parameter (primes ending in 1 are fine, A030430,  but not primes ending in 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='3  Most entries give an ordered list of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' But triangles of numbers are included  by reading them row-by-row: Pascal’s triangle becomes 1, 1,1, 1,2,1, 1,3,3,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=',  A007318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Doubly-infinite square arrays are included by reading them by antidiago-  nals: the standard multiplication table for positive integers becomes 1, 2,2, 3,4,3,  4,6,6,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', A003991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Sequences of fractions are included as a linked pair giving the numerators and denom-  inators separately (the Bernoulli numbers are A027641/A027642).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Important individual  real numbers are included by giving their decimal or continued fraction expansions (for  π see A000796 and A001203).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' A relatively small number of sequences of nonintegral  3The OEIS Wiki, which has a great deal of useful information about the database, has a section  listing additional examples of “what not to submit”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  5  real numbers are included by rounding them to the nearest integer, or by taking floors  or ceilings (the imaginary parts of the zeros of Riemann’s zeta function give A002410).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Two less obvious sources for sequences are binomial coefficient identities and number-  theoretic inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The values of either side of the identity  n  ∑  k=0  (2n  k )  2  = 1  2(4n  2n) − 1  2(2n  n )  2  [7, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='68)] give A036910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' From the inequality σ(n) < n√n for n > 2, [10, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='b],  we get the integer sequence ⌊n√n⌋ − σ(n), A055682.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The point is that if you want to  know if this inequality is known, you look up the difference sequence, and find A055682  and a reference to the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Many more sequences of these two types should be added  to the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='2  How the database is used  The main applications of the database are in identifying sequences or in finding out the  current status of a known sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Barry Cipra has called it a mathematical analogue  of a “fingerprint file”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' You encounter a number sequence, and wish to know if anyone  has ever come across it before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  If your sequence is in the database, the reply will give a definition, the first 50 or  so terms, and, when available, formulas, references, computer code for producing the  sequence, links to any relevant web sites, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Figures 3 and 4 show what happens if you submit 1,2,5,14,42,132,429, the first few  Catalan numbers, one of the most famous sequences of all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Figure 3: The result of submitting 1,2,5,14,42,132,429 to the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' This figure  shows the banner at the top of the reply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' There are 26 matches, ranked in order of  importance, the top match being the one we want, the Catalan numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' A shortened  version of the top match is shown in the next figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  6  TheOEisissupportedbythemanygenerousdonorstotheOEisFoundation  013627  THE ON-LINE ENCYCLOPEDIA  20  OF INTEGER SEOUENCES  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  10221121  founded in 1964 by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Sloane  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='14,42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='132,429  Search  Hints  (GreetingsfromTheOn-LineEncyclopediaofIntegerSequences!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=')  Search:seq:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='429  Displaying1-10of26resultsfound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  page 1 2 3  Sort:relevanceIreferencesInumberImodifiedIcreated  Format:longIshortIdataA000108  Catalan numbers: C(n) = binomial(2n,n)/(n + 1) = (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='/(n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' (n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  (Formerly M1459 N0577)  DATA  1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  COMMENTS  These were formerly sometimes called Segner numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  A very large number of combinatorial interpretations are known -  see references, esp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Stanley, Catalan Numbers, Camb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  This is probably the longest entry in the OEIS, and rightly so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  The solution to Schr¨oder’s first problem: number of ways to insert n pairs  of parentheses in a word of n + 1 letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', for n = 2 there are 2 ways:  ((ab)c) or (a(bc));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' for n=3 there are 5 ways: ((ab)(cd)), (((ab)c)d), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  REFERENCES  The large number of references and links demonstrates the ubiquity  of the Catalan numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Alter, Some remarks and results on Catalan numbers, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 109-132  in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Louisiana Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Combinatorics, Graph Theory and  Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 2, edited R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Mullin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Bona, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', Handbook of Enumerative Combinatorics, CRC Press, 2015  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Comtet, Advanced Combinatorics, Reidel, 1974, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Conway & R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Guy, The Book of Numbers, Springer, 1995, 96-106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  LINKS  Robert G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Wilson v, Table of n, a(n) for n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='.1000  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Bernhart, Catalan, Motzkin and Riordan numbers, Disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=',  Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 204, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 1-3 (1999), 73-112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Brown, Historical Note on a Recurrent Combinatorial Problem,  Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Monthly,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 72, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 9 (1965), 973-977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  FORMULA  Recurrence: a(n) = 2 ∗ (2 ∗ n − 1) ∗ a(n − 1)/(n + 1) with a(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  MAPLE  A000108 := n-¿binomial(2*n, n)/(n+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  MATHEMATICA  A000108[n ] := (2 n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='/n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='/(n+1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  PARI  a(n)=binomial(2*n, n)/(n+1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  KEYWORD  core,nonn,easy,nice  AUTHOR  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Sloane  Figure 4: The entry for the Catalan number A000108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The full entry has over 750 lines,  which have been edited here to show samples of the different fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  I could have chosen a simpler example, like the Fibonacci numbers, but I have a  particular reason for choosing the Catalan numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' When the OEIS was new, people  would sometimes say to me that they had a sequence they were trying to understand,  and would I show them how to use the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  At least twice when I used the  Catalan sequence as an illustration, they said, why, that is my sequence, how on earth  did you know?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' It was no mind-reading trick, the Catalan numbers are certainly the  most common sequence that people don’t know about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' This entry is the longest—and  7  one of the most important—in the whole database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  If we do not find your sequence in the database, we will send you a message inviting  you to submit it (if you consider it is of general interest), so that the next person who  comes across it will be helped, and your name will go on record as the person who  submitted it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  The second main use of the database is to find out the latest information about a  particular sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Of course we cannot hope to keep all 360000 entries up-to-date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' But when a new  paper is published that mentions the OEIS, Google will tell us, and we then add links to  that paper from any sequence that it mentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' People have told us that this is one of  the main ways they use the OEIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' After all, even a specialist in (say) permutation groups  cannot keep track of all the papers published worldwide in that area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' And if a paper in  a physics journal happens to mention a number-theoretic sequence, for example, that is  unlikely to be noticed by mathematicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  There are also many other ways in which the database has proved useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  For example, it is an excellent source of problems to work on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The database is con-  stantly being updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Every day we get thirty to fifty submissions of new sequences,  and an equal number of comments on existing entries (new formulas, references, ad-  ditional terms, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  The new sequences are often sent in by non-mathematicians,  and are a great source of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' You can see the current submissions at https:  //oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='org/draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Often enough you will see a sequence that is so interesting you  want to drop everything and work on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' And remember that we are always in need of  more volunteer editors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' In fact anyone who has registered with the OEIS can suggest  edits, you do not even need to be an official editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' We have been the source of many  international collaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  There is also an educational side: several people have told us that they were led into  mathematics through working as an editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Here is a typical story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Subject: Reminiscence from a young mathematician  I wanted to relay a bit of nostalgia and my heartfelt thanks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Back in the late 1990s, I was a high  school student in Oregon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' While I was interested in mathematics, I had no significant mathematically  creative outlet until I discovered the OEIS in the course of trying to invent some puzzles for myself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' I  remember becoming a quite active contributor through the early 2000s, and eventually at one point, an  editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' My experience with the OEIS, and the eventual intervention of one of my high school teachers,  catalyzed my interest in studying mathematics, which I eventually did at [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='] College.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' I went on to a  Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' in algebraic geometry at the University of [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='], and am currently at [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  I wanted to thank you for seriously engaging with an 18-year old kid, even though I likely submitted  my fair share of mathematically immature sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' I doubt I would have become a mathematician  without the OEIS!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  A less-obvious use of the database is to quickly tell you how hard a problem is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' I use  it myself in this way all the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Is the sequence “Catalan” or “Collatz”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' If a sequence  comes up in your own work, or when reviewing someone else’s work, it is useful to know  right away if this is a well-understood sequence, like the Catalan numbers, or if it is one  of the notoriously intractable problems like the Collatz or 3x + 1 problem (A006577).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  8  Finally, the OEIS is a welcome escape when you feel the world is falling apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Take  a look at Scott Shannon’s drawings of stained glass windows in A331452;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' or Jonathan  Wild’s delicate illustrations of the ways to draw four circles (in A250001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' or ´Eric An-  gelini’s “1995” puzzle (A131744) or any of his “lexicographically earliest sequences”  (A121053, A307720, and many more);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' or find better solutions to the Stepping Stones  Problem (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='5,A337663).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' You can find brand new problems at any hour of the day or  night by looking at the stack of recent submissions: but beware, you may see a problem  there that will keep you awake for days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Or search in the database for phrases like “It  appears that .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..”, or “Conjecture: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..”, or “It would be nice to know more!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='3  Layout of a typical entry  This is a good place to mention some of the features of an OEIS entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Most of the  fields (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 3 and 4) are self-explanatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' At the top it tells you how many matches  were found to your query (26 in the example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' These are ranked in order of importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  The DATA section shows the start of the sequence, usually enough terms to fill a few  lines on the screen (typically 300 to 500 decimal digits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Often one wants more terms  than are shown, and the first link in the entry will point to a plain text file with perhaps  10000 or 20000 terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' That file will have a name like b001006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='txt, and is called the  “b-file” for the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Some entries also have much larger tables, giving a million or  more terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  If you click the “graph” button near the top of the reply, you will be shown two plots  of the sequence, and if you click the “listen” button, you could listen to the sequence  played on an instrument of your choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The default instrument is the grand piano, and  the terms of the sequence would be mapped to the 80 keys by reducing the numbers  mod 80 and adding 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  I conclude this section with a philosophical comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  When you are seriously trying to analyze a sequence, and are prepared to spend  any amount of time needed (searching for a formula or recurrence, for instance), you  need all the help you can get, which is why we provide the b-files and other data files,  and why we give computer programs in so many languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' This is also the reason we  give as many references and links as possible for a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Even if the reference is to  an ancient or obscure journal, or one that has been accused as being “predatory”, we  still give the reference, especially for sequences that are not well-understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The same  thing holds for formulas, comments, and cross-references to other sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' When you  are desperate, you will accept help from anywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' And do not forget “Superseeker”!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='4  Arrangement of the entries  The entries in the database are (virtually) arranged in two different ways, the first  essentially chronological, the second lexicographic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  9  The first is by their absolute identification number, or A-number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='4 Once the collec-  tion reached a few hundred entries, I sorted them into lexicographic order and numbered  them A1, A2, A3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='. A1 gives the number of symmetry groups of order n, A2 is the  famous Kolakoski sequence, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' This numbering is still used today, only A1 has  become A000001, A2 is A000002, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', and as each new submission comes in it gets a  number from the stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Current sequences are being issued numbers around A360000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Rejected A-numbers are recycled, so there are no gaps in the order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' We reached 100000  entries in 2004, and 250000 in 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The present growth rate is about 12000 new entries  each year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  The second arrangement is a kind of lexicographic ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  First I describe an  idealized, theoretical, lexicographic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Sequences of nonnegative numbers can be  arranged in lexicographic (or dictionary) order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  For example, sequences beginning  1,2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' come before 1,2,5,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', 1,2,4,3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', 1,3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', but after 1,2,3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='.  Also  1,2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' comes after the two-term sequence 1,2 (because blanks precede numbers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  More formally, we compare the two sequences term-by-term, and in the first position  where they differ whichever is smaller (or blank) is the lexicographically earlier sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  For sequences with negative terms, we ignore the signs and sort according to the  absolute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Here is the actual ordering used in the OEIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The sequences are arranged (virtually)  into a version of lexicographic order, according to the following rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' First, delete all  minus signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Then find the first term that is greater than 1, and discard all the terms  before it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' What’s left determines its position in the lexicographic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' For example,  to place −1,0,1,1,2,1,17,3,2,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' in the ordering, we would ignore the terms before  the underlined 2, and consider the sequence as beginning 2,1,17,3,2,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='.  Sequences that contain only 0s, 1s and −1s are sorted into lexicographic order by  absolute value and appear at the beginning of the ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The first sequence in the  database is therefore the zero sequence A000004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  In this way every sequence has a unique position in the ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The sequences  have been sorted in this way since the 1960s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' For the first ten years the punched card  entries were physically sorted into this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  When you look at an OEIS entry, A005132 say (the subject of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='1), towards  the bottom you will see two lines like5  Sequence in context: A277558 A350578 A335299 * A064388 A064387 A064389  Adjacent sequences: A005129 A005130 A005131 * A005133 A005134 A005135  which tell you the three entries immediately before and after that entry in the lexico-  graphic ordering, and the three entries before and after it in the A-numbering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The  asterisks represent the sequence you are looking at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The first group can be useful if you  are uncertain about a term in your sequence, the second in case you want to look at  other sequences submitted around that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Today the sequences are actually stored internally in an SQLite database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' However,  4The sequences in the 1973 and 1995 books were numbered N0001, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', and M0001, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  5If you don’t see these, click on the A-number at the top of the entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  10  the punched card format has been so useful that when you view a sequence, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 4,  it is still presented to you in something very like the old punched card format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='5  Summary: “A Handbook of Integer Sequences” today  – Now The On-Line Encyclopedia of Integer Sequences or OEIS: https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='org  – Accurate information about 360000 sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  – Definition, formulas, references, links, programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' View as list, table, graph, music!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  – Traffic: 1 million hits/day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  – 30 new entries, 50 updates every day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  – Often called one of best math sites on the Web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Fingerprint file for mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  – Street creds: 10000 citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  – A moderated Wiki, owned by OEIS Foundation, a 501(c)(3) public charity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  – Uses: to see if your sequence is new, to find references, formulas, programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  – Catalan or Collatz?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' (Very easy or very hard?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=')  – Source of fascinating research problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='6 low-hanging fruit from recent submis-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  – Accessible (free, friendly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  – Fun (1,2,4,6,3,9,12,8,10,5,15,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Interesting, educational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  – Addictive (better than video games).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  – Has led many people into mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  – One of the most successful international collaborations, a modest contribution  towards world peace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  – Need editors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  4  Some favorite sequences  I’m sometimes asked what my favorite sequence is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' This is a difficult question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' I’m  tempted to reply by saying: If you were the keeper of the only zoo in the world, how  would you answer that question?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' (Because that is roughly the situation I’m in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=') Would  you pick one of the exotic animals, a giraffe, a kangaroo, or a blue whale?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Or one of the  essential animals, like a horse, a cow, or a duck?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' If the question came from a visiting  alien, of course, there is only one possible answer: a human being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  6Look for “Conjecture”, “It appears that”, “It would be nice to”, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  11  For sequences, the essential ones are the primes, the powers of 2, the Catalan num-  bers, or (especially if the question came from an alien with no fingers or toes), the  counting sequence 0, 1, 2, 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' (A001477).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  But here I’ll mention a few that are fairly exotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The Recam´an and Gijswijt se-  quences have simple recursive definitions, yet are astonishingly hard to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='1  Recam´an’s sequence (A005132)  This remarkable sequence has resisted analysis for over 30 years, even though we have  computed an astronomical number of terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' It was contributed to the database by  Bernardo Recam´an Santos in 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  The definition is deceptively simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The first term is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' We now add or subtract 1,  then we add or subtract 2, then add or subtract 3, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The rule is that we always  first try to subtract, but we can only subtract if that leaves a nonnegative number that  is not yet in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Otherwise we must add.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Here is how the sequence starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' We have the initial 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' We can’t subtract 1, because  that would give a negative number, so we add 1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' So the second term is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' We can’t  subtract 2 from 1, so we add it, getting the third term 1+2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Again we can’t subtract  3, for that would give 0, which has already appeared, so we add 3, getting the fourth  tern 3 + 3 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Now we must add or subtract 4, and this time we can subtract, because 6−4 = 2, and  2 is nonnegative and a number that hasn’t yet appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' So at this point the sequence  is 0,1,3,6,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Then it continues 7(= 2 + 5), 13(= 7 + 6),20(= 13 + 7),12(= 20 − 8), and so  on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The first 16 terms are  0,1,3,6,2,7,13,20,12,21,11,22,10,23,9,24,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  When adding rather than subtracting, repeated terms are permitted (42 is repeated  at the 24th term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Edmund Harriss has found an elegant was to draw the sequence as a spiral on the  number line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Start at 0, and when we subtract n, draw a semicircle of diameter n to  the left from the last point, or to the right if we are adding n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Draw the semicircles  alternately below and above the horizontal axis so as to produce a smooth spiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  The main question about this sequence is: Does every positive number appear?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  What makes this sequence so interesting is that certain numbers (for reasons we do not  understand) are extremely reluctant to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 4 does not appear until 131 steps, and  19 takes 95734 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  A group of us at AT&T Bell Labs worked on this sequence in 2001, and developed a  way to greatly speed up the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Allan Wilks used it to compute the first 1015  terms, and found that 2406 (which had been missing for a long time) finally appeared  at step 394178473633984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  At this point the smallest missing number was 852655 = 5⋅31⋅5501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Benjamin Chaffin  has continued this work, and in 2018 reached 10230 terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' However, 852655 was still  12  Figure 5: Harriss’s drawing of the first 64 terms of Recam´an’s sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' (The tiny  initial semicircle, at the extreme left, is below the axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' It has diameter 1 and joins the  points 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' It continues as a semicircle of diameter 2, above the axis, joining the  points 1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=')  missing, and there has been no progress since then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Thirty years ago I thought that every number would eventually appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Now I  am not so sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' My current belief is that there are two possibilities: either there are  infinitely many numbers that never appear, and 852655 just happens to be the smallest  of them, but has no other special property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' A similar phenomenon appears to occur  when iterating various number-theoretic functions—see the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Or, every  number will eventually appear (just as presumably every one of Shakespeare’s plays will  eventually appear in the expansion of π in base 60), although we may never be able  to extend the sequence far enough to hit 852655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' For the latest information about this  sequence (or any other sequence mentioned in this article), consult the OEIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Open question: Does 852655 appear in A005132?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='2  Iteration of number-theoretic functions  Many mysterious sequences arise from the iteration of number-theoretic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' A  classic problem concerns the iteration of the function f(n) = σ(n) − n, the sum of the  “aliquot parts” of n (see Guy [8, §B6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' For an initial value of n, what happens to the  trajectory n,f(n),f(f(n)),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' All n < 276 terminate by entering a cycle (such n are  called “perfect”, “amicable”, or “sociable” numbers), or reaching a prime, then 1, then  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' But it appears likely that n = 276 and perhaps infinitely many even numbers, will  never terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The trajectory of 276 is A008892, and the b-file gives the first 2140  13  terms, term 2140 being a 213-digit number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The continuation of the trajectory has  stalled at term 2051, where a 202-digit number is waiting to be factored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' A098007 gives  the number of distinct terms in the trajectory of n, or −1 if the trajectory is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  The value of A098007(276) is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  If indeed 276 does go to infinity, it is natural to ask, how did 276 know it was destined  to be the first immortal number under the map f?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The answer may be that there are  infinitely many immortal numbers, and 276 just happens to be the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' It got lucky,  that’s all!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Just as 852655 got lucky in Recam´an’s problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  A similar question, also discussed by Guy [8, §B41], which has received much less  attention, concerns the map g(n) = (σ(n) + φ(n))/2, where φ(n) is the Euler totient  function A000010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The trajectory may end at 1, a prime, or a fraction, or it may increase  monotonically to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Sequence A292108 gives the number of steps in the trajectory,  or −1 if the trajectory is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' All numbers n < 270 have finite trajectories, but it  appears that 270 goes increases forever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The trajectory of 270 is A291789.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Andrew  Booker has given a heuristic argument showing that almost all numbers go to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  What makes 270 the first immortal number under g?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Again I suspect it just got lucky!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Open questions: Does the trajectory of 276 under f increase forever?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' What about  the trajectory of 270 under g?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='3  Gijswijt’s sequence (A090822)  For this sequence it will be helpful to remember that chemists do not write H − H − O,  they write H2O, they do not write AlAlAlSOOOOSOOOO, they write Al3(SO4)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  We will apply a similar compression to sequences of numbers, except that we indicate  repetition by superscripts rather than by subscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  For this problem, when we look at a sequence of numbers, we want to write it in the  form XY Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='Y , or XY k, where X and Y are themselves sequences of numbers, X can  be missing, and the exponent k is as large as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  For example, we can write 1,2,2,2,2 as XY k, where X = 1, Y = 2, and k = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The  highest k we can achieve for a sequence is called its “curling number”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' So 1,2,2,2,2  has curling number 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Think of an animal with its head looking to the left, with a very  curly tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' X represents the head and body of the animal, and Y k represents the curls  in its tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Consider the sequence 3,2,4,4,2,4,4,2,4,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' We could take X = 3,2,4,4,2,4,4,2 and  Y = 4, getting XY 2, with k = 2, or we could take X = 3, Y = 2,4,4, getting XY 3, with  k = 3, which is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' So this sequence has curling number 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Remember that X may be missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' So the sequence with a single term 99, say, can  be written as Y 1 where Y is the number 99, and it has curling number 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' (The notion  of curling number is independent of the base in which the numbers are written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=')  We are now ready to define Dion Gijswijt’s absolutely brilliant sequence, which he  sent to the OEIS in 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  14  The rule for finding the next term is simple: it is the curling number of the sequence  so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' And you start with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' That’s the sequence!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  So let’s construct it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' We start with 1, and the curling number of 1 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' So now we  have 1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' This has curling number 2, so now we have 1,1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' At each step we recompute  the curling number, and make that the next term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Here are the first few generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  1  1 1  1 1 2  1 1 2 1  1 1 2 1 1  1 1 2 1 1 2  1 1 2 1 1 2 2 (we took Y = 1 1 2)  1 1 2 1 1 2 2 2  1 1 2 1 1 2 2 2 3  and we have found the first 3, at the 9th term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' After a while, a 4 appears at term 220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  But Gijswijt was unable to find a 5, and left that question open when he submitted  the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Some Bell Labs colleagues computed many millions of terms, but no 5  appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Finally, over the course of a long weekend, Fokko van der Bult (a fellow student of  Gijswijt’s in Amsterdam) and I independently showed that there is a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' In fact there  are infinitely many 5’s, but the first one does not appear until about term 101023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The  universe would be cold long before any computer search would find it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  In the paper we wrote about the sequence [4], we also conjectured that the first time  a number N > 4 appears is at about term  2 ↑ (2 ↑ (3 ↑ (4 ↑ (5 ↑ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' ↑ (N − 1))))),  where the up-arrows (↑) indicate exponentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' This is a tower of exponents of height  N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  A very recent manuscript by a student of Gijswijt’s, Levi van de Pol, still under  review, has extended our work, and may have proved the above conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  I cannot resist adding a further comment about curling numbers, which if true shows  that the Gijswijt sequence is in a sense universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  The Curling Number Conjecture asserts that if any finite starting sequence is ex-  tended by the rule that the next term is the curling number of the sequence so far, then  eventually the curling number will be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  If true, this implies that if the starting sequence contains no 1s, then the sequence  eventually becomes Gijswijt’s sequence [5, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' In fact I conjecture that this is true  for any starting sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Open question: Is the Curling Number Conjecture true?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  15  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='4  Lexicographically Earliest Sequences  Although there is no space to discuss them in detail, let me just mention that there  are many fascinating and difficult sequences in the OEIS whose definition has the form  “Lexicographically Earliest Sequence of distinct positive numbers with the property that  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..”, where now we are using lexicographic in its pure sense, as defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  A favorite example is the EKG (or ECG) sequence A064413, whose definition is the  lexicographically earlier infinite sequence of distinct positive numbers with the property  that each term after the first has a nontrivial common factor with the previous term [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Other L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' examples are the Yellowstone permutation A098550 [2], the Enots Wolley  sequence A336957 (the name suggests the definition), and the Binary Two-Up sequence  A354169 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Open question: Show that the terms of the Enots Wolley sequence are precisely 1,  2, and all numbers with at least two distinct prime factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='5  The Stepping Stones Problem (A337663)  This lovely problem was invented in 2020 by two undergraduates, Thomas Ladouceur  and Jeremy Rebenstock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' You have an infinite chessboard, and a handful of brown stones,  which are worth one point each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' You also have an infinite number of white stones, of  values 2, 3, 4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', one of each value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Suppose you have n brown stones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' You start by  placing them anywhere on the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Now you place the white stones, trying to place  as many as you can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The rules are that you can only place a white stone labeled k on  a square if the values of the stones on the eight squares around it add up to k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' And  you must place the white stones in order, first 2, then 3, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' You stop when  you cannot place the next higher-numbered white stone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The goal is to maximize the  highest value that you place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Call this a(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  9  5  10  11  4  1  12  8  3  2  16∗  6  1  15  13  7  14  Figure 6: A solution to the Stepping Stones problem for two starting stones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The high  point a(2) = 16 here is indicated by an asterisk, as it is in the next three tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Say we start with n = 2 brown stones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' There are infinitely many squares where they  can be placed, but it turns out that the best thing is to place them so they are separated  diagonally by a single blank square, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Now we start trying to place the white  16  stones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The 2 stone has to go between the two brown (or 1) stones, and then the 3 goes  on a square adjacent to the 1 and the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' There is now a choice for where the 4 goes, but  the choice shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 6 is the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' (After we have placed the 4, the neighbors of the  3 no longer add up to 3, but that is OK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' It is only when we place the 3 that its neighbors  must add to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=') Continuing in this way, we eventually reach 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' There is nowhere to  place the 17, so we stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Ladouceur and Rebenstock showed, using a computer and  considering all possible arrangements, that 16 is the highest value that can be attained  with two starting stones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' So a(2) = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  This is clearly a hard problem, since the number of possibilities grows rapidly with  the number of brown stones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Only six terms of this sequence are known: a(1) through  a(6) are 1,16,28,38,49,60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' A solution for n = 4 found by Arnauld Chevallier is shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' There are lower bounds for larger values of n which may turn out to be optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  For n = 7,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=',10 the current best constructions give 71,80,90,99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' See A337663 for the  latest information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  35  18  36  23  21  32  17  1  14  9  12  20  34  16  15  5  4  8  26  27  31  10  1  3  19  25  1  28  11  2  6  33  29  24  13  22  1  7  37  30  38∗  Figure 7: A solution to the Stepping Stones problem for four starting stones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  We don’t know how fast a(n) grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' There have been a series of upper and lower  bounds, initiated by Robert Gerbicz and Andrew Howroyd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The simple linear construc-  tion shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 8 shows that a(n) ≥ 6(n − 1) for n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  1  2  3  4  5  6  7  8  9  1  1  1  10  18∗  17  16  15  14  13  12  11  Figure 8: Every additional 1 on the middle row increases the number of white stones  by 6, showing that a(n) ≥ 6(n − 1) for n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  17  By combining the constructions of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 6 and 8, Menno Verhoeven obtained a(n) ≥  6n + 3 for n ≥ 3 (Table 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  25  24  1  26  23  27  22  28  21  1  29  20  30  19  31  9  5  10  11  18  1  32  4  1  17  33∗  12  8  3  2  16  6  1  15  13  7  14  Figure 9: Combining the constructions of of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 6 and 8 gives a(n) ≥ 6n + 3 for n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  The case n = 5 is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' For other values of n, adjust the height of the “chimney” on  the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  The best lower bound for large n is due to Robert Gerbicz, who has shown by a  remarkable extension of the construction in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 8 and 9 that limn→∞ a(n)/n > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' (A  preliminary version of his bound gives a(n) > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='0128n−5621 for all n, although the exact  values of the constants have not been confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=') In his construction the “chimney” on  the right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 9 gets expanded into a whole trellis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  One might think that with a sufficiently clever arrangement, perhaps extending the  construction in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 8 so that the path wraps around itself in a spiral, one could achieve  large numbers with only a few starting stones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' But a simple counting argument due  to Robert Gerbicz shows this is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The current best upper bound is due to  Jonathan F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Waldmann, who has shown that a(n) < 79n + C for some constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' See  A337663 for the latest information, including proofs of of the results mentioned here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Open question: Improve the upper and lower bounds on a(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='6  Stained glass windows  In 1998 Poonen and Rubinstein [12] famously determined the numbers of vertices and  cells in the planar graph formed from a regular n-gon by joining every pair of vertices  by a chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' The answers are in A006561 and A007678.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Lars Blomberg, Scott Shannon,  and I have studied versions of this question when the regular n-gon is replaced by other  18  Figure 10: A 4×2 grid of squares with every pair of boundary points joined by a chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  The graph has 213 vertices and 296 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  The cells are color-coded to distinguish  triangles (red), quadrilaterals (yellow), and pentagons (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  polygons, for instance by a square in which n equally-spaced points are placed along each  side and each pair of boundary points is joined by a chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' We also studied rectangles,  triangles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' In most cases we were unable to find formulas for the numbers of vertices  or cells, but we collected a lot of data, and the graphs, when colored, often resemble  stained glass windows (see [3] and the illustrations in A331452 and other sequences  cross-referenced there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='7 So we consoled ourselves with the motto: if we can’t solve it,  make art!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  The most promising case to analyze seemed to be the n × 2 grid (although we did  not succeed even there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Open question: How many vertices and cells are there in the graph for the n×2 grid,  7There is no fee for downloading images in the OEIS, but if you use any of them, please credit the  source!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  19  as illustrated for n = 4 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 10?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Sequences A331763 and A331766 give the first 100  terms, yet even with all that data we have not found a formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  The case of an n × n grid seems even harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Figure 11 shows the 6 × 6 graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Sequences A331449 and A255011 give the numbers of vertices and cells for n ≤ 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  A334699 enumerates the cells by number of sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Figure 11: A 6×6 grid with every pair of boundary points joined by a chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' There are  4825 vertices and 6264 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  In the summer of 2022 Scott Shannon and I considered several other families of planar  graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' I cannot resist showing one of Shannon’s graphs, a 16 × 16 grid, illustrating the  16th term of A355798 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' There are 61408 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Although Shannon has calculated  40 terms of this sequence, again no formula is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  20  Figure 12: Scott Shannon’s “Magic Carpet” graph, illustrating A355798(16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='7  If I had more space  If I had had more space I would also have discussed some very interesting sequences  arising from:  – Dissecting a square to get a regular n-gon (A110312).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  – Gerrymandering (A341578, A348453, and many others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  – In how many ways can circles overlap?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' (A250001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  – The Inventory sequence A342585.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  – Kaprekar’s junction numbers (A006064, [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  – The kissing number problem (A001116, A257479).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  – The neural network problem that started it all (A000435).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  – Squares in the plane (A051602).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  And, maybe, meta-sequences such as A051070 (a(n) is the nth term of An) and  A107357 (the nth term is 1 + the nth term of An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  21  A final comment: there are many videos on the Internet of talks I have given about  sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' There are over twenty videos that Brady Haran and I have made that have  appeared on the Youtube Numberphile channel (and have been viewed over eight million  times).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' See for example “Terrific Toothpick Patterns”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  5  Acknowledgments  I would like to thank some good friends who have helped me and the OEIS over the  years: David L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Applegate, William Cheswick, Russ Cox, Susanna S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Cuyler, Harvey  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Dale, Ronald L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Graham, Richard K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Guy, Marc LeBrun, John Riordan, and Doron  Zeilberger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  There are many active volunteer editors, and it is impossible to thank them all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  But I would like to give particular thanks to J¨org Arndt, Michael S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Branicky, Michael  De Vlieger, Amiram Eldar, Charles R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Greathouse IV, Maximilian F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Hasler, Alois P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Heinz, Andrew Howroyd, Sean A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Irvine, Michel Marcus, Richard J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Mathar, Peter  Munn, Hugo Pfoertner, Kevin Ryde, Jon E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Schoenfield, R´emy Sigrist, and Chaiwah  Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  I also thank the members of the Board of Trustees of the OEIS Foundation, past  and present, for all their help, both to me personally and to the OEIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  Figure credits: Figure 1(c): Clifford A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Pickover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Figure 5: Edmund Harriss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Fig-  ures 6, 7, 8, and 9 are based on communications from Thomas Ladouceur and Jeremy  Rebenstock, Arnauld Chevallier, Skylark Xentha Murphy-Davies, and Menno Verho-  even, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Figures 10 and 11: Lars Blomberg and Scott Shannon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Figure 12:  Scott R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Shannon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Other figures: the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  References  [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Alekseyev and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
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+page_content=' Sloane, On Kaprekar’s Junction Numbers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Com-  binat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' and Number Theory, 2023, in press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Applegate, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Havermann, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Selcoe, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Shevelev, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Sloane, and  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Zumkeller, The Yellowstone permutation, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Integer Seqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=', 18 (2015), #15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  [3] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Blomberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Shannon, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Sloane, Graphical enumeration and  stained glass windows, 1: Rectangular grids, Integers, Ron Graham Memorial Vol-  ume 21A (2021), #A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  [4] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' van de Bult, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Gijswijt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Linderman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Sloane, and Allan  Wilks, A slow-growing sequence defined by an unusual recurrence, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Integer Seqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=',  10 (2007), #07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content='  [5] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' Chaffin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQfZQTY/content/2301.03149v1.pdf'}
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diff --git a/ONE2T4oBgHgl3EQfrAiX/content/tmp_files/2301.04045v1.pdf.txt b/ONE2T4oBgHgl3EQfrAiX/content/tmp_files/2301.04045v1.pdf.txt
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+Experimental characterization of an ultra-broadband dual-mode 
+symmetric Y–junction based on metamaterial waveguides 
+Raquel Fernández de Caboa,*, Jaime Vilasa,b, Pavel Chebenc, Aitor V. Velascoa, David González-
+Andraded 
+a Instituto de Óptica Daza de Valdés, Consejo Superior de Investigaciones Científicas (CSIC), 121 Serrano, Madrid 28006, Spain 
+b Alcyon Photonics S.L., 11 Génova, Madrid 28004, Spain 
+c National Research Council Canada, 1200 Montreal Road, Bldg. M50, Ottawa K1A 0R6, Canada 
+d Centre de Nanosciences et de Nanotechnologies, CNRS, Université Paris-Saclay, Palaiseau 91120, France 
+* Corresponding author: r.fernandez@csic.es 
+ARTICLE INFO 
+ 
+Keywords:  
+Silicon photonics 
+Subwavelength grating 
+metamaterial 
+Power splitter  
+Y-junction 
+Ultra-broadband 
+Fabrication tolerant 
+ 
+ 
+ 
+ 
+ 
+ 
+ABSTRACT
+ 
+Silicon photonic integrated circuits routinely require 3-dB optical power dividers with 
+minimal losses, small footprints, ultra-wide bandwidths, and relaxed manufacturing 
+tolerances to distribute light across the chip and as a key building block to form more 
+complex devices. Symmetric Y–junctions stand out among other power splitting 
+devices owing to their wavelength-independent response and a straightforward design. 
+Yet, the limited resolution of current fabrication methods results in a minimum feature 
+size (MFS) at the tip between the two Y–junction arms that leads to significant losses 
+for the fundamental mode. Here we propose to circumvent this limitation by leveraging 
+subwavelength metamaterials in a new type of ultra-broadband and fabrication-
+tolerant Y–junction. An exhaustive experimental study over a 260 nm bandwidth 
+(1420–1680 nm) shows excess loss below 0.3 dB for the fundamental transverse-
+electric mode (TE0) for a high-resolution lithographic process (MFS ~ 50 nm) and less 
+than 0.5 dB for a fabrication resolution of 100 nm. Subwavelength Y–junctions with 
+deterministically induced errors of ±10 nm further demonstrated robust fabrication 
+tolerances. Moreover, the splitter exhibits excess loss lower than 1 dB for the first-
+order transverse-electric mode (TE1) within a 100 nm bandwidth (1475–1575 nm), 
+using high-resolution lithography.  
+1. Introduction 
+Photonic integrated circuits (PICs) built on the silicon-
+on-insulator (SOI) platform benefit from high modal 
+confinement, small footprints, energy efficiency and 
+large-scale production, whilst driving-down costs thanks 
+to the compatibility with complementary metal-oxide-
+semiconductor (CMOS) fabrication processes [1–3]. 
+These compelling advantages substantially extend the 
+scope of photonic integration beyond telecom and 
+datacom to emerging applications with a far-reaching 
+impact. These include 5G mobile communications [4], 
+the Internet of Things [5], quantum photonics [6], light 
+detection and ranging [7], spectrometry [8,9] and 
+biochemical sensing [10], also enabling lab-on-a-chip 
+solutions [11]. 
+The complexity leap of the aforementioned 
+applications requires an increasing number of on-chip 
+components that take advantage of either multimode or 
+broadband operations. Specifically, 3-dB optical power 
+splitters are key components extensively used in light 
+distribution or as building blocks for more intricate 
+arrangements, including optical switches, multiplexers 
+or integrated Mach-Zehnder interferometers [12,13]. 
+Sequentially concatenated 3-dB power splitters are often 
+utilized to implement 1×N dividers [14,15], requiring 
+compact and low-loss designs. For datacom applications 
+such as mode-division multiplexing [16] or multitarget 
+sensing [17], power splitters with broad bandwidths are 
+required. Different power division architectures have 
+been reported based, among others, on symmetric Y–
+junctions [18,19], multimode interference (MMI) 
+couplers [20–22], inverse tapers [23], adiabatic tapers 
+[24], directional and adiabatic couplers [25–27], slot 
+waveguides [28] and photonic crystal structures [29–31]. 
+MMI devices offer good fabrication tolerances and 
+compact footprints, and their operational bandwidth can 
+be optimized through geometry design [20,21] or 
+partially shallowly etched regions [22]. Inverse tapers 
+[23] provide efficient mode evolution but typically 
+present narrow bandwidths. Adiabatic tapers [24] exhibit 
+wider bandwidths, but the performance of transverse-
+electric (TE) polarized light has larger fluctuations across 
+the operation bandwidth due to the higher sensitivity to 
+sidewall roughness. Conversely, the bandwidth of 
+conventional directional couplers is restricted due to the 
+strong wavelength-dependence of the evanescent 
+coupling. Asymmetric [25] and bent [26] directional 
+
+couplers have been demonstrated with reduced 
+wavelength dependence, but it is not sufficient for ultra-
+broadband optical systems. Additional power division 
+architectures with low losses over extensive operational 
+bandwidth also include adiabatic couplers [27], slot 
+waveguides [28] photonic crystal power splitters [29–
+31], inverse design methods [32,33], pixelated meta-
+structures [34], and deep neural networks [35].  
+Amongst these power splitters, symmetric Y–
+junctions are one of the most common alternatives given 
+their 
+polarization- 
+and 
+wavelength-independent 
+response, and their straightforward design. Symmetric 
+Y–junctions consist of a stem waveguide ramifying into 
+two arms of the same width. However, these structures 
+typically present significant loss for fundamental modes 
+at the junction, especially as the tilt angle between the 
+two branching arms increases [36]. Two basic 
+mechanisms are responsible for this loss: the wavefront 
+mismatch due to the abrupt tilt angle, and the 
+transformation of the mode profile in the interface 
+between the stem and the arms [37]. In order to reduce 
+the effect of the tilt angle, s-bend shaped waveguides can 
+be employed for the arms [38], whereas the junction 
+region can be tapered to ensure an approximately 
+adiabatic mode evolution. Despite these design 
+optimizations, minimum feature size (MFS) limitations 
+of fabrication technologies lead to an imperfect tip 
+between the two Y–junction arms. This MFS constraint 
+causes considerable losses on the fundamental mode as 
+the maximum of its power profile coincides with the 
+junction tip [39]. Conversely, this same effect is 
+negligible for the first-order mode, which presents a 
+minimum of its power profile at the waveguide center. 
+Several approaches to optimized Y–junction designs 
+have been proposed to circumvent the effect of the MFS 
+at the tip, including tapered and slotted waveguides 
+[19,27,28], core size optimization [40] or particle swarm 
+optimization algorithms [18]. Nevertheless, ultra-
+broadband, low-loss and fabrication-tolerant solutions 
+are still sought after. 
+Subwavelength gratings (SWG) provide a powerful 
+tool for improving the performance of a wide range of 
+photonic devices [41–45]. SWGs are based on periodic 
+grating waveguides with a period (Λ) significantly 
+smaller than the operating wavelength (λ), hence 
+behaving as a homogenous metamaterial (i.e. a medium 
+with tailored optical properties). This behavior inhibits 
+diffraction and enables refractive index and dispersion 
+engineering [44]. Subwavelength engineering has 
+become a strong design method for the realization of 
+integrated silicon photonics components, including  
+fiber-chip couplers [41], reconfigurable filters [46] and 
+gradient-index lenses [47]. SWG metamaterials have 
+also been successfully applied to several power splitters 
+such as asymmetric directional couplers [48], three-guide 
+directional couplers [49], inverse tapers [45,50], slot 
+adiabatic waveguides [51], MMI devices [52–54], 
+waveguide crossings [55] and Y–junctions [56]. 
+Recently, we proposed an architecture for a high-
+performance and fabrication-tolerant SWG Y–junction 
+[57]. However, this previous study only covered 
+simulation results and preliminary measurements for 
+TE0. 
+In this paper, we present a comprehensive 
+experimental study of a 
+dual-mode Y–junction 
+engineered with subwavelength metamaterials for 
+deconfinement of the fundamental mode near the 
+junction tip and mitigating losses. Specifically, we 
+greatly expand our original study [57] by including 
+accurate TE0 mode measurements through cascaded 
+splitters and TE1 mode measurements through auxiliary 
+mode multiplexers, for both SWG and conventional Y-
+junctions. We also measure biased devices in order to 
+study the fabrication tolerances of our device considering 
+both 50 nm and 100 nm MFSs. Our power splitter yields 
+a measured fundamental transverse-electric mode (TE0) 
+excess loss (EL) of less than 0.3 dB considering a high-
+resolution fabrication process with MFS of 50 nm and 
+below 0.5 dB for a resolution scenario with MFS of 100 
+nm within the 1420–1680 nm wavelength range. 
+Moreover, the splitter exhibits excess loss lower than 1 
+dB for the first-order transverse-electric mode (TE1) 
+within a 100 nm bandwidth (1475–1575 nm), in the high-
+resolution process. 
+2. Device design 
+Our proposed SWG Y–junction, presented in Fig. 1(a), 
+operates according to similar principles as a conventional 
+symmetric Y–junction, illustrated in Fig. 2(a). The 
+structure of the SWG and conventional Y–junctions 
+comprise two single-mode output arms and a multimode 
+input stem waveguide that supports the first two TE 
+modes. When TE0 is injected into the stem waveguide, 
+its power is equally divided into two in-phase TE0 modes, 
+one in each output arm. When the stem waveguide is 
+excited with TE1, two TE0 modes of equal power but with 
+a relative phase difference of π are generated in the two 
+output arms [39]. Therefore, notice that the Y-junction 
+operates simultaneously as a splitter and a mode 
+converter for TE1 mode. In the conventional symmetric 
+Y–junction, MFS limitations at the junction tip results in 
+significant TE0 loss penalty. By applying SWG 
+engineering, modal confinement near the junction tip is 
+reduced and the TE0 mode transition at the stem-arm 
+interface is smoothed, significantly reducing losses. 
+Both splitters were designed for a 220-nm-thick Si 
+core. Our SWG Y–junction includes an input strip 
+waveguide of width Ws = 1.2 m, optimized to avoid a 
+weak confinement of the Bloch–Floquet TE1 mode that 
+would lead to high TE1 excess losses (ELTE1) due to 
+substrate leakage or mode radiation. Using an adiabatic 
+input taper of length Lti = 10 m, the strip waveguide is 
+progressively adapted to a subwavelength stem 
+waveguide of the same width and a length Lc = 13 m. 
+The SWG stem splits into two SWG single-mode arms 
+of width W = 500 nm and length Lb = 12.3 m. Each 
+SWG arm is shaped as an s-bend to avoid abrupt tilt 
+angles at the junction tip. After the SWG s-bends, each 
+arm waveguide transforms into a strip output waveguide 
+by means of an output taper of length Lto = 6 m. The 
+
+final separation between arms (Ha) is 1.5 m. An initial 
+arm offset (Hoff) is included to account for two alternative 
+MFS scenarios. We considered an MFS of 50 nm for 
+high-resolution fabrication processes and an MFS of 100 
+nm.  
+In both SWG arms and stem, Λ was set to 220 nm to 
+prevent Bragg reflection while maintaining SWG feature 
+sizes above the target MFS. In order to minimize mode 
+mismatch at the junction, we set different duty cycles in 
+the stem (DCs =as/Λ) and the arms (DCa=aa/Λ), where 
+as and aa are the lengths of the SWG silicon segments in 
+the stem and arms, respectively. For DCs = 0.5, the best 
+EL balance between TE0 and TE1 modes was achieved 
+for DCa = 0.6 when considering an MFS of 50 nm and 
+for DCa = 0.55 with MFS of 100 nm. The design 
+procedure followed is described in further detail in [57]. 
+Notice that fabrication tolerances and the effect of 
+temperature variations was also studied, showing 
+negligible performance degradation for ±10 K. Figures 
+1(b) and 1(c) show scanning electron microscope (SEM) 
+images of the fabricated SWG Y–junctions for MFS 50 
+nm and 100 nm, respectively.  
+In order to compare the performance of our SWG 
+device with a conventional splitter, we designed 
+conventional Y–junctions with different MFSs (Fig. 
+2(a)). These devices also comprise an input multimode 
+stem waveguide of width W0 = 1 m, which is adapted to 
+the width at the junction (Wt = 2W + Hoff) by means of a 
+taper of length Lt = 4 m. The two single-mode s-shaped 
+output arms maintain the same arm width (W), length (Lb) 
+and final separation (Ha) as in the SWG splitter. The 
+initial offset between the arms (Hoff) was also included to 
+consider the MFSs of 50 nm and 100 nm. Additionally, 
+we also included a third study-case with Hoff = 0 nm, 
+which ideally would result in a perfect tip. SEM images 
+of the fabricated conventional Y–junctions for Hoff of 0 
+nm, 50 nm and 100 nm can be seen in Figs. 2(b), 2(c) and 
+2(d), respectively. It can be observed that conventional 
+Y–junctions with Hoff = 0 nm and Hoff = 50 nm, as 
+fabricated, present comparable tip dimensions. This 
+leads to almost analogous performance for both devices, 
+as discussed in more detail in the following sections. 
+ 
+Fig. 2. Conventional Y–junction schematic (a), and SEM images of the tip for three resolutions:  ideal 0 nm (b), and realistic 50 nm (c) and 100 nm (d). 
+  
+ 
+Fig. 1. Subwavelength Y–junction schematic (a) and SEM images for devices with an MFS of 50 nm (b) and 100 nm (c). 
+ 
+
+TEi
+TE
++
+W
+(b)
+(c)
+(d)
+462nm
+468nm
+455nm
+947nm
+956nm
+42nm
+993nm
+nm
+200nm
+200nm
+200nmTEi
+TE0
+(b)
+C
+114nm
+134nm
+117nm
+127nm
+493nm
+486nm
+100nm
+200nm
+1157nm
+200nm
+L1163nm3. Fabrication and experimental characterization 
+The device was fabricated using electron-beam 
+lithography in a commercial foundry [58]. The SOI wafer 
+has a silicon layer thickness of 220 nm and a 2-µm-thick 
+buried oxide (BOX). The mask pattern was defined by 
+exposing the resist to a 100 keV electron-beam 
+lithography system, followed by an anisotropic reactive 
+ion etching process that transfers the pattern to the Si 
+layer. A SiO2 cladding with a thickness of 2.2 µm was 
+deposited by chemical vapor deposition. Finally, a deep 
+etch process was applied to smooth the chip facets, 
+allowing efficient fiber-chip edge coupling by using 
+high-efficiency broadband SWG edge couplers [41]. 
+Experimental characterization was carried out using 
+two tunable lasers to sweep the wavelength from 1420 
+nm to 1680 nm, coupled to a three-paddle fiber 
+polarization controller, a linear polarizer and a half-wave 
+plate. TE polarized light was coupled into the chip using 
+a lensed polarization-maintaining optical fiber. Light at 
+the chip output was collected by a 40× microscope 
+objective, directed to a Glan-Thompson polarizer, and 
+focused onto a germanium photodetector. 
+3.1 Fundamental transverse-electric mode (TE0) 
+In [57], 3D finite-difference time-domain simulations 
+were performed for the SWG Y-junction, which 
+provided excess loss for the TE0 (ELTE0) mode below 0.3 
+dB in a 350 nm bandwidth for the worst-case MFS 
+scenario of 100 nm. Given the challenge of measuring 
+losses of this order of magnitude in a stand-alone 
+configuration, we implemented cascaded structures with 
+multiple stages. That is, 1 to 4 stages of concatenated Y–
+junctions: 1×2, 1×4, 1×8 and 1×16 structures. SEM 
+image of a 1×16 structure and the different stages are 
+shown in Fig. 3(a), with a close-up view of the SWG Y–
+junction in Fig. 3(b). The 50 nm MFS is within the limit 
+of the fabrication resolution offered by the foundry and, 
+as it can be seen in the SEMs, the device was fabricated 
+correctly. Reference waveguides with the same length 
+and number of bends as the cascaded structures were also 
+included to determine Y–junction excess loss. 
+The measured excess loss for different cascaded Y–
+junctions is presented in Fig. 4. The ELTE0 of the SWG 
+splitter designed for MFS = 50 nm (𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,50) is plotted in 
+Fig. 4(a) and for MFS = 100 nm (𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,100) in Fig. 4(c). 
+The ELTE0 of the conventional Y–junction is shown in 
+Fig. 4(b) for MFS = 50 nm (𝐸𝐿𝑇𝐸0
+𝐶𝑜𝑛𝑣,50) and for MFS = 100 
+nm (𝐸𝐿𝑇𝐸0
+𝐶𝑜𝑛𝑣,100) in Fig. 4(d). Conventional Y–junctions 
+with Hoff = 0 nm were also measured but have not been 
+included in Fig. 4 for clarity, since they are very similar 
+to results with Hoff = 50 nm. This similarity is caused by 
+the MFS limitation of the fabrication process and verifies 
+our initial assumption on e-beam fabrication MFS. That 
+is, even when the nominal design includes a perfect tip 
+(Hoff = 0), experimental MFS limitations will induce an 
+imperfect tip, in this case, similar to the design of Hoff = 
+50 nm. Notwithstanding, results for this Hoff are included 
+in Figs. 5 and 8 for direct comparison. For both MFSs of 
+50 nm and 100 nm, the SWG splitter has a substantially 
+reduced ELTE0 compared to the conventional Y–junction.  
+Figure 4 shows the overall excess loss increment as 
+more cascaded stages are included. To obtain the ELTE0 
+relative to a single Y-junction, we performed a linear 
+regression with the measured losses in each cascaded 
+stage. Figure 5 shows the resulting ELTE0 per splitter, that 
+is, the average loss per cascaded splitter calculated 
+through the slope of the linear regression. The SWG Y–
+junction (solid line) exhibits a flat response in the entire 
+measured bandwidth from 1420 nm to 1680 nm, with an 
+𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,50 lower than 0.3 dB and 𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,100 below 0.5 dB in 
+the full 260 nm spectrum. By contrast, the conventional 
+Y–junction 
+(dotted 
+line) 
+shows 
+a 
+performance 
+degradation towards shorter wavelengths, resulting in a 
+loss penalty, especially for an MFS of 100 nm (see Fig. 
+5(b)). Considering the high-resolution fabrication (MFS 
+ 
+Fig. 4. Excess loss of the 1×2, 1×4, 1×8 and 1×16 cascaded structures, 
+for SWG Y–junction with 50 nm MFS (a), conventional Y–junction 
+with 50 nm MFS (b), SWG Y–junction with 100 nm MFS (c), and 
+conventional Y–junction with 100 nm MFS (d). 
+ 
+Fig. 3. SEM images of the SWG Y–junction in cascaded configuration 
+1×16 (a) and inset of the device (b). 
+
+SWG.MFS = 50
+over
+3
+1x2
+1x2
+2.5
+1x4
+2.5
+1x4
+1x8
+1x8
+2
+1x16
+2
+1x16
+1.5
+(aB)
+1.5
+n
+1
+EL
+1
+0.5
+0.5
+0
+0
+-0.5
+0.5
+1420
+1485
+1550
+1615
+1680
+1420
+1485
+1550
+1615
+1680
+Wavelength(nm)
+Wavelength (nm)
+6
+6
+1x2
+1x2
+5
+1x4
+1x4
+1x8
+1x8
+1x16
+1x16
+4
+(dB)
+dB)
+m
+73
+2
+0
+0
+1420
+1485
+1550
+1615
+1680
+1420
+1485
+1550
+1615
+1680
+Wavelength(nm)
+Wavelength(nm)(a)
+20μm
+4cascaded stages
+3cascaded stages
+2 cascaded stages
+istage
+(b)
+3μm= 50 nm), our device consistently outperforms the 
+conventional Y–junction, with 𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,50 < 0.23 dB in a 
+215 nm bandwidth (1420 nm – 1635 nm) as presented in 
+Fig. 5(a). As previously explained, it can be seen that 
+excess losses for conventional Y-junctions with an Hoff = 
+0 nm and Hoff = 50 nm are very similar, only differing in 
+a mean deviation of less than 0.06 dB. As the MFS 
+increases to 100 nm, the negative impact on the 
+performance of the conventional Y–junction is more 
+pronounced, with an 𝐸𝐿𝑇𝐸0
+𝐶𝑜𝑛𝑣,100 above 0.57 dB in the 1420 
+nm –1680 nm window. The SWG device, on the other 
+hand, yields improved performance over the entire 
+measured bandwidth with 𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,100 below 0.46 dB.   
+Devices with deterministically induced dimension 
+variations (Δδ) of ±10 nm were incorporated in the mask 
+layout to measure the robustness of the SWG Y–junction 
+to fabrication errors. Figure 6(a) shows 𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,50, 
+demonstrating that performance is preserved despite the 
+presence of geometric variations, and even exhibiting a 
+slight improvement for over-etching deviations (i.e., Δδ 
+= -10 nm, SEM shown in Fig. 6(c)). In contrast, for the 
+MFS of 100 nm (Fig. 6(b)), 𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,100 is slightly improved 
+for Δδ = +10 nm (SEM shown in Fig. 6(f)). For Δδ = -10 
+nm (SEM shown in Fig. 6(d)), 𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,100performance 
+degrades towards longer wavelengths. The largest 
+fabrication bias was observed in waveguide width, 
+narrowing the designed stem waveguide (Ws = 1200 ± 10 
+nm) by approximately 40 nm.  
+3.2 First-order transverse-electric mode (TE1) 
+In order to characterize the TE1 mode division, a mode 
+multiplexer [59] was included in combination with the 
+SWG Y–junction, as schematically shown in Fig. 7. 
+When TE0 is injected through the upper input port of the 
+mode multiplexer, the TE0 mode is generated at the 
+output multimode waveguide. When the lower input port 
+is excited with TE0, mode evolution results in TE1 at the 
+mode multiplexer output. Two mode multiplexers in 
+back-to-back were used as reference to extract TE1 mode 
+excess loss of the Y–junctions.  
+Figure 8(a) shows ELTE1 measurements for MFS = 50 
+nm in both SWG (𝐸𝐿𝑇𝐸1
+𝑆𝑊𝐺,50) and conventional (𝐸𝐿𝑇𝐸1
+𝐶𝑜𝑛𝑣,50) 
+Y–junctions, as well as for the conventional Y–junction 
+with Hoff = 0 nm. Likewise, ELTE1 for MFS = 100 nm in 
+ 
+Fig. 6. Tolerances to fabrication errors of Δδ = ±10 nm for the SWG 
+Y–junction with MFS of 50 nm (a) and 100 nm (b). Excess loss per 
+splitter was measured through linear regression of cascaded stages. 
+SEM images of devices with Δδ = -10 nm for an MFS of 50 nm (c) 
+and 100 nm (d). SEM images of devices with Δδ = +10 nm for an MFS 
+of 50 nm (e) and 100 nm (f). 
+ 
+Fig. 7. Schematic of the structure employed for the characterization of TE1 mode (a) and SEM images of the mode multiplexer (b). 
+ 
+Fig. 5. EL per splitter measured through linear regression of the 
+response of four cascaded stages for SWG (solid line) and conventional 
+(dotted line) Y–junctions, for MFS of 50 nm (a) and 100 nm (b). Results 
+obtained for the conventional splitter with ideal resolution (MFS = 0 
+nm) are also shown in panel (a). 
+
+b)1.5
+1.5
+Nominal
+20=-10nm
+Nominal
+A=-10nm
+1
+40=+10nm
+1
+A0=+10nm
+【dB]
+0.5
+0.5
+0
+-0.5
+-0.5
+1420
+1485
+1550
+1615
+1680
+1420
+1485
+1550
+1615
+1680
+Wavelength(nm)
+Wavelength(nm)
+(C)
+103nm
+125nm
+483nm
+(d)
+105nm
+116nm
+-479nm
+56nm
+f107nm
+L1146nm
+200nm
+-1146nm
+200nm
+(e)
+123nm
+147nm
+507nm
+(f).124nm
+136m
+-497nm
+37nm
+f88nm
+L1168nm
+200nm
+1166nm
+200nmTEi
+TE
+(b)
+2μm
+15.16μm
+5.04μm
+0.50μm
+0.15μm7
+0.74μm
+0.15μmr
+0.25μm
+0.52μmConv MFS = O nm
+..
+Conv MFS = 50 nm
+SWG MFS =50 nm
+1
+(ap)
+0.5
+0
+1420
+1485
+1550
+1615
+1680
+Wavelength(nm)
+ConvMFS= 100 nm
+SWG MFS = 100 nm
+(ap)
+0.5
+0
+1420
+1485
+1550
+1615
+1680
+Wavelength(nm)SWG (𝐸𝐿𝑇𝐸1
+𝑆𝑊𝐺,100) and conventional (𝐸𝐿𝑇𝐸1
+𝐶𝑜𝑛𝑣,100) Y–
+junctions are depicted in Fig. 8(b). As previously 
+mentioned, 𝐸𝐿𝑇𝐸1
+𝐶𝑜𝑛𝑣,50 and 𝐸𝐿𝑇𝐸1
+𝐶𝑜𝑛𝑣,100 are negligible due to 
+TE1 mode odd symmetry, with a power minimum at the 
+center of the multimode stem. Our device exhibits an 
+𝐸𝐿𝑇𝐸1
+𝑆𝑊𝐺,50 below 1 dB over a 100 nm bandwidth ranging 
+from 1475 to 1575 nm. Within a 170 nm bandwidth 
+(1420 – 1590 nm), 𝐸𝐿𝑇𝐸1
+𝑆𝑊𝐺,50 and 𝐸𝐿𝑇𝐸1
+𝑆𝑊𝐺,100 only increase by 
+0.5 dB. Compared to the conventional Y–junction, the 
+degradation of the TE1 mode for our SWG Y–junction 
+arises from the selection of the stem waveguide width of 
+WS = 1200 nm, as a compromise between ELTE0 and 
+ELTE1. Note that increasing the width of the SWG stem 
+waveguide results in a stronger modal confinement that 
+prevents TE1 mode radiation, but increasingly penalizes 
+TE0 due to the resulting mode profile at the junction tip 
+[57]. Nevertheless, the performance of the proposed 
+device compares very favorably to state-of-the-art 
+higher-order mode power splitters (see Table 1).  
+SWG Y–junctions with Δδ = ±10 nm were also 
+included in combination with the mode multiplexer 
+structures to study fabrication tolerances for TE0 and TE1 
+mode division mux/demux and are shown in Figs. 9(a) 
+and (b). Figure 9(a) shows that 𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,50 is lower than 0.5 
+dB for both under-etching and over-etching errors in the 
+full 1420 nm – 1680 nm bandwidth. Figure 9(b) shows 
+that 𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,100 remain <1 dB for Δδ = -10 nm and <0.7 dB 
+for Δδ = +10 nm, for the same bandwidth. These results 
+corroborate the robustness of the SWG Y–junction to 
+fabrication errors for TE0 mode, shown in section 3.1 
+(see Figs. 6(a) and (b)). The performance for TE1 mode 
+division is more sensitive to fabrication errors in the 
+width of the SWG stem owing to the weaker confinement 
+of the Bloch-Floquet TE1 mode compared to the Bloch-
+Floquet TE0 mode. While under-etching results in 
+𝐸𝐿𝑇𝐸1
+𝑆𝑊𝐺,50 < 2.1 dB and 𝐸𝐿𝑇𝐸1
+𝑆𝑊𝐺,100 < 1.7 dB for the full 
+bandwidth, over-etching errors are negligible at shorter 
+wavelengths and increase towards longer wavelengths. 
+4. Conclusions 
+The detailed experimental study conducted in this work 
+demonstrates the broadband performance and relaxed 
+fabrication tolerances of SWG-based Y–junctions. Two 
+resolutions were investigated to account for different 
+fabrication processes, namely MFS of 50 nm and 100 
+nm. Accurate measurements in cascaded splitters 
+demonstrate excess loss for the fundamental TE mode 
+lower than 0.3 dB for MFS = 50 nm and below 0.5 dB 
+for MFS = 100 nm, in a 260 nm bandwidth (1420 nm – 
+1680 nm). Characterization of the first-order TE mode 
+was performed in combination with a mode multiplexer, 
+showing excess loss lower than 1 dB over a 100 nm 
+ 
+Fig. 8. ELTE1 measurements for SWG (solid) and conventional (dotted) 
+Y–junctions for MFS of 50 nm (a) and 100 nm (b). Conventional Y–
+junction (dashed) with MFS = 0 nm is also shown in panel (a). 
+ 
+Fig. 9. Tolerances to fabrication errors of the SWG Y–junction for 
+TE0 (dashed line) and TE1 (solid line) polarizations, for MFS of 50 nm 
+(a) and 100 nm (b). 
+Table 1. Experimental performance comparison of state-of-the-art multimode power dividers. 
+Ref. 
+Design method 
+EL (dB) 
+Bandwidth (nm) 
+MFS (nm) 
+Length (µm) 
+Functionality 
+[33] 
+Inverse 
+design 
+subwavelength 
+axisymmetric 
+1.5 
+60 
+30 
+2.88 
+2-mode splitter 
+[34] 
+Pixelated meta-structure 
+1.5 
+40 
+30 
+4.5 
+3-mode splitter 
+[60] 
+MMI coupler 
+0.76 
+60 
+1000 
+86.5 
+2-mode splitter 
+[61] 
+Tapered directional coupler 
+0.7 
+30 
+200 
+25 
+2-mode splitter 
+[62] 
+Densely-packed waveguide array 
+1 
+28 
+110 
+46 
+2-mode splitter 
+This work 
+SWG Y–junction 
+1 
+100 
+50 
+41.3 
+2-mode splitter & converter 
+This work 
+SWG Y–junction 
+1.5 
+170 
+100 
+41.3 
+2-mode splitter & converter 
+ 
+
+a
+2.5
+-Conv MFS = O nm
+...
+Conv MFS = 50 nm
+2
+SWG MFS = 50 nm
+B
+1.5
+1
+0.5
+0
+-0.5
+1420
+1470
+1520
+1570
+1620
+Wavelength (nm)
+b)
+2.5
+Conv MFS = 100 nm
+SWG MFS = 100 nm
+2
+1.5
+1
+0.5
+0
+-0.5
+1420
+1470
+1520
+1570
+1620
+Wavelength(n-TE
+2.5
+Nominal
+48=-10nm
+2
+14§=+10nm
+(dB)
+1.5
+0.5
+-0.5
+1420
+1470
+1520
+1570
+1620
+Wavelength (nm)
+TE
+TEi
+2.5
+Nominal
+145=-10nm
+40=+10nm
+B
+d
+0.5
+0
+-0.5
+1420
+1470
+1520
+1570
+1620
+Wavelength(nm)bandwidth (1475 nm – 1575 nm) for the 50 nm MFS. 
+SWG Y–junctions with deterministically induced errors 
+of -10 nm and +10 nm were also measured to analyze 
+resilience 
+to 
+over- 
+and 
+under-etching 
+errors. 
+Experimental results demonstrate robust fabrication 
+tolerances for the fundamental TE mode. Our SWG-
+engineered metamaterial Y–junction opens up promising 
+prospects for improving performance of diverse silicon 
+photonic integrated circuits where power splitters are 
+ubiquitous component, such as on-chip high bandwidth 
+communication systems and broadband spectroscopic 
+systems. 
+Credit authorship contribution statement 
+Raquel Fernández de Cabo: Software, Formal 
+Analysis, Validation, Data curation, Writing – original 
+draft. Jaime Vilas: Experimental validation, Writing – 
+review & editing. Pavel Cheben: Conceptualization, 
+Writing – review & editing. Aitor V. Velasco: 
+Conceptualization, Resources, Funding acquisition, 
+Supervision, Writing – review & editing. David 
+González-Andrade: Methodology, Supervision, Project 
+administration, Writing – review & editing. 
+Declaration of Competing Interest 
+The authors declare that they have no known competing 
+financial interests or personal relationships that could 
+have appeared to influence the work reported in this 
+paper.  
+Acknowledgements 
+This work has been funded by the Spanish Ministry of 
+Science 
+and 
+Innovation 
+(RTI2018-097957-B-C33, 
+RED2018-102768-T, 
+PID2020-115353RA-I00); 
+the 
+Spanish State Research Agency and the European Social 
+Fund 
+Plus 
+under 
+grant 
+PRE2021-096954; 
+the 
+Community of Madrid – FEDER funds (S2018/NMT-
+4326); the Horizon 2020 research and innovation 
+program under Marie Sklodowska-Curie grant No. 
+101062518; European Union – NextGenerationEU, 
+through the Recovery, Transformation and Resilience 
+Plan (DIN2020-011488). 
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf,len=1129
+page_content='Experimental characterization of an ultra-broadband dual-mode  symmetric Y–junction based on metamaterial waveguides  Raquel Fernández de Caboa,*, Jaime Vilasa,b, Pavel Chebenc, Aitor V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Velascoa, David González-  Andraded  a Instituto de Óptica Daza de Valdés, Consejo Superior de Investigaciones Científicas (CSIC), 121 Serrano, Madrid 28006, Spain  b Alcyon Photonics S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=', 11 Génova, Madrid 28004, Spain  c National Research Council Canada, 1200 Montreal Road, Bldg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' M50, Ottawa K1A 0R6, Canada  d Centre de Nanosciences et de Nanotechnologies, CNRS, Université Paris-Saclay, Palaiseau 91120, France  Corresponding author: r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='fernandez@csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='es  ARTICLE INFO  Keywords:  Silicon photonics  Subwavelength grating  metamaterial  Power splitter  Y-junction  Ultra-broadband  Fabrication tolerant  ABSTRACT  Silicon photonic integrated circuits routinely require 3-dB optical power dividers with  minimal losses, small footprints, ultra-wide bandwidths, and relaxed manufacturing  tolerances to distribute light across the chip and as a key building block to form more  complex devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Symmetric Y–junctions stand out among other power splitting  devices owing to their wavelength-independent response and a straightforward design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Yet, the limited resolution of current fabrication methods results in a minimum feature  size (MFS) at the tip between the two Y–junction arms that leads to significant losses  for the fundamental mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Here we propose to circumvent this limitation by leveraging  subwavelength metamaterials in a new type of ultra-broadband and fabrication-  tolerant Y–junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' An exhaustive experimental study over a 260 nm bandwidth  (1420–1680 nm) shows excess loss below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='3 dB for the fundamental transverse-  electric mode (TE0) for a high-resolution lithographic process (MFS ~ 50 nm) and less  than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5 dB for a fabrication resolution of 100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Subwavelength Y–junctions with  deterministically induced errors of ±10 nm further demonstrated robust fabrication  tolerances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Moreover, the splitter exhibits excess loss lower than 1 dB for the first-  order transverse-electric mode (TE1) within a 100 nm bandwidth (1475–1575 nm),  using high-resolution lithography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Introduction  Photonic integrated circuits (PICs) built on the silicon-  on-insulator (SOI) platform benefit from high modal  confinement, small footprints, energy efficiency and  large-scale production, whilst driving-down costs thanks  to the compatibility with complementary metal-oxide-  semiconductor (CMOS) fabrication processes [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  These compelling advantages substantially extend the  scope of photonic integration beyond telecom and  datacom to emerging applications with a far-reaching  impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' These include 5G mobile communications [4],  the Internet of Things [5], quantum photonics [6], light  detection and ranging [7], spectrometry [8,9] and  biochemical sensing [10], also enabling lab-on-a-chip  solutions [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  The complexity leap of the aforementioned  applications requires an increasing number of on-chip  components that take advantage of either multimode or  broadband operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Specifically, 3-dB optical power  splitters are key components extensively used in light  distribution or as building blocks for more intricate  arrangements, including optical switches, multiplexers  or integrated Mach-Zehnder interferometers [12,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Sequentially concatenated 3-dB power splitters are often  utilized to implement 1×N dividers [14,15], requiring  compact and low-loss designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' For datacom applications  such as mode-division multiplexing [16] or multitarget  sensing [17], power splitters with broad bandwidths are  required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Different power division architectures have  been reported based, among others, on symmetric Y–  junctions [18,19], multimode interference (MMI)  couplers [20–22], inverse tapers [23], adiabatic tapers  [24], directional and adiabatic couplers [25–27], slot  waveguides [28] and photonic crystal structures [29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  MMI devices offer good fabrication tolerances and  compact footprints, and their operational bandwidth can  be optimized through geometry design [20,21] or  partially shallowly etched regions [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Inverse tapers  [23] provide efficient mode evolution but typically  present narrow bandwidths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Adiabatic tapers [24] exhibit  wider bandwidths, but the performance of transverse-  electric (TE) polarized light has larger fluctuations across  the operation bandwidth due to the higher sensitivity to  sidewall roughness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Conversely, the bandwidth of  conventional directional couplers is restricted due to the  strong wavelength-dependence of the evanescent  coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Asymmetric [25] and bent [26] directional  couplers have been demonstrated with reduced  wavelength dependence, but it is not sufficient for ultra-  broadband optical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Additional power division  architectures with low losses over extensive operational  bandwidth also include adiabatic couplers [27], slot  waveguides [28] photonic crystal power splitters [29–  31], inverse design methods [32,33], pixelated meta-  structures [34], and deep neural networks [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Amongst these power splitters, symmetric Y–  junctions are one of the most common alternatives given  their  polarization-  and  wavelength-independent  response, and their straightforward design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Symmetric  Y–junctions consist of a stem waveguide ramifying into  two arms of the same width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' However, these structures  typically present significant loss for fundamental modes  at the junction, especially as the tilt angle between the  two branching arms increases [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Two basic  mechanisms are responsible for this loss: the wavefront  mismatch due to the abrupt tilt angle, and the  transformation of the mode profile in the interface  between the stem and the arms [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' In order to reduce  the effect of the tilt angle, s-bend shaped waveguides can  be employed for the arms [38], whereas the junction  region can be tapered to ensure an approximately  adiabatic mode evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Despite these design  optimizations, minimum feature size (MFS) limitations  of fabrication technologies lead to an imperfect tip  between the two Y–junction arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' This MFS constraint  causes considerable losses on the fundamental mode as  the maximum of its power profile coincides with the  junction tip [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Conversely, this same effect is  negligible for the first-order mode, which presents a  minimum of its power profile at the waveguide center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Several approaches to optimized Y–junction designs  have been proposed to circumvent the effect of the MFS  at the tip, including tapered and slotted waveguides  [19,27,28], core size optimization [40] or particle swarm  optimization algorithms [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Nevertheless, ultra-  broadband, low-loss and fabrication-tolerant solutions  are still sought after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Subwavelength gratings (SWG) provide a powerful  tool for improving the performance of a wide range of  photonic devices [41–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' SWGs are based on periodic  grating waveguides with a period (Λ) significantly  smaller than the operating wavelength (λ), hence  behaving as a homogenous metamaterial (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' a medium  with tailored optical properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' This behavior inhibits  diffraction and enables refractive index and dispersion  engineering [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Subwavelength engineering has  become a strong design method for the realization of  integrated silicon photonics components, including  fiber-chip couplers [41], reconfigurable filters [46] and  gradient-index lenses [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' SWG metamaterials have  also been successfully applied to several power splitters  such as asymmetric directional couplers [48], three-guide  directional couplers [49], inverse tapers [45,50], slot  adiabatic waveguides [51], MMI devices [52–54],  waveguide crossings [55] and Y–junctions [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Recently, we proposed an architecture for a high-  performance and fabrication-tolerant SWG Y–junction  [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' However, this previous study only covered  simulation results and preliminary measurements for  TE0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  In this paper, we present a comprehensive  experimental study of a  dual-mode Y–junction  engineered with subwavelength metamaterials for  deconfinement of the fundamental mode near the  junction tip and mitigating losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Specifically, we  greatly expand our original study [57] by including  accurate TE0 mode measurements through cascaded  splitters and TE1 mode measurements through auxiliary  mode multiplexers, for both SWG and conventional Y-  junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' We also measure biased devices in order to  study the fabrication tolerances of our device considering  both 50 nm and 100 nm MFSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Our power splitter yields  a measured fundamental transverse-electric mode (TE0)  excess loss (EL) of less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='3 dB considering a high-  resolution fabrication process with MFS of 50 nm and  below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5 dB for a resolution scenario with MFS of 100  nm within the 1420–1680 nm wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Moreover, the splitter exhibits excess loss lower than 1  dB for the first-order transverse-electric mode (TE1)  within a 100 nm bandwidth (1475–1575 nm), in the high-  resolution process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Device design  Our proposed SWG Y–junction, presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 1(a),  operates according to similar principles as a conventional  symmetric Y–junction, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' The  structure of the SWG and conventional Y–junctions  comprise two single-mode output arms and a multimode  input stem waveguide that supports the first two TE  modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' When TE0 is injected into the stem waveguide,  its power is equally divided into two in-phase TE0 modes,  one in each output arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' When the stem waveguide is  excited with TE1, two TE0 modes of equal power but with  a relative phase difference of π are generated in the two  output arms [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Therefore, notice that the Y-junction  operates simultaneously as a splitter and a mode  converter for TE1 mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' In the conventional symmetric  Y–junction, MFS limitations at the junction tip results in  significant TE0 loss penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' By applying SWG  engineering, modal confinement near the junction tip is  reduced and the TE0 mode transition at the stem-arm  interface is smoothed, significantly reducing losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Both splitters were designed for a 220-nm-thick Si  core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Our SWG Y–junction includes an input strip  waveguide of width Ws = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='2 \uf06dm, optimized to avoid a  weak confinement of the Bloch–Floquet TE1 mode that  would lead to high TE1 excess losses (ELTE1) due to  substrate leakage or mode radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Using an adiabatic  input taper of length Lti = 10 \uf06dm, the strip waveguide is  progressively adapted to a subwavelength stem  waveguide of the same width and a length Lc = 13 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  The SWG stem splits into two SWG single-mode arms  of width W = 500 nm and length Lb = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='3 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Each  SWG arm is shaped as an s-bend to avoid abrupt tilt  angles at the junction tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' After the SWG s-bends, each  arm waveguide transforms into a strip output waveguide  by means of an output taper of length Lto = 6 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' The  final separation between arms (Ha) is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' An initial  arm offset (Hoff) is included to account for two alternative  MFS scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' We considered an MFS of 50 nm for  high-resolution fabrication processes and an MFS of 100  nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  In both SWG arms and stem, Λ was set to 220 nm to  prevent Bragg reflection while maintaining SWG feature  sizes above the target MFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' In order to minimize mode  mismatch at the junction, we set different duty cycles in  the stem (DCs =as/Λ) and the arms (DCa=aa/Λ), where  as and aa are the lengths of the SWG silicon segments in  the stem and arms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' For DCs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5, the best  EL balance between TE0 and TE1 modes was achieved  for DCa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='6 when considering an MFS of 50 nm and  for DCa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='55 with MFS of 100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' The design  procedure followed is described in further detail in [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Notice that fabrication tolerances and the effect of  temperature variations was also studied, showing  negligible performance degradation for ±10 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Figures  1(b) and 1(c) show scanning electron microscope (SEM)  images of the fabricated SWG Y–junctions for MFS 50  nm and 100 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  In order to compare the performance of our SWG  device with a conventional splitter, we designed  conventional Y–junctions with different MFSs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' These devices also comprise an input multimode  stem waveguide of width W0 = 1 \uf06dm, which is adapted to  the width at the junction (Wt = 2W + Hoff) by means of a  taper of length Lt = 4 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' The two single-mode s-shaped  output arms maintain the same arm width (W), length (Lb)  and final separation (Ha) as in the SWG splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' The  initial offset between the arms (Hoff) was also included to  consider the MFSs of 50 nm and 100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Additionally,  we also included a third study-case with Hoff = 0 nm,  which ideally would result in a perfect tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' SEM images  of the fabricated conventional Y–junctions for Hoff of 0  nm, 50 nm and 100 nm can be seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 2(b), 2(c) and  2(d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' It can be observed that conventional  Y–junctions with Hoff = 0 nm and Hoff = 50 nm, as  fabricated, present comparable tip dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' This  leads to almost analogous performance for both devices,  as discussed in more detail in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Conventional Y–junction schematic (a), and SEM images of the tip for three resolutions:  ideal 0 nm (b), and realistic 50 nm (c) and 100 nm (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Subwavelength Y–junction schematic (a) and SEM images for devices with an MFS of 50 nm (b) and 100 nm (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  TEi  TE  +  W  (b)  (c)  (d)  462nm  468nm  455nm  947nm  956nm  42nm  993nm  nm  200nm  200nm  200nmTEi  TE0  (b)  C  114nm  134nm  117nm  127nm  493nm  486nm  100nm  200nm  1157nm  200nm  L1163nm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Fabrication and experimental characterization  The device was fabricated using electron-beam  lithography in a commercial foundry [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' The SOI wafer  has a silicon layer thickness of 220 nm and a 2-µm-thick  buried oxide (BOX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' The mask pattern was defined by  exposing the resist to a 100 keV electron-beam  lithography system, followed by an anisotropic reactive  ion etching process that transfers the pattern to the Si  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' A SiO2 cladding with a thickness of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='2 µm was  deposited by chemical vapor deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Finally, a deep  etch process was applied to smooth the chip facets,  allowing efficient fiber-chip edge coupling by using  high-efficiency broadband SWG edge couplers [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Experimental characterization was carried out using  two tunable lasers to sweep the wavelength from 1420  nm to 1680 nm, coupled to a three-paddle fiber  polarization controller, a linear polarizer and a half-wave  plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' TE polarized light was coupled into the chip using  a lensed polarization-maintaining optical fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Light at  the chip output was collected by a 40× microscope  objective, directed to a Glan-Thompson polarizer, and  focused onto a germanium photodetector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='1 Fundamental transverse-electric mode (TE0)  In [57], 3D finite-difference time-domain simulations  were performed for the SWG Y-junction, which  provided excess loss for the TE0 (ELTE0) mode below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='3  dB in a 350 nm bandwidth for the worst-case MFS  scenario of 100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Given the challenge of measuring  losses of this order of magnitude in a stand-alone  configuration, we implemented cascaded structures with  multiple stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' That is, 1 to 4 stages of concatenated Y–  junctions: 1×2, 1×4, 1×8 and 1×16 structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' SEM  image of a 1×16 structure and the different stages are  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 3(a), with a close-up view of the SWG Y–  junction in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' The 50 nm MFS is within the limit  of the fabrication resolution offered by the foundry and,  as it can be seen in the SEMs, the device was fabricated  correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Reference waveguides with the same length  and number of bends as the cascaded structures were also  included to determine Y–junction excess loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  The measured excess loss for different cascaded Y–  junctions is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' The ELTE0 of the SWG  splitter designed for MFS = 50 nm (𝐸𝐿𝑇𝐸0  𝑆𝑊𝐺,50) is plotted in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 4(a) and for MFS = 100 nm (𝐸𝐿𝑇𝐸0  𝑆𝑊𝐺,100) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  The ELTE0 of the conventional Y–junction is shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 4(b) for MFS = 50 nm (𝐸𝐿𝑇𝐸0  𝐶𝑜𝑛𝑣,50) and for MFS = 100  nm (𝐸𝐿𝑇𝐸0  𝐶𝑜𝑛𝑣,100) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Conventional Y–junctions  with Hoff = 0 nm were also measured but have not been  included in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 4 for clarity, since they are very similar  to results with Hoff = 50 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' This similarity is caused by  the MFS limitation of the fabrication process and verifies  our initial assumption on e-beam fabrication MFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' That  is, even when the nominal design includes a perfect tip  (Hoff = 0), experimental MFS limitations will induce an  imperfect tip, in this case, similar to the design of Hoff =  50 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Notwithstanding, results for this Hoff are included  in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 5 and 8 for direct comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' For both MFSs of  50 nm and 100 nm, the SWG splitter has a substantially  reduced ELTE0 compared to the conventional Y–junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Figure 4 shows the overall excess loss increment as  more cascaded stages are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' To obtain the ELTE0  relative to a single Y-junction, we performed a linear  regression with the measured losses in each cascaded  stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Figure 5 shows the resulting ELTE0 per splitter, that  is, the average loss per cascaded splitter calculated  through the slope of the linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' The SWG Y–  junction (solid line) exhibits a flat response in the entire  measured bandwidth from 1420 nm to 1680 nm, with an  𝐸𝐿𝑇𝐸0  𝑆𝑊𝐺,50 lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='3 dB and 𝐸𝐿𝑇𝐸0  𝑆𝑊𝐺,100 below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5 dB in  the full 260 nm spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' By contrast, the conventional  Y–junction  (dotted  line)  shows  a  performance  degradation towards shorter wavelengths, resulting in a  loss penalty, especially for an MFS of 100 nm (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  5(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Considering the high-resolution fabrication (MFS  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Excess loss of the 1×2, 1×4, 1×8 and 1×16 cascaded structures,  for SWG Y–junction with 50 nm MFS (a), conventional Y–junction  with 50 nm MFS (b), SWG Y–junction with 100 nm MFS (c), and  conventional Y–junction with 100 nm MFS (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' SEM images of the SWG Y–junction in cascaded configuration  1×16 (a) and inset of the device (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  SWG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='MFS = 50  over  3  1x2  1x2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  1x4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  1x4  1x8  1x8  2  1x16  2  1x16  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  (aB)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  n  1  EL  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  1420  1485  1550  1615  1680  1420  1485  1550  1615  1680  Wavelength(nm)  Wavelength (nm)  6  6  1x2  1x2  5  1x4  1x4  1x8  1x8  1x16  1x16  4  (dB)  dB)  m  73  2  0  0  1420  1485  1550  1615  1680  1420  1485  1550  1615  1680  Wavelength(nm)  Wavelength(nm)(a)  20μm  4cascaded stages  3cascaded stages  2 cascaded stages  istage  (b)  3μm= 50 nm), our device consistently outperforms the  conventional Y–junction, with 𝐸𝐿𝑇𝐸0  𝑆𝑊𝐺,50 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='23 dB in a  215 nm bandwidth (1420 nm – 1635 nm) as presented in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' As previously explained, it can be seen that  excess losses for conventional Y-junctions with an Hoff =  0 nm and Hoff = 50 nm are very similar, only differing in  a mean deviation of less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='06 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' As the MFS  increases to 100 nm, the negative impact on the  performance of the conventional Y–junction is more  pronounced, with an 𝐸𝐿𝑇𝐸0  𝐶𝑜𝑛𝑣,100 above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='57 dB in the 1420  nm –1680 nm window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' The SWG device, on the other  hand, yields improved performance over the entire  measured bandwidth with 𝐸𝐿𝑇𝐸0  𝑆𝑊𝐺,100 below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='46 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Devices with deterministically induced dimension  variations (Δδ) of ±10 nm were incorporated in the mask  layout to measure the robustness of the SWG Y–junction  to fabrication errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Figure 6(a) shows 𝐸𝐿𝑇𝐸0  𝑆𝑊𝐺,50,  demonstrating that performance is preserved despite the  presence of geometric variations, and even exhibiting a  slight improvement for over-etching deviations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=', Δδ  = -10 nm, SEM shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 6(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' In contrast, for the  MFS of 100 nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 6(b)), 𝐸𝐿𝑇𝐸0  𝑆𝑊𝐺,100 is slightly improved  for Δδ = +10 nm (SEM shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 6(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' For Δδ = -10  nm (SEM shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 6(d)), 𝐸𝐿𝑇𝐸0  𝑆𝑊𝐺,100performance  degrades towards longer wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' The largest  fabrication bias was observed in waveguide width,  narrowing the designed stem waveguide (Ws = 1200 ± 10  nm) by approximately 40 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='2 First-order transverse-electric mode (TE1)  In order to characterize the TE1 mode division, a mode  multiplexer [59] was included in combination with the  SWG Y–junction, as schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  When TE0 is injected through the upper input port of the  mode multiplexer, the TE0 mode is generated at the  output multimode waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' When the lower input port  is excited with TE0, mode evolution results in TE1 at the  mode multiplexer output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Two mode multiplexers in  back-to-back were used as reference to extract TE1 mode  excess loss of the Y–junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Figure 8(a) shows ELTE1 measurements for MFS = 50  nm in both SWG (𝐸𝐿𝑇𝐸1  𝑆𝑊𝐺,50) and conventional (𝐸𝐿𝑇𝐸1  𝐶𝑜𝑛𝑣,50)  Y–junctions, as well as for the conventional Y–junction  with Hoff = 0 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Likewise, ELTE1 for MFS = 100 nm in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Tolerances to fabrication errors of Δδ = ±10 nm for the SWG  Y–junction with MFS of 50 nm (a) and 100 nm (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Excess loss per  splitter was measured through linear regression of cascaded stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  SEM images of devices with Δδ = -10 nm for an MFS of 50 nm (c)  and 100 nm (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' SEM images of devices with Δδ = +10 nm for an MFS  of 50 nm (e) and 100 nm (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Schematic of the structure employed for the characterization of TE1 mode (a) and SEM images of the mode multiplexer (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' EL per splitter measured through linear regression of the  response of four cascaded stages for SWG (solid line) and conventional  (dotted line) Y–junctions, for MFS of 50 nm (a) and 100 nm (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Results  obtained for the conventional splitter with ideal resolution (MFS = 0  nm) are also shown in panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  b)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  Nominal  20=-10nm  Nominal  A=-10nm  1  40=+10nm  1  A0=+10nm  【dB]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  1420  1485  1550  1615  1680  1420  1485  1550  1615  1680  Wavelength(nm)  Wavelength(nm)  (C)  103nm  125nm  483nm  (d)  105nm  116nm  479nm  56nm  f107nm  L1146nm  200nm  1146nm  200nm  (e)  123nm  147nm  507nm  (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='124nm  136m  497nm  37nm  f88nm  L1168nm  200nm  1166nm  200nmTEi  TE  (b)  2μm  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='16μm  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='04μm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='50μm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='15μm7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='74μm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='15μmr  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='25μm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='52μmConv MFS = O nm  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='.  Conv MFS = 50 nm  SWG MFS =50 nm  1  (ap)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  0  1420  1485  1550  1615  1680  Wavelength(nm)  ConvMFS= 100 nm  SWG MFS = 100 nm  (ap)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  0  1420  1485  1550  1615  1680  Wavelength(nm)SWG (𝐸𝐿𝑇𝐸1  𝑆𝑊𝐺,100) and conventional (𝐸𝐿𝑇𝐸1  𝐶𝑜𝑛𝑣,100) Y–  junctions are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 8(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' As previously  mentioned, 𝐸𝐿𝑇𝐸1  𝐶𝑜𝑛𝑣,50 and 𝐸𝐿𝑇𝐸1  𝐶𝑜𝑛𝑣,100 are negligible due to  TE1 mode odd symmetry, with a power minimum at the  center of the multimode stem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Our device exhibits an  𝐸𝐿𝑇𝐸1  𝑆𝑊𝐺,50 below 1 dB over a 100 nm bandwidth ranging  from 1475 to 1575 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Within a 170 nm bandwidth  (1420 – 1590 nm), 𝐸𝐿𝑇𝐸1  𝑆𝑊𝐺,50 and 𝐸𝐿𝑇𝐸1  𝑆𝑊𝐺,100 only increase by  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Compared to the conventional Y–junction, the  degradation of the TE1 mode for our SWG Y–junction  arises from the selection of the stem waveguide width of  WS = 1200 nm, as a compromise between ELTE0 and  ELTE1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Note that increasing the width of the SWG stem  waveguide results in a stronger modal confinement that  prevents TE1 mode radiation, but increasingly penalizes  TE0 due to the resulting mode profile at the junction tip  [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Nevertheless, the performance of the proposed  device compares very favorably to state-of-the-art  higher-order mode power splitters (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  SWG Y–junctions with Δδ = ±10 nm were also  included in combination with the mode multiplexer  structures to study fabrication tolerances for TE0 and TE1  mode division mux/demux and are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 9(a)  and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Figure 9(a) shows that 𝐸𝐿𝑇𝐸0  𝑆𝑊𝐺,50 is lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  dB for both under-etching and over-etching errors in the  full 1420 nm – 1680 nm bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Figure 9(b) shows  that 𝐸𝐿𝑇𝐸0  𝑆𝑊𝐺,100 remain <1 dB for Δδ = -10 nm and <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='7 dB  for Δδ = +10 nm, for the same bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' These results  corroborate the robustness of the SWG Y–junction to  fabrication errors for TE0 mode, shown in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='1  (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 6(a) and (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' The performance for TE1 mode  division is more sensitive to fabrication errors in the  width of the SWG stem owing to the weaker confinement  of the Bloch-Floquet TE1 mode compared to the Bloch-  Floquet TE0 mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' While under-etching results in  𝐸𝐿𝑇𝐸1  𝑆𝑊𝐺,50 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='1 dB and 𝐸𝐿𝑇𝐸1  𝑆𝑊𝐺,100 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='7 dB for the full  bandwidth, over-etching errors are negligible at shorter  wavelengths and increase towards longer wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Conclusions  The detailed experimental study conducted in this work  demonstrates the broadband performance and relaxed  fabrication tolerances of SWG-based Y–junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Two  resolutions were investigated to account for different  fabrication processes, namely MFS of 50 nm and 100  nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Accurate measurements in cascaded splitters  demonstrate excess loss for the fundamental TE mode  lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='3 dB for MFS = 50 nm and below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5 dB  for MFS = 100 nm, in a 260 nm bandwidth (1420 nm –  1680 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Characterization of the first-order TE mode  was performed in combination with a mode multiplexer,  showing excess loss lower than 1 dB over a 100 nm  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' ELTE1 measurements for SWG (solid) and conventional (dotted)  Y–junctions for MFS of 50 nm (a) and 100 nm (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Conventional Y–  junction (dashed) with MFS = 0 nm is also shown in panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Tolerances to fabrication errors of the SWG Y–junction for  TE0 (dashed line) and TE1 (solid line) polarizations, for MFS of 50 nm  (a) and 100 nm (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Experimental performance comparison of state-of-the-art multimode power dividers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Design method  EL (dB)  Bandwidth (nm)  MFS (nm)  Length (µm)  Functionality  [33]  Inverse  design  subwavelength  axisymmetric  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  60  30  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='88  2-mode splitter  [34]  Pixelated meta-structure  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  40  30  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  3-mode splitter  [60]  MMI coupler  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='76  60  1000  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  2-mode splitter  [61]  Tapered directional coupler  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='7  30  200  25  2-mode splitter  [62]  Densely-packed waveguide array  1  28  110  46  2-mode splitter  This work  SWG Y–junction  1  100  50  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='3  2-mode splitter & converter  This work  SWG Y–junction  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  170  100  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='3  2-mode splitter & converter  a  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  Conv MFS = O nm  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Conv MFS = 50 nm  2  SWG MFS = 50 nm  B  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  1420  1470  1520  1570  1620  Wavelength (nm)  b)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  Conv MFS = 100 nm  SWG MFS = 100 nm  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  1420  1470  1520  1570  1620  Wavelength(n-TE  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  Nominal  48=-10nm  2  14§=+10nm  (dB)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  1420  1470  1520  1570  1620  Wavelength (nm)  TE  TEi  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  Nominal  145=-10nm  40=+10nm  B  d  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='5  1420  1470  1520  1570  1620  Wavelength(nm)bandwidth (1475 nm – 1575 nm) for the 50 nm MFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  SWG Y–junctions with deterministically induced errors  of -10 nm and +10 nm were also measured to analyze  resilience  to  over-  and  under-etching  errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Experimental results demonstrate robust fabrication  tolerances for the fundamental TE mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Our SWG-  engineered metamaterial Y–junction opens up promising  prospects for improving performance of diverse silicon  photonic integrated circuits where power splitters are  ubiquitous component, such as on-chip high bandwidth  communication systems and broadband spectroscopic  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Credit authorship contribution statement  Raquel Fernández de Cabo: Software, Formal  Analysis, Validation, Data curation, Writing – original  draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Jaime Vilas: Experimental validation, Writing –  review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Pavel Cheben: Conceptualization,  Writing – review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Aitor V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' Velasco:  Conceptualization, Resources, Funding acquisition,  Supervision, Writing – review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' David  González-Andrade: Methodology, Supervision, Project  administration, Writing – review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Declaration of Competing Interest  The authors declare that they have no known competing  financial interests or personal relationships that could  have appeared to influence the work reported in this  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  Acknowledgements  This work has been funded by the Spanish Ministry of  Science  and  Innovation  (RTI2018-097957-B-C33,  RED2018-102768-T,  PID2020-115353RA-I00);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  the  Spanish State Research Agency and the European Social  Fund  Plus  under  grant  PRE2021-096954;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  the  Community of Madrid – FEDER funds (S2018/NMT-  4326);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' the Horizon 2020 research and innovation  program under Marie Sklodowska-Curie grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  101062518;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content=' European Union – NextGenerationEU,  through the Recovery, Transformation and Resilience  Plan (DIN2020-011488).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
+page_content='  References  [1]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONE2T4oBgHgl3EQfrAiX/content/2301.04045v1.pdf'}
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+arXiv:2301.00842v1  [math.DS]  2 Jan 2023
+Skewed Anosov flows are orbit equivalent to Reeb-Anosov
+flows in dimension 3
+Marty Th´eo
+Abstract
+We prove that in dimension 3, Anosov flows which are R-covered and skewed are orbit
+equivalent to Reeb-Anosov flows. We characterize the existence of an invariant contact form
+or of a Birkhoff section with a given boundary, in terms of linking numbers between two
+invariant signed measures. Furthermore, we prove the existence of open book decompositions
+with one boundary component for Reeb-Anosov flows.
+Introduction
+Geodesic flows on hyperbolic surfaces have been studied systematically using two approaches:
+they are classical examples of hyperbolic flows and are the Reeb flows of the natural contact
+structures on the corresponding manifolds.
+Anosov generalized hyperbolic geodesic flows in
+what is now known as Anosov flows. In dimension 3, other families of Anosov flows have been
+studied: suspension of linear Anosov diffeomorphisms of the torus, Anosov flows built from Dehn
+surgeries along periodic orbits (also called Fried-Goodman surgeries [Fri83, Goo83]), and Anosov
+flows built from gluing hyperbolic plugs [BBY17].
+Except for geodesic flows, very few Anosov flows were known to be of the Reeb type. Foulon
+and Hasselblatt [FH13] produced Reeb-Anosov flows from surgeries on hyperbolic geodesic flows.
+Their construction extends a previous construction of surgery by Handel and Thurston [HT80].
+Recently Salmoiraghi [Sal22] produced a larger family of Reeb-Anosov flows using bi-contact
+structures. His construction extends the surgery operation introduced by Goodman.
+Fenley and Barbot [Fen94, Bar01] independently introduced the key notion of orbit space
+to understand topological properties of Anosov flows. One remarkable family of Anosov flows
+is characterized by its orbit space: the skewed R-covered Anosov flows. Barbot [Bar01] proved
+that Reeb-Anosov flows are skewed R-covered flows. The main result of the present paper is the
+converse implication:
+Theorem A. Let φ be an Anosov flow on an oriented, closed, 3-dimensional manifold. Suppose
+that φ is R-covered and positively skewed. Then φ is orbit equivalent to a smooth Reeb-Anosov
+flow. Additionally if α is the corresponding contact form, then α ∧ dα is positive.
+The theorem gives an answer to Barbot/Barthelm´e’s conjecture [Bar17]. By combining the
+theorem and previous works, we obtain a series of equivalences, represented in Table 1. Birkhoff
+sections and linking numbers between invariant measures are introduced later in the introduction.
+We denote by M0
+p(φ) the set of null-cohomologous φ-invariant probability measures.
+Corollary B. In a given column in Table 1, two cells are equivalent to each other. The columns
+correspond to exclusive natures for Anosov flows.
+1
+
+Positively twisted
+Flat flows
+Negatively twisted
+Others
+1
+positively skewed
+R-covered
+R-covered with trivial
+bi-foliated plane
+negatively skewed
+R-covered
+non
+R-covered
+2
+∃ Birkhoff section with
+positive boundary
+∃ Birkhoff section
+with no boundary
+∃ Birkhoff section with
+negative boundary
+other
+3
+∃µ ∈ M0
+p(φ) with
+positive linking numbers
+with all M0
+p(φ)
+M0
+p(φ) = ∅
+∃µ ∈ M0
+p(φ) with
+negative linking numbers
+with all M0
+p(φ)
+other
+4
+∃ψ ≃ φ, ∃α contact,
+ψ-invariant, α ∧ dα > 0
+∃ψ ≃ φ, ∃α ̸= 0
+ψ-invariant, dα = 0
+∃ψ ≃ φ, ∃α contact
+ψ-invariant, α ∧ dα < 0
+other
+Table 1: Tetrachotomy for Anosov flows between twisted flows, flat flows and others.
+Each
+line corresponds to one notion, each column corresponds to equivalent properties. Line 1: the
+topology of the bi-foliated plane. Line 2: the existence of a particular Birkhoff section. Line 3: the
+existence of a particular null-cohomologous invariant probability measure. Line 4: the existence
+of an orbit equivalent Anosov flow, for which there exists a particular differential 1-form. It can
+be rephrased, for columns 1 to 3, by the existence of a particular orbit equivalent geodesible
+Anosov flow.
+In the columns 1 and 3, the ambient manifold is oriented. Reversing the orientation ex-
+changes the role of these two columns. We should discuss the already existing equivalences.
+The tetrachotomy was first discovered by Barbot [Bar95] and Fenley [Fen94] simultaneously
+(corresponding to the line 1). The equivalence between the lines 1 and 2 is proven in [ABM22].
+For the first column, Barbot [Bar01] proved that the 4th cell implies the 1st cell. Barbot does
+not specify a sign for α∧dα, but it can be recovered. We give a second proof of this implication in
+Section 5.3, with the sign. The main theorem of this article of the converse implication: the 1st
+cell implies the 4th cell. We additionally introduce the 3rd line which is intersecting on its own.
+The equivalence between the 2nd and 3rd cells is more elementary and in the spirit of the work
+of Ghys [Ghy09] (see Section 5.3).
+For the second column, Solodov1 proved the equivalence between cell 2 and cell 3.
+The
+equivalence between the 2nd and 3rd cells follows from Schwartzman-Sullivan’s Theorem on cross-
+sections [Sch57, Sul76]. The equivalence with the 4th cell is well known and more elementary.
+To prove the main theorem, we roughly follow the following steps. We need to find an orbit
+equivalent Anosov flow which preserves a smooth volume form V , where V would correspond
+to α∧dα for a contact form α. Asaoka constructed an orbit equivalent, volume preserving Anosov
+flow using a Gibbs measure [Asa07].
+Gibbs measures form an important family of invariant
+probability measure, which satisfies some dynamical properties similar to the invariant volume
+forms. Given a Gibbs measure, Asaoka builds a new C1+H differential structure on the ambient
+manifold, for which the Gibbs measure is induced by a H¨older continuous volume form. Then he
+approximates the flow by smooth Anosov flows, smooth for the new differential structure, each
+of them preserving a smooth volume form.
+For an Anosov flow in general, preserving a smooth volume form is not enough to be a
+1V.V. Solodov did not publish the proof. Tha main argument is that actions on R, for which every element
+acts with at most one fixed point, are conjugated to affine actions.
+2
+
+reparametrization of a Reeb-Anosov flow.
+We need two additional conditions, which we ex-
+press in terms of linking number. To an invariant probability measure corresponds an element
+in H2(M, R). Denote by M0(φ) the set of null-cohomologous φ-invariant signed measures.
+Theorem C (Made precise in Theorem 21). Let φ be a transitive Anosov flow in a closed, ori-
+ented, 3-dimensional manifold. There exists a unique continuous bilinear map linkφ : M0(φ) ×
+M0(φ) → R extending the linking number for two disjoint, null-homologous knots. Addition-
+ally linkφ is symmetric.
+The new linking number map extends previous linking numbers given by Ghys2 [Ghy09],
+Arnold [Arn74], Vogel [Vog03] and Prasad [Pra22]. Our definition of the linking number map
+requires the existence of a Markov partition, so it cannot be extended past maybe transitive
+pseudo-Anosov flows. One can rephrase McDuff criterion [Duf87, Theorem 5.2] for contact-type
+differential 2-forms.
+Theorem D (McDuff [Duf87] - Prasad [Pra22]). Let φ be a smooth Anosov flow on a closed,
+oriented 3-dimensional manifold. Then φ is a smooth reparametrization of a Reeb-Anosov flow
+for a contact form α with α ∧ dα > 0, if and only if it preserves a smooth probability measure ν,
+which is null-cohomologous and has positive linking number with all null-cohomologous invariant
+probability measures.
+McDuff’s theorem is written in the language of structural boundaries developed by Sulli-
+van [Sul76].
+Prasad recently interpreted McDuff’s theorem in terms of linking number, for
+right-handed flows on rational homology spheres. We give a proof of the reformulation in Sec-
+tion 4.
+We need to find a Gibbs measure satisfying the two hypothesis in the previous theorem:
+it is null-cohomologous and has positive linking numbers with all null-cohomologous invariant
+probability measures. We first construct an invariant probability measure Lebγ supported on a
+single periodic orbit γ, satisfying these two properties. Then we approximate Lebγ by a sequence
+of null-homologous Gibbs measures. The map linkφ is continuous, so a sub-sequence of Gibbs
+measures has positive linking numbers. Then we adapt Asaoka’s strategy to get a smooth Anosov
+flow preserving a smooth volume form, for which McDuff’s criterion is satisfied.
+It remains to explain the existence of the measure Lebγ. Fried [Fri83] proved the existence of
+Birkhoff section for transitive Anosov flow. Birkhoff section are roughly speaking transverse sur-
+faces, bounded by periodic orbits, which intersect efficiently the orbits of the flow. The Birkhoff
+sections constructed by Fried bring little information on the topology of the ambient manifold
+and of the flow. In a previous work [ABM22], we characterized positively skewed R-covered
+Anosov flows using the existence of Birkhoff sections with positive boundaries. In Section 5, we
+improve this result.
+Theorem E. Let φ be a R-covered and positively skewed Anosov flow on a closed, oriented 3-
+dimensional manifold. Then φ admits an embedded positive Birkhoff section with only one bound-
+ary component, whose corresponding periodic orbit has orientable stable and unstable leaves. In
+particular, it gives an open book decomposition of M, adapted to φ.
+Take a periodic orbit γ bounding a Birkhoff section as in the theorem. Consider the Lebesgue
+measure Lebγ on γ, invariant by the flow. Up to rescaling, Lebγ is a null-cohomologous invari-
+ant probability measure. Its linking number with any null-cohomologous invariant probability
+measure is positive, since γ is the positive boundary of a Birkhoff section. Hence one can apply
+the above discussion on Lebγ.
+2Ghys proposes a general definition for the linking number, but it is not clear why it is well-defined. Florio
+and Hryniewicz proposed a similar definition by replacing limits with limits inferior, for right-handed flows.
+3
+
+We end the introduction by discussing two annex theorems, which we obtain with the same
+techniques. Our smoothing strategy also works for Anosov flows said to by homologically full,
+that is every element in H1(M, Z) is the homology class of a periodic orbit.
+Sharp [Sha93]
+proved that an Anosov flow is homologically full if and only it admits a null-cohomologous Gibbs
+measure. We can smooth the Gibbs measure with Asaoka’s technique.
+Theorem F. Let φ be a homologically full Anosov flow on a closed 3-dimensional manifold M.
+There exists a smooth Anosov flow ψ and a smooth volume form V on M which satisfy the
+following: ψ is orbit equivalent to φ, it preserves V , and the interior product ι
+�
+∂ψt
+∂t
+�
+V is null-
+cohomologous.
+In [Ghy09, CDHR22], a positive linking number condition is used to construct a Birkhoff
+section.
+For Anosov flows, their idea and the linking number map give a criterion for the
+existence of Birkhoff section bounding a specific boundary.
+Theorem G. Let φ be an Anosov flow on a closed, oriented, 3-dimensional manifold M. Let Γ
+be a collection of periodic geodesic, with multiplicities in Z, supposed to be null-homologous
+in H1(M, Q). Then the following statement are equivalent:
+1. there exist n > 0 and a Birkhoff section bounded nΓ,
+2. the invariant Lebesgue measure LebΓ supported on Γ has a positive linking number with
+every null-cohomologous probability measure invariant by φ.
+Notice that the theorem holds even when there exists no null-cohomologous probability mea-
+sure. In which case both conditions are satisfied independently on Γ. The author does not know if
+one can always take n = 1 in the first assertion, assuming that Γ is null-homologous in H1(M, Z).
+We prove the theorem in Section 5, and deduce a new proof of Barbot’s Theorem [Bar01] on
+Reeb-Anosov flows.
+In our strategy, we change the differential structure on the ambient manifold, and the
+parametrization of the flow comes as a consequence of the technique used. It comes as a sur-
+prise since the parametrization of the flow seems to play a more important role than the smooth
+structure. Barbot introduced a notion of topologically contact Anosov flow [Bar01]. Being topo-
+logically contact is invariant by (only continuous) conjugation, so it does not depend on the
+specific choice of the smooth structure on the ambient manifold.
+Question 1. Can we characterize all topologically contact parametrizations of a given Reeb-
+Anosov flow?
+Acknowledgments.
+I am grateful to M Asaoka for the discussion and help regarding the use
+of Gibbs measures. I thank the Max Plank Institute in Bonn for its financial support. I thank P
+Dehornoy and A Boulanger for the discussions on the linking number. I also thank A Rechtman,
+T Barbot, M Postic and V Colin for the various discussions on the subject.
+Contents
+1
+Preliminary
+5
+1.1
+Anosov flows
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+1.2
+Orbit space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+4
+
+1.3
+Reeb flow of a contact form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+1.4
+Invariant measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+1.5
+Invariant 1-cycle
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+2
+Markov partition
+13
+2.1
+Suspension of a Markov partition . . . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+2.2
+Vertical and horizontal Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+16
+2.3
+Word combinatorial
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+17
+3
+Linking number of invariant measures
+19
+3.1
+Linking number linkf
+φ
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+19
+3.2
+Linking function for a Markov partition
+. . . . . . . . . . . . . . . . . . . . . . .
+23
+3.2.1
+Measure zero sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+24
+3.2.2
+Reduction to finitely many orbits . . . . . . . . . . . . . . . . . . . . . . .
+25
+3.2.3
+Linking function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+30
+3.2.4
+Adapted cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+35
+3.2.5
+Linking number linkφ
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+40
+4
+Reeb-like condition for measures
+42
+5
+Birkhoff Section
+44
+5.1
+Fried section
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+45
+5.2
+Birkhoff section with one boundary . . . . . . . . . . . . . . . . . . . . . . . . . .
+47
+5.3
+Birkhoff section with given boundary . . . . . . . . . . . . . . . . . . . . . . . . .
+48
+6
+Reeb-like properties for positively skewed Anosov flow
+50
+6.1
+Reeb-like Gibbs measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+50
+6.2
+C1+H differential structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+56
+6.3
+Smoothing the flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+61
+A Fried-desingularisation
+67
+References
+70
+1
+Preliminary
+We introduce the notions appearing in the main theorem: Anosov flows, orbits space, positively
+skewed R-covered Anosov flows and Reeb flows.
+1.1
+Anosov flows
+Let M be a closed three-dimensional manifold and φ a C1 flow on M. The flow φ is said Anosov
+if there exists a φ-invariant splitting of T M into three line bundles T M = Es ⊕ X ⊕ Eu and two
+real numbers A, B > 0 such that for one/any Riemannian norm ∥.∥ on T M, we have:
+•
+X is tangent to the flow,
+• for all t ≥ 0, ∥dEsφt∥ ≤ A exp−Bt,
+• for all t ≤ 0, ∥dEuφt∥ ≤ A exp−B|t|.
+5
+
+Here dEsφt and dEuφt correspond to the differential dφt restricted to the line bundles Es
+and Eu. The flow is said transitive if there exists an orbit of φ dense inside M. We compare three
+properties for Anosov flows, each of them implies that the flow is transitive: R-covered Anosov
+flows [Bar95], Reeb-Anosov flows [Bar01] and Anosov flows admitting a Birkhoff section [Fri83].
+In the rest of the article, M is supposed oriented and φ is supposed transitive.
+The bundles Es and Eu are integrable in two 1-foliations, which we denote respectively by Fss
+and Fuu. They are called the strong stable and unstable foliations. The plane bundles Es ⊕ X
+and Eu ⊕ X are also integrable in two 2-foliations, invariant by φ. They are denoted by Fs
+and Fu and called weak stable and unstable foliations.
+Every leaf of the foliations Fss and Fuu is homeomorphic to R and each leaf of the foli-
+ations Fs and Fu is homeomorphic to either R/Z × R or an open M¨obius strip if it contains
+a periodic orbit, or R2 otherwise. Through the article, we use the notation F(x) for the leaf
+of the foliation F containing the point x. We also denote by Leaf(F) the set of leaves of the
+foliation F, equipped with the quotient topology.
+Let φ and ψ be two flows on respectively M and N. An orbit equivalence h: (M, φ) →
+(N, ψ) is a homeomorphism h: M → N such that for each orbit arc γ of φ, h(γ) is an orbit arc
+of ψ and h preserves the orientation by the flow. When M and N are oriented, we additionally
+suppose that h preserves the orientation. Important notions in this article are invariant by orbit
+equivalence: the homeomorphism class of the bi-foliated plane, existence of a Birkhoff section
+with positive boundary and linking number between invariant measures.
+Orientations-bundle covering.
+Let M ′ be the manifold given as the set of tuples (x, os
+x, ou
+x)
+with x ∈ M, and os
+x, ou
+x are local orientations of respectively the stable and unstable foliations
+at x. The projection M ′ → M on the first coordinate is a degree 4 covering map. Denote by �
+M
+one connected component of M ′. The projection �π : �
+M → M is a covering map, which is of
+degree one if and only φ has orientable stable and unstable foliations. Recall that M is supposed
+orientable, so �
+M → M is of degree one or two. We call �
+M the orientations-bundle covering
+of φ. We also denote by �φ the lift of φ to �
+M. Notice that �φ is Anosov, its stable and unstable
+foliations are the lifts of the corresponding foliations for φ, and they are orientable.
+Notice that �π∗(π1(�
+M)) is a subgroup of π1(�
+M) of index one or two, so it is normal. We denote
+by sign: π1(M) → {−1, 1} the group morphism obtained as π1(M) ։ π1(�
+M)/�π∗(π1(�
+M)) ֒−→
+{−1, 1}. That is given a simple loop γ based on x, sign([γ]) = 1 if and only if the stable and
+unstable foliations of M are orientable in a neighborhood of γ.
+Notice that sign is a group
+morphism. The map sign is defined so that for a closed and injective loop γ, the stable and
+unstable foliations of φ are orientable on a small neighborhood of γ if and only if sign(γ) = 1.
+1.2
+Orbit space
+Take an Anosov flow φ on a closed 3-manifold M. We denote by πM : �
+M → M the universal
+covering map of M. The flow and the foliations lifts to a flow �φ and foliations �Fs, �Fu, �Fss
+and �Fuu in �
+M.
+We call orbit space of φ, the quotient Pφ of �
+M by orbits of the flow �φ.
+Barbot and Fenley proved [Bar95, Fen94] that Pφ is homeomorphic to a plane.
+We denote
+by πPφ : �
+M → Pφ the projection. The foliations �Fs, �Fu are saturated by �φ, so they project
+in Pφ to two transverse foliations by curves, which we denote by Ls and Lu. If the flow φ is
+smooth, the set Pφ comes with a C∞ structure, for which the foliations Ls and Lu are of class C1
+(see [FH19, Corollary 9.4.11]). We call bi-foliated plane of φ the set (Pφ, Ls, Lu).
+Suppose that M is oriented. We lift the orientation on M to an orientation on �
+M. We
+equip any surface S ⊂ �
+M transverse to the flow �φ, with the orientation which satisfies that S is
+6
+
+Ls
+Lu
+Figure 1: Classification of the bi-foliated plane of R-covered Anosov flows: suspension flows,
+positively skewed and negatively skewed.
+positively transverse to the flow. The projection πPφ restricts to an immersion. We fix on Pφ
+the orientation for which for any positively transverse surface S ⊂ �
+M, the immersion S → Pφ is
+orientation preserving.
+A particular family of Anosov flow are the ones for which the space of leaves Leaf(Fs) (or
+equivalently Leaf(Ls)) is homeomorphic to R. This condition is equivalent to having Leaf(Fu)
+and Leaf(Lu) homeomorphic to R. These flows are said to be R-covered.
+Theorem 2 (Barbot-Fenley [Bar95, Fen94]). Let φ be a R-covered Anosov flow.
+Then φ is
+transitive. Additionally there is a homeomorphism, preserving the orientation and the foliations,
+from the bi-foliated plane of φ to one of the three following cases (illustrated in Figure 1):
+1.
+(R2, H, V ) where H and V are the foliations made of horizontal and vertical lines,
+2.
+(D+, H, V ) where D+ is the diagonal strip {(x, y) ∈ R2, |x − y| < 1},
+3.
+(D−, H, V ) where D− is the anti-diagonal strip {(x, y) ∈ R2, |x + y| < 1}.
+Here R2, D+, D− are equipped with the trigonometric orientations.
+In the first case, an argument from Solodov implies that φ is topologically equivalent to a
+suspension of a linear Anosov diffeomorphism on the torus. In the second and third case, M
+is additionally orientable, and we say that φ is respectively positively skewed and negatively
+skewed (they are sometime call twisted instead of skewed).
+Notice that these two cases are
+disjoint because of the choice of the orientation. In particular reversing the orientation on M
+switches the rolls of the cases 2 and 3.
+The fundamental group π1(M) acts on �
+M. That action preserves the foliations of �φ, so it
+induces an action π1(M) ↷ Pφ which preserves Ls and Lu. We denote by g·�p the action of π1(M)
+on �
+M, and g ∗ξ the action π1(M) ↷ Pφ. A point ξ in Pφ corresponds to the orbit πM(π−1
+P ({ξ}))
+of φ. Additionally a point ξ invariant by a non-trivial element in π1(M) corresponds to a periodic
+orbit of φ. More precise, let �x ∈ �
+M, x = πM(�x) ∈ M and ξ = πPφ(�x) ∈ Pφ. Then x is φ-periodic
+if and only if there exists g ∈ π1(M) \ {0} such that g ∗ ξ = ξ.
+Suppose x to be periodic,
+then g can be taken as the homotopy class of the orbit of x in π1(M, x) (see [VS73] for the
+non-triviality of that element). Denote by g ∈ π1(M, x) the homotopy class of the orbit of x.
+Then g preserves the leaves Ls(ξ) and Lu(ξ).
+If the stable leaf Fs(x) is orientable, then g
+preserves the orientation on Ls(ξ) and Lu(ξ). Additionally g contracts Lu(ξ) and expends Ls(ξ)
+(see [ABM22]). When Fs(x) is not orientable, then g2 satisfies that property.
+Lemma 3 (Fenley [Fen94]). Let φ be a skewed R-covered Anosov flow. For any periodic orbit γ
+of φ, there exists a periodic orbit δ and n, m ≥ 1 for which γn is anti-homotopic to δm.
+7
+
+In the lemma, γn stands for the orbit γ traveled n times.
+Lemma 4. A skewed R-covered Anosov flow is homologically full.
+Proof. This is a consequence of two facts: the homology class of periodic orbits span H1(M, Z)
+(see [PP86]), and every periodic orbit has a positive multiple which is anti-homotopic to a positive
+multiple of another periodic orbit (see Lemma 3). It follows that there exists a finite family of
+periodic orbits (γi)i, for which the convex hull of the homology class [γi] ∈ H1(M, R) contains 0 in
+its interior. So the flow is homologically full according to the Sharp’s characterization [Sha93].
+■
+1.3
+Reeb flow of a contact form
+A contact form on a 3-dimensional manifold M is a C1 differential form α such that α ∧ dα is
+never zero. In particular α ∧ dα is a volume form on M, so M is orientable. Given a contact
+form α, the plane distribution given by ξ = ker(α) is called a contact structure. Given a contact
+form α, there exists a unique vector field Yα on M satisfying ι(Yα)α = 1 and ι(Yα)dα = 0. This
+vector field is called the Reeb vector field of α. When α is of class C∞, Yα is also of class C∞, so
+it generates a C∞ flow ψα, called Reeb-flow of α. Flows which are both Anosov and Reeb-flows
+are called Reeb-Anosov flows.
+Theorem 5 (Barbot [Bar01]). In dimension 3, any Reeb-Anosov flow is R-covered and skewed,
+and in particular transitive.
+Barbot does not relate the nature positively/negatively skewed of the flow with the orientation
+given by the contact form. We give a new proof of his theorem in Section 5, which additionally
+gives the sign correspondence.
+Given a smooth flow ψ, we say that it is Reeb-like if there exists a smooth reparametrization
+of ψ which is a Reeb-flow of a contact from. It is equivalent to the existence of a contact form α
+on M such that ι(X)α > 0 and ι(X)dα = 0, where X =
+∂ψt
+∂t . Indeed in this case, the flow
+generated by Yα =
+1
+ι(X)αX is a Reeb-flow and a smooth reparametrization of ψ.
+The contact structure of the Reeb flow of a smooth contact form α, that is the plane dis-
+tribution given by ker(α) ⊂ T M, is smooth, transverse to the flow and invariant by the flow.
+We the flow is Anosov, only one plane distribution satisfies this property, the sum Es ⊕ Eu of
+the strong stable and strong unstable directions of the flow. In particular this sum is smooth
+for Reeb-Anosov flows. It is known that when Es ⊕ Eu is of class C1, then either the flow is a
+constant time suspension of an Anosov diffeomorphism of the torus, or Es ⊕ Eu is the kernel of
+an invariant C1 contact form. In general, this plane field is only H¨older, but it can still satisfy
+some topological version of being of the contact type. Barbot call these flows topologically con-
+tact, and proved [Bar01] that these flows are also R-covered and skewed. In Section 4, we give
+a condition for an Anosov flow to be Reeb-like, in terms of linking number.
+1.4
+Invariant measures
+We give our notation for invariant measures and current in this section.
+In this article, ex-
+cept when stated otherwise, the term measure refers to signed measure. We denote by Mp(φ)
+and M(φ) the sets of φ-invariant probability measure and φ-invariant signed measure on M. We
+equip these sets with the weak topology a sequence of measures (µn)n converges toward µ∞ if
+for any continuous function f : M → R, the integral
+�
+M fµn converges toward
+�
+M fµ∞. The
+set Mp(φ) is compact for the weak topology.
+8
+
+Remark 6. A consequence of Banach-Steinhaus Theorem is the following. Given a sequence of
+signed measure µn converging toward a µ∞, the total variation of µn is bounded by some constant
+in dependent on n.
+A transverse measure µ is the data for any compact embedded surface S transverse to φ,
+of a finite measure µS on S, such that for any compact embedded surfaces S1, S2 transverse to φ,
+and any holonomy map h: U1 ⊂ S1 → U2 ⊂ S2 along the flow, h∗µS1 and µS2 coincide on h(U1).
+Given a measure µ ∈ M(φ), we denote by µ⊥ the corresponding transverse measure. It is defined
+by µ⊥(A) = 1
+ǫµ(φ[0,ǫ](A)) for any measurable set A ⊂ S transverse to φ, and ǫ > 0 small enough
+(which depend on S). Denote by Lebφ the unique measure on orbits of φ in M, satisfying that
+for every x ∈ M and ǫ > 0 small enough, φ[0,ǫ](x) = ǫ. Given a transverse measure µ⊥, we
+denote by µ⊥ ⊗ Lebφ the invariant measure obtained locally as a product of µ⊥ and Lebφ along
+the orbits of φ. The maps µ �→ µ⊥ and µ⊥ �→ µ⊥ ⊗ Lebφ are continuous and inverse to each
+other.
+Any orbit equivalence h: (M, φ) → (N, ψ) induces a natural homeomorphism µ⊥ �→ h∗µ⊥
+between the sets of transverse measures. There is a natural homeomorphism Θh : M(φ) → M(ψ)
+between the set of invariant measures of the two flows. For a measure µ ∈ M(φ), Θh(µ) is the
+measure h∗(µ⊥) ⊗ψ Lebψ. We introduce later the cohomology class and the linking number of
+signed measures. These notions are invariant by orbit equivalence.
+Remark 7. Since they do not depend on the parametrization of the flow nor on the smooth
+structure on the manifold, transverse measures is the natural notion to define the cohomology
+class and the linking number. For convenience, we still use invariant measures instead, but we
+keep in mind the underlining transverse measures.
+Cohomology class of an invariant measure
+Given an invariant measure, there is a natural
+current associated to that measure. We introduce currents and use it to define the cohomology
+class of an invariant measure. See [DR84] for more background on currents. Denote by ∧ the
+wedge product, by ι(X)ω the inner product of ω by X, by ·∩ the algebraic intersection and by ·
+the integration of a differential k-form on a k-chain.
+Denote by n = 3 the dimension of M. A k-current is a linear form on the set Ωk(M) of
+smooth k-differential form on M, which is continuous for the C∞ topology on Ωk(M). Given
+a k-current C and a smooth vector field Y on M, we define the currents dC and ι(Y )C to be
+the k − 1 and k + 1 currents given respectively by:
+•
+dC(α) = C(dα),
+•
+ι(Y )C(α) = C(ι(Y )α).
+Given an invariant measure µ, the map Ω0(M) = C∞(M, R) → R given by f �→
+�
+M f(x)µ(x)
+is a 0-current, which we denote by Cµ.
+In particular if X is the vector field generating the
+flow φ, ι(X)Cµ is a 1-current, which corresponds to the transverse measure µ⊥. For a (n − k)
+differential form β (only supposed continuous), we denote by Cβ the k-current defined by Cβ(α) =
+�
+M α ∧ β.
+The differential map d on currents induces a current homology class given by ker(d)/ im(d).
+DeRham [DR84] proved that the current homology is equivalent to the simplicial homology.
+Additionally, if β is a continuous differential form, the homology class of Cβ is Poincar´e-dual to
+the cohomology class of β. For that reason for a k-current C, we denote by [C] ∈ Hn−k(M, R)
+the Poincar´e-dual of the homology class of C. More precisely [C] is equal to the cohomology
+class [ω] for a smooth (n−k)-differential form ω, if and only if for all smooth differential k-form α,
+one has C(α) =
+�
+M α ∧ ω.
+9
+
+Lemma 8. For any invariant measure µ ∈ M(φ), we have d(ι(X)Cµ) = 0.
+Proof. Let µ ∈ M(φ) be an invariant measure.
+For any smooth function f : M → R, we
+have Cµ(φ∗
+t f) = Cµ(f). It follows that LXCµ(f) = Cµ(LXf) = 0. Here LX is the Lie derivative
+along the vector filed X = ∂φt
+∂t , that is LXf = ι(X)(df). Therefore we have d(ι(X)Cµ) = 0.
+■
+We denoted by [µ]φ ∈ H2(M, R) the cohomology class of ι(X)Cµ.
+Lemma 9. The map µ ∈ M(φ) → [µ]φ ∈ H2(M, R) is continuous.
+Proof. Using Poincar´e duality, the cohomology class of a measure µ ∈ M(φ) is determined by the
+integrals
+�
+M ι(X)αkdµ for a finite family (αk)k of closed 1-form for which ([αk])k spans H1(M, R).
+Since the functions ι(X)αk are continuous, the maps µ �→ �
+M ι(X)αkdµ are also continuous.
+Hence the map µ �→ [µ]φ is continuous.
+■
+Lemma 10. Let V be an invariant volume form and µV be the invariant measure induced by V .
+Then ι(X)V is closed and [µV ]φ = [ι(X)V ] in H2(M, R).
+Proof. Take a closed 1-form α. We have 0 = ι(X)(α ∧ V ) = (ι(X)α) ∧ V − α ∧ (ι(X)V ). It
+follows that:
+[µ]φ · [α] =
+�
+M
+ι(X)αdµV
+=
+�
+M
+(ι(X)α) ∧ V
+=
+�
+M
+α ∧ (ι(X)V )
+= [ι(X)V ] · [α]
+The equality holds for all [α] ∈ H1(M, R), so [µ]φ = [ι(X)V ].
+■
+Recall some homology/cohomology homomorphism. Take a homeomorphism h: M → N.
+Given a 1-chain in M, c = �
+i aifi, where ai ∈ R and fi is a continuous map from [0, 1] → M, we
+denote by h∗c the 1-chain in N given by h∗c = �
+i aih ◦ f. We also denote by h∗ : H1(M, R) →
+H1(N, R) the corresponding map.
+We view the DeRham cohomology space as the dual of H1, in the sens that H1(M, R) is natu-
+rally identified with HomR(H1(M, R), R), where the map is given by integrating a closed 1-form
+on a C1 1-cycle. Denote by h∗ : H1(N, R) → H1(M, R) the map given by h∗(α)(c) = α(h∗(c))
+for α ∈ HomR(H1(N, R), R) and c ∈ H1(M, R). By Poincar´e duality, H1(M, R) and H2(M, R)
+are dual. So h induces a map h−∗ : H2(M, R) → H2(N, R) given by h−∗[ω] · [α] = [ω] · h∗[α].
+Here h∗α is not well-defined, only h∗[α] is.
+Proposition 11 (Invariance of the cohomology class). Let h: M → N be an orbit equivalence
+between two flows φ and ψ. Then for any µ in M(φ), we have [Θh(µ)]ψ = h−∗[µ]φ.
+Essentially, the cohomology class is a topological data which depends on the transverse mea-
+sure corresponding to µ.
+Lemma 12. For any closed 1-forms α and β on M and N with [α] = h∗[β], we have:
+�
+M
+ι
+�∂φt
+∂t
+�
+αdµ =
+�
+N
+ι
+�∂ψt
+∂t
+�
+βd(Θh(µ))
+10
+
+Proof of Proposition 11. We temporarily admit the above lemma. Take α, β as in the lemma.
+Then we have:
+h−∗[µ]φ · [β] = [µ]φ · h∗[β] = [µ]φ · [α]
+=
+�
+M
+ι
+�∂φt
+∂t
+�
+αdµ
+=
+�
+N
+ι
+�∂ψt
+∂t
+�
+βd(Θh(µ))
+= [Θh(µ)]ψ · [β]
+This equation is true for all [β] ∈ H1(M, R), which implies [Θh(µ)]ψ = h−∗[µ]φ.
+■
+Proof of Lemma 12. Take a subset Y ⊂ M topologically transverse to φ, that is no short orbit
+arc intersects Y twice. Take also a continuous map T : Y → R∗
++, so that the map Y ×[0, 1) → M
+given by (x, s) �→ φsT (x) is injective. Denote by φ[0,T )(Y ) the union of φ[0,T (x))(x) for x in Y .
+Take a continuous map uY : φ[0,T )(Y ) → �
+M lifting the universal covering map.
+Claim 1. There exists a finite partition of M of the form (φ[0,T i)(Y i))i of M, for some sets Y i ⊂ M
+topologically transverse to φ and some continuous maps T i : Y i → R∗
++.
+For g ∈ π1(M), and θ ∈ H1(M, R) we denote by θ(g) the integral of α on any C1 closed
+loop representing g. A continuous map f : �
+M → R is said θ-equivariant if f(g · q) = θ(g) + f(q)
+for all g ∈ π1(M) and q ∈ �
+M.
+Fix a point p ∈ �
+M.
+Given a closed 1-form α, we consider
+the [α]-equivariant map fα uniquely defined by fα(p) = 0 and dfα = π∗
+Mα. Then we have:
+�
+φ[0,T )(Y )
+ι
+�∂φt
+∂t
+�
+αdµ =
+�
+x∈Y
+��
+s∈[0,T (x))
+ι
+�∂φt
+∂t
+�
+α(φs(x))ds
+�
+dµ⊥
+=
+�
+Y
+(fα(uY (φT (Y )) − fα(uY (x)))) dµ⊥
+=
+�
+φT (Y )
+fα ◦ uY dµ⊥ −
+�
+Y
+fα ◦ uY dµ⊥
+�
+M
+ι
+�∂φt
+∂t
+�
+αdµ =
+�
+i
+�
+φ[0,T )(Y )
+ι
+�∂φt
+∂t
+�
+αdµ
+=
+�
+i
+��
+φT i (Y i)
+fα ◦ uY idµ⊥ −
+�
+Y i fα ◦ uY idµ⊥
+�
+=
+�
+i
+�
+φT i(Y i)−Y i fα ◦ uY idµ⊥
+Here we write
+�
+A−B fdµ =
+�
+A fdµ −
+�
+B fdµ.
+Claim 2. Given a θ-equivariant map f : �
+M → R, the quantity �
+i
+�
+φT i(Y i)−Y i f ◦uY idµ⊥ depends
+on θ but not on the specific choice of θ-equivariant map f.
+Assume the claim. Denote by �h: �
+M → �
+N a lifting map of h: M → N, and by h∗ : π1(M) →
+π1(N) the map induced by h. By assumption, we have [α] = h∗[β], so the map fβ ◦ �h is a [α]-
+equivariant map. It follows that we have the equality of the integrals:
+�
+M
+ι
+�∂φt
+∂t
+�
+αdµ =
+�
+N
+ι
+�∂ψt
+∂t
+�
+βd(Θh(µ))
+11
+
+Proof of Claim 1.
+Take a smooth triangulation T of M for which every face of positive codi-
+mension is transverse to φ. We also suppose each simplex small.
+For a 3-dimensional simplex S of T , denote by YS the set of point x ∈ ∂S for which φ(0,ǫ)(x)
+is included in S for some ǫ > 0. Also define the map TS : YS → R∗
++ as the first return map
+from YS to ∂S \ YS. Notice that YS and ∂S \ YS are transverse to the flow. Therefore, TS is
+well-defined and continuous, as restriction of a first return time from YS to ∂S \YS. Additionally,
+the family (φ[0,TS)(YS))S form a finite partition of M.
+□
+Proof of Claim 2.
+Take two θ-equivariant maps f1, f2 : �
+M → R. Then f1 −f2 is invariant by the
+action of π1(M). Therefore, there exists a continuous map f3 : M → R such that f1−f2 = f3◦πM.
+We have:
+�
+i
+�
+φT i (Y i)−Y i(f1 − f2) ◦ uY idµ⊥ =
+�
+i
+�
+φT i (Y i)−Y i f3dµ⊥
+The latter is equal to zero thanks to a telescoping argument. Indeed, the family of non-empty
+intersections of the form φT j(Y j) ∩ Y i gives a partition of Y i when j varies, and a partition
+of φT j(Y j) when i varies. So we have:
+�
+i
+�
+φT i (Y i)−Y i f3dµ⊥ =
+�
+i
+�
+j
+��
+φT i (Y i)∩Y j f3dµ⊥ −
+�
+Y i∩φT i (Y j)
+f3dµ⊥
+�
+= 0
+□
+■
+Flux.
+One can interpret [µ]φ as the flux of φ for the measure µ. That is [µ]φ · [S] corresponds
+to the quantity of flow going through a closed surface S, relatively to µ. In [CHL08], the authors
+establish a connection between first return maps of Reeb flows and area preserving diffeomor-
+phisms having zero flux. More precisely fix a compact surface S with non-empty boundary, and a
+diffeomorphism f : S → S preserving an area form ω. Consider the kernel of the action of f∗ −id
+on H1(S, Z). The flux of f is a map flux: ker(f∗ − id) → R, where fluc([c]) =
+�
+d ω, where d is
+a 2-chain bounding f(c) − c. It is proven in the same article that if f is the first return map of
+a Reeb flow, then it has zero flux.
+Consider the suspension N = S × R/(x, t + 1) ≃ (f(x), t) of the map f. We lift ω to a
+2-form in N, denoted by ωN. One can verify that H2(N, R) ≃ ker(f∗ − id) ⊂ H1(S, R), where an
+element [c] ∈ ker(f∗ − id) is in correspondence with the element [c × [0, 1] + d × 0] ∈ H2(N, R),
+where d a 2-chain in S bounded f(c) − c. Then the map f has zero flux if and only the integral
+of ωN over all element in H2(N, R) is zero. That is [ωN] = 0 in H2(N, R) by Poincar´e duality.
+For the suspension with boundary case, it follows from [CHL08] that the suspension flow on N
+is a reparametrization of a Reeb-flow if and only if the preserving 2-form ωN is exact.
+1.5
+Invariant 1-cycle
+Let γ be a periodic orbit, we denote by Lebγ the unique invariant measure on γ whose total
+mass is the length of γ for φ.
+That is Lebγ is the Lebesgue measure on γ obtained as the
+pullback f ∗
+γ Leb, where fγ : R/T Z �→ γ is map given by fγ(t) = φt(x) for some x ∈ γ of period T .
+A map M → R is called an invariant 1-cycle if it is invariant by φt and is zero outside
+finitely many periodic orbits. The support of an invariant 1-cycle is the union of these periodic
+12
+
+orbits. Take a simplicial 1-chain c = �n
+i=1 λifi, where fi : [0, 1] → M is an immersion with
+image in a single orbit. The map defined by gc(x) = �
+i λ(x ·∩[(fi)|[0,1)]) is a 1-cycle if and only
+if ∂c = 0. In that case we identify c and gc. Similarly given a finite union Γ of periodic orbits
+of M and a: Γ → R, �
+γ∈Γ a(γ)γ is identified with the invariant 1-cycle g given by g(x) = a(γ)
+if x ∈ γ ∈ Γ, g(x) = 0 otherwise. An invariant 1-cycle is said positive if the corresponding map
+has non-negative values. Given an invariant 1-cycle Γ = �
+γ a(γ)γ, we define the invariant signed
+measure LebΓ = �
+γ a(γ) Lebγ.
+Lemma 13. The cohomology class [LebΓ]φ is Poincar´e dual to [Γ] ∈ H1(M, R).
+Proof. For any closed 1-form β and any periodic orbit γ of period T , we have:
+[Lebγ]φ · [β] =
+�
+M
+ι(X)β Lebγ =
+� T
+0
+ι(X)β(φt(x))dt =
+�
+γ
+β = [β] · [γ]
+■
+Lemma 14. Given a measure µ ∈ M(φ) and a finite union of periodic orbit ∆ ⊂ M, there
+exists a sequence of invariant 1-cycles Γn with rational coefficients, such that LebΓn converges
+toward µ, and so that for all n:
+• the support of Γn is disjoint from ∆,
+• [LebΓn]φ = [µ]φ.
+Additionally if ∆ is empty and µ is a probability measure, Γn can be taken so that LebΓn is a
+probability measure.
+It is essentially a variation of a well known result on the density of finitely supported measures.
+This lemma follows from similar Lemmas 41 and 42, about symbolic dynamic.
+2
+Markov partition
+Here φ is supposed to be a transitive Anosov flow. We call Markov cuboid for φ a compact
+subset R ⊂ M for which there exists a homeomorphism f : [0, 1]3 → R, such that for all x, y ∈
+[0, 1], we have:
+• f(x × [0, 1]2) is included in a weak unstable leaf,
+• f([0, 1] × y × [0, 1]) is included in a weak stable leaf.
+Similarly a Markov rectangle is the image of a topological embedding f : [0, 1]2 → M, topo-
+logically transverse to φ, which sends the horizontal and vertical foliations in [0, 1]2 to the weak
+stable and unstable foliations in M. Take a Markov cuboid R and a map f : [0, 1]3 → R given
+as above, and suppose that for every x, y ∈ [0, 1], we have t ∈ [0, 1] �→ f(x, y, t) is orientation
+preserving, for the orientation given by the flow. We define the sets:
+• ∂sR = f([0, 1] × {0, 1} × [0, 1]) the stable boundary of R,
+• ∂uR = f({0, 1} × [0, 1] × [0, 1]) the unstable boundary of R,
+• ∂suR = ∂sR ∪ ∂uR,
+• ∂−R = f([0, 1]2 × 0), ∂−R = f([0, 1]2 × 1), which are Markov rectangles. We call the
+transverse boundaries of R.
+13
+
+• we denote Int ∂ǫR for ǫ ∈ {s, u, su, −, +}, the image by f of the sames set, where [0, 1] is
+replaced by (0, 1). That is Int ∂ǫR is the interior of the rectangles corresponding to ∂ǫR.
+The weak stable and unstable foliations on M induce stable and unstable foliations on R,
+on ∂−R and on ∂+R. Denote by W s
+R and W u
+R the stable and unstable foliations on ∂−R. Given
+a Markov rectangle R, a subset U ⊂ R is said to be a vertical (resp. horizontal) sub-rectangle
+of R if it is connected and is the union of unstable (resp. stable) leaves on R. A Markov
+partition P is a finite family of Markov cuboids {R1, . . . , Rn} satisfying:
+1. ∪iRi = M,
+2. for any distinct i, j, the interior of Ri and Rj are disjoint,
+3. for any i, j, either the intersection (Int ∂+Ri) ∩ (Int ∂−Rj) is empty, or ∂+Ri ∩ ∂−Rj is a
+horizontal sub-rectangle of ∂+Ri and a vertical sub-rectangle of ∂−Rj.
+Any transitive Anosov flows admits a Markov partition (see [Rat69, Rat73]). Our definition
+of Markov partition is more restrictive than the usual definition, in the sense that a Markov
+cuboid R have a nice transverse boundary ∂+R. By taking Markov sub-partitions, any Markov
+partition in the usual sense can be transformed in a Markov partition in our sense. We de-
+fine ∂P, ∂sP, ∂uP, ∂suP, ∂−P, ∂+P as the union of the corresponding boundary of the Markov
+cuboids of the Markov partition P.
+Symbolic dynamic.
+Here we fix a Markov partition P = {R1, . . . , Rn}. We call the transition
+matrix of P the matrix A ∈ Mn(Z), containing a 1 in position (i, j) if Int(∂+Ri) ∩ Int(∂−Rj)
+is non-empty, and a zero otherwise. We call a P-word any sequence u: I ⊂ Z → P satisfying
+Aj−i
+ui,uj > 0 whenever i and j are in I. The set of bi-infinite P-words (that is sequences u: Z → P)
+is denoted by ΣP. For two P-words u, v: I ⊂ Z → P, we denote by dP the distance between u
+and v, defined by:
+dP(u, v) =
+�
+i∈I
+ui̸=vi
+2−|i|
+It induces on ΣP the topology for which cylinders form a basis of the topology. Here cylinders
+are the subsets of the form {u ∈ ΣP, ui = vi for all i ∈ I}, where I is a finite subset of Z and
+v: I → P is a P-word. We denote by σ the shift map on ΣP, given by σ(u)i = ui+1. Notice
+that σ is 2-Lipchitz.
+Similar to Anosov flows, σ admits a stable foliation and an unstable foliation, denote by W s
+and W u. Here W s(v) (resp. W u(v)) is the set of sequence w ∈ ΣP with wi = vi for all i large
+enough (resp. for all i ≤ i0 for some i0 ∈ Z). The shift map divides the distances by 2 in each
+stable leaf, and multiplies the distances by 2 on each unstable leaf. We additionally call the local
+stable and unstable leaves of v the sets defined by:
+W s
+l (v) = {w ∈ ΣP, wi = vi for all i ≥ 0}
+and
+W u
+l (v) = {w ∈ ΣP, wi = vi for all i ≤ 0}
+A cyclic P-word is a map u: Z/nZ → P such that we have Aui,ui+1 > 0 for all i ∈ Z/nZ.
+We call n the length of u. We say that u is primitive if it is not k-periodic for some k < n.
+Given a cyclic P-word u, we denote by u: Z → P the P-word given by ui = u(i mod n). For two
+cyclic P-words u, v of length n, m such that u0 = v0, we denote by uv or u ◦ v the concatenation
+14
+
+of u and v, which is the cyclic P-word of length n + m given for i ∈ �0, n − 1� by (uv)i = ui, and
+for i ∈ �n, m + n − 1� by (uv)i = vi−n.
+Given x ∈ M and u ∈ ΣP, we say that x admits an itinerary (un, tn)n∈Z if (tn)n is an
+increasing sequence in R, such that for any n ∈ Z and s ∈ [tn, tn+1], the point φs(x) lies in the
+cuboid un. We also say that x admits an itinerary along u. It implies that tn converges to −∞
+and +∞ when n goes to −∞ and +∞. Every point x ∈ M admits an itinerary, but there is no
+uniqueness in general. Additionally every u in ΣP corresponds to an itinerary of a point in M.
+Given a set I ⊂ Z and a P-word u: I → P, we say that x ∈ M admits a short itinerary along u
+if x admits an itinerary of the form (vn, tn)n where v is a bi-infinite P-word extending u.
+Let R be a Markov cuboid of a Markov partition.
+Take a finite P-word u: �0, n� → P
+satisfying u0 = R, and denote by Ru the set of points in ∂−R admitting a short itinerary
+along u. We denote by T u : Ru → R∗
++ the smooth map obtained the following way. For z ∈ Ru,
+the point z admits an itinerary of the form (wi, ti)i∈Z such that w�0,n� = u. We denote by
+T u(z) = �n−1
+i=0 ti, which satisfies that φT u(z)(z) is in ∂−un. The element T u(z) does not depend
+on the specific choice of w. We also denote by σu : Ru → un the map given by σu(x) = φT u(x)(x).
+Lemma 15. The following holds:
+• Ru is a closed horizontal sub-rectangle of ∂−R.
+• σu(Ru) is a closed vertical sub-rectangle of ∂−un.
+Proof. When n = 1, it is a consequence of the definition of Markov partitions. When n ≥ 2, it
+can be verified by induction on n.
+■
+We denote by πP : ΣP → M the encoding map given by πP(u) = x, where x is the unique
+point in M which admits an itinerary of the form (un, tn)n for some sequence (tn)n satisfying
+t0 = 0. In particular πP(u) lies in the rectangle ∂−u0. The map πP is well-defined, continuous,
+and sends the foliations W s
+l , W u
+l to the foliations W s
+R, W u
+R in each rectangle ∂−R for R ∈ P.
+2.1
+Suspension of a Markov partition
+Fix a Markov partition P of φ. Given u in ΣP, denote by TP(u) ∈ R+ the first return time
+from πP(u) ∈ ∂−u0 to πP ◦ σ(u) ∈ ∂+u0 ∩ ∂−u1 for φ. The map TP : ΣP → R+ is continuous
+since πP is continuous. We denote by (MP, φP) the suspension flow for the map σ and the
+time function TP. That is MP is the compact quotient ΣP × R/ ≃ where ≃ is the relation
+equivalence induced by (u, s) ≃ (σ(u), s−TP(u)). The flow φP is defined by φP
+t (u, s) = (u, t+s).
+Lemma 16. The map hP : MP → M given by hP(u, s) = φs(πP(u)) is well-defined and is a
+semi-conjugation between φP and φ. That is hP ◦ φP
+t = φt ◦ hP. Additionally it is continuous,
+and is injective on the set of point u satisfying that the orbit of hP(x) is disjoint from ∂suP.
+Denote by M(σ) and M(φP) the set of signed measures on ΣP and MP which are invariant
+by σ and φP. For any measure µ ∈ M(σ), we denote by µ ⊗ LebP ∈ M(φP) the measure which
+coincide locally with the product of µ and the Lebesgue measure on R. We define ΘP : M(σ) →
+M(φ), the surjective map given by ΘP(µ) = h∗
+P(µ ⊗ LebP). The map ΘP is used in Section 3 to
+relate a linking number between φ-invariant measures with a linking number between σ-invariant
+measures.
+Given a cyclic P-word u, Denote by �γu the periodic orbit of (u, 0) for φP, and by γu the
+periodic orbit of πP(u) for φ. We call the realization of u the map obtained as a restriction
+of πP to �γu → γu.
+15
+
+Lemma 17. Let u be a cyclic P-word. When πP(u) is on the stable or unstable boundary of P
+and have non-orientable stable and unstable leaves, the realization of u is a degree two covering
+map. Otherwise, it is a homeomorphism.
+Lemma 18. Let u, v be two cyclic P-words with u0 = v0. Then either u = v or πP(u) and
+πP(v) are on distinct stable and unstable leaves of W s,u
+R .
+Sketch of proof, for Lemma 17 and 18. The realization map is a cover since hP is a semi-
+conjugation. We give a sketch, the details are let to the reader.
+Take ǫ0 > 0, so that for each cuboid R, the two connected components of ∂sR are at distance
+at least 2ǫ, and similarly for ∂uR. Given a point x and 0 < ǫ < ǫ0, an ǫ-quadrant at x is defined
+to be a close set containing x, of the form C, where C is a connected component of B(x, ǫ)\∂suP.
+An ǫ1-quadrant and an ǫ2-quadrant are said equivalent if they contain a common ǫ3-quadrant
+for some ǫ3 > 0. An equivalence class of ǫ-quadrant is called a germ of quadrant at x.
+Given a periodic point x ∈ M and t ∈ R, we claim that φt gives a bijection between the
+germs of quadrant at x and the germs of quadrant at φt(x). To see that, notice that the stable
+boundary ∂sP is invariant by φt for t > 0, and ∂uP is invariant by φt for t < 0. Therefore
+for periodic points x ∈ ∂suP of periodic T , there exists a neighborhood U ⊂ ∂suP of x which
+satisfies φt(U) ⊂ ∂suP for any t ∈ [−T, T ]. It implies the claim.
+One can verify that the map sending (u, s) ∈ MP with s ∈ [0, TP(u)) to the germ of quadrant
+at hP(u, s) which intersects Int u0 ∩ B(hP(u, s), ǫ), is injective. If T denote the period of x, the
+map φT permutes the germ of quadrant at x. This permutation is the identity either when x
+is in the interior of a Markov cuboid, or if it has orientable stable and unstable leaves. In the
+remaining case, φT exchanges the germ of quadrant which are opposite relatively to x. So each
+germ of quadrant are 2T periodic.
+The injectivity of the previous map also proves the second Lemma (stable and unstable leaves
+contain at most one periodic orbit).
+■
+2.2
+Vertical and horizontal Order
+Here we suppose that the foliations Fss and Fuu are oriented.
+Each curve c in Fs or Fu,
+transverse to the flow, is endowed with an orientation, induced by the first return map from the
+curve c to any strong stable or unstable leaf. The set of leaves Leaf(W u
+R) is homeomorphic to a
+segment, and is endowed with a natural orientation coming from the orientation on Fss. Denote
+by ≤h the order on Leaf(W u
+R), coming from its orientation. If x, y are in R, we write x ≤h y
+if W u
+R(x) ≤h W u
+R(y), and x <h y if W u
+R(x) <h W u
+R(y) (<h is not the strict order on R coming
+from ≤h). We similarly define the orders ≤v on Leaf(W s
+R) and on R. The orders ≤h and ≤v
+are called the horizontal and vertical orders. Here we picture the stable foliation to be made of
+horizontal leaves, oriented to the right, and the unstable foliation to be made of vertical leaves
+oriented upward.
+Lexical Order in ΣP.
+We define in the same manner horizontal and vertical orders on ΣP. We
+do not simply lift the orders on R, because the map ΣP → M is not injective. Given some cuboids
+R0, R1, R2 in P, we say that R1 is below R2 relatively to R if Int ∂+R0 intersects both Int ∂−R1
+and Int ∂−R2, and the intersection Int ∂+R0 ∩ Int ∂−R1 is inferior to Int ∂+R0 ∩ Int ∂−R2 for
+the order ≤v on R.
+Denote by Σ+
+P and Σ−
+P the sets of P-word of the form u: �0, +∞� → P and respectively
+u: � − ∞, 0� → P. They correspond to the sets of leaves of the foliations W s
+l and W u
+l . They
+come with the distances induced by dP. We define on Σ+
+P the order ≤h defined by u ≤h v if and
+only if either u = v, or there exists n ≥ 0 satisfying that ui = vi for all 0 ≤ i ≤ n, and un+1 is
+16
+
+on the left of vn+1 relatively to vn = un. In particular, u and v are comparable for ≤h if and
+only if u0 = v0. We define similarly an order ≤h on Σ−
+P.
+Denote by p+, p− the natural maps ΣP → Σ+
+P and ΣP → Σ−
+P. We define the order ≤h, ≤v
+and strict order <h, <v on ΣP by pulling back the respective orders by the maps p+ and p−. One
+can verify that for u ∈ ΣP, the leaf W s
+R(πP(u)) ⊂ u0 depends only on p+(u). We denote by Σ+
+P,R
+the set of u ∈ Σ+
+P satisfying u0 = R. The map πP induces a map πs
+P : Σ+
+P,R → Leaf(W s
+R), given
+by πs
+P(u) = W s
+R(πP(v)) for any v ∈ ΣP with p+(v) = u. It follows from the construction that πs
+P
+is increasing for the orders ≤v.
+2.3
+Word combinatorial
+We prove a lemma which is essential to build primitive periodic orbits of an Anosov flow. Let A
+be a finite set and A∗ the free monoid generated by A. That is elements in A∗ are finite words
+(possibly empty) whose letters are inside A. For u ∈ A∗, we denote by |u| the length of u. We
+also denote by u ≤ v if u is a prefix of w, that is v ∈ u · A∗.
+Lemma 19. Let two elements x, y ∈ A∗ which are not powers of a common element in A∗.
+Then for all n, m ≥ 1, the cyclic P-word x2ny2m is not a power of another element.
+To prove the lemma, we use the following fact, which can be found in [Lot97, Corollary
+6.2.5]. If two words x, y ∈ A∗ satisfy a non-trivial relation, then they are power of a common
+word. By satisfy non-trivial relation, one should understand that the unique monoid morphism
+f : {a, b}∗ → A, satisfying f(a) = x and f(b) = y, is not injective. In particular the hypothesis
+of the lemma implies that xk and yl are distinct for all k, l ≥ 1.
+Proof. Assume the lemma true for n = m = 1 and use it to prove it for any n. Take u = xn and
+v = ym. If u and v are powers on a common a word, then uk = vl for some integers k, l ≥ 1,
+then x and y satisfy a non-trivial relation, so they are powers of a common element. It is not
+possible by assumption. So one can apply the case n = m = 1 to u and v, which proves the
+lemma for any other n, m.
+We prove now the case n = m = 1. Suppose that x2y2 = zk for some z ∈ A∗ and k ≥ 2.
+Without lost of generality, we can suppose |x| ≥ |y|. We consider several cases for the lengths
+of x and z, and prove that they are all impossible.
+Case 1. Suppose that |x| = |z|. Since zk and x2y2 are equal and start with z and x, one
+has z = x. It follows that some power of x is equal to some power of y, which is impossible.
+Case 2. Suppose that |x| > |z|. Since x2 ≤ zk, there exist u, v prefixes of z and integers
+1 ≤ a, b < k such that x = zau and x2 = zbv. Notice that b ≥ 2a ≥ a+1. We claim that xz = zx.
+Indeed z ≤ x, so xz ≤ x2 ≤ zk. Similarly we have zx = za+1u, so zx ≤ zbu ≤ zk when b = a + 1,
+and zx ≤ za+2 ≤ zb ≤ zk when b > a+1. Hence zx and xz have the same length and are prefixes
+of the same word. So xz = zx. It follows that x = tc and z = td for some t ∈ A∗ and c, d ≥ 1.
+Then we have y2 = tkd−2c, which contradicts that x and y are not power of a common element.
+We suppose now that |z| > |x| Sine 4|x| ≥ 2|x| + 2|y| = k|z|, it follows that k is equal
+to 2 or 3. Case 3. Suppose that |z| > |x| and k = 2. One has x ≤ z ≤ x2 ≤ z2, so there
+exists u, v ∈ A∗ such that x2 = zv and z = xu. It follows that x = uv. We have z2 = uvuuvu
+and x2y2 = uvuvy2, so vy2 = uvu. Thus y and u have the same length, and they are both suffix
+of the same word, so y = u. Then uvu = vy2 = vuu, so uv = vu. Hence u and v are power of a
+common word, which contradicts the assumption on x, y.
+Case 4. Suppose that |z| > |x| and k = 3. Similarly we can write x = uv and z = uvu.
+Notice that we have 2|y| = 3|z| − 2|x| = 4|u| + |v| ≤ 2|x| = 2|u| + 2|v|, so |v| ≥ 2|u|. On has
+17
+
+x2 = uvuv ≤ z2 = uvuuvu, so v ≤ uv. Denote by w the suffix of v satisfying uv = vw = x. Since
+le length of w is equal to |u| ≤ |v|/2, there exists an integer p ≥ 2 and t ∈ A∗ with |t| < |w| such
+that v = twp. We can write:
+z3 = uvuuvuuvu
+= uvuvwuvwu
+= x2wvw2u
+Hence y2 = wvw2u = wtwp+2u. Since we have |w| = |u| and |t| < |w| < (p+2)|w|, the middle
+index in wtwp+2u is inside the term wp+2 . So there exist two words a, b ∈ A∗ and q, r ∈ N such
+that w = ab, q + r + 1 = p + 2 and y = wtwqa = bwru. We can not have r = 0, otherwise
+|wtwqa| ≥ (q + 1)|w| ≥ 4|w| > 2|w| ≥ |bu|. Hence the words wtwqa bwru are equal and start
+with ab and ba. So ab = ba, and they are power of a common element s ∈ A∗. It follows the
+sequence of equalities:
+u = ba ∈ {s}∗
+wtwqa = bwru ∈ {s}∗ so t ∈ {s}∗
+v = twp ∈ {s}∗
+uv = vw so u ∈ {s}∗
+Hence x, y are powers of s, which is not possible. Therefore the last case is also impossible.
+■
+Lemma 20. Let x, y be two cyclic P-words with x0 = R = y0, such that x ̸= y. Then the
+point xy lies horizontally and vertically strictly between the points x and y.
+Proof. We mimic a classical argument used to find invariant points in the orbit space.
+We
+prove the result for the vertical order only. Denote by x′ = p+(x), y′ = p+(y). We suppose
+that x′ <v y′, the other case is similar.
+Consider the two maps fx, fy : Σ+
+P,R → Σ+
+P,R satisfying that fx(w) is the element in Σ+
+P
+obtained by concatenating x to the left of w, and similarly for fy. That is fx starts with the
+same first |x| elements than x, and σ|x| ◦ fx = id. Denote by d+
+P the distance on Σ+
+P,R given
+for a ̸= b by:
+d+
+P(a, b) = 2− inf{i≥0,ai̸=bi}
+Notice that fx and fx are increasing and Lipchitz for d+
+P and for the Lipchitz constants
+2−|x| and 2−|y|. Hence the map fx ◦ fy as a unique fixed point, which is p+(xy). The goal is
+to prove that the fixed point is between x′ and y′. Given two points a, b ∈ ΣP,R, we denote
+by [a, b]v the set of points between a and b for the order ≤v. For any point w ∈ [a, b]v, we have
+d+
+P(a, w) ≤ d+
+P(a, b). Indeed if n equals inf{i ≥ 0, ai ̸= bi}, then by definition of ≤h, w have the
+same (n − 1) coefficients than a. It follows that d+
+P(a, w) ≤ 2−n = d+
+P(a, b).
+Suppose that fx([x′, y′]v) is not included in [x′, y′]v.
+Then there exists w in [x′, y′]v for
+which fx(w) is not in [x′, y′]v. Since fx is increasing, we have x′ = fx(x′) ≤v fx(w). It follows
+that fx(w) > y′ since ≤v is a total order. So y′ lies in [x′, fx(w)v]. It implies the following
+inequality, which is a contradiction:
+2−|x|d+
+P(x′, w) ≥ d+
+P(x′, fx(w)) ≥ d+
+P(x′, y′) ≥ d+
+P(x′, w) > 0
+So [x′, y′]v is invariant by fx. Similarly it is invariant by fy and by fx ◦ fy. The set [x′, y′]v is
+compact and invariant by the contracting map fx ◦ fy, so fx ◦ fy admits fixed point in [x′, y′]v.
+By uniqueness of the fixed point, we have x′ <h p+(xy) <h y′.
+■
+18
+
+3
+Linking number of invariant measures
+We fix one transitive Anosov flow φ on a closed oriented 3-dimensional manifold M. Except in
+the following theorem, we also suppose that the stable and unstable foliations of φ are orientable.
+We extend the notion of linking number for invariant measures of φ. We construct successively
+maps linkff
+φ , linkf
+φ, linkφ which gives the linking number of two invariant measures under the
+hypothesis that respectively two, one or none of them are supported on finitely many periodic
+orbits.
+Consider Γ1 and Γ2 two invariant 1-cycles. We suppose that they are null-homologous and
+supported on disjoint sets of periodic orbits. We denote by linkff
+φ (LebΓ1, LebΓ2) the linking
+number between the corresponding invariant 1-cycles. That is if S1 is a 2-chain in M with ∂S1 =
+Γ1, then one has:
+linkff
+φ (LebΓ1, LebΓ2) = S1 ·∩Γ2
+Denote by M0(φ) the set of measures ν ∈ M(φ) which are null-cohomologous.
+Theorem 21. Let φ be a transitive Anosov flow on an oriented 3-manifold. There exists a unique
+continuous bilinear map linkφ : M0(φ) × M0(φ) → R which satisfies linkφ(ν, µ) = linkff
+φ (ν, µ)
+for all measures ν, µ ∈ M0(φ) which are supported on disjoint finite unions of periodic orbits,
+Additionally the map linkφ is symmetric.
+The proof is technically simpler when the foliations of φ are orientable, and can be deduced on
+the general case from the orientable case. Recall that M0
+p(φ) is the set of null-cohomologous in-
+variant probability measures. As a consequence of the continuity and the compactness of M0
+p(φ),
+we obtain the following.
+Corollary 22. Suppose that M0
+p(φ) is not empty. The map M0
+p(φ) → R defined as follows is
+well-defined and continuous:
+µ �→
+min
+ν∈M0p(φ) linkφ(µ, ν)
+The proof of the theorem follows two steps. First consider a null-homologous invariant 1-
+cycle Γ and a measure µ ∈ M0(φ). We define the linking number linkf
+φ(LebΓ, µ) roughly as the
+integral of the transverse measure µ⊥ on a 2-chain S bounding Γ. We actually lift S and µ in
+the manifold obtained by blowing up M along each periodic orbit in the support of Γ. Then the
+linking number linkf
+φ(LebΓ, µ) is well-defined for all µ, even when µ charges the support of LebΓ.
+In a second part, we use Markov partitions to give a precise description of the linking number.
+We build a map Σ2
+P → R which has a similar role to the linking form Given by Gauss and
+DeRham [DR84]. The difficulty is to prove that this map indeed computes linking numbers, and
+that its integral for measures ν ⊗ µ is continuous in (ν, µ).
+3.1
+Linking number linkf
+φ
+Fix a null-homologous invariant 1-cycle Γ. We define the linking number between LebΓ and
+any µ ∈ M0(φ).
+Blowup flow.
+Take a set ∆ ⊂ M, obtained as union of periodic orbits, containing the support
+of Γ. We consider the blowing up M∆ of M along each orbit of ∆, constructed bellow.
+Denote by X = ∂φt
+∂t the vector field generating φ. Given a periodic orbit γ in ∆, we denote
+by Nγ the normal vector bundle to γ. It is obtained as the set of equivalence classes of vectors v
+based on a point p in γ, non-parallel to X(p), up to the equivalence relation v1 ≃ v2 if there
+19
+
+exist a > 0, b ∈ R such that v1 = av2 + bX(p). The set M∆ is the union of M \ ∆ and of the
+circle bundle ∪γNγ, equipped with the atlas induced by M and by the angular sectors around ∆.
+We define the map π∆ : M∆ → M by π∆(x) = x for x ∈ M \ ∆, and π∆(v) = x if v is a vector
+based on x ∈ ∆.
+For a periodic orbit γ ⊂ ∆, we denote by Tγ = π−1
+∆ (γ) ⊂ M∆ the boundary component
+corresponding to γ. Since M is supposed orientable, Tγ is a torus. If the flow φ is of class Ck,
+k ≥ 1, it lifts to a Ck−1 flow φ∆ on M∆, by setting φ∆
+t (v) = dφt(v) for a vector v based on a
+point in ∆. We denote by X∆ = ∂φ∆
+t
+∂t
+the vector fields generating φ∆.
+Notice that on each torus Tγ, the restriction flow φ∆
+|Tγ has one or two periodic orbits corre-
+sponding to the stable direction, and one or two corresponding to the unstable direction. We
+refer to them as the stable and unstable orbits of Tγ. Every other orbit converges in the past
+toward one stable orbit, and in the future toward one unstable orbit. From this remark follows
+the lemma.
+Lemma 23. For any measure µ ∈ M(φ∆), the measure µ|Tδ is supported on the stable and
+unstable orbits of Tγ.
+For a given measure µ ∈ M(φ), one can build a measure µ∆ ∈ M(φ∆) so that π∗
+Pµ∆ = µ and
+whose total variation is the same as the total variation of µ. To do this, consider the invariant
+measure ν given by ν(A) = µ(A ∩ ∆). For any γ ⊂ ∆, take a measure νγ ∈ M(φ∆), supported
+by one of the stable orbits in Tγ, so that π∗
+Pνγ coincides with ν on γ. Then the measure µ∆ can
+be obtained as the sum of the measures νγ and of (((πP)| Int M∆)−1)∗µ.
+Linking number.
+Given an invariant measure µ ∈ M(φ∆), we denote by [µ]φ∆ ∈ H2(M∆, R)
+the cohomology class of µ for φ∆. Denote by M0(φ∆) the set of measure µ ∈ M(φ∆) satisfy-
+ing [π∗
+∆µ]φ = 0. We consider the Poincar´e duality P∆ : H1(M∆, R) → H2(M∆, ∂M∆, R). For a
+smooth map f : M∆ → R/Z, the element P∆([df]) is the relative homology class of the surfaces
+obtained as f −1(x) for any regular value x of f.
+Lemma 24. Given a measure µ ∈ M(φ∆) with [π∗
+∆µ]φ = 0, and ω ∈ H1(M∆, R), the quan-
+tity [µ]φ∆ · ω depends only on µ and on (π∆)∗(∂P∆(ω)) ∈ H1(∆, R).
+Proof. Let ω ∈ H1(M∆, R) be a 1-form which satisfies (π∆)∗(∂P∆(ω)) = 0. Denote by i the
+inclusion ∆
+i∗
+֒−→ M, so that H2(M, R)
+i∗
+−→ H2(M, Γ, R)
+∂−→ H1(Γ, R) is exact. Since ∂(π∆)∗(P∆(ω))
+is equal to (π∆)∗(∂P∆(ω)) = 0, there exists δ in H2(M, R) satisfying (π∆)∗(P∆(ω)) = i∗(δ).
+Denote by Q: H1(M, R) → H2(M, R) the Poincar´e duality.
+Similarly to P∆, Q can be
+described by Q([df]) = [f −1(x)] for any smooth map f : M → R/Z and any regular value x of f.
+Using this characterization, one can verify that the following diagram commute.
+H1(M, R)
+H2(M, R)
+H2(M, ∆, R)
+H1(M∆, R)
+H2(M∆, ∂M∆, R)
+Q
+π∗
+∆
+i∗
+P∆
+(π∆)∗
+It follows that (π∆)∗ ◦ P∆(ω) = (π∆)∗ ◦ P∆ ◦ π∗
+∆ ◦ Q−1(δ). The map (π∆)∗ so injective so we
+20
+
+have ω = π∗
+∆ ◦ Q−1(δ). Take α a smooth closed 1-form on M with [α] = Q−1(δ). Then we have:
+[µ]φ∆ · ω =
+�
+M∆
+ι(X∆)(π∗
+∆α)dµ
+=
+�
+M
+ι(X)αd(π∗
+∆µ)
+= [π∗
+∆µ]φ · α
+= 0
+since [π∗
+∆µ]φ = 0
+■
+Given µ ∈ M(φ∆) satisfying [π∗
+∆µ]φ = 0, we define link∆
+Γ (µ) = [µ]φ∆ · [α] for any smooth
+differential 1-form α with (π∆)∗(∂P∆(ω)) = [Γ] ∈ H1(Γ, R).
+Notice that (µ ∈ M0(φ∆)) �→
+link∆
+Γ (µ) is continuous.
+Lemma 25. Let µ ∈ M(φ∆) be an invariant measure, γ be a periodic orbit in ∆ and α be a closed
+1-form on Tγ. Then the integral
+�
+Tγ ι(X∆)αdµ depends only on µ(Tγ) and on [α] ∈ H1(Tγ, R).
+Proof. Thanks to Lemma 8, the integral
+�
+Tγ ι(X∆)αdµ does not depend on the choice of the closed
+1-form α in its cohomology class. Denote by dt the closed 1-form on γ satisfying ι
+�
+∂φt
+∂t
+�
+dt = 1.
+We denote by β⊥ = π∗
+∆dt and by β∥ a closed 1-form which is not null-cohomologous and such
+that the stable and unstable periodic orbits on Tγ are tangent to ker(β∥). Then ι(X∆)β∥ is zero
+on the support of µ|Tγ (see Lemma 23), so one has:
+�
+Tγ
+ι(X∆)β∥dµ = 0
+and
+�
+Tγ
+ι(X∆)β⊥dµ =
+�
+Tγ
+π∗
+∆(ι(X)dt)dµ
+=
+�
+Tγ
+1dµ
+= µ(Tγ)
+The lemma follows by linearity and from the fact that [β⊥] and [β∥] form a basis of H1(Tγ, R).
+■
+Lemma 26. Given a null-homologous invariant 1-cycle Γ and a measure µ ∈ M0(φ∆), the
+quantity link∆
+Γ (µ) depends only on Γ and π∗
+∆(µ) ∈ M0(φ). In particular, it does not depend
+on ∆.
+Proof. Consider a closed 1-form α and two measures µ1, µ2 ∈ M0(φ∆) satisfying π∗
+∆(µ1) =
+π∗
+∆(µ2). Since π∆ is injective outside ∆, the measure µ1 and µ2 coincide outside ∪γTγ. Ad-
+ditionally their restrictions to Tγ have the same weight. Hence according to Lemma 25, the
+integral of ι(X∆)α on Tγ is the same for µ1 and µ2. It follows that [µ1]φ∆ · [α] = [µ2]φ∆ · [α], and
+that the linking numbers are equal.
+We now prove that it does not depend on ∆. Denote ∆1 and ∆2 two choices of union of
+periodic orbit containing Γ. Denote by ∆3 = ∆1 ∪ ∆2. Take µ ∈ M0(φ), µ1 ∈ M0(φ∆1), µ3 ∈
+M0(φ∆3) satisfying π∗
+∆1µ1 = µ = π∗
+∆3µ3. Denote by π: M∆3 → M∆1 the blow-down projection,
+which satisfies π∆1◦π = π∆3. According to the previous case, we have link∆1
+Γ (π∗µ3) = link∆1
+Γ (µ1).
+21
+
+Fix a closed 1-form α on M∆1 satisfying (π∆1)∗◦(∂P∆1[α]) = [Γ] as in the definition of link∆1
+Γ .
+Notice that π∗α is a closed 1-form on M∆3 satisfying the same equality (π∆3)∗◦(∂P∆3[π∗α]) = [Γ].
+Hence one has:
+link∆3
+Γ (µ3) =
+�
+M∆3
+ι(X∆3)π∗αdµ3
+=
+�
+M∆1
+ι(X∆1)αdπ∗µ3
+= link∆1
+Γ (π∗µ3)
+= link∆1
+Γ (µ1)
+It follows that link∆1
+Γ (µ1) = link∆2
+Γ (µ2).
+■
+Definition 27. For a null-homologous invariant 1-cycle Γ in M, and µ ∈ M0(φ), we de-
+fine linkf
+φ(LebΓ, µ) as link∆
+Γ (ν) for any union of periodic orbit ∆ containing the support of Γ,
+and any measure ν ∈ M0(φ∆) satisfying π∗
+∆ν = µ.
+Denote by support(µ) the support of a measure µ.
+Proposition 28. Let ∆ be a finite union of periodic orbits.
+The map linkf
+φ is continuous
+on {ν ∈ M0(φ), support(µ) ⊂ ∆} × M0(φ).
+Additionally, we have linkf
+φ(LebΓ1, LebΓ2) =
+linkff
+φ (LebΓ1, LebΓ2) for any null-homologous invariant 1-cycle Γ1, Γ2 with disjoint support.
+Proof. Take a sequence (νn, µn)n in {ν ∈ M0(φ), support(µ) ⊂ ∆} × M0(φ), converging to-
+ward (ν∞, µ∞). Denote by Γn and Γ∞ the invariant 1-cycle corresponding to νn and ν∞. As
+explain in Remark 6, the total variations of νn and µn are bounded by some C > 0.
+We take a sequence (µ∆,n)n in M0(φ∆) so that π∗
+∆µ∆,n = µn and the total variation of µ∆,n is
+at most C. We can decompose µ∆,n as a difference of two invariant positive measures µ+
+∆,n, µ−
+∆,n,
+so that each of them as total mass at most C. By compactness, the sequence (µ+
+∆,n, µ−
+∆,n)n
+accumulates toward a pair of positive measures (µ+
+∆,∞, µ−
+∆,∞), which satisfies π∗
+P(µ+
+∆,∞−µ−
+∆,∞) =
+µ∞. So we have linkf
+φ(ν∞, µ∞) = link∆
+Γ∞(µ+
+∆,∞) − link∆
+Γ∞(µ−
+∆,∞).
+Since ∆ is a finite union of periodic orbits, there exists a continuous map Γ �→ ωΓ, so
+that ωΓ is in H1(M∆, R) and satisfies (π∆) ∗ (∂P∆(ωΓ)) = [Γ] ∈ H1(∆, R).
+The quanti-
+ties link∆
+Γn(µ±
+∆,n) = [µ±
+∆,n]φ∆ · ωΓn converges toward link∆
+Γ∞(µ±
+∆,∞). It follows that linkf
+φ(νn, µn)
+converges toward linkf
+φ(ν∞, µ∞).
+The last point is a consequence of Poincar´e duality. Take two invariant 1-cycles Γ1 and Γ2,
+an invariant 1-cycle �Γ2 = �
+i miγi in M∆ with π∗(�Γ2) = Γ2. Take a closed 1-form α on M∆ and
+a 2-chain S satisfying [S] = P∆[α] and (π∆)∗(∂S) = Γ1, then we have:
+�
+M∆
+ι(X∆)αd Leb�Γ2 =
+�
+i
+mi
+�
+γi
+α
+=
+�
+i
+miγi ·∩S
+by duality
+= Γ2 ·∩(π∆)∗(S)
+= linkff
+φ (LebΓ1, LebΓ2)
+■
+22
+
+As a consequence, one obtain the self-linking number of a null-homologous invariant 1-cycle
+as the linking number between a Seifert surface of the 1-cycle and the stable direction. The
+following lemma is used as a definition by Ghys [Ghy09].
+Lemma 29 (Self-linking number). Let Γ = �
+i miγi be a null-homologous invariant 1-cycle, S ⊂
+M be a 2-chain bounding Γ. For each i, denote by γs
+i a 1-cycle in Fs(γi) \ γi homologous to γi
+in Fs(γi), and Γs = �
+i miγs
+i . Then we have:
+linkf
+φ(LebΓ, LebΓ) = S ·∩Γs
+Proof. Consider ∆ the support of Γ and M∆ the blowing up manifold. We lift S to a 2-chain S∗ ∈
+H2(M∆, ∂M∆, R). We also lift γs
+i to a 1-cycle γs∗
+i
+in M∆. By hypothesis, there exists a 1-cycle δs∗
+i
+in the stable direction of the torus Tγi which is homologous to γs∗
+i . Then δs∗
+i
+is an invariant
+1-cycle and (π∆)∗δs∗
+i
+= γi. Denote by α a closed 1-form on M∆, dual to S∗. We have:
+linkf
+φ (LebΓ, LebΓ) = link∆
+Γ
+��
+i
+mi Lebδs∗
+i
+�
+=
+�
+i
+mi
+�
+M∆
+ι(X∆)αd Lebδs∗
+i
+=
+�
+i
+miS∗ ·∩δs∗
+i
+=
+�
+i
+miS∗ ·∩γs∗
+i
+= S∗ ·∩
+��
+i
+miγs∗
+i
+�
+= S ·∩Γs
+■
+3.2
+Linking function for a Markov partition
+In this subsection, we prove Theorem 21, by extending the linking number for two invariant
+measures. First we transfer the problem to a question on invariant measure by a sub-shift using
+a Markov partition. Given an invariant measure µ on ΣP, we associate to µ a measure N(µ)
+supported on finitely many periodic orbits. Then we give a measurable function L: Σ2
+P → R
+such that computing the linking number between µ − R(µ) and ν corresponds to integrating L
+for the measure µ ⊗ ν. We use that relation to define the linking number for any two measures,
+knowing what is the linking number between two invariant measures finitely supported. Then
+we prove the continuity of the linking number.
+We suppose that φ is an Anosov flow whose stable and unstable foliations are orientable.
+The main theorem in the non orientable case is proven using a degree two covering. We fix a
+Markov partition P for the flow φ. Denote by (R1, . . . , Rp) the family of Markov cuboids in P.
+We fix two orientations on the stable and unstable foliations, and consider the induced vertical
+and horizontal orders on ΣP.
+Invariant measure of σ.
+Recall that we denote by ΘP : M(σ) → M(φ) the map sending
+σ-invariant measures to their corresponding φ-invariant measure. For µ ∈ M(σ), we denote
+by [µ]φ = [ΘP(µ)]φ ∈ H2(M, R) the cohomology class corresponding to µ. We denote by M0(σ)
+the set in measure µ ∈ M(σ) satisfying [µ]φ = 0.
+23
+
+Definition 30. For two measures ν, µ ∈ M0(σ) finitely supported, we define the linking number
+of ν and µ by:
+linkff
+σ (ν, µ) = linkf
+φ(ΘP(ν), ΘP(µ))
+3.2.1
+Measure zero sets
+Because of the Portmanteau Lemma, it is useful to determine some sets which have zero measures
+for all measures in M(σ).
+A set X ⊂ ΣP is said to have uniformly zero measure if for
+all µ ∈ M(σ), one has |µ|(X) = 0. Notice that it is enough to satisfy the property on probability
+measures.
+Lemma 31 (Portmanteau Lemma for almost continuous functions). Let X be a metrizable
+topological space endowed with the Borel algebra. Let Y ⊂ X be a measurable set and f : X → R
+a bounded function, which is continuous on X \ Y . Given a sequence (µk)k of signed measures
+on X, weakly converging toward a measure ν, one has:
+lim sup
+k→+∞
+����
+�
+X
+fdµk −
+�
+X
+fdν
+���� ≤ 2|ν|(Y ) · ∥f∥∞
+A particular case is when |ν|(Y ) = 0, where we have convergence of the integrals of f.
+Proof. Take a distance d compatible with the topology on X. For t ∈ R+, denote by h+
+t : X → R
+the map defined by h+
+t (x) = supy∈X(f(y) − td(x, y)). The map h+
+t is well-defined since f is
+bounded, and t-Lipchitz as supremum of t-Lipchitz maps. Notice that we have f ≤ ht ≤ sup f ≤
+∥f∥∞ for all t. It follows that:
+lim sup
+k
+�
+X
+fdµk ≤ lim sup
+k
+�
+X
+h+
+t dµk =
+�
+X
+h+
+t dν
+For any x in X \ Y , f is continuous at x so when t goes to +∞, h+
+t (x) is decreasing and
+converges toward f(x). Hence for all k one has:
+lim sup
+t→+∞
+�
+X
+h+
+t dν =
+�
+X
+�
+lim sup
+t
+h+
+t
+�
+dν
+according to Fatou’s Lemma
+=
+�
+Y
+�
+lim sup
+t
+h+
+t
+�
+dν +
+�
+X\Y
+�
+lim sup
+t
+h+
+t
+�
+dν
+≤ |ν|(Y ) · ∥f∥∞ +
+�
+X\Y
+fdν
+≤ |ν|(Y ) · ∥f∥∞ +
+�
+X
+fdν −
+�
+Y
+fdν
+≤
+�
+X
+fdν + 2|ν|(Y ) · ∥f∥∞
+It follows that:
+lim sup
+k
+�
+X
+fdµk −
+�
+X
+fdν ≤ 2|ν|(Y ) · ∥f∥∞
+Using the map h−
+t (x) = infy(f(y) + td(x, y)), the same arguments prove that:
+− lim inf
+k
+��
+X
+fdµk −
+�
+X
+fdν
+�
+≤ 2|ν|(Y ) · ∥f∥∞
+■
+24
+
+We prove in the following two lemmas that two families of sets have uniformly zero measures.
+Define the sets:
+En+
+R
+= {u ∈ ΣP, un = R, and for all k > n, uk ̸= R}
+and
+En−
+R
+= {u ∈ ΣP, un = R, and for all k < n, uk ̸= R}
+We also define E±
+R = ∪n∈Z(En−
+R
+∪ En+
+R ), the set of P-word containing a R at some index,
+and admitting a half sub-word without R.
+Lemma 32. The set E±
+R has uniformly zero measure.
+Proof. Take a measure ν ∈ M(σ). By definition, we have σm(En+
+R ) = E(n+m)+
+R
+and En+
+R
+∩
+Em+
+R
+= ∅ when n ̸= m. Since ν is σ-invariant, one has ν(ΣP) ≥ �
+n∈Z ν(En+
+R ) ≥ �
+n ν(E0+
+R ).
+The measure ν is finite so ν(En+
+R ) = 0 for all n. Similarly ν(En−
+R ) = 0 and ν(E±
+R) = 0.
+■
+A second family of sets with uniformly zero measures are the union of stable and unstable
+leaves, after removing the periodic points. Take a point v ∈ ΣP. Its strong stable manifold W s(v)
+is the set of point w such that d(σn(v), σn(w)) −−−−−→
+n→+∞ 0. Notice that W s(v) is the set of P-words
+which coincide with v for large enough indexes.
+Lemma 33. Let ws ⊂ ΣP be a strong stable leaf. If ws contains a periodic point u, then it has
+a unique periodic point u and ws \ {u} has uniformly zero measure. Otherwise ws has uniformly
+zero measure.
+Proof. Suppose that u, v ∈ ws are periodic. Denote by n a common multiple of the periods of u
+and v. Then the distance between σkn(u) = u and σkn(v) = v goes to zero when k goes to +∞.
+Hence we have u = v.
+Suppose that ws contains a periodic point u, of period N ≥ 1.
+Denote by ws
+n = {v ∈
+ΣP, vn−1 ̸= un−1, ∀k ≥ n, vk = uk}, so that ws \ {u} = ∪nws
+n. The sets ws
+n are pairwise disjoints
+and σN(ws
+n) = ws
+n−N. So using the same argument as in the previous proof, the sets ws
+n−N(v)
+have uniformly zero measures. Then ws \ {u} has uniformly zero measure.
+Suppose now that ws contains no periodic point. If there would exist two distinct integers n <
+m ∈ Z and an element u in σn(ws) ∩ σm(ws), then u would be (m − n)-periodic after a certain
+index.
+So the limit limk σk(m−n)+n(u) would exist and be in ws.
+It is not possible so for
+any n ̸= m, we have σn(ws) ∩ σm(ws) = ∅. Then the previous argument shows that ws has
+uniformly zero measure.
+■
+3.2.2
+Reduction to finitely many orbits
+We denote by Prime(P) the set of periodic orbits of σ intersecting every rectangle at most once.
+That is u ∈ ΣP lies in Prime(P) if and only if it is periodic of period some T ≥ 1, and for
+any R ∈ P there is at most one i ∈ �1, T � such that ui = R. Notice that Prime(P) has a finite
+cardinal.
+Fix a cuboid R ∈ P. We call R-reduction map the map NR : ΣP → ΣP defined as follows.
+For n, m ∈ Z with n ≤ m, we denote Sn,m = {u ∈ ΣP, un = R = um, R ̸∈ u�n+1,m−1�}. Then
+we define:
+NR(u) =
+�
+u|�n,m−1�
+if u belongs in Sn,m for some n ≤ 0 and m ≥ 1
+u
+otherwise
+25
+
+In the first case, u|�n,m−1� is the unique periodic P-word of period m − n, which coincides
+with u on the set of indexes �n, m�. Notice that for all u ∈ ΣP, NR(u) coincides with u for the
+indexes 0 and 1. After giving a formula for N ∗
+Rµ for finitely supported measures µ, we prove the
+following. The map N ∗
+R induces a continuous map M(φ) → M(φ), which additionally preserves
+the cohomology class of measures on M(φ).
+Let u be a cyclic P-word. Denote by ηu ∈ M(σ) the measure �|u|−1
+k=0 1σk(u), which is the
+unique invariant measure supported on the orbit of u, normalized by ηu(u) = 1.
+Lemma 34. Let u be a cyclic P-word with u(0) = R. Denote by v1, . . . , vn the unique family of
+cyclic P-word which satisfies v1v2 · · · vn = u and v−1
+j (R) = {0} for all j. Then we have:
+N ∗
+Rηu =
+n
+�
+j=1
+ηvj
+Proof. If k ∈ �0, |u| − 1� satisfies �j−1
+i=1 |vi| ≤ k < � i = 1j|vi| for some j ∈ �1, n�, then we
+have NR ◦ σk(u) = σk−�j−1
+i=1 |vi|(vj). It follows that:
+N ∗
+R1σi(u) = 1NR◦σk(u)
+= 1
+σk−�j−1
+i=1 |vi|(vj)
+Hence we have �|u|−1
+k=0 N ∗
+R1σk(u) = �
+j
+�|vj|−1
+k=0
+N ∗
+R1σk(vj) = �
+j N ∗
+Rηvj.
+■
+Lemma 35. The map NR : ΣP → ΣP is continuous on ΣP \ E±
+R, and is measurable.
+Proof. The topology on ΣP is generated by the family of open sets of the form Cv = {u ∈
+ΣP, u|�−m,m� = v} for all m ∈ N and for P-words v: �−m, m� → P. We determine the reciprocal
+image of these sets by NR.
+When the letter R does not appear inside v, we have N −1
+R (Cv) = Cv. Suppose that v contains
+the letter R in position n1 ≤ 0 and n2 ≥ 1. We take n1 maximal and n2 minimal with these
+properties. When v is the restriction to [−m, m] of the periodic P-word of period v|�n1,n2−1�,
+then we have N −1
+R (Cv) = Cv|�n1,n2�. When v is not such a restriction, we have N −1
+R (Cv) = ∅.
+Lastly suppose that v contains the letter R in position n ≤ 0, but it does not contain this
+letter on positive positions. The last case is similar to this one. We can take n ≤ 0 maximal
+with vn = R. For u ∈ ΣP, the image NR(u) is in Cv if and only if one of the two following cases
+are satisfied:
+•
+u contains no R on positive positions, and u|�−m,m� = v. The set of such word u is a
+closed subset of ΣP, and is contained inside E±
+R.
+• For some k > m, u contains a R on position k, contains no R on positions [1, k −
+1], u|�−m,m� = v and u|�k−n−m,k� coincides with v|�−m,n�. For k fixed, the set of such
+word u is open.
+Hence in any case, N −1
+R (Cv) is a union of open sets and of one closed set contained in E±
+R.
+So it is measurable and continuous outside E±
+R.
+■
+Lemma 36. For all µ ∈ M(σ),the pushed forward measure N ∗
+Rµ is invariant by σ.
+26
+
+Proof. We prove that µ((NR ◦ σ)−1(A)) = µ((σ ◦ NR)−1(A)) by considering the intersection
+between (NR ◦ σ)−1(A) and two subspaces of ΣP, which are S1,1 = {u ∈ ΣP, u1 = R} and its
+complementary.
+For any element u in ΣP \ S1,1, we have σ ◦ NR(u) = NR ◦ σ(u). Indeed, when u contains
+no coefficient equal to R for all positive indexes or for all non-negative indexes, then NR(u) = u
+and σ(u) satisfies the same condition. When there exist n ≤ 0 maximal and m ≥ 2 minimal such
+that un = R = um, it follows from the definition that σ ◦ NR(u) = NR ◦ σ(u). Hence:
+µ((NR ◦ σ)−1(A) \ S1,1) = µ((σ ◦ NR)−1(A) \ S1,1)
+Consider a P-words u ∈ S1,1 with u1 = R. Denote by E±
+R = ∪n∈Z(En+
+R ∪ En−
+R ) the set of P-
+words which contains the letter R somewhere and admits an infinite half sub-word without R.
+According to Lemma 32, we have µ(E±
+R) = 0. Notice that S1,1 \ E±
+R = (⊔n≤0(Sn,1)) \ E±
+R =
+⊔m≥2(S1,m) \ E±
+R. For n ≤ 0, we define the homeomorphism ˜gn : Sn,1 �→ S1,2−n by ˜gn(u) =
+σ−n−1(u). Notice that NR ◦ σ(˜gn(u)) = σ ◦ NR(u), so ˜gn restricts to a homeomorphism gn : (σ ◦
+NR)−1(A) ∩ Sn,1 \ E±
+R �→ (NR ◦ σ)−1 ∩ S1,2−n \ E±
+R. It follows that:
+µ((σ ◦ NR)−1(A) ∩ S1,1) = µ((σ ◦ NR)−1(A) ∩ (S1,1 \ E±
+R))
+=
+�
+n≤0
+µ((σ ◦ NR)−1(A) ∩ (Sn,1) \ E±
+R)
+=
+�
+n≤0
+µ
+�
+σ−n−1 �
+(σ ◦ NR)−1(A) ∩ (Sn,1) \ E±
+R
+��
+since µ is σ-invariant
+=
+�
+n≤0
+µ
+�
+(NR ◦ σ)−1(A) ∩ (S1,2−n) \ E±
+R
+�
+= µ((NR ◦ σ)−1(A) ∩ S1,1 \ E±
+R)
+= µ((NR ◦ σ)−1(A) ∩ S1,1)
+Hence one has σ∗(N ∗
+Rµ)(A) = µ(N −1
+R
+◦ σ−1(A)) = µ(σ−1 ◦ N −1
+R (A)) = µ(N −1
+R (A)) =
+(N ∗
+Rµ)(A), concluding.
+■
+Lemma 37. The map N ∗
+R : M(σ) → M(σ) is continuous.
+Proof. Consider a sequence of measures µk ∈ M(σ) weakly converging toward the measure ν ∈
+M(σ).
+Take a continuous map g : ΣP → R.
+The map g ◦ NR : ΣP → R is bounded, mea-
+surable and continuous outside the set E±
+R which has uniformly zero measure. According to
+the Portmanteau Lemma 31, the sequence of integrals
+�
+gd(N ∗
+Rµk) =
+�
+g ◦ NRdµk converges
+toward
+�
+gd(N ∗
+Rν), which proves the continuity of N ∗
+R.
+■
+It is not essential for the proof of the main theorem to prove now that µ and N ∗
+Rµ are
+cohomologous.
+We prove latter a stronger statement when µ is supported by finitely many
+orbits. To be complete, we add a sketch of proof of the cohomologous statement in the general
+case.
+Proposition 38. For every measure µ ∈ M(σ), we have [N ∗
+Rµ]φ = [µ]φ. Additionally we have
+|N ∗
+Rµ|(ΣP) ≤ |µ|(ΣP).
+The statement on the total variation is clear. For x, y ∈ P, denote by Ax,y the set {v ∈
+ΣP, v0 = x, v1 = y}. Notice that for all u ∈ ΣP, NR(u) and u have the same coefficients at the
+indexes 0 and 1. Hence for ν = N ∗
+Rµ − µ, we have ν(Ax,y) = 0 for all x, y ∈ P. The proposition
+follows from the next lemma.
+27
+
+Lemma 39. For any measure ν ∈ M(σ) satisfying ν(Ax,y) = 0 for all x, y ∈ P, we have [ν]φ = 0.
+Proof. Take a smooth closed 1-form α on M.
+For each Markov cuboid R ∈ P, we denote
+by fR : R → R a smooth function for which dfR coincides with α. We denote by iR : ∂−R → ∂+R
+the map obtained by pushing ∂−R along the flow.
+Recall that MP is the suspension set over ΣP and hP : MP → M is the corresponding semi-
+conjugation. Denote by T : ΣP → R+ the suspension time and by M R
+P = {(x, t) ∈ ΣP × R, 0 ≤
+t < T (x)} the subset of MP corresponding to the cuboid R. We have:
+�
+M
+ι(X)αdΘ(ν) =
+�
+MP
+(ι(X)α) ◦ hPd(ν ⊗ Leb)
+=
+�
+R∈P
+�
+MR
+P
+(dfR(X)) ◦ hPd(ν ⊗ Leb)
+Using the local product of MP, the integral
+�
+MR
+P (dfR(X)) ◦ hPd(ν ⊗ Leb) is equal to
+�
+ΣR(fR ◦
+iR − fR) ◦ hP(x, 0)dν(x). We denote by f +, f −: ΣP → R the maps whose restrictions to ΣR are
+given respectively by fR ◦ iR ◦ πP and fR ◦ πP. Then:
+�
+M
+ι(X)αdΘ(ν) =
+�
+R∈P
+�
+ΣP,R
+(f +
+R − f −
+R )dν
+=
+�
+ΣP
+(f + − f −)dν
+=
+�
+ΣP
+(f + − f − ◦ σ)dν
+Notice that in a neighborhood of a point u ∈ ΣP, f + and f − ◦ σ are obtained from integrating α
+on the Markov sub-rectangle given by ∂+u0 ∩ ∂−u1. It follows that f + − f − ◦ σ is constant on
+each set Ax,y. Therefore the integral �
+Ax,y(f +−f −◦σ)dν is zero. Since the non empty sets of the
+form Ax,y give a partition of ΣP, we have
+�
+ΣP (f +−f −◦σ)dν. It follows that
+�
+M ι(X)αdΘ(ν) = 0
+for all α, and that [ν]φ = 0.
+■
+Denote by (R1, . . . , Rp) the cuboids of the Markov partition P. Consider the product of the
+maps �p
+i=1 fRi : ΣP → ΣP.
+Lemma 40. For any µ ∈ M(σ), the support of (�p
+i=1 fRi)∗ µ is included inside Prime(P).
+Proof. For a measure µ ∈ M(σ), denote by S(µ) ⊂ ΣP the support of µ. First notice that for
+any i and u ∈ ΣP, NRi(u) is either in Prime(P) or is equal to u. It follows that given a mea-
+sure µ ∈ M(σ), we have S(N ∗
+Riµ) ∪ Prime(P) ⊂ S(µ) ∪ Prime(P). Hence S(
+��k
+i=1 fRi
+�∗
+µ) ∪
+Prime(P) is a decreasing sequence in k.
+Take µ ∈ M(σ). Denote by AR the union of Prime(P) and of the set of P-words of the
+form u: Z → P \ R. For any u in ΣP, the element v = NR(u) is in one of the three cases:
+1.
+v is in Prime(P), if u is already in Prime(P), or if un = R = um for some n < 0, m ≥ 1,
+2.
+v = u contains no R if u contains no R,
+3.
+v = u if either u|�1,+∞� of u|�−∞,−1� contains no R.
+The set of u in the third case has uniformly zero measure according to Lemma 32.
+So
+the support S(N ∗
+Rµ) is included in AR. It follows from the above discussion that the support
+of (�p
+i=1 fRi)∗ µ is included in ∩RAR, which is equal to Prime(P).
+■
+28
+
+Density of finitely supported measures.
+Lemma 41. Given ν ∈ M(σ), there exists a sequence of measures (νn)n converging toward ν,
+so that for all n:
+•
+νn is finitely supported,
+•
+[νn]φ = [ν]φ,
+•
+|νn|(ΣP) ≤ |ν|(ΣP).
+Additionally if ν is positive, νn can be taken positive.
+Proof. For n ≥ 1, we denote by Pn the set of P-words of the form u: �−n, n� → P.
+An
+element u ∈ Pn correspond to a sub-cuboid of u−n, consisting of points admitting a short
+itinerary along σ−n(u).
+Then Pn corresponds to the Markov partition of φ given by these
+cuboids.
+Denote by Σn the set of bi-infinite Pn-words, and σn the shift map on Σn. We have a natural
+map πn : Σn → ΣP, so that for v ∈ Σn, πn(v) and i ∈ Z, we have (πn(v))i = (vi)0. Notice
+that πn is continuous bijective, so ν ∈ M(σn) → π∗
+nν ∈ M(σ) is continuous and bijective.
+For n and R ∈ Pn, we denote by N n
+R : Σn → Σn the R-reduction map for the Markov
+partition Pn of φ.
+Given n, we fix an order on Pn and denote by Nn the product of the
+maps N n
+Ri for all Ri ∈ Pn. Here the product is ordered using the previously fixed order.
+We claim that the sequence πn ◦ Nn ◦ π−1
+n
+converges uniformly toward the identity. Notice
+that for any u ∈ Σn, we have Nn(u)0 = u0. It follows that for v ∈ ΣP, πn ◦ Nn ◦ π−1
+n (v) coincide
+with v on �−n, n�. Hence πn ◦ Nn ◦ π−1
+n (v) converges uniformly toward v.
+Take ν ∈ M(σ) and define the invariant measure νn = (πn ◦ Nn ◦ π−1
+n )∗ν. From the claim
+follows that the sequence νn converges toward ν.
+The second and third points follow from
+Proposition 38.
+■
+Lemma 42. Given a measure ν ∈ M(σ) and a finite set ∆ ⊂ ΣP, there exists a sequence of
+measures (νn)n converging toward ν, so that for all n:
+•
+νn has a finite support which is disjoint from ∆,
+•
+[νn]φ = [ν]φ.
+Proof. Given a cyclic P-word w, denote by ηw the invariant measure �|w|−1
+i=0
+1σi(w). Thanks to
+Lemma 41, it is enough to prove the lemma when we have ν = ηv for some cyclic P-word v.
+We fix a cyclic P-word u containing all cuboids of P, so that u0 = v0. We additionally suppose
+that u and v are not power of a common cyclic P-word. The lemma follows from the two claims:
+Claim 1. The sequence ( 1
+kηvku)k≥0 converges weakly toward the measure ηv.
+Claim 2. The measure 1
+k(ηvku − ηu) is cohomologous to ηv.
+Proof of Claim 1.
+The sequence σi(vku) converges toward σi(v) when k goes to +∞ and
+√
+k ≤
+i ≤ k|v| −
+√
+k. So the Dirac measures on σi(vku) converges toward the Dirac measures on σi(v)
+for all sequences of pairs (k, i) with k → +∞. Hence 1
+kηvku converges toward ηv.
+□
+Proof of Claim 2.
+According to Lemma 34, we have N ∗
+R(ηvku − ηu) = kN ∗
+Rηv, and the map N ∗
+R
+preserves the cohomology according to Proposition 38.
+□
+■
+29
+
+u
+w
+σ|u|(w)
+w′
+σ|u|(w′)
+x
+x′
+R−
+u
+R+
+u
+Figure 2: Illustration of (w, x) ∈ R−
+u and (w′, x′) ∈ R+
+u .
+3.2.3
+Linking function
+Denote by ΣR the set of u ∈ ΣP satisfying u0 = R.
+We construct a measurable func-
+tion LR : ΣR × ΣR → R, such that for all measures ν, µ ∈ M(σ) supported on finitely many
+orbits, one has linkσ(ν − N ∗
+Rν, µ) = �
+Σ2
+R LRdν ⊗ µ. Then we prove the continuity in (ν, µ).
+Denote by PrimeR the set of cyclic P-word u with u0 = R and ui ̸= R for all i ̸= 0.
+For u ∈ PrimeR, We denote by Σu the set of w ∈ ΣP which satisfy wk = uk for all k ∈ �0, |u|�.
+Notice that for w ∈ Σu, the element σ|u|(w) is in ΣR. For u ∈ PrimeR, we define the two
+following measurable sets:
+R+
+u = {(w, x) ∈ Σu × ΣR, such that:
+u <h x and σ|u|(w) ≤v x <v w}
+R−
+u = {(w, x) ∈ Σu × ΣR, such that:
+u <h x and w ≤v x <v σ|u|(w)}
+The lemma follow from the definition. We call linking function the function LR : Σ2
+P → R
+defined by:
+LR =
+�
+u∈P rimeR
+�
+1R+
+u − 1R−
+u
+�
+Here 1X is the characteristic map of the set X. From the definitions follows the next lemma.
+Lemma 43. For any u ∈ PrimeR, the following holds:
+1. For any (w, x) ∈ R+
+u , we have w <v u,
+2. For any (w, x) ∈ R−
+u , we have w >v u,
+3.
+R+
+u ∩ R−
+u = ∅,
+4.
+LR is measurable,
+5.
+∥LR∥∞ = 1.
+The coming theorem and property are critical to define the linking number between two
+invariant measures.
+30
+
+Theorem 44. For ν, µ ∈ M(σ) be finitely supported, with [µ]φ = 0. Then we have:
+linkff
+σ (ν − N ∗
+Rν, µ) =
+�
+Σ2
+R
+LRdν ⊗ µ
+The proof of the theorem is postponed to next subsection.
+Proposition 45. The map M(σ)×M(σ) → R given by (ν, µ) �→
+�
+Σ2
+R LRd(ν ⊗µ) is continuous.
+To prove the proposition, we first estimate the integral
+�
+Σ2
+R LRd(ν ⊗ µ) in three estimations
+lemmas.
+Lemma 46 (Estimation Lemma 1). For any ǫ > 0, there exists A ∈ N, such that for all
+measures ν, µ ∈ M(σ), we have:
+|ν ⊗ µ|
+
+
+�
+u∈P rimeR,|u|≥A
+R+
+u
+
+ ≤ ǫ|ν ⊗ µ|(Σ2
+P)
+For u in PrimeR and k ≥ 0, we denote by uk the concatenation of k copies of u, with u0 = R
+by convention. We define the sets:
+Σu,k = Σuk \ Σuk+1
+and
+R+
+u,k = {(w, x) ∈ R+
+u , w ∈ Σu,k}
+Recall that we denote W s
+l (v) = {w ∈ ΣP, for all k ≥ 0, wk = vk}. Notice that Σu \ W s
+l (u)
+is the union of Σuk \ Σuk+1 for all k ≥ 1. Given w in W s
+l (u), we do not have σ|u|(w) <v w nor
+σ|u|(w) >v w, so no (w, x) can be in R+
+u ∪ R−
+u . Therefore the sets R+
+u and R−
+u are the unions of
+the set R+
+u,k and R−
+u,k for all k ≥ 1.
+Lemma 47 (Estimation Lemma 2). For all ǫ > 0 and u ∈ PrimeR, there exists B ∈ N such
+that for all measures ν, µ ∈ M(σ), we have:
+|ν ⊗ µ|
+
+ �
+k≥B
+R+
+u,k
+
+ ≤ ǫ|ν ⊗ µ|(Σ2
+P)
+Lemma 48 (Estimation Lemma 3). Given u ∈ PrimeR and k ≥ 1, the quantity (ν ⊗ µ)(R+
+u,k)
+is continuous in (ν, µ) ∈ M(σ) × M(σ).
+Proof of Proposition 45. Assume the three estimation lemmas. Let (νn, µn)n be a sequence of
+measures in M(σ), converging toward a measure (ν∞, µ∞). We should prove that �
+Σ2
+R LRd(νn ⊗
+µn) converges toward
+�
+Σ2
+R LRd(ν∞ ⊗ µ∞).
+According to Remark 6, the total variations of the measures νn ⊗ µn are bounded by some
+C > 0. Take ǫ > 0 and A ∈ N given by the first estimation lemma for ǫ. Denote by N ∈ N the
+number of element u ∈ PrimeR with |u| < A. Take B ∈ N the maximal integer given by the
+31
+
+second estimation lemma, applied for all u with |u| < A and for ǫ/N. We have:
+lim sup
+n→+∞
+�
+νn ⊗ µn
+�
+�
+u∈P rimeR
+R+
+u
+��
+= lim sup
+n
+
+νn ⊗ µn
+
+
+�
+|u|<A,k<B
+R+
+u,k
+
+
++ νn ⊗ µn
+
+ �
+|u|≥A
+R+
+u
+
+
++νn ⊗ µn
+
+
+�
+|u|<A,k≥B
+R+
+u,k
+
+
+
+
+≤ lim sup
+n
+
+νn ⊗ µn
+
+
+�
+|u|<A,k<B
+R+
+u,k
+
+
+
+
++ Cǫ + N · Cǫ
+N
+≤ ν∞ ⊗ µ∞
+
+
+�
+|u|<A,k<B
+R+
+u,k
+
+ + 2Cǫ
+according to the estimation Lemma 3
+≤ ν∞ ⊗ µ∞
+��
+u
+R+
+u
+�
++ 4Cǫ for all ǫ > 0
+≤ ν∞ ⊗ µ∞
+��
+u
+R+
+u
+�
+lim inf
+n
+�
+νn ⊗ µn
+�
+�
+u∈P rimeR
+R+
+u
+��
+≥ ν∞ ⊗ µ∞
+��
+u
+R+
+u
+�
+Hence (νn ⊗ µn)(∪uR+
+u ) converges toward (ν∞ ⊗ µ∞)(∪uR+
+u ). The same statement holds
+for R−
+u . Therefore the integrals
+�
+Σ2
+R LRd(νn ⊗ µn) converges toward
+�
+Σ2
+R LRd(ν∞ ⊗ µ∞).
+■
+Now we prove the estimation lemmas.
+Proof of Lemma 46. Given A ∈ N, denote by LA the union of the sets Σu for u ∈ PrimeR
+with |u| ≥ A. Notice that if w is in LA, then w0 = R and wk ̸= R for all k ∈ �1, A − 1�.
+Hence for two distinct indexes i, j ∈ �0, A − 1�, the set σi(LA) and σj(LA) are disjoint. For a
+measure ν ∈ M(σ), we have:
+A|ν|(LA) =
+A−1
+�
+i=0
+|ν|(σi(LA)) ≤ |ν|(ΣP)
+So |ν|(LA) ≤ |ν|(ΣP)
+A
+. We take A satisfying 1 ≤ Aǫ. Then we have:
+|ν ⊗ µ|
+�
+∪|u|≥AR+
+u
+�
+≤ |ν ⊗ µ| (LA × ΣP) ≤ ǫ|ν ⊗ µ|(Σ2
+P)
+■
+32
+
+Proof of Lemma 47. Take (w, x) in R+
+u,k. Then both w and σ|u|(w) are in Σuk−1 \ Σuk+1. By
+definition, we have σ|u|(w) ≤v x <v w, so we also have x ∈ Σuk−1. According to Lemma 43, we
+have x <v w <v u. So if x is in Σuk+1, u is in Σuk+1, so w is also in Σuk+1. Since w is not is
+this set, x lies to Σuk−1 \ Σuk+1. Take B > 1. Notice that we have σ|u|(Σuk) ⊂ Σuk−1, so given i
+with 0 ≤ 2i < B − 1, one has:
+(id ×σ2i)(R+
+u,k) ⊂ (Σuk \ Σuk+1) × (Σuk−1−2i \ Σuk+1−2i)
+It follows that the sets (id ×σ2i)(∪k≥BR+
+u,k) for 0 ≤ 2i < B − 1 are pairwise disjoint. For
+any ν, µ ∈ M(σ), we have (ν ⊗ µ)
+�
+(id ×σ2i)(∪k≥BR+
+u,k)
+�
+= (ν ⊗ µ)
+�
+∪k≥BR+
+u,k
+�
+, so using a
+similar argument to above, we have:
+(ν ⊗ µ)
+�
+∪k≥BR+
+u,k
+�
+≤ |ν ⊗ µ|(Σ2
+P)
+⌊ B−1
+2 ⌋
+■
+Denote by Sk = {w ∈ Σu,k, w <v u, u <h w} for k ≥ 1.
+Lemma 49. We have Sk <v Sk+1 <v u for all k ≥ 1.
+Proof. Take w1 ∈ Sk and w2 ∈ Sk+1. Suppose that w2 ≤v w1. So w1 lies vertically between w1
+and u, which start by uk+1. By definition of the vertical order, w1 starts by uk+1, which is not
+possible since Σu,k = Σuk \ Σuk+1.
+■
+We prove the estimation Lemma 3: ν ⊗ µ(R+
+u,k) is continuous in (ν, µ) ∈ M(σ)2. The idea
+is to use the Portmanteau Lemma. The boundary of R+
+u,k can have a positive measure, so we
+need a first step before using the Portmanteau Lemma. For that we notice that a measure ν ⊗ µ
+charges ∂R+
+u,k only on pairs (z, z) and (z, σ|u|(z)), where z is periodic. We cut a neighborhood
+of (z, z) in R+
+u,k and glue it on (z, σ|u|(z)), to remove these points from the boundary of ∂R+
+u,k.
+Then we use the Portmanteau Lemma.
+Proof of Lemma 48. Let (νn, µn)n be a sequence of measures in M(σ) converging toward a
+measure (ν∞, µ∞). We use a Portmanteau Lemma to prove the convergence of the integral.
+Take ǫ > 0 and k ≥ 1. We can take a subset Xǫ ⊂ Sk for which the sum of |ν∞|({x}) for x ∈
+Sk \ Xǫ atom of |ν∞|, is less that ǫ. That is:
+�
+x∈Sk\Xǫ
+|ν∞|({x}) ≤ ǫ
+For each z ∈ Xǫ, we take a small neighborhood Tz ⊂ ΣP of z. We can take it such that the
+following are satisfied:
+Claim 1. There exist neighborhoods Tz of the points z ∈ Xǫ satisfying the following:
+1.
+∂Tz = ∅,
+2. the sets Tz and the sets σ|u|(Tz) all z ∈ Xǫ are pairwise disjoint,
+3.
+σ|u|(Tz) <v σ(Tz),
+4. σ|u|(Tz) >h u for all z ∈ Xǫ.
+33
+
+Proof of Claim 1.
+We take Tz = {w ∈ ΣP, for all |k| ≤ N, wk = zk} for some N > 0. It is open
+and close, so it has no boundary. If we prove, the points 2, 3 and 4 when replacing Tz by z, then
+points 2 to 4 will be satisfied when taking N large enough. If z is in Xǫ ⊂ Sk, then σ|u|(z) is not
+in Σuk, so it is not in Sk. Hence the set Xǫ and σ|u|(Xǫ) are pairwise disjoint. Since z is in Σu
+and z <v u, we have σ|u|(z) <v z according to Lemma 49. For z ∈ Xǫ, we have u <h z, and u
+and z coincide on �0, |u|�. Hence we have u <h σ|u|(z). Therefore the points 2 to 4 are satisfied
+for large N.
+□
+We define the sets:
+T =
+� �
+z∈Xǫ
+Tz × Tz
+� �
+R+
+u,k
+and
+R′ =
+�
+R+
+u,k \ T
+�� �
+id ×σ|u|�
+(T )
+Claim 2.
+R+
+u,k \ T and (id ×σ|u|)(T ) are disjoint, so ν∞ ⊗ µ∞(R+
+u,k) = ν∞ ⊗ µ∞(R′)
+Proof of Claim 2.
+When (w, x) is in R+
+u,k we have σ|u|(w) ≤v x. When (w, x) is in (id ×σ|u|)(T ),
+we have (w, σ−|u|(x)) is in T ⊂ R+
+u,k, so σ−|u|(x) <v w and x <v σ|u|(w).
+□
+We now prove that the boundary of R′ ⊂ Σ2
+P has small measure for ν∞ ⊗ µ∞. Notice that
+for z in Xǫ, the point (z, z) is in the interior of T , so it is not in the adherence of R′.
+Claim 3. The set Tz × σ|u|(Tz) is included in R′.
+Assume the claim. Denote x ≡v w when we have x ≤v w ≤v x, that is x ∈ W s
+l (w). The
+boundary ∂R′ is contained in union of the three sets:
+1.
+{(w, x) ∈ Σu,k × ΣR, x ≡h u, u <h x}
+2.
+{(w, x) ∈ (Σu,k \ Xǫ) × ΣR, x ≤v u, u ≤h x, x ≡v w}
+3.
+{(w, x) ∈ (Σu,k \ Xǫ) × ΣR, x ≤v u, u ≤h x, x ≡v σ|u|(w)}
+The measure for |ν∞ ⊗ µ∞| of the first set is zero (see Lemma 33). For the two other sets,
+we have:
+|ν∞ ⊗ µ∞| ({(w, x) ∈ (Σu,k \ Xǫ) × ΣR, x ≤v u, u ≤h x, x ≡v w})
+=
+�
+w∈Σu,k\Xǫ
+��
+x∈W s
+l (w)
+1x≤vu,u≤hxd|µ∞|(x)
+�
+d|ν∞|(w)
+=
+�
+w∈Σu,k\Xǫ
+w periodic
+��
+x∈W s
+l (w)
+1x≤vu,u≤hxd|µ∞|(x)
+�
+d|ν∞|(w) according to Lemma 33
+=
+�
+w∈Σu,k\Xǫ
+w periodic
+1w≤vu,u≤hw|µ∞|({w})d|ν∞|(w)
+=
+�
+w∈Sk\Xǫ
+|µ∞|({w})d|ν∞|(w)
+≤ ǫ|ν∞|(ΣP)
+34
+
+A similar argument shows that the third sets has measure at most ǫ|ν∞|(ΣP). It follows from
+the Portmanteau Lemma 31 that we have:
+lim sup
+n
+|νn ⊗ µn(R′) − ν∞ ⊗ µ∞(R′)| ≤ 2ǫ|ν∞|(ΣP)
+This holds for all ǫ > 0, and ν⊗µ(R′) = ν⊗µ(R+
+u,k) for all ν, µ ∈ M(σ). Therefore νn⊗µn(R+
+u,k)
+converges in n toward ν∞ ⊗ µ∞(R+
+u,k).
+Proof of Claim 3.
+Recall that for x ∈ σ|u|(Tz), we have u <h x. If (w, x) is in Tz × σ|u|(Tz)
+and satisfies w ≤v σ−|u|(x), then we have σ|u|(w) ≤v x <v w.
+The second equation is due
+to σ|u|(Tz) <v σ(Tz). In that case, (w, x) lies in R+
+u,k and not in T , so (w, x) ∈ R′.
+If (w, x) ∈ Tz × σ|u|(Tz) satisfies σ−|u|(x) <v w, we also have σ|u|(w) ≤v σ−|u|(x).
+In-
+deed σ|u|(w) belongs to Sk−1, σ−|u|(x) belongs to Sk, and Sk >v Sk−1 according to Lemma 49.
+Hence (w, σ−|u|(x)) is in both R+
+u,k and T 2
+z , so it lies in T . Hence (w, x) is in R′.
+□
+■
+3.2.4
+Adapted cells
+We prove Theorem 44, that is the linking number linkff
+σ (N ∗
+Rν − ν, µ) is equal to
+�
+Σ2
+R LRdν ⊕ µ
+when µ and ν are finitely supported. Denote by ∆n the n-dimensional simplex. We call n-chains,
+the abstract finite sums of continuous maps ∆n → ∂−R, with coefficients in R. We also denote
+by ∂ the boundary map. A differential embedding ∆1 → ∂−R is said adapted if it is either
+transverse to the stable and the unstable foliations, or is tangent to one of them. A 1-chain is
+said adapted if it is the sum of adapted embeddings ∆1 → ∂−R. For λi ∈ R and fi : ∆2 → ∂−R,
+the 2-chain c = �n
+i=1 λifi is said adapted if the maps fi are orientation preserving immersions
+and ∂fi are adapted. Here ∆2 is endowed with the anti-clockwise orientation.
+Recall that W s
+R and W u
+R are the stable and unstable foliations on ∂−R. We identify the bi-
+foliated rectangle (∂−R, W s
+R, W u
+R) with a sub rectangle of (R2, H, V ), with a C1 diffeomorphism
+preserving the orientations of the foliations.
+Let C be an adapted 2-chain.
+We define the
+support function of C has SuppC : ∂−R → R defined for x ∈ R ⊂ R2 by SuppC(x) =
+limt→0+ C ·∩(x + (−t, t2)).
+Let c be an adapted 1-cycle.
+We define Suppc = SuppC for any
+2-chain C satisfying ∂C = c.
+Remark 50. Given an adapted 2-cell C, if either ∂C = 0 or C is contained in finitely many
+stable leaves and one unstable leave, then SuppC = 0.
+Reducing a cyclic P-word
+We fix a primitive cyclic P-word r with r0 = R. Denote by ηr
+the measure ηr = �|r|−1
+k=0 1σk(r). We construct an adapted 2-cell to prove the Theorem 44 for
+the measure ηr. Write r = s1 ◦ . . . ◦ sn where all si are cyclic words such that (si)−1(R) = {0}.
+According to Lemma 34, we have N ∗
+Rηr = �n
+j=1 ηsj.
+We suppose that n ≥ 2. For i ∈ �0, n − 1�, we denote by ri = si ◦ . . . ◦ sn ◦ s1 ◦ . . . ◦ si−1.
+Notice that ri and si are different since n > 1. So thanks to Lemma 18, we can find a smooth
+curve ai : [0, 1] → ∂−R, from πP (ri) to πP (si), which is additionally transverse to the stable
+and unstable foliations. Since the set of periodic orbits is countable, we can take the curve ai
+such that its interior is disjoint from all period orbits.
+Recall that when x ∈ ∂−R admits an itinerary starting with u0, u1, · · · , u|u| = u0, then T u(x)
+is the time realizing this itinerary on x, and σu(x) = φT u(x)(x). Denote by bi : [0, 1] → ∂−R the
+curve given by bi(t) = σsi ◦ ai(1 − t), and define the adapted 2-chain c = �
+i(ai + bi).
+35
+
+Suppc = 1
+Suppc = −1
+a1
+b1
+a2
+b2
+a3
+b3
+Figure 3: An adapted cell c = �
+i(ai + bi) and its support Suppc on the left. On the right is
+represented the corresponding 2-chain in M, bounded the invariant 1-cycle.
+Lemma 51. For any cyclic P-word r and any measure µ ∈ M0(σ) finitely supported, so that
+the support of ΘP(µ) is disjoint from the support of ΘP(ηr), we have:
+linkff
+σ (ηr − N ∗
+Rηr, µ) =
+�
+ΣR
+Suppc ◦ πPdµ
+The map Suppc is used to build a 2-chain bounding the invariant 1-cycle corresponding
+to ηr − N ∗
+Rηr. It comes with a choice for the curves ai and bi. We remove later that choice by
+replacing c by another adapted 1-cycle. We denote by vi = ai(0) = bi−1(1), wi = ai(1) = bi(0).
+Proof. Given a periodic P-word u, we denote by γu ⊂ M the invariant 1-cycle corresponding to
+the periodic orbit of πP(u). According to Lemma 17, the realization of u has degree one (here
+the foliations are orientable). Hence we have ΘP(ηu) = Lebγu. We also write Γ = γr − �
+i γsi.
+The Lemma is clear when r does not contain R, since N ∗
+Rηr = ηr in that case. Otherwise
+write r = s1 ◦ . . . ◦ sn as above. When n = 1 we also have N ∗
+Rηr = ηr.
+We suppose now n ≥ 2. Consider the curves ai, bi and c = �
+i(ai + bi) as above. Denote
+by C an adapted 2-chain in ∂−R, such that ∂C = c. We also denote by ei ⊂ M the 2-chain
+obtained as the image of the map [0, 1] × [0, 1] → M, (s, t) �→ φtT si ◦ai(s)(ai(s)). Its boundary
+is ∂ei = ai + φ[0,T si(wi)](wi) + bi − φ[0,T si(vi)i](vi). Define a 2-chain by S = �
+i ei − C, illustrated
+in Figure 3.
+Claim 1. We have ∂S = Γ.
+Take an invariant 1-cocycle Γ′ = �
+γ m(γ)γ, whose support is disjoint from the support of Γ.
+By definition of the linking number linkff
+φ , we have linkff
+φ (LebΓ, LebΓ′) = S ·∩Γ′. By construction
+of ai, ei is disjoint from Γ′, so:
+linkff
+φ (LebΓ, LebΓ′) = C ·∩Γ′
+=
+�
+γ
+m(γ)
+�
+x∈γ∩∂−R
+d ·∩x
+=
+�
+γ
+m(γ)
+�
+x∈γ∩∂−R
+Suppc(x)
+=
+�
+∂−R
+Suppc Leb⊥
+Γ′
+=
+�
+ΣR
+Suppc ◦ πPdµ
+36
+
+The last equality is due to the following. The measures Leb⊥
+Γ′ and π∗
+Pµ coincide on the interior
+of ∂−R. On the boundary of ∂−R, the support supc is equal to zero, maybe outside finitely
+many points wi and vi. These points are in the support of ηr, so by hypothesis they are not in
+the support of µ. Therefor we have
+�
+∂−R Suppcdπ∗
+Pµ =
+�
+∂−R Suppc Leb⊥
+Γ′.
+Proof of Claim 1.
+Notice that vi = πP(ri), so we have:
+φT si (vi)(vi) = φT si (πP(ri))
+= πP(σ|si|(ri))
+= πP(ri+1)
+= ai+1(0)
+Similarly we have φT si (wi)(wi) = wi. By construction, we have:
+∂S =
+�
+i
+∂ei − ∂C
+=
+�
+i
+�
+ai + bi + φ[0,T si(wi)](wi) − φ[0,T si (vi)i](vi)
+�
+− c
+=
+�
+i
+φ[0,T si(vi)i](vi) −
+�
+i
+φ[0,T si(wi)](wi)
+= φ[0,T r(v1)](v1) −
+�
+i
+φ[0,T si(wi)](wi)
+= γr −
+�
+i
+γsi
+□
+■
+We denote by w−
+i = W s
+R(vi) ∩ W u
+R(wi) and w+
+i = W s
+R(vi+1) ∩ W u
+R(wi). We write [a, b]s the
+stable segment in ∂−R between two points a, b on a same stable leaf, and similarly for [a, b]u.
+Denote by □c the adapted 1-cycle given by �
+i([vi, w−
+i ]s + [w−
+i , w+
+i ]u + [w+
+i , vi+1]s), which we
+call the square model of c (see Figure 4).
+Lemma 52. For any µ ∈ M(σ), we have
+�
+ΣR(Suppc − Supp□c) ◦ πPdµ = 0.
+Proof of Lemma 52. Denote di = [vi, w−
+i ]s + [w−
+i , w+
+i ]u + [w+
+i , vi+1]s − ai − bi, which is a closed
+adapted 1-chain. We denote by fi : ∆2 → ∂−R an adapted 2-simplex whose boundary is ai +
+[wi, w−
+i ]s + [w−
+i , vi]s. The image of fi lies in the smallest horizontal rectangle of ∂−R containing
+both wi = πP(si) and vi = πP(ri). Hence according to Lemma 15, the image of fi is inside Rsi.
+Hence σsi ◦ fi is well-defined and ∂σsi ◦ fi = −bi + [wi, w+
+i ]u + [w+
+i , vi+1]s.
+The 1-chain ∂(σsi ◦ fi − fi) − di is equal to [wi, w+
+i ]u − [wi, w−
+i ]u − [w−
+i , w+
+i ]u. According to
+Remark 50, the support of the previous adapted 1-chain is zero. Therefore, we have Suppdi =
+Suppσsi◦fi − Suppfi.
+Let µ ∈ M(σ) be an invariant measure. Since σsi ◦fi is obtained from fi by pushing along the
+φ, and the φ preserves the orientations of the foliations W s
+R, W u
+R, we have
+�
+Suppσsi◦fi ◦ πPdµ =
+�
+Suppfi ◦ πPdµ. It follows that
+�
+Suppdi ◦ πPdµ = 0. Notice that we have �
+i di = □c − c, so
+the integral
+�
+ΣR(Suppc − Supp□c) ◦ πPdµ is equal to zero.
+■
+We now prove that the integral
+�
+Σ2
+R Ld(ν ⊗ µ) computes linking numbers.
+37
+
+□c
+c
+v1
+v2
+w1
+w−
+1
+w+
+1
+w−
+1
+w+
+1
+x1
+x2
+x3
+{x ∈ R, v2 ≤v x <v v1, w1 <h x}
+{x ∈ R, v3 ≤v x <v v1, w3 <h x}
+Fu(w1)
+Figure 4: Adapted cell c = �(ai + bi) and its corresponding square model □c on the left.
+In dashed filling are two sets of equal areas, used in the proof of Lemma 52.
+On the right
+is represented the set of points vertically between vi and vi+1, and on the right of wi. These
+sets are uses to relate the support function of □c and the linking function GR, in the proof of
+Theorem 44.
+Proof of Theorem 44. Notice that if ν(ΣR) = 0, then N ∗
+Rν = ν and the two terms in the equality
+are zero. By linearity, it is enough to prove the result when ν is equal to the measure ηr for some
+primitive cyclic P-word r with r0 = R. We use the notations si, ri used in the above discussion.
+Suppose that n = 1. Then N ∗
+Rηr = ηr so linkff
+σ (ηr − N ∗
+Rηr, µ) = 0. Fix u ∈ PrimeR.
+If u ̸= r, then ηr ⊗ µ(R±
+u ) = 0 since no points w ∈ Σu are in the support of ηr. If u = r,
+then we also have ηr ⊗ µ(R±
+r ) = 0, since if w is in Σu and in the support of ηr, then w = u
+and there is no x ∈ ΣR satisfying w ≤v x <v σ|u|(w) = w or σ|u|(w) ≤v x <v w. Hence we
+have
+�
+Σ2
+P LRdηr ⊗ µ = 0 = linkff
+σ (ηr − N ∗
+Rηr, µ).
+Now suppose that n ≥ 2. We denote by Iu
++ the unstable boundary component of ∂−R, which
+is on the right, and by xi = W s(vi) ∩ Iu
++ the projection of vi on Iu
++, in parallel to W s
+R. We define
+the adapted 2-cycle fi by fi = [w−
+i , w+
+i ]u + [w+
+i , xi+1]s + [xi+1, xi]u + [xi, w−
+i ]s.
+Claim 1. We have �
+i Suppfi = Supp□c (illustrated in Figure 4).
+Claim 2. For any µ ∈ M(σ),
+�
+ΣR Suppfi ◦ πPdµ = (1ri ⊗ µ)(R+
+si) − (1ri ⊗ µ)(R−
+si)
+Assume the two claims for now, and take µ ∈ M(φ) finitely supported, so that the support
+of ΘP(µ) is disjoint from the support of ΘP(ηr). Take u ∈ PrimeR. Recall the definition:
+R+
+u = {(w, x) ∈ Σu × ΣR, such that:
+u <h x and σ|u|(w) ≤v x <v w}
+38
+
+Notice that (1ri ⊗ µ)(R±
+u ) = 0 when u ̸= si. It follows that:
+�
+ΣR
+LRd(ηr ⊗ µ) =
+�
+i
+�
+ΣR
+LRd(1ri ⊗ µ)
+=
+�
+i
+�
+u∈P rimeR
+�
+(1ri ⊗ µ)(R+
+u ) − (1ri ⊗ µ)(R−
+u )
+�
+=
+�
+i
+�
+(1ri ⊗ µ)(R+
+si) − (1ri ⊗ µ)(R−
+si)
+�
+=
+�
+i
+�
+ΣR
+Suppfi ◦ πPdµ
+according to Claim 2.
+=
+�
+ΣR
+Supp□c ◦ πPdµ
+according to Claim 1.
+= linkff
+σ (ηr − N ∗
+Rηr, µ) according to Lemmas 51, 52
+Which implies linkff
+σ (ηr − N ∗
+Rηr, µ) =
+�
+ΣR LRd(ηr ⊗ µ).
+Proof of Claim 1.
+One can verify that the closed 1-chain �
+i fi − □c is a chain contained in the
+union of the unstable leaf Iu
++ and of finitely many stable leaves containing the points vi. Essen-
+tially the parts in [w−
+i , w+
+i ]u cancel each other. Hence its support function is Supp(�
+i fi−□c) = 0
+(see Remark 50).
+□
+Proof of Claim 2.
+Notice that for (w, x) ∈ R+
+si to be in the support of (1ri ⊗ µ), we must
+have w = ri and ri+1 <v ri. Similarly for (w, x) ∈ R−
+si to be in the support of the same measure,
+we must have ri <v ri+1. Hence the measure of either R+
+si or R−
+si is zero.
+Consider the case when ri+1 <v ri.
+By construction, we have w+
+i
+<v w−
+i .
+So fi is the
+boundary of a Markov rectangle Ri ⊂ ∂−R with the induced orientation. Hence the support of
+the support of fi is equal to:
+Suppfi(x) =
+1
+if x is in the interior of the rectangle Ri
+1
+if x ∈ (w+
+i , xi+1]s ∪ [xi+1, xi)u
+0
+otherwise
+So the integral �
+ΣR Suppfi ◦ πPdµ is equal to the measure for µ of the set of x ∈ ΣR satisfying:
+πP(si) = wi <h πP(x)
+πP(ri+1) = vi+1 ≤v πP(x) <v vi = πP(ri)
+According to Lemma 18 this condition on x is equivalent to having:
+si <h x
+σ|si|(ri+1) ≤v x <v ri
+Hence we have
+�
+ΣR Suppfi ◦ πPdµ = µ(R+
+si). In the case ri+1 >v ri, the same argument shows
+that
+�
+ΣR Suppfi ◦ πPdµ = −µ(R−
+si).
+□
+■
+39
+
+3.2.5
+Linking number linkφ
+We compose the reduction maps for all Markov cuboids of P to reduce an invariant measure
+on one supported by a fixed finite set. We then prove Theorem 21. Denote by R1, · · · , Rp the
+Markov cuboids in P and by N k = NRk ◦ . . . ◦ NR1.
+Definition 53. For ν, µ ∈ M(σ) such that [ν]φ = [µ]φ = 0, we define the linking number
+between ν and µ by:
+linkσ(ν, µ) = linkf
+φ(ΘP(N
+∗
+ρν), ΘP(µ)) +
+p−1
+�
+k=1
+�
+Σ2
+P
+LRkd(N
+∗
+kν) ⊗ µ
+Lemma 54. The map linkσ is continuous on M0(σ) × M0(σ).
+Proof. The maps ν �→ N
+∗
+ρµ and ΘP are continuous (see Lemma 37).
+The measure N
+∗
+ρν is
+supported by a subset of Prime(P) (see Lemma 40), which is finite. Denote by ∆ ⊂ M the
+union of the periodic orbits containing πP(Prime(P)). The map linkf
+φ is continuous on the set
+of pairs of measures (νφ, µφ) ∈ M0(φ) × M0(φ) for which ∆ contains the support of νφ (see
+Proposition 28). Therefore the term linkf
+φ(ΘP(N
+∗
+ρν), ΘP(µ)) is continuous. The second term is
+continuous according to Proposition 45.
+■
+Lemma 55. For any ν, µ ∈ M0(σ) with finite supports, so that ΘP(µ) and ΘP(ν) have disjoint
+support, we have:
+linkσ(ν, µ) = linkff
+σ (ν, µ)
+Proof. Suppose first that the support of µ is also disjoint from Prime(P). Then for all k, the
+support of ΘP(N
+∗
+kν) is disjoint from the support of ΘP(µ). Hence according to Theorem 44, we
+have
+�
+Σ2
+P LRkd(N
+∗
+kν) ⊗ µ = linkff
+σ (N ∗
+k ν − N ∗
+k+1ν, µ). It follows that:
+linkff
+σ (ν, µ) = linkff
+σ (N
+∗
+ρν, µ) +
+p−1
+�
+k=1
+linkff
+σ (N ∗
+kν − N ∗
+k+1ν, µ)
+= linkf
+φ(ΘP(N
+∗
+ρν), ΘP(µ)) +
+p−1
+�
+k=1
+�
+Σ2
+P
+LRkd(N
+∗
+kν) ⊗ µ
+= linkσ(ν, µ)
+The general case follows from the density of measures in the first case (see Lemma 42) and
+from the continuity of linkσ.
+■
+Lemma 56. There exists a continuous bilinear map linkφ : M0(φ)×M0(φ) → R such that for all
+measure ν, µ ∈ M0(σ), one has linkσ(ν, µ) = linkφ(ΘPν, ΘPµ). Additionally linkφ is continuous
+on M0(φ) × M0(φ).
+Proof. It is enough to prove that for any measures ν1, ν2, µ ∈ M0(σ) with ΘP(ν1) = ΘP(ν2), we
+have linkσ(ν1, µ) = linkσ(ν2, µ). Once it is proven, we define the map linkφ by linkφ(νφ, µφ) =
+linkσ(νσ, µσ) for any measures νσ, µσ ∈ M0(σ) in the reciprocal images of νφ, µφ by the surjec-
+tion ΘP : M(σ) → M(φ).
+To prove the claim, take ν1, ν2, µ ∈ M0(σ) with ΘP(ν1) = ΘP(ν2). Denote by Γ the set
+of periodic orbits of φ, contained in the stable/unstable boundary of P. The map ΘP is injec-
+tive outside the reciprocal image of the set ∪γ∈Γ(Fs(γ) ∪ Fu(γ)) for the map πP. Hence the
+measures νi coincide on this set.
+40
+
+According to Lemma 33, the measure Θ(νi) charges the set π−1(∪γ∈Γ(Fs(γ) ∪ Fu(γ))) only
+on Γ. It follows that ν1 − ν2 has a finite support. Suppose that µ is finitely supported, and
+that the support of ΘP(µ) is disjoint from the support of Θ(ν1 − ν2). According to Lemma 55,
+we have linkσ(ν1 − ν2, µ) = linkff
+σ (ν1 − ν2, µ) = linkf
+φ(ΘP(ν1 − ν2), ΘP(µ)) = 0 since ΘP(ν1 −
+ν2) = 0. By continuity of linkσ and density of the measures satisfying that assumption, we
+have linkσ(ν1, µ) = linkσ(ν2, µ) for all measures µ ∈ M0(σ).
+■
+Proof of Theorem 21. Lemma 42 implies that we can approximate any pair of invariant mea-
+sures (ν, µ) ∈ M0(φ) by sequence of pairs (νn, µn) ∈ M0(φ), where νn and µn are supported
+by disjoint and finite unions of periodic orbits. Therefore the uniqueness and symmetry of the
+map linkφ follow from the continuity.
+When the stable and unstable foliations of φ are orientable, the theorem follows from the
+above discussion (see Lemmas 55 and 56).
+Suppose that the stable and unstable foliations are not orientable. We denote by �π: �
+M → M
+the bundle-orientations covering (which is of degree 2), and by g the non-trivial deck transforma-
+tion of �
+M → M. Lift φ to a flow ψ on �
+M. We denote by linkψ : M0 (ψ)2 → R the linking number
+map obtained in the orientable case. Take a measure µ ∈ M0(φ) and denote by µ ◦ �π ∈ M(ψ)
+the measure which coincides with ((�π|U)−1)∗µ on each set U ⊂ �
+M where �π is invertible. Notice
+that µ ◦ �π is the unique ψ and g invariant measure lifting 2µ.
+Take a closed 1-form �α on �
+M. Since �α + g∗�α is invariant by the action of g, there exists a
+closed 1-form α on M satisfying �π∗α = �α + g∗�α. The flow ψt and the measure µ◦ �π are invariant
+by g, so we have:
+�
+�
+M
+ι
+�∂ψt
+∂t
+�
+�αdµ ◦ �π = 1
+2
+��
+�
+M
+ι
+�∂ψt
+∂t
+�
+�αdµ ◦ �π +
+�
+�
+M
+ι
+�∂ψt
+∂t
+�
+g∗�αdµ ◦ �π
+�
+= 1
+2
+�
+�
+M
+ι
+�∂ψt
+∂t
+�
+�π∗αdµ ◦ �π
+=
+�
+M
+ι
+�∂φt
+∂t
+�
+αdµ = 0
+It follows that µ◦�π is null-cohomologous. Take K ⊂ �
+M a measurable subset such that (K, g·K) is
+a partition of �
+M. Then the projection K → M is bijective. Given a continuous map f : �
+M → R,
+one has:
+�
+�
+M
+fdµ ◦ �π =
+�
+K
+fdµ ◦ �π +
+�
+g·K
+fdµ ◦ �π
+=
+�
+M
+f ◦ (�πK)−1dµ +
+�
+M
+f ◦ (g · �πK)−1dµ
+=
+�
+M
+
+
+�
+y∈�π−1(x)
+f(x)
+
+ dµ(x)
+It follows that µ �→ µ◦ �π is continuous. For ν, µ ∈ M0 (ψ), we define linkφ(ν, µ) = 1
+2 linkψ(µ◦
+�π, ν ◦ �π). By construction, it is continuous. Take Γ1, Γ2 two invariant 1-cycles for φ. Take S1 a
+2-chain on M, bounding Γ1. Lift Γi to an invariant 1-cycle �Γi of ψ, which satisfies g · �Γi = �Γi.
+41
+
+Also lift S1 to a 2-chain �S1 in �
+M, bounding �Γ1. According to the orientable case, we have:
+linkφ(LebΓ1, LebΓ2) = 1
+2 linkψ(LebΓ1 ◦�π, LebΓ2 ◦�π)
+= 1
+2 linkψ(Leb�Γ1, Leb�Γ2)
+= 1
+2
+�S1 ·∩�Γ2
+= S1 ·∩Γ2
+= linkff
+φ (LebΓ1, LebΓ2)
+■
+Corollary 57 (Linking number is invariant by orbit equivalence). For any orbit equivalence
+h: (M, φ) → (N, ψ) between two Anosov flows on oriented closed 3-dimensional manifolds, we
+have linkφ = linkψ ◦(Θh, Θh).
+Proof. By continuity and density of finitely supported orbits (see Lemma 42), it is enough to
+prove it for pairs of measures LebΓ1, LebΓ2 ∈ M0(φ), for two null-homologous invariant 1-
+cycles Γ1, Γ2 with disjoint supports.
+Given a periodic orbit γ ⊂ M, we have Θh(Lebγ) =
+h∗ Leb⊥
+γ ⊗ Lebψ = Lebh(γ). Therefore we have Θh Lebγ = Lebh(γ), and Θh LebΓi = Leb h∗(Γi).
+The linking number of disjoint 1-chains is invariant by preserving orientation homeomorphism,
+so the linking numbers linkψ(Θh LebΓ1, Θh LebΓ2) and linkφ(LebΓ1, LebΓ2) are equal.
+■
+4
+Reeb-like condition for measures
+We prove Theorem D. An invariant measure µ for φ is said to satisfy the Reeb-like condition
+if the two following conditions are satisfied:
+• [µ]φ = 0 in H2(M, R),
+• minν link(µ, ν) > 0, where the minimum is taken over all invariant probability measure ν
+such that [ν]φ = 0.
+Notice that the Reeb-like condition is invariant by orbit equivalence of φ. That is if h: (M, φ) →
+(N, ψ) is an orbit equivalence between two transitive Anosov flows, then a measure µ ∈ M(φ)
+satisfies the Reeb-like condition for φ if and only if Θh(µ) satisfies the Reeb-like condition for ψ
+(see Proposition 11 and Corollary 57).
+Let α be a C1 differential 1-form such that ι(X)(dα) = 0. The 2-form dα is invariant by φ,
+so it induces a transverse measure for φ. Denote by να the invariant measure whose transverse
+measure ν⊥
+α corresponds to dα.
+Lemma 58. We have [να]φ = 0. Additionally for any invariant measure µ ∈ M0(φ), we have:
+linkφ(µ, να) =
+�
+M
+ι(X)αµ
+Prasad [Pra22] define the linking number with a smooth measure as the integral appearing
+in the above lemma, for homology spheres. Theorem D is a consequence of McDuff’s Theo-
+rem [Duf87, Theorem 5.2], except for the signs. That is in McDuff’s theorem, there is no relation
+between the sign of the linking number and the sign of the reparametrization (McDuff formula-
+tion allows one to reverse the direction of the flow). For completeness, we give a sketch of proof
+for the reformulation, without using McDuff’s theorem.
+42
+
+Proof. It is enough to verify the equality for measures of the form LebΓ for null-homologous
+invariant 1-cycles Γ. The lemma follows from continuity of linkφ and from the density Lemma 14.
+Take a smooth 1-form β which satisfies ι(X)β = 1. Then the 3-form V = dα ∧ β is invariant
+by the flow. Indeed we have:
+ι(X)V = (ι(X)dα) ∧ β + dα ∧ (ι(X)β) = dα
+Therefore we have LX(dα ∧ β) = 0. If νV denote the invariant measure corresponding to the
+invariant 3-form V , then ν⊥
+v corresponds to ι(X)V = dα, so νV = να. It follows that [να]φ =
+[ι(X)V ] = 0 (see Lemma 10).
+We denote by ∆ the support of Γ.
+Denote by α∆ = π∗
+∆α and β∆ = π∗
+∆β and take a
+rational 2-chain S in MΓ si that (π∆)∗(∂S) = Γ. We can take a closed 1-form ω on MΓ so that
+P∆([ω]) = [S]. Take a closed 1-form ω on M∆ satisfying (π∆)∗(∂P∆[ω]) = [Γ]. Then we have:
+0 = ι(X∆)(ω ∧ dα∆ ∧ β∆)
+= (ι(X∆)ω) ∧ (dα∆ ∧ β∆) − ω ∧ (ι(X∆)dα∆) ∧ β∆ − ω ∧ dα∆ ∧ (ι(X∆)β∆)
+= (ι(X∆)ω)(dα∆ ∧ β∆) − ω ∧ dα∆
+From the definition link∆
+Γ follows:
+linkf
+φ(να, LebΓ) =
+�
+M∆
+(ι(X∆)ω)(dα∆ ∧ β∆)
+=
+�
+M∆
+ω ∧ dα∆
+=
+�
+S
+dα∆
+by Poincar´e duality
+=
+�
+(π∆)∗(S)
+dα
+=
+�
+Γ
+α
+=
+�
+M
+ι(X)αd LebΓ
+■
+Lemma 59. Let E be a finite dimensional vector space and F ⊂ E × R be a compact convex set
+such that F ∩ (0 × (−∞, 0]) = ∅. Then there exist η > 0 and a linear map f : E → R satisfying:
+F ⊂ {(x, t) ∈ E × R, t ≥ f(x) + η}
+Proof. Since F is convex and compact, there exists a hyperplane H ⊂ E × R separating F
+and (0, 0). Additionally H can be chosen so that H ∩ (0 × R) = {(0, c)} for some c > 0. Then
+there exists a linear map f : E → R satisfying that (x, y) ∈ E × R lies in H if and only if
+y = f(x) + c, and f(0, 1) > 0. Then for all (x, y) ∈ F, we have y > f(x) + c.
+■
+Sketch proof of Theorem D. Suppose that a smooth reparametrization ψ of φ is the Reeb of a
+contact form α with α ∧ dα > 0. Denote by να ∈ M(ψ) the ψ-invariant measure correspond-
+ing to α ∧ dα.
+For any measure µ ∈ M0
+p(ψ), we have linkψ(να, µ) =
+�
+M ι
+�
+∂ψt
+∂t
+�
+αdµ = 1.
+43
+
+Since α ∧ dα is positive, να is a positive multiple of a probability measure. Therefore to να cor-
+responds a smooth φ-invariant probability measure which satisfies the Reeb-like condition (since
+the cohomology class and the linking numbers are invariant by topological equivalence).
+Suppose now that there exists a smooth probability measure νV satisfying the Reeb-like
+condition. Here V is a smooth a positive volume form, invariant by φ. According to Lemma 10,
+the cohomology class of ι(X)V is equal to [νV ]φ = 0.
+So there exists a smooth differential
+1-form β on M such that dβ = ι(X)V . We transform β in a contact form in two steps.
+Apply Lemma 59 to the convex F = {([µ]φ,
+�
+M ι(X)βdµ) ∈ H2(M, R) × R, µ ∈ Mp(φ)}. We
+obtain that there exists an element [δ] ∈ H1(M, R), where δ is a closed 1-form, and c > 0 such
+that for all µ ∈ Mp(φ), we have
+�
+M ι(X)(β + δ)dµ =
+�
+M ι(X)βdµ + [µ]φ · [δ] ≥ c. Then by
+averaging β + δ along the flow, one obtain a differential 1-form α =
+1
+T
+� T
+0 φ∗
+t (β + δ)dt, which
+satisfies ι(X)α > 0 and dα = ι(X)V . It follows that φ is Reeb-like.
+■
+5
+Birkhoff Section
+We give a brief overview on partial sections and Birkhoff sections.
+Here the manifold M is
+supposed oriented.
+We give a variation of the usual definition of partial sections using the
+orientation on M.
+A partial section of φ is a pair (S∗, f) where S∗ is a compact C∞ oriented surface and f
+is a smooth immersion of S∗ in M, which satisfies that f| Int S∗ is positively transverse the flow,
+and f(∂S∗) is a finite union of periodic orbits. It implies that each boundary component of S∗ is
+sent to a periodic orbit γ of φ, and f : γ∗ → γ is a covering map. We usually denote by S = f(S∗),
+and identify the immersed surface S with (S∗, f). A Birkhoff section is a partial section (S∗, f)
+embedded in its interior, such that there exists T > 0 for which any orbit arc of φ of length T
+intersects S.
+Proposition 60 (Fried-desingularisation [Fri82]). Let φ be a smooth flow on a 3-dimensional
+closed manifold M. Let S1 be a Birkhoff section and S2 be a partial section for φ. Then there
+exists a Birkhoff section for φ, relatively homologous to [S1] + [S2] in H2(M, ∂S1 ∪ ∂S2, R).
+Fried introduced a desingularisation and used it (together with the Fried sections we define
+below) to prove that any transitive Anosov flow admits a Birkhoff section. Fried description is
+not precise, we write a precise proof of the proposition in the Appendix A, which additionally
+works for non-Anosov flows.
+Sign of boundary components.
+Fix a partial section S = (S∗, f). The orientation on the
+surface S∗ induces an orientation on its boundary, so that the normal direction going outside S∗
+plus the positive direction tangent to the boundary, induce the orientation on S∗. We orient
+the orbits of the flow with the orientation given by the flow. A boundary component γ∗ is said
+positive if f : γ∗ → f(γ) is orientation preserving. A partial section is said positive if all its
+boundary components are positive.
+Theorem 61 (Asaoka-Bonatti-Marty [ABM22]). An Anosov flow is R-covered and positively
+skewed if and only if it admits a positive Birkhoff section.
+Given a positive Birkhoff section S as in the last theorem, we show later that the probability
+measure Leb∂S has positive linking numbers with all null-cohomologous probability measures.
+In the last section, we need more than what is prescribed by the theorem, namely we need a
+44
+
+u1
+u2
+v1v2
+v2v1
+γ1
+γ2
+γw
+Figure 5: Illustration of a 2-chain modeling a Fried Section. In green is represented a transverse
+polygon (corresponding to f(P) ⊂ ∂−R in the proof of Lemma 62). In orange and yellow are
+represented two tangent rectangles. They can be smoothed to obtain a Fried section.
+Birkhoff section with only one boundary component. Given a partial section (S∗, f), we view ∂S
+as an invariant 1-cycle. It is given by:
+∂S =
+�
+γ∗⊂∂S∗
+deg(γ∗ → f(γ∗))f(γ∗)
+Here deg(γ∗ → f(γ∗)) is the algebraic degree of the map γ∗
+f−→ f(γ∗). Note that S is positive
+if and only if ∂S is an invariant 1-cycle with positive coefficients.
+5.1
+Fried section
+Fried constructed [Fri83] a type of partial section which we call Fried section. We give a more
+precise definition to additionally control the boundary orbits.
+Let P be a markov partition, u1, u2 be two distinct, primitive cyclic P-words, which start
+with the same letter R. Denote by γi the periodic orbit of φ corresponding to ui. We denote
+by Ni ∈ {1, 2} the degree of the realization of ui, given in Lemma 17. Fix two integers k1, k2 ∈
+N≥1. We write w = uk1
+1 uk2
+2
+and denote by γw ⊂ M the periodic orbit corresponding to w. If γi
+has non orientable leaves, we suppose that kiNi is even.
+Lemma 62. Suppose that u1 is smaller than u2 for both the vertical and horizontal orders. Then
+there exist m ≥ 1 and a partial section S satisfying ∂S = (k1N1γ1 + k2N2γ2) − mγw.
+Suppose that u1 <v u2 and u2 <h u1.
+Then there exist m ≥ 1 and a partial section S
+satisfying ∂S = −(k1N1γ1 + k2N2γ2) + mγw.
+Suppose that k1 and k2 are both even, then m = 1 holds in the two previous statement.
+We call Fried section the partial section S given in the lemma. It is illustrated in Figure 5.
+Proof. We prove the first case, the second case is similar. We use the notation Ru, σu and φT u
+from Section 2. That is Ru is the set of elements x ∈ ∂−R having a short itinerary along u, σu
+is the map Ru → R corresponding to pushing along the flow along the itinerary u, and T u is
+the time we push along φ to go from x to σu(x).
+We write vi = uki
+i . Denote by P the 4-gon given by [0, 1]2, equip with the anti-clockwise
+orientation. By assumption, either γi has orientable leaves, or ki is even, or Ni = 2 (and πP(ui)
+lies in the boundary of the Markov partition P). So for j ∈ {1, 2}, πP(v1v2) = σv2(πP(v2v1))
+and πP(v2v1) = σv1(πP(v1v2)) are on the same side relatively to πP(uj), for the horizontal
+and the vertical order. In particular πP(v1v2) and πP(v2v1) are in the Markov sub-rectangle
+45
+
+of ∂−R which admits πP(u1) and πP(u2) as opposite corners. Therefore, we can find an immer-
+sion f : P → ∂−R ⊂ M such that:
+•
+(0, 0), (1, 0), (1, 1) and (0, 1) are respectively sent to πP(u1), πP(v1v2), πP(u2), πP(v2v1),
+• the restrictions of f to each edge of P is transverse to the stable and unstable foliations,
+•
+σv1 ◦ f(t, 0) = f(0, t) for every t ∈ [0, 1],
+•
+σv2 ◦ (t, 1) = f(1, t) for every t ∈ [0, 1].
+In the 3rd point, f(t, 0) lies in the smaller horizontal rectangle containing f(0, 0) = πP(v1)
+and f(1, 0) = πP(v1v2), so f(t, 0) lies in the horizontal sub-rectangle Rv1. Therefore σ|v1|◦f(t, 0)
+is well-defined, and similarly for the 4th point.
+Denote by Q the blow up of P along its four corners, and byπQ : Q → P be the blow-
+down projection.
+We still denote by (t, 0) and (0, t) the points in ∂Q corresponding to lifts
+of (t, 0), (0, t) ∈ ∂P, for t ∈ (0, 1).
+Denote by (0+, 0) and (0, 0+) the limits in ∂Q of (t, 0)
+and (0, t) when t goes to 0. Take τ : Q → R a smooth map, and consider the map g : Q → M
+given by:
+g(x) = φτ(x)(f ◦ πQ(x))
+We choose the map τ on the edges corresponding to [0, 1] × 0 and 0 × [0, 1] so that τ(t, 0) =
+T v1 ◦f(t, 0)+τ(0, t). Then we have g(t, 0) = g(0, t). That is the image of g is an immersed 8-gon,
+whose edges corresponding to [0, 1] × 0 and 0 × [0, 1] coincide. Similarly we choose τ on [0, 1] × 1
+and 1×[0, 1], so that the corresponding edges in g(Q) coincide. We additionally choose τ so that
+the surface obtained by gluing the coinciding edges, is smooth. Denote by S that surface.
+The surface S is constructed by pushing im f ⊂ ∂−R along the flow, so it is transverse to the
+flow in its interior.
+It is in fact positively transverse.
+Indeed we have πP(u1) <v πP(u2),
+so πP(v1v2) <v πP(v2v1).
+Similarly we have πP(v1v2) >h πP(v2v1).
+It follows that the
+curve f(∂[0, 1]2) is going anti-clockwise.
+So f preserves the orientation and S is positively
+transverse in its interior.
+By construction, S has some potential boundaries only on the images by g of the blow-up
+corners of Q, which are sent in the orbits going through the corners of f([0, 1]2), namely γ1, γ2
+and γw. We need to determine the boundary of S.
+Recall the definition of the realization of a primitive cyclic P-word u. We constructed in
+Section 2.1 a suspension flow φP for the homeomorphism σ: ΣP → ΣP, with the time function
+corresponding to the first return time from ∂−R to ∂+R. To a cyclic P-word u corresponds a
+periodic orbit �γu of φP. The realization is the covering map, from the orbit �γu, to the φ-periodic
+orbit of corresponding to u. Here we additionally call the length of the realization of u, the
+length of the orbit �γu (counted with multiplicity if u is a multiple of a primitive cyclic P-word).
+We have τ(t, 0) = T v1 ◦ f(t, 0) + τ(0, t) for t ∈ (0, 1), so at the limit we have τ(0+, 0) =
+T v1 ◦ f(0, 0) + τ(0, 0+). Therefore T v1 ◦ f(0, 0) is the length of the boundary component of S
+corresponding to the blow-up of the corner (0, 0) (view as an orbit traveled multiple times).
+The time T v1 ◦ f(0, 0) is equal to k1 times the length of the realization of u1. So the boundary
+component of S in γ1 has multiplicity k1N1 ̸= 0 (recall that the realization of ui has degree Ni).
+Therefore we can choose τ so that S is additionally immersed on the boundary component cor-
+responding to the edge [(0, 0+), (0+, 0)] ⊂ ∂Q. Similarly we can choose τ so that S is everywhere
+immersed, the boundary component in γ2 has multiplicity k2N2, and the boundary component
+in γw has negative multiplicity.
+To be more specific, the length of the boundary of S in γw, is given by −T v1 ◦ f(1, 0) − T v2 ◦
+f(0, 1) = −T v1v2 ◦ f(1, 0). Suppose that k1 and k2 are both even. Then w = uk1
+1 uk2
+2
+is not the
+46
+
+ξ1
+ξ2
+ξ
+L++(ξ)
+L−−(ξ)
+Ls
++(ξ1)
+Lu
++(ξ1)
+Figure 6: Lozenge with corners ξ1 and ξ2 on the left. The two lozenges with a corner in ξ on the
+right. On the right, the flow is R-covered and positively skewed.
+power of a smaller cyclic P-word (see Lemma 19). Additionally γw is not in the boundary of P
+(since it intersects the interior of the Markov partition ∂−R in f(1, 0)), so the realization of w is
+of degree one. By definition of the realization, the length of the realization of w is T w ◦ f(1, 0),
+so the multiplicity of S in γw is -1.
+■
+5.2
+Birkhoff section with one boundary
+We prove Theorem E: when the flow is positively skewed, it admits a Birkhoff section with
+only one boundary component, with multiplicity one and the corresponding periodic orbit lies
+in orientable stable and unstable leaves. For that, we follow the proof of Theorem 61 given
+in [ABM22].
+A key notion to prove the Theorem are lozenges. Recall that (Pφ, Ls, Lu) is the bi-foliated
+plane of φ.
+Fix a pair of orientations on Ls and Lu so that the orientation on Ls plus the
+orientation on Lu gives the orientation on Pφ. For x ∈ Pφ, we denote by Ls
++(x) the open half
+leaves of x (excluding x), so that the orientation goes from x to Ls
++(x). We define similarly
+Ls
+−(x) = Ls(x) \ (Ls
++(x) ∪ {x}), Lu
++(x) and Lu
+−(x). For ξ ∈ Pφ and σ, ǫ ∈ {−, +}, we define the
+quadrant Qσǫ(ξ) as the set of points ζ ∈ Pφ satisfying Ls
+σ(ξ) ∩ Lu(ζ) ̸= ∅ and Lu
+ǫ (ξ) ∩ Ls(ζ) ̸= ∅.
+A lozenge L ⊂ Pφ is a subset of the form Q++(ξ1) = Q−−(ξ2) or Q+−(ξ1) = Q−+(ξ2), for
+two points ξ1, ξ2 ∈ Pφ. In this article, only the lozenges of the form Q++(ξ1) = Q−−(ξ2) are
+used, which we call positive lozenges. The points ξ1 and ξ2 are respectively called the corners
+(−, −) and (+, +) of L. We use lozenges later to construct positive Birkhoff sections.
+When the flow is R-covered and positively skewed, each point ξ ∈ Pφ bounds two positive
+lozenges. We denote by L++(ξ) the lozenge for which ξ is a corner (−, −), and L−−(ξ) the other
+lozenge.
+Let K ⊂ M be a compact subset of the orbit space Pφ such that for each point x ∈ M, there
+exists a lift �x ∈ �
+M whose image πPφ(�x) is in K. For example, one can take a big compact ball
+in �
+M and take its image in Pφ. Every point in K admits an open neighborhood lying inside
+one lozenge. So by compactness, there exists a finite family of lozenges (Li)i∈I covering K.
+In [ABM22], these lozenges are used to build a positive Birkhoff section, whose boundary is
+the union of the periodic orbits corresponding to the corners of the lozenges Li. We prove the
+theorem above by choosing carefully the family of lozenges.
+Proof of Theorem E. We denote by K and (Li) the compact and lozenges introduced in the
+previous paragraph.
+One can take the family (Li)i so that there exist open subsets (Ui)i∈I
+47
+
+with U i ⊂ Li such that K is included in ∪iUi.
+Denote by �
+M the orientation bundle covering and �φ the lifted flow. φ.
+We also denote
+by π�
+M : �
+M → �
+M the universal covering map of �
+M.
+Take ǫ > 0 small. Since the flow φ is Anosov and transitive, �φ is also Anosov and transitive.
+So by using the closing Lemma, there exists a periodic orbit �γ of �φ such that any point �
+M
+is a distance at most ǫ from �γ. If ǫ is chosen small enough, one can find for every i ∈ I, a
+point ξi ∈ πPφ(π−1
+�
+M (�γ)) close to the corner −, − of the lozenges Li, so that L++(ξi) contains U i.
+Denote by g1 = 1, g2, . . . , gn some elements in π1(�
+M) satisfying gi ∗ ξ1 = ξi. Also denote by ξ′
+i
+the corner +, + of the lozenge L++(ξi). Since �φ has orientable stable and unstable foliations,
+the lozenge L++(ξi) is equal to gi ∗ L++(ξ1), so gi ∗ ξ′
+1 = ξ′
+i.
+Denote by γ and δ the periodic orbits in M corresponding to the points ξ1 and ξ′
+1 in M ′. The
+proof of the implication ⇐ of [ABM22, Theorem 1] gives a positive Birkhoff section S1 bounded
+by γ and by γ′.
+Take a Markov partition P of φ. Recall that �γ is ǫ-dense in �
+M. Up to taking ǫ > 0 smaller,
+we can suppose that the orbits γ and γ′ intersect a same Markov rectangle ∂−R of P, in two
+points p, q satisfying p <v q and q <h p, that is p is bellow and on the right of q.
+Denote by n and m the multiplicity of S1 along γ and γ′. According to Lemma 62, there
+exist a partial section S2 and a periodic orbit δ such that ∂S2 = δ − 2nγ − 2mγ′. According to
+Proposition 60, there exists a Birkhoff section S3 relatively homologous to 2[S1]+[S2] in H2(M, γ∪
+γ′ ∪ δ, R). The boundary of 2[S1] + [S2] is equal to δ, with multiplicity one. Hence S3 has only
+one boundary component in δ, with multiplicity one, so it is a positive Birkhoff section.
+Notice that δ is homologous to 2(nγ + mγ′). Recall that the map sign: π1(M) → {−1, 1} is
+a group morphism, which measures the orientability of the stable and unstable foliations, on a
+neighborhood of a simple closed loop. Since H1(M, R) is the Abelianization of π1(M) and {−1, 1}
+is Abelian, it factorizes into a group morphism �
+sign: H1(M, R) → {−1, 1}. So �
+sign(δ) = 1. By
+definition of the function sign, the stable and unstable leaves of δ are orientable.
+■
+5.3
+Birkhoff section with given boundary
+We prove Theorem G: given Γ, there exists a Birkhoff section bounding NΓ for some N ∈ N, if
+and only if LebΓ has positive linking numbers with all the measures in M0
+p(φ). As a corollary,
+we obtain a second proof of Barbot’s Theorem 5.
+Given an oriented compact 3-manifold N and a flow ψ on N, a global section of ψ is an
+oriented compact surface S ⊂ N which is positively transverse to the flow, with ∂S ⊂ ∂N and
+intersecting all orbits of ψ. It follows from compactness that S intersects every orbit arc of
+length T for some T > 0.
+We start by constructing a Birkhoff section from a global section on a blow up manifold.
+Take ∆ ⊂ M a finite union of periodic orbits.
+Denote by M∆ the blow up manifold of M
+along ∆, by π∆ : M∆ → M the blow down projection, and by φ∆ the lifted flow on M∆.
+Lemma 63. Let S ⊂ M∆ a smooth global section of φ∆.
+Then there exist a smooth func-
+tion T : S → R satisfying that the image π∆ ◦ φT (S) is a Birkhoff section of φ.
+Proof. In this proof, all isotopies are isotopies along the flow. There are two types of boundary
+components for S. The curves in ∂S ∩ Tγ, for some γ ⊂ ∆, which are homologous to a fiber of
+Tγ → γ, and the other. We call these boundary components of type 1 and type 2. We need to
+prove that up to an isotopy on S, the projection π∆ restricts to a nice map on S. Here ’nice
+map’ means that for each boundary component δ ⊂ Tγ of type 1, δ is sent to a single point in γ
+and the image of S is smooth close to that point. And for each boundary component δ of type
+2, π∆ restricts to an immersion close to δ. Then the image of S is a Birkhoff section.
+48
+
+Take a boundary component δ ⊂ ∂S∩Tγ of type 1. Then δ is isotopic to one fiber (see [Fri82]).
+In fact there exists a global isotopy of Tγ sending ∂S ∩ Tγ to a finite union of fibers of Tγ → γ.
+We can extend the global isotopy on Tγ to a global isotopy on M∆. Denote by S1 the image
+of the surface S by the global isotopy along the flow. We can take the global isotopy so that
+π∆(S1) is an embedded and smooth surface, close to the orbits of type 1.
+Take a boundary component δ ⊂ ∂S1 ∩ Tγ of type 2.
+Then similarly δ is isotopic to a
+smooth curve δ′ in Tγ, transverse to the flow, and transverse to the foliation on Tγ by fibers of
+Tγ → γ. Similarly there exists a global isotopy on M∆, whose image of S1 is a surface S2, and
+which satisfies that π∆(S2) is immersed close to the orbits of type 2. Then π∆(S2) is a Birkhoff
+section.
+■
+Proof of Theorem G. Let Γ be a null-homologous invariant 1-cycle with integer coefficients.
+Denote by ∆ the support of Γ. Suppose first that there exists a Birkhoff section S bounding NΓ
+with N ≥ 1. We lift S to a smooth surface S∗ ⊂ M∆ whose boundary is inside ∂M∆. Up to an
+isotopy along the flow for S, we can suppose that there exists ǫ > 0 for which there is no arc of
+length 2ǫ from S∗ to itself. In particular S∗ is positively transverse to φ∆. Take a non-decreasing
+smooth map h: [−ǫ, ǫ] → R such that h ≡ −1/2 close to −ǫ, h ≡ 1/2 close to ǫ,
+Then define a smooth map f : M∆ → R/Z by:
+f(x) =
+h(t)
+(mod 1)
+when φ∆
+−t(x) lies in S∗ for some t ∈ [−ǫ, ǫ]
+1/2
+(mod 1)
+otherwise
+By construction we have (π∆)∗(∂P∆[df]) = NΓ and ι(X∆)df ≥ 0. Hence for any positive
+measure µ ∈ M(φ∆) with [π∗
+∆µ]φ = 0, we have:
+linkf
+φ(LebΓ, π∗
+∆µ) = 1
+N
+�
+M∆
+ι(X∆)dfdµ
+= 1
+N
+�
+φ∆
+[−ǫ,ǫ](S∗)
+ι(X∆)dfdµ
+= 1
+N
+�
+S∗
+�� ǫ
+−ǫ
+ι(X∆)df ◦ φt(x)dt
+�
+dµ⊥(x)
+= 1
+N
+�
+S∗ (h(ǫ) − h(−ǫ)) dµ⊥(x)
+= 1
+N µ⊥(S∗) > 0
+So LebΓ has only positive linking numbers with M0
+p(φ).
+Suppose now that LebΓ has a
+positive linking number with any null-cohomologous invariant probability measure of φ. Fix
+an element ω1 ∈ H1(M∆, R) such that (π∆)∗(∂P∆(ω1)) is equal to [Γ] ∈ H1(∆, R). We can
+choose ω1 ∈ H2(M∆, Q) with rational coefficients.
+Denote by F ⊂ H2(M, R) × R be the set {([π∗
+∆µ]φ, [µ]φ∆ · ω1), µ ∈ Mp(φ∆)}. By assump-
+tion, we have [µ]φ∆ · ω1 = linkφ(LebΓ, π∗
+∆µ) > 0 for all measure µ ∈ M0
+p(φ∆). Therefore the
+intersection F ∩ (0 × (−∞, 0]) is empty. According to Lemma 59, there exists η > 0 and a linear
+map f : H2(M, R) → R satisfying that for all µ ∈ Mp(φ∆), we have [µ]φ∆ · ω1 ≥ f([π∗
+∆µ]φ) + η.
+By Poincar´e duality, f corresponds to an element ω2 ∈ H1(M, R). Since Mp(φ∆) is compact,
+we can perturbate f so that ω2 has rational coefficients. Take an integer N ∈ N>0 satisfying
+that N(ω1 − π∗
+∆ω2) have integer coefficients. Then for all measures µ ∈ Mp(φ∆), we have:
+[µ]φ∆ · N(ω1 − π∗
+∆ω2) ≥ η > 0
+49
+
+It follows from Schwartzman-Fried-Sullivan Theory (see [Fri82, Sul76]), that φ∆ admits a
+global section S whose relative homology class [S] ∈ H2(M∆, ∂M∆, R) is dual to N(ω1 − π∗
+∆ω2).
+According to Lemma 63, the global section can be taken so that π∆(S) is a Birkhoff section of φ.
+Additionally the boundary of π∆(S) is homologous to (π∆)∗(∂P∆(N(ω1 − π∗
+∆ω2))) = N[Γ] ∈
+H1(∆, R). Therefore NΓ bounds a Birkhoff section.
+■
+We should discuss a particular case. When there do not exist any null-cohomologous invariant
+probability measure, Sullivan [Sul76] proved that the flow admits a global section. Since the
+flow is Anosov, the first return map for that global section is conjugated to a linear Anosov
+diffeomorphism on the torus. In that case, every null-homologous invariant 1-cycle, with integer
+coefficient, bounds a Birkhoff section (see [Mar21, Section 4.2]). Hence Theorem G is a type
+of generalization of this suspension case, where we have less control over the multiplicity at the
+boundary.
+Proof of the remaining equivalences in the Table B. We prove the implications in the first colum
+of the table: cell 4 =⇒ cell 2 ⇐⇒ cell 3.
+Suppose that there exists ψ orbit equivalent to φ, which preserves a contact form α with
+α ∧ dα > 0 (cell 4). Then according to Theorem D, the ψ-invariant probability measure induced
+by α ∧ dα has the Reeb-like property. The Reeb-like property is invariant by orbit equivalence,
+so there exists a φ-invariant probability measure with the Reeb-like property.
+Namely it is
+null-cohomologous and has only positive linking number with null-cohomologous φ-invariant
+probability measures (cell 3).
+The implication cell 2 (there exists a positive Birkhoff section) =⇒ cell 3 follows from the
+above discussion (Theorem G).
+Suppose that there exists a measure µ ∈ M0
+p(φ) with the Reeb-like property (cell 3). Accord-
+ing to Lemma 14, there exists a sequence of null-homologous invariant 1-cycles Γn, with rational
+coefficients, so that LebΓn converges toward µ. By continuity of the linking number, for large n,
+LebΓn has positive linking numbers with measures in M0
+p(φ). It follows from Theorem G that
+there exists a positive Birkhoff section (cell 2).
+■
+6
+Reeb-like properties for positively skewed Anosov flow
+Gibbs measures are a family of invariant probability measures which satisfy similar properties
+than invariant smooth volume forms. On one hand a Gibbs measure corresponds to a volume
+form in some other differential structure on the manifold. On the other hand Gibbs measures
+offer some control, which we use to obtain the Reeb-like property.
+6.1
+Reeb-like Gibbs measure
+We introduce some thermodynamic formalism we need later, more can be found in [Bow08]. A
+H¨older continuous map p: M → R is call a H¨older potential. Denote by P(M) the set of
+H¨older potentials on M, which we equip with the topology given by the infinity norm ∥.∥∞.
+Denote by hµ(φ) the measured entropy of φ relatively to an invariant probability measure µ.
+Given a H¨older potential p: M → R the pressure of p is the real number:
+P(p) =
+sup
+µ∈Mp(φ)
+�
+hµ(φ) +
+�
+M
+pdµ
+�
+Any invariant probability measure µ for which the supremum is achieved is call a Gibbs
+measure of p (or the equilibrium state of p). It is known that for Anosov flows, Gibbs measures
+50
+
+exist and are unique for H¨older potentials [Bow08]. Additionally a Gibbs measure has no atom
+and charges every non-empty open sets. Given a H¨older potential p, we denote by µp its Gibbs
+measure.
+Theorem 64 (Sharp [Sha93]). Let φ be an Anosov flow. The φ is homologically full, then φ
+admits a null-cohomologous Gibbs measure.
+Sharp actually proves that four properties on Anosov flows are equivalent, two of them are
+the ones cited above.
+Theorem 65. Let φ be an Anosov flow. If φ is R-covered and positively skewed, then φ admits
+a Gibbs measure satisfying the Reeb-like condition.
+To prove the theorem, we obtain some control on a Gibbs measure, and use the continuity
+of µp in p to obtain implicitly the Gibbs measure in the theorem.
+Lemma 66. The maps p ∈ P(M) �→ P(p) ∈ R and p ∈ P(M) �→ µp ∈ Mp(φ) are continuous.
+Proof. The pressure is continuous as supremum of 1-Lipchitz maps. Take (pn)n a sequence of
+H¨older potentials converging toward a H¨older potential p∞.
+By compactness of Mp(φ), the
+sequence µpn accumulates to a measure ν ∈ Mp(φ). Then we have:
+����
+�
+M
+pndµpn −
+�
+M
+p∞dν
+���� ≤
+�
+M
+|pn − p∞| dµpn +
+����
+�
+M
+p∞dµpn −
+�
+M
+p∞dν
+����
+≤ ∥pn − p∞∥∞ +
+����
+�
+M
+p∞dµpn −
+�
+M
+p∞dν
+����
+−−−−−→
+n→+∞ 0
+Since the entropy hµ(φ) is upper semi-continuous in µ, the entropy hν(φ) is not less than the
+limit inf of the entropies hµpn (φ). Hence we have:
+P(p∞) = lim inf
+n
+P(pn)
+= lim inf
+n
+�
+hµpn (φ) +
+�
+M
+pndµpn
+�
+≤ hν(φ) +
+�
+M
+p∞dν ≤ P(p∞)
+Hence ν is the Gibbs measure of p∞, so the map p �→ µp is continuous.
+■
+Null-cohomologous Gibbs measures.
+We first reprove Sharp Theorem, by constructing a
+null-homologous Gibbs measure. Our construction allows us to have more control on the linking
+numbers. The proof goes as follows. Lemma 68 gives a way to control the Gibbs measure. Given
+a H¨older potential which charges mostly one periodic orbit, its Gibbs measure charge mostly a
+neighborhood of that orbit. We apply this lemma on enough periodic orbits, to obtain a better
+control on the Gibbs measure’s cohomology class. To go from one to several orbits, we need a
+topological property, stated in Lemma 69.
+51
+
+Here we suppose that the flow φ is homologically full. Denote by n the dimension of H1(M, R).
+We construct a map f : Rn −→ P(M)
+[µ·]φ
+−−−→ H1(M, R) ≃ Rn, such that for r large enough, f has
+no zero on the sphere of radius r, and induces a map ∂B(0, r) → Sn−1 of degree one.
+Fix a family of periodic orbits γ+
+1 , · · · , γ+
+p , such that ([γ+
+1 ], · · · , [γ+
+p ]) is a basis of H1(M, R)
+(see [PP86] for the existence). According to Lemma 3, for each index i there exists a periodic
+orbit γ−
+i , homologous to −[γ+
+i ]. Take two families of closed neighborhoods U +
+i
+and U −
+i
+of γ+
+i
+and γ−
+i , which we take pairwise disjoint. Then fix H¨older potentials g+
+i and g−
+i such that gǫ
+i ≡ 1
+on γǫ
+i and gǫ
+i ≡ 0 outside U ǫ
+i , for ǫ ∈ {+, −}. Define the map p: Rn → P(M) given by:
+p(x) =
+�
+1≤i≤n
+xi>0
+xi · g+
+i +
+�
+1≤i≤n
+xi<0
+|xi| · g−
+i
+For each x ∈ Rn, the map p(x) is a H¨older potential, so it admits a Gibbs measure µp(x). Take
+a basis (ωi)i of H1(M, R) which is Poincar´e dual to the basis ([γ+
+i ])i. More precisely ωi · [γ+
+j ]
+equals 1 when i = j, and it equals 0 otherwise. We define the map fp: Rn → Rn defined by
+fp(x) =
+�
+[µp(x)]φ ∧ ω1, · · · , [µp(x)]φ ∧ ωn
+�
+Proposition 67. There exists r > 0 such that f −1
+p (0) ⊂ [−r, r]n, and such that the map
+(x ∈ ∂[−r, r]n) �→
+�
+fp(x)
+∥fp(x)∥2 ∈ Sn−1�
+is of degree 1. Additionally there exists x ∈ (−r, r)n satis-
+fying fp(x) = 0.
+Theorem 64 is a consequence of the proposition, since fp(x) = 0 if and only if [µp(x)]φ = 0.
+To prove the proposition, we first need some control on the Gibbs measure. Denote by htop the
+topological entropy.
+Lemma 68. Let γ be a periodic orbit of the flow, C2 > C1 > 0 be two real numbers, U ⊂ M
+be a neighborhood of γ, and p be a H¨older potential. Suppose that p ≤ C2 with equality on γ,
+and p ≤ C1 outside U. Then we have:
+µp(M \ U) ≤ htop(φ)
+C2 − C1
+Proof. By definition a Gibbs measure maximizes the pressure, so we have:
+C2µp(M \ U) + C2µp(U) = C2 =
+�
+M
+pd Lebγ
+≤
+�
+M
+pd Lebγ +hLebγ(φ)
+≤
+�
+M
+pdµp + hµp(φ)
+≤ C1µp(M \ U) + C2µp(U) + htop(φ)
+It follows that we have (C2 − C1)µp(M \ U) ≤ htop(φ).
+■
+For x ∈ Rn, we denote by
+xi its ith coordinates, such that x = (x1, · · · , xn). Similarly
+for a map f with values in Rn, we write f = (f1, · · · , fn).
+For C ∈ R+ and x ∈ Rn, we
+define Cx = {i, |xi| > C}. Also denote by sign(x) ∈ {−1, 1} the sign of a real number x ̸= 0.
+Lemma 69. Take 0 < C < 1 and ǫ > 0. Let f : [−1, 1]n → Rn be a continuous map such that
+for all x ∈ ∂[−1, 1]n the following holds:
+52
+
+•
+�
+i∈Cx |fi(x)| > nǫ,
+• for all i ∈ Cx, sign(xi)fi(x) ≥ −ǫ,
+• for all j ̸∈ Cx, |fj(x)| ≤ ǫ.
+Then the map G: ∂[−1, 1]n → ∂[−1, 1]n given by G(x) =
+fp(x)
+∥fp(x)∥∞ is well-defined and of degree 1.
+In particular the exists x in (−1, 1)n with fp(x) = 0.
+Proof. For x in ∂[−1, 1]n, we have n∥fp(x)∥∞ ≥ �
+i |fi(x)| > nǫ so ∥fp(x)∥∞ > ǫ. Hence G(x)
+is well-defined. For t ∈ R+, we define the continuous map gt : R → R by:
+gt(x) =
+
+
+
+
+
+0
+|x| ≤ t
+x − t
+if x > t
+x + t
+if x < −t
+and the continuous map gn
+t : Rn → Rn, (x1, · · · , xn) �→ (gt(x1), · · · , gt(xn)).
+Then for all x
+in ∂[−1, 1]n, the following holds:
+1. for all t ∈ [0, ǫ], ∥gn
+t ◦ fp(x)∥∞ > 0,
+2. for all i ∈ Cx, sign(xi) · gn
+ǫ ◦ fi(x) ≥ 0,
+3. for all j ̸∈ Cx, gn
+ǫ ◦ fj(x) = 0,
+4. for all i, sign(xi) · gn
+ǫ ◦ fi(x) ≥ 0,
+The last item is a consequence of the two previous ones. Denote by S the unit sphere of Rn for
+the 1-norm ∥.∥1, and by N1, N∞ : Rn\{0} → S the normalization maps given by N1(x) = x/∥x∥1
+and N∞(x) = x/∥x∥∞. Consider the homotopy ht : S → S defined by:
+ht(x) = N1 ◦ gn
+t ◦ f ◦ N∞(x)
+The sphere S admits a natural triangulation, whose maximal dimensional faces are given by
+the sets {x ∈ S, ∀i, sixi ≥ 0} for some s1, · · · , sn ∈ {−1, 1}. The points 2 and 3 imply that any
+face F of S (of any dimension) is let invariant by hǫ. Hence using a convex interpolation, hǫ is
+homotopic to the identity on S. Since the map IdS is of degree one, hǫ is also of degree 1, and
+so is h0 since the degree is invariant by homotopy. Hence x ∈ ∂[−1, 1]n �→ x/∥f∥2 ∈ Sn−1 ⊂ Rn
+is of degree one.
+■
+Proof of Proposition 67. Recall that (ωi)i is the dual base to ([γ+
+i ])i. Hence we can take a closed
+1-form αi whose cohomology class is ωi and such that the following holds:
+•
+ι(X)αi ≡ 0 on U +
+j ∪ U −
+j whenever j ̸= i,
+•
+ι(X)αi is positive on U +
+i
+and negative on U −
+i .
+We fix D > 0 such that for all i, |ι(X)αi| is larger than D on U +
+i ∪ U −
+i . Take ǫ > 0, r > 0, and
+η = htop(φ)
+r
+. We suppose ǫ small and r large so that we have D(1−η) > 2nǫ and η∥ι(X)α∥∞ < ǫ.
+Fix a point x in Rn satisfying ∥x∥∞ > r and take 0 < B < ∥x∥∞ − r. We start by estimating
+the measure µp(x). We define the sets B+ = {1 ≤ i ≤ n, |xi| > B} and the set:
+Vx =
+�
+i∈B+
+U sign(xi)
+i
+53
+
+Claim 1. We have µp(x)(M \ Vx) < η.
+Proof of Claim 1.
+The function p(x) = �
+i |xi| · gsign(xi)
+i
+is bounded by ∥x∥∞ on all M, since
+the maps g±
+i
+have disjoint supports and ∥g±
+i ∥∞ = 1. Let j be such that |xj| = ∥x∥∞. Then
+for z ∈ γsign(xj)
+j
+, we have gsign(xj)
+j
+= 1 and p(x)(z) = ∥x∥∞.
+For z ∈ M \ Vx, the point z is in at most one set U ±
+j . If z belongs to U ±
+j , then j is not in B+,
+so |xj| ≤ B. Hence we have p(x)(z) = �
+i |xi| · g±
+i (z) ≤ B. It follows from Lemma 68 that:
+µp(x)(M \ Vx) ≤
+htop(φ)
+∥x∥∞ − B
+< htop(φ)
+r
+< η
+□
+To estimate fp(x), we apply Lemma 69 on the map x ∈ [−1, 1]n �→ fp(rx), with the con-
+stant C = B/r. For that it is enough to prove the three following points for x ∈ [−r, r]n:
+1. for j ̸∈ B+,
+��[µp(x)]φ · [αj]
+�� ≤ ǫ
+2. for j ∈ B+, sign(xj)[µp(x)]φ · [αj] ≥ −ǫ
+3.
+�
+i∈B+
+��[µp(x)]φ · [αj]
+�� > nǫ
+Take j ̸∈ B+, so that |xj| ≤ B. We have ι(X)αj ≡ 0 on Vx and µp(x)(M \ Vx) ≤ η, so:
+��[µp(x)]φ · [αj]
+�� =
+����
+�
+M
+ι(X)αjdµp(x)
+����
+≤
+����
+�
+Vx
+ι(X)αjdµp(x)
+���� +
+�����
+�
+M\Vx
+ι(X)αjdµp(x)
+�����
+≤ η∥ι(X)α∥∞ ≤ ǫ
+Similarly for j ∈ B+, with |xj| > B, we have:
+[µp(x)]φ · [αj] =
+�
+M
+ι(X)αjdµp(x)
+=
+�
+U
+sign(xj )
+j
+ι(X)αjdµp(x) +
+�
+Vx\U
+sign(xj )
+j
+ι(X)αjdµp(x)
++
+�
+M\Vx
+ι(X)αjdµp(x)
+By hypothesis, the sign of ι(X)αj on U sign(xj)
+j
+is sign(xj).
+Additionally ι(X)αj is zero
+on Vx \ U sign(xj)
+j
+and
+���
+�
+M\Vx ι(X)αjdµp(x)
+��� ≤ η∥ι(X)α∥∞. So one has:
+sign(xj) · ι(X)µp(x)(αj) ≥ −η∥ι(X)α∥∞ ≥ −ǫ
+54
+
+Finally we have:
+�
+i∈B+
+��ι(X)µp(x)(αj)
+�� =
+�
+i∈B+
+����
+�
+M
+ι(X)αjdµp(x)
+����
+≥
+�
+i∈B+
+�����
+�
+Vx
+ι(X)αjdµp(x)
+���� −
+�����
+�
+M\Vx
+ι(X)αjdµp(x)
+�����
+�
+≥
+�
+i∈B+
+�
+Dµp(x)
+�
+U sign(xi)
+i
+�
+− ∥ι(X)α∥∞µp(x)(M \ Vx)
+�
+≥ D(1 − η) − nη∥ι(X)α∥∞
+> nǫ
+■
+Gibbs measures with positive linking number.
+We deform the Gibbs measure obtained
+in Proposition 67 to find another Gibbs measure which has a positive linking number with every
+null-cohomologous invariant probability measure.
+According to Lemma E, there exists a Birkhoff section bounded by a single orbit δ. The
+measure Lebδ satisfies the Reeb-like property according to Theorem G. We extend the map p
+given in the previous subsection. Take the sets U ±
+i
+disjoint from δ, and take U0 to be a compact
+neighborhood of δ, disjoints from each set U ±
+i . Also take a map g0 : M → [0, 1] such that g0 ≡ 1
+on δ and g0 ≡ 0 outside U0.
+We additionally suppose that g0 < 1 outside δ and g±
+i
+< 1
+outside γ±
+i . We define the maps q: Rn × R → P(M) and fq : Rn × R → Rn by:
+q(x, z) = p(x) + z · g0(x)
+and
+fq(x, t) = ([µq(x,t)]φ ∧ ωi)i
+Lemma 70. There exists a sequence (xn, zn)n in Rn × R such that ∥(xn, zn)∥∞ goes to +∞,
+and such that for all n, we have zn ≥ 0 and [µq(xn,zn)] = 0.
+Lemma 71. For any sequence (xn, zn)n given by the previous lemma, the sequence µq(xn,zn)
+converges toward
+Lebδ
+Lebδ(M).
+Proof of Lemma 70. Suppose that all points (x, z) with x ≥ 0 and with [µq(x,z)] = 0 remain in a
+bounded region. Take r > 0 large, and define the map h: ∂[−r, r]n × [0, 1] → Sn−1 by h′(x, t) =
+fq((1 − t)x, rt) and h(x, t) =
+h′(x,t)
+∥h′(x,t)∥2 . For r > 0 large enough, h is well-defined, continuous
+and is a homotopy between a degree one map (see Lemma 67) and a constant map. This is not
+possible, concluding.
+■
+Proof of Lemma 71. Consider a sequence (xn, zn) as in Lemma 70. By compactness of Mp(φ),
+the sequence µq(xn,zn) accumulates to some measure ν ∈ Mp(φ).
+Up to extraction, we can
+suppose that xn
+i have the same sign for all n. Up to reversing the roles of γ+
+i
+and γ−
+i , we can
+suppose that xn
+i is non-negative for all i and all n.
+We should prove that ν is supported by δ ∪i γ+
+i . Take ǫ > 0 smaller than 1. Define the open
+subset Aǫ ⊂ M as the set of point x ∈ M satisfying g0(x) < 1−ǫ and g+
+i < 1−ǫ for all i ∈ �1, n�.
+55
+
+The union of ∪ǫ<1Aǫ is equal to M \ (δ ∪i γ+
+i ). Denote by Nn = ∥(xn, zn)∥∞. According to
+Lemma 68, we have:
+µq(xn,zn)(Aǫ) ≤
+htop(φ)
+Nn − Nn(1 − ǫ)
+≤ htop(φ)
+Nnǫ
+−−−−−→
+n→+∞ 0
+Since Aǫ is open, at the limit we have ν(Aǫ) = 0. So ν(M \ (δ ∪i γ+
+i )) = 0. Hence the measure ν
+is of the form a0 Lebδ + �
+i ai Lebγ+
+i for some non-negative numbers ai. Since the cohomology
+class of a measure is continuous in the measure, we have [ν]φ = 0. According to Lemma 13,
+the Poincar´e dual of [ν]φ ∈ H2(M, R) is a0[δ] + �
+i ai[γ+
+i ] ∈ H1(M, R). The orbit δ is null-
+homologous and [ν]φ = 0, so one has �
+i ai[γ+
+i ]. The family ([γ+
+i ])1≤i≤n is a basis of H1(M, R),
+so the numbers ai for i ≥ 1 are null.
+Hence ν is the unique invariant probability measure
+supported on δ, that is
+Lebδ
+Lebδ(M).
+The sequence (µq(xn,zn))n in Mp(φ) has a unique accumulating point, which is
+Lebδ
+Lebδ(M).
+Therefore it converges toward that measure.
+■
+Proof of Theorem 65. According to Lemmas 70 and 71, there exists a sequence of null-cohomologous
+Gibbs measure µq(xn,zn) which accumulates toward
+Lebδ
+Lebδ(M). By construction, Lebδ has positive
+linking number with all invariant probability measures.
+The linking number is continuous. So it follows from Corollary 22, that for n large enough, the
+measure µq(xn,zn) has positive linking numbers with all null-cohomologous invariant probability
+measures. Namely µq(xn,zn) have the Reeb-like property for n large enough.
+■
+6.2
+C1+H differential structure
+Fix a H¨older potential p: M → R.
+We construct a class C1+H differential structure on M,
+and a reparametrization of φ, which is of class C1+H and Anosov for that differential structure.
+Additionally the Gibbs measure of φ, associated to p, is induced by a H¨older volume form for that
+differential structure. This as already be done by E Cawley [Caw91] for Anosov diffeomorphisms
+and done by Asaoka [Asa07] for codimension one Anosov flows.
+We need an additional control on the invariant volume form, so we use the local product of
+Gibbs measures given by Haydn [Hay94], and follow Asaoka’s work. The end of the section gives
+some control on the invariant measure, in terms of positive linking number.
+Theorem 72 (Haydn [Hay94]). Fix a H¨older potential p of φ. There exists positive measures µs
+p
+and µu
+p on the strong stable and unstable foliations (that is a measure on each leaf) satisfying
+the properties:
+1. The measure µp is locally equal to a positive multiple of the product measure µs
+p⊗µu
+p ⊗Lebφ,
+2.
+φ∗
+t µs
+p = e−Ftµs
+p and φ∗
+t µu
+p = eFtµu
+p, where Ft(x) = � t
+0(f ◦ φs(x) − P(f))ds,
+3. For any points x, x′ such that x′ ∈ Fss(x), denote the holonomy map h: Fss
+l (x) → Fss
+l (x′)
+from x to x′ along the weak unstable foliation, we have µs
+p = eωh∗µs
+p, where ω is the
+H¨older function given by ω =
+� +∞
+0
+(f ◦ φt ◦ h − f ◦ φt) dt. A similar statement holds for
+the measure µu
+p.
+56
+
+4. The measure µs
+p, restricted on a strong stable leaf, is finite on compact segments and positive
+on non-empty open subsets.
+Remark 73. The measure µs
+p is a measure on each leaves of the strong stable foliation. We can
+also interpret it as a pseudo-transverse measure to the flow, on each weak stable leaf. That is
+given a point x ∈ M, I ⊂ Fss(x) a strong stable segment and a continuous function T : I → R,
+the measure µs
+p induces the measure (φT )∗ �
+eFT µs
+p
+�
+on the curve φT (I).
+We equip the orbit space with a family of measures on the stable and unstable foliations.
+For p ∈ �
+M, we denote ξp the point πPφ(p), and by θs
+p : Fss(�π(p)) → Ls(ξp) the lift/projection
+map, defined as follows. The maps �π : �
+M → M and πPφ : �
+M → Pφ induce two homeomor-
+phisms �Fss(p) → Fss(�π(p)) and �
+Fss(p) → Ls(πPφ(p)).
+We define θs
+p as the product of the
+corresponding homeomorphisms Fss(�π(p))
+�π−1
+−−→ �Fss(p) → Ls(ξp).
+Then define µs
+p = (θs
+p)∗µs
+p, which is a measure on Ls(ξp). The measure µs
+p induces a transverse
+measure to the unstable foliation, on an open subset of Pφ obtained as the union of unstable
+leaves intersecting Ls(ξp).
+Thanks to Theorem 72, the transverse measure induced by µs
+p is
+multiplied by a H¨older continuous function when p varies.
+Fix an orientation on Ls(ξp). We denote by
+� b
+a µs
+p the integral of µs
+p on the unstable seg-
+ment [a, b] ⊂ Ls(ξp), with a negative sign if the orientation on Ls(ξp) goes from b to a.
+Lemma 74 (Asaoka, Lemma 42 in [Asa07]). The map ηs
+p : Ls(Q(p)) → R, ξ �→
+� ξ
+ξp µs
+p is a
+bi-H¨older homeomorphism.
+Lemma 75 (Asaoka, Lemma 43 in [Asa07]). Given a holonomy map along the weak unstable
+foliation hp,q : I ⊂ �Fss
+l (p) → J ⊂ �Fss
+l (q), the map ηs
+q ◦ hp,q ◦ (ηs
+p)−1 is of class C1+H.
+The two lemmas also hold for the measures on unstable leaves. Using these lemmas, we define
+a C1+H differential structure on Pφ transverse to Lu. We denote by A⊥
+p the atlas on Pφ whose
+charts are given by the maps (ηs
+p, ηu
+p). Thanks to the two lemmas, the transition maps are of
+Class C1+H. So A⊥
+p induces a C1+H differential structure on the surface Pφ. From the definition
+follow the next lemma.
+Lemma 76. The action π1(M) ↷ Pφ is of class C1+H for the atlas A⊥
+p .
+The Gibbs measure µp induces a transverse measure µ⊥
+p on M. It lifts to a measure on Pφ
+which is invariant by the action π1(M) ↷ Pφ. We also denote by µ⊥
+p the corresponding measure
+on Pφ.
+Lemma 77. The measure µ⊥
+p on Pφ is induced by a H¨older area form on Pφ, for the differential
+structure A⊥
+p .
+Proof. It follows from Theorem 72 that µ⊥
+p is locally a product of ηs
+p and ηs
+q, multiplied by a
+positive function f (corresponding to the functions in points 2 and 3 in the theorem). Since
+the foliation Fss and Fuu are H¨older for the initial differential structure (see
+cite[Corollary
+9.4.11]Fisher19), the function f is H¨older for the initial differential structure on Pφ. According
+to Lemma 74, the function f is also H¨older for the atlas A⊥
+p , so µ⊥
+p is H¨older regular.
+■
+From transverse atlas to atlas.
+Given a compact surface S ⊂ M (only continuously em-
+bedded), topologically transverse to the flow, the atlas A⊥
+p induces a C1+H differential structure
+on S (using a lift of S to �
+M and the projection to Pφ). For that reason on view the atlas A⊥
+p
+on Pφ as a transverse atlas to φ.
+57
+
+Proposition 78. There exist a C1+H atlas Ap on M and a continuous reparametrization ψ of
+the flow φ which satisfy the following:
+•
+ψ is of class C1+H for Ap,
+• for any C1+H compact surface S, topologically transverse to ψ, the C1+H differential struc-
+tures on S induced by Ap and A⊥
+p are the same,
+• if id: (M, φ) → (M, ψ) is seen as an orbit equivalence, the measure Θid(µp) is induced by
+a H¨older volume form for Ap.
+Given a continuous and positive function f : M → R∗
++, there exists a unique flow ψf on M,
+differentiable in the direction of the flow (for the initial smooth structure), and for which we
+have f ∂ψf
+t
+∂t = ∂φt
+∂t . The flow ψf can also be characterized by the following formula. For every t ∈
+R, we have:
+φt(x) = ψ� t
+0 f◦φs(x)ds(x)
+We construct an atlas on M whose charts are given by flow boxes of the flow ψf for some func-
+tion f. Take a compact surface S ⊂ M, supposed to be topologically embedded and transverse
+to φ, and T > 0. If the map a: S × [0, 1] → M defined by a(x, s) = ψf
+T (x)(x) is an embedding,
+the image of a is called a flow box for S.
+Fix T > 0 small and take a finite family of compact surfaces (Si)1≤i≤n. We suppose that
+the set φ[0,T ](Si) is a flow box for the surface Si for all i. We also suppose that the union of the
+interior of the flow boxes φ[0,T/2](Si) is equal to M.
+Notice that for any continuous function f : M → [1, 2], the set ψf
+[0,T ](Si) is a flow box con-
+taining φ[0,T/2](Si), and is included in φ[0,T ](Si). In particular these flow boxes cover M. We
+denote by af
+i : Si × [0, T ] → ψf
+[0,T ](Si) the parametrization of the flow box for the flow ψf. That
+is af
+i (x, s) = ψf
+s (x).
+The map af
+i induces a C1+H differential structure on Bi, given as the image of the product
+between A⊥
+p and the standard smooth structure on [0, 1]. Take two surfaces Si, Sj whose flow
+boxes intersect. There is a maximal set Ui,j ⊂ Si and a continuous function T f
+i,j : Ui,j → R+,
+so that for any point (x, s) ∈ Si × [0, 1] the point af
+i (x, s) lies in Sj if and only if x is in Ui,j
+and s = T f
+i,j(x). We call the map Ti,j a short return time.
+Lemma 79. Suppose that for all i, j ∈ I, the map T f
+i,j is of class C1+H for the atlas A⊥
+p
+on Ui,j ⊂ Si. Then all transition maps of the form (af
+i )−1 ◦ af
+j are of class C1+H where they are
+well-defined. In particular the set of charts (ai)i form a C1+H atlas on M, compatible with the
+transverse atlas A⊥
+p .
+Proof. The map (af
+i )−1 ◦ af
+j coincides with the map (x, s) �→ (ψf
+Ti,j(x)(x), s + T f
+i,j(x)). The map
+(x ∈ Ui,j ⊂ Si) �→ (ψf
+Ti,j(x)(x) ∈ Sj) is C1+H, by definition of the C1+H differential structure
+on Si and Sj. So the map (af
+i )−1 ◦ af
+j is as regular as T f
+i,j.
+■
+We now prove the proposition. The idea is to define locally f to make the short return times
+of class C1+H. Given one short return time, one can build by hand f to make that short return
+time regular. To extend f globally, we build it piece by piece. At each step, the short return
+maps which are well-defined are regular. And we choose the pieces where we define f, so that it
+is afterward possible to extend it by keeping short return times regular.
+58
+
+Given i, j and a continuous function g : Ui,j ⊂ Si → [1, 2], consider the map f defined
+on {φt(x), x ∈ Ui,j, t ∈ [0, T 1
+i,j(x)] or [T 1
+i,j(x), 0]} by f(φt(x)) = T 1
+i,j(z)g(z). For this function f,
+T f
+i,j is well-defined on Ui,j and coincides with g, so it is as regular as g. To define a function f
+guarantying the regularity of all maps T f
+i,j, we define f inductively using this idea.
+Proof of Proposition 78. For all indexes i < j, up to removing a small open neighborhood
+of Si ∩ Sj to Sj, we can suppose that the surface Si and Sj are disjoint. It can be done while
+preserving that the interior of the flow boxes φ(0,T/2)(Int Sk) cover M. Indeed the parts of the
+flow boxes we remove, are cover by either the flow boxes of Si or by the interior of the flow boxes
+which cover Si ∩ Sj. Then all surfaces Si are disjoint.
+For i, j ≤ n, we consider pairs of the form (x, t) ∈ Si × R satisfying t > 0, that φt(x) lies
+in Sj, and that the orbit arc φ(0,t)(x) is disjoint from ∪k Int Sk. We say that (x, t) is an adjacent
+pair for (i, j). For m ≥ 0, we define Ai,j,m as the set of adjacent pairs (x, t) for (i, j), which
+satisfy that the curve φ(0,t)(x) intersects ∪k∂Sk in exactly m points. The surfaces Sk are called
+in between surfaces. We suppose that the surfaces (Sk)k are in general positions, so that the
+adherence of the sets Ai,j,m = ∅ are compact manifolds of dimension 2 − m. In particular Ai,j,m
+is finite for m = 2, and empty for m > 2. Notice that all points in M \ ∪k Int Sk is on a unique
+orbit arc of the form φ(0,t)(x), for an adjacent pair (x, t).
+To prove that all short return times are C1+H, it is enough to prove that for every adjacent
+pair (x, t) for (i, j), and every in between surfaces Sk for (x, t), the short return times T f
+i,j and T f
+i,k
+are C1+H on a neighborhood of x. Indeed all short return times are locally sums of the short
+return times described above.
+Now we construct f inductively. For each adjacent pair (x, t) ∈ Ai,j,2, choose a small flow
+box neighborhood U ⊂ M of φ[0,t](x). Denote by Sk1, Sk2 the two in between surfaces for the pair
+(x, t). Choose the function fU : U → [1, 2], such that the short return times T f
+i,j, T f
+i,k1 and T f
+i,k2
+are well-defined and of class C1+H close to x. Notice that the boundary of the 1-dimensional
+sub-manifold Ai,j,1 is Ai,j,2. Take a small neighborhood V of the set:
+U ∪
+�
+i,j
+
+
+�
+(x,t)∈Ai,j,1
+φ[0,t](x)
+
+
+Up to taking U and V smaller, we can extend fU to a function fV : V → [1, 2] satisfying the
+following. For every adjacent pair (x, t) ∈ ∪i,jAi,j,1 and in between surfaces Sk for (x, t), the
+short return times T f
+i,j, T f
+i,k are well-defined and C1+H close to x. Then we can extend f on
+all M so that it additionally satisfies that: for every adjacent pair (x, t) ∈ ∪i,jAi,j,0, the short
+return time T f
+i,j is well-defined and C1+H close to x.
+Denote by ψ = ψf the reparametrization of φ corresponding to f. It follows from the above
+discussion that the flow boxes for ψ, and the surfaces Sk, induce a C1+H atlas Ap, compatible
+with the atlas A⊥
+p . The atlas Ap is constructed using flow boxes for ψ, so ψ is of class C1+H
+for Ap. For a similar reason, Θid(µp) is H¨older regular.
+■
+Lemma 80. There exists T > 0 for which we have
+� T
+0 (p ◦ φs(x) − P(p))ds < 0 for all x ∈ M.
+This statement is already known for symbolic dynamic.
+We give a proof using only the
+definition of the pressure.
+Proof. Take a measure µ ∈ Mp(φ). By definition of the pressure, we have:
+�
+M
+(p − P(p))dµ ≤ −hµ(φ) ≤ 0
+59
+
+The equality potential holds only when µ is the Gibbs measure of φ for the H¨older potential f,
+in which case we have hµ(φ) > 0. Hence
+�
+M(p − P(p))dµ < 0 holds in all cases. Suppose that for
+all N ∈ N, there exists xN ∈ M for which we have � N
+0 (p◦φs(xN)−P(p))ds ≥ 0. Denote by νn the
+pull back of the Lebesgue measure on [0, T ], by the map t ∈ [0, N] �→ φt(xN) ∈ M, rescaled to be
+a probability measure. By compactness of the set of probability measures on M, the sequence νn
+accumulates to some probability measure µ. By hypothesis, we have
+�
+M(p − P(p))dνN ≥ 0 for
+all N, so at the limit
+�
+M(p − P(p))dµ ≥ 0. Take a continuous function f : M → R and t ∈ R.
+For any N ∈ N we have:
+����
+�
+M
+(f ◦ φt − f)dνN
+���� =
+�����
+1
+N
+� N
+0
+(f ◦ φt+s(xN) − f ◦ φt(xN))ds
+�����
+=
+����
+1
+N
+� t
+0
+(f ◦ φN+s(xN) − f ◦ φt(xN))ds
+����
+≤ 2t
+N ∥f∥∞
+Hence the integral
+�
+M(f ◦ φt − f)dνN converges toward
+�
+M(p ◦ φt − p)dµ = 0. It follows that µ
+is invariant by φ, which contradicts that
+�
+M(p − P(p))dµ is negative. Therefore for some T > 0,
+the integral
+� T
+0 (p ◦ φs(x) − P(p))ds is negative for all x ∈ M.
+■
+Lemma 81. The flow ψ is Anosov for the atlas Ap.
+Write by Y = ∂ψt
+∂t . Take m a continuous metric on M, for the atlas Ap. By construction
+of the product atlas Ap, the foliation Fs is of class C1 for the atlas Ap. Denote by Ecs the
+invariant plane field tangent to Fs, for Ap. We denote by | detm(dcsψt)| the absolute value of
+the determinant of the restriction of dψt to Ecs, computed with the metric m.
+Proof. We first prove that | detm(dcsψt)| decreases exponentially in t ≥ 0. Call stable curve any
+curve which remains in a single leaf of Fs, and topologically transverse to the flow. Denote by µs
+p
+the measure on the leaves of Fss coming from the decomposition of the measure µp in a local
+product (see Theorem 72). As discuss previously, µs
+p induces a measure on every stable curve.
+We define a similar measure using the metric m.
+For a point p and a vector v ∈ TpM
+tangent to Fs, we denote by ∥v∥⊥
+m the minimum of ∥v + tY (p)∥m for t ∈ R. For a C1 stable
+curve γ : [0, 1] → M and A ⊂ [0, 1] measurable, define the measure νs
+m(A) by:
+νs
+m(A) =
+�
+A
+∥γ′(t)∥⊥
+m dt
+It defines a measure νs
+m on C1 stable curves. Additionally it is of the Lebesgue type. By
+definition of the atlas Ap, the measure µs
+p is also of the Lebesgue type on C1 stable curves.
+Therefore there exists a continuous function g : M → R, which satisfies µs
+p = egνs
+m.
+Lemma 80 and Theorem 72 imply that the measure ψ∗
+−tµs
+p decreases exponentially in t > 0,
+with uniform constants. By compactness, g is bounded. So ψ∗
+−tνs
+m also decreases exponentially
+in t > 0. Therefore there exists A, B > 0 for which | detm(dcsψt)| ≤ Ae−Bt for all t ≥ 0.
+We use a cone argument to decompose Ecs as a sum of two invariants line bundles. Take a
+bi-valued continuous section U of Ecs, so that for every p ∈ M, U(p) consist of two opposites
+vectors not parallel to Y (p). We can change the metric m so that (Y, U) gives an orthonormal
+basis of each plane of Ecs. Take T > 0 satisfying Ae−T B < 1. Denote by a: M → R, b: M → R+
+two continuous maps satisfying dψT (U) = aY + bU. Then we have:
+|detm(dcsψT )| =
+����det
+�
+1
+a
+0
+b
+����� = b
+60
+
+It follows that b < 1. Take λ >
+max a
+1−max b and consider the cone filed C ⊂ Ecs defined for p ∈ M
+by C(p) = {xU(p) + yY (p), |y| ≤ λ|x|}. The image cone ψT (C) is the set of points of the form:
+x′b(p)U(ψT (p)) + (y′ + x′a(p))Y (ψT (p))
+with |y′| ≤ λ|x′|. We claim that ψT (C) contains C. Indeed given (x, y) ̸= 0 with |y| ≤ λ|x|, we
+have:
+xU(ψT (p)) + yY (ψT (p)) = x′b(p)U(ψT (p)) + (y′ + x′a(p))Y (ψT (p))
+for x′ =
+1
+b(p)x and y′ = y − a(p)
+b(p) x
+Then we have:
+|y′|
+|x′| = |b(p)y − a(p)x|
+|x|
+≤ b(p)|y|
+|x| + a(p)
+≤ λ max b + max a < λ
+Therefore the cone field C is invariant by ψ−T for large t > 0, and we have a uniform expo-
+nential expansion of ψ−T on C, so by a classical argument of invariant cone fields, there exists
+a continuous invariant line bundle Es ⊂ C, disjoint from the image of Y , which is exponen-
+tially contracted by ψt for t > 0. Similarly there exists an invariant line bundle Eu which is
+exponentially contracted by ψt for t < 0, and with T M = Es ⊕ Eu ⊕ RY . It follows that ψ is
+Anosov.
+■
+6.3
+Smoothing the flow
+In this section φ is supposed homologically full, p is a H¨older potential so that µp is null-
+cohomologous, ψ is a reparametrization of φ, of class C1+H and Anosov for the atlas Ap. We also
+denote by Y = ∂ψt
+∂t the vector field generation ψ, and νp = cste·Θid(µp) the ψ-invariant probabil-
+ity corresponding to µp. The measure νp is also null-cohomologous (see to Proposition 11), and
+is induced by a H¨older volume form. When φ is additionally R-covered and positively skewed,
+we additionally suppose that νp satisfies the Reeb-like condition.
+Plan of proof for Theorem A. Suppose φ to be a positively skewed R-covered Anosov flow in an
+oriented closed 3-manifold M. According to Theorems 65, φ admits a H¨older potential p: M → R
+whose Gibbs measure µp satisfies the Reeb-like condition. In section 6.2, we construct a C1+H
+atlas Ap on M and a reparametrization ψ of φ, which is of class C1+H and Anosov for the
+atlas Ap. Additionally, the Gibbs measure νp of ψ, corresponding to µp, is given by a H¨older
+volume form V for the atlas Ap.
+We follow Asaoka’s ideas [Asa07] to smooth the flow ψ. The atlas Ap can be extended in
+a class C∞ atlas A∞
+p , for which ψ is generated by a C1+H vector field Y (Lemma 84). Then
+we approximate the H¨older volume form V , corresponding to νp, by smooth volume form Vn.
+The vector field Y can be approximated in the C1 topology, be a smooth vector field Yn which
+preserve Vn (see Lemma 87, where V and Vn appear as eλV and eλnV ). Since Anosov flows
+are C1 structurally stable, the vector field Yn generates an Anosov flow ψn, orbit equivalent to ψ
+and to φ. Additionally it preserves a smooth volume form Vn.
+Suppose that ι(Yn)Vn is null-cohomologous for n large enough. Since the linking number is
+continuous (see Corollary 22), for n large, the probability measure induced by Vn has positive
+61
+
+linking numbers with all null-cohomologous ψn-invariant probability measure. Hence it satisfies
+the Reeb-like property. It follows from Theorem D that ψn is a reparametrization of a Reeb-
+Anosov flow. Hence φ is orbit equivalent to a Reeb-Anosov flow.
+In general, ι(Yn)Vn is not null-cohomologous, but its cohomology class converges toward the
+cohomology class of νp, that is zero. We perturbate Yn so that ι(Yn)Vn is null-cohomologous.
+For that we remark that given a smooth volume form W, we can find smooth vector fields Z
+preserving W, so that ι(Z)W represent any element ω ∈ H2(M, R). Additionally Z can be taken
+with a C1 norm bounded by Cste(W)∥ω∥. The idea is to replace Yn by Yn + Zn in the above
+argument, for a suitable vector field Zn satisfying [ι(Zn)Vn] = −[ι(Yn)Vn].
+We need Yn+Zn to converges toward Y in the C1 topology. We do not control the dependence
+of the C1 norm of Zn on W = Vn. For that reason, we add a step before extending the atlas Ap
+is a smooth atlas. We fix a subset U ⊂ M which we call generating family of tori (U corresponds
+to ∪iUi in the coming definition).
+If U is small, we can choose the atlas A∞
+p
+so that the
+measure νp is smooth inside U (see Lemma 83). We approximate the volume form V by smooth
+volume forms Vn, so that Vn and V coincide inside U. We approximate Y by vector fields Yn,
+Yn preserves Vn. The map H2(U, R) → H2(M, R) is surjective, so we can take a vector field Zn
+as above, but supported inside U (see Lemma 82). Since Zn is chosen for a fixed volume form V
+inside U, its C1 norm converges toward zero. Then Yn + Zn converges toward Y . It follows that
+for n large enough, Yn + Zn generates a smooth Anosov flow, orbit equivalent to φ, and which
+satisfies the Reeb-like property. In particular, it is a Reeb-Anosov flow.
+■
+The proof of Theorem F follows the same plan, except for the consideration on the linking
+number.
+Denote by D2 the unit disc in R2. A generating family of tori is a finite family of disjoint
+compact subsets (Ui)1≤i≤n of M, such that there exists some homeomorphisms fi : D2×S1 → Ui,
+satisfying that the homology classes of the curves fi(0 × S1) form a basis of H1(M, R). When M
+is a R-homology sphere, generating families of tori are empty families by conventions. Given a
+generating family of tori (Ui)i, a generating sub-family of torus is a generating family of tori (Vi)i
+with Vi ⊂ Ui for all i.
+The purpure of generating families of tori is to have a subset U ⊂ M for which H1(U, R) →
+H1(M, R) is surjective, but which is small enough so that we can extend the C1+H structure Ap
+on M, in a smooth structure for which the flow and the invariant volume form are smooth
+inside U. Denote by Λ∞(TM) the set of C∞ vector fields on M. Take a norm ∥.∥ on H1(M, R).
+Lemma 82. Let M be a smooth closed 3-manifold with a generating family of tori (Ui)1≤i≤n.
+Fix a volume form V on M, supposed to be smooth on each Ui. Then there exist A > 0 and a
+continuous map v: H2(M, R) → Λ∞(TM), such that for each x ∈ H2(M, R), the following holds:
+• v(x) ≡ 0 outside ∪iUi,
+• v(x) preserves V , that is Lv(x)V = 0,
+• ∥v(x)∥C2 ≤ A∥x∥,
+• [ι(v(x))V ] = x in H2(M, R).
+Proof. We prove a similar result for the manifold N = D2 × S1, and push it forward to M. We
+denote by (x, y) the coordinates on D2 ⊂ R2, by θ the coordinate on S1, and V0 = dx ∧ dy ∧ dθ.
+Take a function h: D2 → R such that h ≡ 0 near ∂D2 and
+�
+D2 h(x, y)dx ∧ dy = 1. We define
+on N the vector field Y = h(x, y)∂θ.
+62
+
+One has LY (V0) = d(ι(Y )V0) = d(h(x, y)dx ∧ dy) = 0, so the vector field Y preserves the
+volume form V0. Let S ⊂ N be an oriented compact surface whose boundary is inside ∂N.
+We claim that the integral
+�
+S ι(Y )V0 is equal to the algebraic intersection S ·∩(0 × S1).
+To
+prove that, notice that ι(X)V0 is closed and equal to zero close to ∂N, so
+�
+S ι(X)V0 depends
+only on the homology class of S relatively to ∂N. The equality �
+S ι(X)V0 = S ·∩(0 × S1) holds
+when S = D2 × {∗}, so it holds for any S by linearity.
+Up to taking a generating sub-family of tori of (Ui), we can suppose that there exist some
+smooth parametrizations fi : N → Ui. Applying Moser’s lemma, we can take fi such that f ∗
+i V =
+λiV0 for some λi > 0. Denote Yi =
+1
+λi dfi(Y ), which we extend to a smooth vector field on M
+by setting Yi ≡ 0 outside Ui. We have LYiV = 0. Additionally for any closed surface S ⊂ M,
+the integral
+�
+S ι(Yi)V is equal to S ·∩γi, where γi is the closed curve corresponding to fi(0 × S1).
+Therefore [ι(Yi)V ] ∈ H2(M, R) is Poincar´e-dual to [γi] ∈ H1(M, R).
+For any x ∈ H2(M, R), we write x = �
+i xi[γi] and define v(x) = �
+i xiYi.
+Then we
+have [ι(v(x))V ] = x and ∥v(x)∥C2 ≤ Cste∥x∥ maxi ∥Yi∥C2
+■
+We denote by f ∗Y the vector field x �→ df(f −1(x))(Y ◦ f −1(x)), conjugated to Y by a
+diffeomorphism f.
+Lemma 83. There exists a generating family of tori (Ui)i and an embedding f : ∪i Ui → R3 of
+class C1+H for the atlas Ap, such that
+•
+f ∗νp and f ∗Y are smooth measure on each Ui,
+•
+f(Vi) is smooth,
+Proof. Construct first a generating family of tori by taking a family of closed C1+H curves γi
+whose homology classes form a basis of H1(M, R). These curves can be taken embedded and
+disjoint. Then fix a family of disjoint compact neighborhoods (Ui)i of the curves γi.
+Inside Ui, thicken γi in a C1+H compact annulus Ai, transverse to the flow ψ. We take a C1+H
+embedding gi : Ai → R2. The measure νp is given by a H¨older volume form V on M, and Y is
+H¨older, so i(Y )V is a H¨older differential 2-form on Ai. Since Ai is transverse to Y , i(Y )V is
+additionally an area form on Ai. Hence (g−1
+i
+)∗(i(Y )V ) is a H¨older area form on gi(Ai).
+We can suppose that Ai has be taken such that gi(Ai) has a smooth boundary in R2.
+In [DM90, Zeh77] is proved a generalization of Moser’s Lemma, which state that there exists
+a C1+H diffeomorphism hi : gi(Ai) → gi(Ai) such that (hi ◦ gi)∗νp is given by a smooth volume
+form on gi(Vi). Now define the map fi : ψ[0,ǫ](Ai) → R3 by fi(ψt(x)) = (h ◦ g(x), t) ∈ R2 × R.
+Since ψ is of class C1+H, fi is also of class C1+H. By construction df(Y ) = ∂z is constant. Take
+a closed simple curve δi in ψ[0,ǫ](Ai) homotopic to γi inside Ui such that f(δi) is smooth, and
+take a neighborhood Vi of δi inside ψ[0,ǫ](Ai), such that fi(Vi) is smooth. The map f defined
+as f|Vi ≡ fi + ci for some constants ci ∈ R3 satisfies the lemma.
+■
+Regularity of the generator.
+Lemma 84. There exists a generating sub-family of tori Ui and a C∞ smooth structure A∞
+p
+on M such that:
+•
+A∞
+p extends the C1+H structure Ap,
+•
+Y is of class C1+H for A∞
+p ,
+•
+νp is smooth inside each Ui.
+63
+
+To prove the lemma, we adapt Hart’s technique to smooth the generator of a flow. Take a
+space E = Rm for some m > 0 and Ψ a flow of class C1+H on E. Denote by Z the generating
+vector field of Ψ, which we suppose to be compactly supported. In general, Z is only of class CH.
+In [Har83], Hart built a C1 diffeomorphism on E, which conjugates Z with a C1 vector field. His
+method, which we adapt bellow, actually give a C1+H vector field. Denote by f ∗Ψt = f ◦Ψt◦f −1
+the flow conjugated to Ψ by f. In particular ∂f ∗Ψt
+∂t
+= f ∗ ∂Ψt
+∂t .
+Let U, V, W ⊂ R3 be three open sub-sets such that W ⊂ V , V ⊂ U, V ∩ ∂U = ∅ and U
+is compact. Take a C1+α flow Ψ on U, generated by a Cα vector field Z. Let T > 0 be such
+that ΦT (V ) ⊂ U, and c: U → [0, T ] be a C2+α function such that c is constant on W, c > 0 on V
+and c ≡ 0 outside V . Define the map Fc : U → R3 by:
+Fc(x) =
+� 1
+0
+Ψc(x)s(x)ds
+Lemma 85. When ∥c∥C1+α is small enough, Fc is well-defined on all U and satisfies:
+•
+Fc is a C1+α diffeomorphism on U,
+•
+Fc ≡ id on Fc(U \ V ),
+•
+F ∗
+c Z is of class C1+H on Fc(W),
+• if Z is of class C1+α on a neighborhood of x ∈ U, then F ∗
+c Z is of class C1+α on a neigh-
+borhood of Fc(x).
+Additionally Fc convergences toward the identity for the C1+α topology, when ∥c∥C1+α goes to
+zero.
+Proof. For all x ∈ U, either x is in V , or not.
+In both case, Ψc(x)s(x) is well-defined for
+all s ∈ [0, 1]. So Fc is well-defined. For all x ∈ U, we have:
+dFc(x) =
+� 1
+0
+dΨc(x)s(x)ds + dc(x)
+� 1
+0
+Z ◦ Ψc(x)s(x)ds
+It follows that ∥Fc − id ∥C1+α is finite and converges to zero when ∥c∥C1+α goes to zero. Since
+the set of C1 embedding is open in the set of C1 map, Fc is a C1+α diffeomorphism on its image
+when T is small. Notice that Fc coincides with the identity map on U \ V ⊃ ∂U. So for T small,
+we have Fc(U) = U and Fc : U → U is a C1+H diffeomorphism. For x in U, we have:
+dFc(x)(Z(x)) =
+� 1
+0
+Z ◦ Ψc(x)s(x)ds + (dc(x)(Z(x)))
+� 1
+0
+Z ◦ Ψc(x)s(x)ds
+= (dc(x)(Z(x)) + 1)Ψc(x)(x) − x
+c(x)
+when c(x) ̸= 0
+Since we have c(x) ̸= 0 and dc ≡ 0 on W, dFc(Z) is of class C1+α on W. When Z is of class C1+α
+on a neighborhood of x, dc(Z) is also C1+α on x. So dFc(Z) is of class C1+H on x.
+■
+Lemma 86 (Extended Hart’s lemma [Har83]). Let Ψ be a C1+α flow on a smooth compact
+manifold M, and suppose that ∂Ψt
+∂t
+is C1+α on a neighborhood of a compact set K ⊂ M. Then
+there exists a C1+α diffeomorphism f : M → M such that f|K = idK and ∂f ∗Ψt
+∂t
+is of class C1+α.
+64
+
+Proof. For U ⊂ M, we denote by O(U) some open subset containing U. Let (A, B1, · · · , Bn) be
+an open covering of M such that K ⊂ A, K ∩Bi = ∅, Z is of class C1+α on O(A), and such that
+there exists C∞ embedding from each Bi to R3. We prove by induction on n that there exists
+a C1+α diffeomorphisms f : M → M such that f ∗Z is of class C1+H on M. If n = 0, f = id has
+this property.
+Let n ≥ 0 for which the induction hypothesis is true. Take C an open neighborhood of B1,
+disjoint from K and g : C → R3 be a C∞ embedding with pre-compact image. We take U = im(g),
+and V, W ⊂ U such that g(B1) ⊂ W, W ⊂ V and V ⊂ U. Denote by h: U → U the C1+α
+diffeomorphism denoted by Fc in Lemma 85, for some small function c: U → R.
+Define a
+map l: M → M by l ≡ id on M \ C and l ≡ g−1 ◦ h ◦ g on C.
+The map h is equal to the identity on a neighborhood of ∂U, so l is of class C1+α on all M.
+We can take h arbitrarily C1 close to the identity so that h(W) contains g(B1). Then l∗Z is of
+class C1+α on a neighborhood of B1. Similarly we can take h such that l∗Z is of class C1+α on
+a neighborhood of A. Indeed g∗Z is of class C1+α on a neighborhood of g(A ∩ C). So for x in a
+smaller neighborhood of g(A ∩ C), (h ◦ g)∗Z is C1+α on h(x). If ∥c∥C1 is small enough, for the
+previous reason h(O(g(A ∩ C))) contains g(A ∩ C). Hence (h ◦ g)∗Z is C1+α on a neighborhood
+of g(A∩C). So l∗Z is C1+α on a neighborhood of A∩B1, and also on A∩(M \B1) by hypothesis.
+The map l is a C1+α diffeomorphism such that l∗Z is C1+α on a neighborhood of A ∪ B1.
+According to the induction hypothesis, there exists a C1+α diffeomorphism j : M → M such
+that j∗(l∗Z) is C1+α on M, and with j|K = idK. So f = j ◦ i satisfies the property at the rank n,
+which conclude the induction.
+■
+Proof of Lemma 84. Let (Vi)i be a generating family of tori. Take a C1+H map f : ∪i Vi → R3
+given by Lemma 83, so that f ∗νp is smooth. Take n ≥ 2 and consider a C1+H map g : M →
+R3 × R2n which extends f × 0 on ∪iVi. If g is chosen generic outside each Vi, then it is an
+embedding.
+Denote by G2n the Grassmannian of 2n-dimensional vector sub-spaces of R2n+3. Take a C1+H
+map p: M → G2n so that for each x ∈ M, p(x) is transverse to dg(TxM), the tangent space
+of g(M) at g(x). We denote by E = {(x, u) ∈ M × R2n+3, u ∈ p(x)} the vector bundle over M.
+Define a C1+H map h: E → R2n+3 by h(x, u) = g(x)+u. Since p(x) is transverse to dg(TxM), the
+map h is injective on a neighborhood U of the zero section, that is on a neighborhood of {(x, 0) ∈
+E}.
+Take a smooth sub-manifold N ⊂ U ⊂ R2n+3 obtained by a C1+H-small perturbation
+of g(M). Since h is C1+H, h−1(N) is the image of a C1+H section s of the bundle E → M. Then
+map h ◦ s: M → R2n+3 is of class C1+H and its image is a smooth sub-manifold.
+For a set I ⊂ �1, 2n + 3� of cardinal 3, denote by πI : R2n+3 → R3 the forgetting map
+corresponding to the indexes not in I. Consider the atlas A whose charts are of the form (πI ◦
+h◦s)U for all I ⊂ �1, 2n + 3� of cardinal 3, and U ⊂ M open on which πI ◦h◦s is an embedding.
+Since N is smooth, the transition maps are smooth. So A is a C∞ atlas on M, which additionally
+extends Ap. Since g(∪iVi) is smooth, we can smooth N only outside that set, so that h◦s coincides
+with g on ∪iVi. Then π�1,3� ◦ h ◦ s coincides with f on that set.
+We now regularize Y and νp.
+Take a generating sub-family of tori (Ui)i of (Vi)i, such
+that Ui ⊂ Int Vi for all i, and denote by K = ∪iUi.
+According to Lemma 86 there exists
+a C1+H diffeomorphism l of M, such that l|K = idK and such that l∗Y is of class C1+H. Since l
+coincides with the identity inside each Ui, (l ◦ h)∗νp are smooth inside each Ui. The atlas A∞
+p is
+obtained by conjugating the chart of A with l.
+■
+Smoothing the generator.
+For this subsection, we use the notation from Lemma 84 and
+consider the atlas denoted by A∞
+p . According to the lemma, there exists a generating family
+of tori (Ui)i, on which νp is smooth. Take α ∈ (0, 1) such that Y is of class C1+α and such
+65
+
+that νp is induced by an α-H¨older volume form on M. Let V be a smooth volume form on M
+and λ: M → R be the function which satisfies νp = eλV . It is α-H¨older continuous on M and
+smooth on each tori Ui. Take λk : M → R a sequence of C∞ function, which approximates λ for
+the Cα topology, and which coincide with λ on each Ui.
+Lemma 87 (Asaoka 2008 [Asa07]). There exists a sequence of C∞ vector fields Yk on M such
+that:
+•
+(Yk)k converges toward Y in the C1+α topology,
+•
+Yk preserves eλkV for all k,
+Denote by ψk the flow generated by Yk. By C1 structural stability of Anosov flows, ψk is
+Anosov and orbit equivalent to φ for k large enough (see [KKW91]). Denote by hn : (M, ψk) →
+(M, ψ) an orbit equivalence so that (hn)n converges toward id for the C0 topology. Also denote
+by µk the probability measure induced by eλkV , invariant by ψk.
+Lemma 88. The sequence of measures Θhn(µn) ∈ Mp(ψ) converges toward νp.
+Proof. We prove that the sequence of transverse measures h∗
+k(µ⊥
+k ) converges toward ν⊥
+p , which
+implies the lemma. Recall that (λk)k converges toward λ, so (µk)k converges toward νp.
+Let S ⊂ M be a compact surface, transverse to Y and Yk for all k. So there exists ǫ > 0 for
+which the maps ψk are all injective on the set [0, ǫ] × S → M. Then for any k, we have µ⊥
+k (S) =
+1
+ǫµk(ψk
+[0,ǫ](S)) ≤ 1
+ǫ.
+Take a continuous function f : S → R so that f ≡ 0 on a neighborhood of ∂S. We can
+extend f to a continuous function M → R, such that f ≡ 0 on a neighborhood of ∂S inside M.
+Fix three neighborhoods U1, U2, U3 ⊂ S of ∂S for which fUi = 0 and U i ⊂ Int Ui+1.
+Since the sequence (hk)k converges toward the identity, there exist for k large enough an
+embedding gk : h−1
+k (S \ U1) → S, obtained as a small isotopy along the flow ψk from h−1
+k (S \ U1)
+to S. For large enough, the image of gk contains U2. Since hk converges toward id and Yk
+converges toward Y , the map gk ◦ h−1
+k
+converges toward the inclusion S \ U1 ֒→ S. Hence for k
+large we have S \ U2 ⊂ gk ◦ h−1
+k (S \ U1). Therefore for large k, the following holds:
+�
+S
+fdh∗
+k(µ⊥
+k ) =
+�
+S\U1
+fdh∗
+k(µ⊥
+k )
+=
+�
+h−1
+k (S\U1)
+f ◦ hkdµ⊥
+k
+=
+�
+gk◦h−1
+k (S\U1)
+f ◦ hk ◦ g−1
+k dµ⊥
+k
+=
+�
+S\U2
+f ◦ hk ◦ g−1
+k dµ⊥
+k
+����
+�
+S
+fdν⊥
+p −
+�
+S
+fdh∗
+k(µ⊥
+k )
+���� ≤
+�����
+�
+S
+fdν⊥
+p −
+�
+S\U2
+f ◦ hk ◦ g−1
+k dµ⊥
+k
+�����
+≤
+�����
+�
+S\U2
+fd
+�
+ν⊥
+p − µ⊥
+k
+�
+����� +
+�
+S\U2
+��f − f ◦ hk ◦ g−1
+k
+�� dµ⊥
+k
+≤
+�����
+�
+S\U2
+fd
+�
+ν⊥
+p − µ⊥
+k
+�
+����� + µ⊥
+k (S)∥f − f ◦ hk ◦ g−1
+k ∥S\U2
+−−−−−→
+k→+∞ 0
+66
+
+The first term converges toward zero since µk converges toward νp. The second term converges
+toward zero since we have µ⊥
+k (S) ≤ 1
+ǫ, and (f ◦hk◦g−1
+k )k converges toward f on S\U2. Therefore
+the measure h∗
+k(µ⊥
+k ) converges toward ν⊥
+p .
+■
+Lemma 89. There exists a sequence Zk of C∞ vector fields on M such that:
+•
+Zk converges toward zero in the C1+α topology,
+•
+Zk preserves eλkV for all k,
+•
+ι(Yk)eλkV and ι(Zk)eλkV are cohomologous.
+Proof. Let v: H2(M, R) → Λ∞(T M) be the map constructed in Lemma 82 for the volume
+form eλV , which is smooth inside each Ui.
+Notice that for x ∈ H2(M, R), v(x) is a vector
+field supported inside the union of the tori Ui, where λ and λk coincide.
+Hence the vector
+field v(x) preserves the volume forms eλkV . Now take Zk = v([ι(Yk)eλkV ]). By hypothesis, we
+have [ι(Zk)eλkV ] = [ι(Yk)eλkV ], which converges toward zero according to Lemmas 88 and 10. It
+follows that Zk converges toward zero for the C2 topology, and so also for the C1+α topology.
+■
+Corollary 90. For k large enough, the smooth vector field Yk − Zk generates an Anosov flow,
+orbit equivalent to φ, which preserves the smooth volume form eλkV . Additionally the probability
+measure induced by eλkV satisfies the Reeb-like property for k large enough.
+The corollary and Theorem D give a proof to Theorem A.
+Proof. Denote by ψn the flow generated by the vector field Yn − Zn. The vector fields (Yk − Zk)
+converges toward Y for the C1 topology. So by structural stability of Anosov flows [KKW91],
+for k large enough, ψn is Anosov and is orbit equivalent to φ. Denote by ηn the ψn-invariant
+probability measure induced by ι(Yk − Zk)eλkV . It follows from the previous lemma that ηn
+is null-cohomologous. Take hn a sequence of orbit equivalences between ψn and φ, so that hn
+converge toward the identity map. According to Lemma 88, the measure Θhn(ηn) converges
+toward νp. So for n large enough, the minimum of linkφ(Θhn(ηn), ν) with all ν ∈ M0
+p(φ), is
+positive (see Corollary 22). It follows from Lemma 57 that the measure ηn satisfies the Reeb-like
+property.
+■
+A
+Fried-desingularisation
+We prove Proposition 60: given a Birkhoff section S1 and a partial section S2, there exists a
+Birkhoff section homologous to [S1] + [S2] in H2(M, ∂S1 ∪ ∂S2, R). Take ∆ = ∂S1 ∪ ∂S2, M∆
+the blow up manifold and fix a Riemannian metric m on M∆. We denote by D the diame-
+ter of M∆ for m, and by ∥ · ∥ one norm on H2(M∆, ∂M∆, R). Define the normalization map
+N : H2(M∆, ∂M∆, R) → H2(M∆, ∂M∆, R) given by N(0) = 0 and N(x) = x/∥x∥ when x ̸= 0.
+For x ∈ M∆ and T ≥ 0, denote by γT
+x : [0, T + 1] → M∆ one map given by γT
+x (t) = φ∆
+t (x),
+for t ∈ [0, T ], and (γT
+x )|[T,T +1] is smooth path from φT (x) to x, whose length is at most D. The
+choice of the curve (γT
+x )|[T,T +1] has no influence on the following notion. The accumulating points
+of elements of the form N([γT
+x ]), for T −→ +∞, are called asymptotic directions. Schwartzman
+and Fried [Sch57, Fri82] proved that a compact surface is relatively homologous to a global section
+of flow if and only if it intersects algebraically positively every asymptotic direction.
+Proof of Proposition 60. We lift the surface Si to a surface �Si inside M∆. We write �S = �
+S1 ∪ �
+S2.
+Since S1 is a Birkhoff section, there exists T0 > 0 for which every orbit arc of length T0 of φ∆
+intersects �
+S1. Recall that Si is positively transverse to the φ. Denote by C1 > 0 the supremum
+67
+
+of the algebraic intersection of �S and a smooth curve of length D, which is finite. Take C2 > 0
+for which every closed C1 curve γ ⊂ M∆ satisfies ∥[γ]∥ ≤ C2lengthm(γ). Also take C3 > 0 the
+maximum of
+��� ∂φ∆
+t
+∂t
+���
+m, so that the length for m of an orbit arc φ[0,T ](x) is at most C3T .
+Take x ∈ M∆ and T > 0. When x is not in the boundary of M∆, the algebraic intersection
+between γT
+x and �S is at least φ∆
+[0,T ](x) ·∩([�
+S1] + [�
+S2]) − C1 ≥
+T
+T0 − 1 − C1. By continuity, it also
+holds for x ∈ ∂M∆. So we have:
+N
+�
+[γT
+x ]
+�
+·∩[�S] ≥ [γT
+x ] ·∩[�S]
+∥[γTx ]∥
+≥
+T
+T0 − 1 − C1
+C2(C3T + D)
+lim inf
+T−→+∞ N
+�
+[γT
+x ]
+�
+·∩[�S] ≥
+1
+T0C2C3
+Hence [�S] intersects positively every asymptotic direction. According to Schwartzman-Fried
+Theorem [Fri82], there exists a global section relatively homologous to �S in H2(M∆, ∂M∆, R).
+The existence of a Birkhoff section homologous to [S1] + [S2] follows from Lemma 63.
+■
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+
diff --git a/QNAyT4oBgHgl3EQf7voS/content/tmp_files/load_file.txt b/QNAyT4oBgHgl3EQf7voS/content/tmp_files/load_file.txt
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@@ -0,0 +1,3023 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf,len=3022
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='00842v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='DS]  2 Jan 2023  Skewed Anosov flows are orbit equivalent to Reeb-Anosov  flows in dimension 3  Marty Th´eo  Abstract  We prove that in dimension 3, Anosov flows which are R-covered and skewed are orbit  equivalent to Reeb-Anosov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We characterize the existence of an invariant contact form  or of a Birkhoff section with a given boundary, in terms of linking numbers between two  invariant signed measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Furthermore, we prove the existence of open book decompositions  with one boundary component for Reeb-Anosov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Introduction  Geodesic flows on hyperbolic surfaces have been studied systematically using two approaches:  they are classical examples of hyperbolic flows and are the Reeb flows of the natural contact  structures on the corresponding manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Anosov generalized hyperbolic geodesic flows in  what is now known as Anosov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In dimension 3, other families of Anosov flows have been  studied: suspension of linear Anosov diffeomorphisms of the torus, Anosov flows built from Dehn  surgeries along periodic orbits (also called Fried-Goodman surgeries [Fri83, Goo83]), and Anosov  flows built from gluing hyperbolic plugs [BBY17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Except for geodesic flows, very few Anosov flows were known to be of the Reeb type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Foulon  and Hasselblatt [FH13] produced Reeb-Anosov flows from surgeries on hyperbolic geodesic flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Their construction extends a previous construction of surgery by Handel and Thurston [HT80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Recently Salmoiraghi [Sal22] produced a larger family of Reeb-Anosov flows using bi-contact  structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' His construction extends the surgery operation introduced by Goodman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Fenley and Barbot [Fen94, Bar01] independently introduced the key notion of orbit space  to understand topological properties of Anosov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' One remarkable family of Anosov flows  is characterized by its orbit space: the skewed R-covered Anosov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Barbot [Bar01] proved  that Reeb-Anosov flows are skewed R-covered flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The main result of the present paper is the  converse implication:  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let φ be an Anosov flow on an oriented, closed, 3-dimensional manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Suppose  that φ is R-covered and positively skewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then φ is orbit equivalent to a smooth Reeb-Anosov  flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Additionally if α is the corresponding contact form, then α ∧ dα is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The theorem gives an answer to Barbot/Barthelm´e’s conjecture [Bar17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By combining the  theorem and previous works, we obtain a series of equivalences, represented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Birkhoff  sections and linking numbers between invariant measures are introduced later in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We denote by M0  p(φ) the set of null-cohomologous φ-invariant probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In a given column in Table 1, two cells are equivalent to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The columns  correspond to exclusive natures for Anosov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  1  Positively twisted  Flat flows  Negatively twisted  Others  1  positively skewed  R-covered  R-covered with trivial  bi-foliated plane  negatively skewed  R-covered  non  R-covered  2  ∃ Birkhoff section with  positive boundary  ∃ Birkhoff section  with no boundary  ∃ Birkhoff section with  negative boundary  other  3  ∃µ ∈ M0  p(φ) with  positive linking numbers  with all M0  p(φ)  M0  p(φ) = ∅  ∃µ ∈ M0  p(φ) with  negative linking numbers  with all M0  p(φ)  other  4  ∃ψ ≃ φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' ∃α contact,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ψ-invariant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' α ∧ dα > 0  ∃ψ ≃ φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' ∃α ̸= 0  ψ-invariant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' dα = 0  ∃ψ ≃ φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' ∃α contact  ψ-invariant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' α ∧ dα < 0  other  Table 1: Tetrachotomy for Anosov flows between twisted flows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' flat flows and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Each  line corresponds to one notion, each column corresponds to equivalent properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Line 1: the  topology of the bi-foliated plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Line 2: the existence of a particular Birkhoff section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Line 3: the  existence of a particular null-cohomologous invariant probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Line 4: the existence  of an orbit equivalent Anosov flow, for which there exists a particular differential 1-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It can  be rephrased, for columns 1 to 3, by the existence of a particular orbit equivalent geodesible  Anosov flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  In the columns 1 and 3, the ambient manifold is oriented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Reversing the orientation ex-  changes the role of these two columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We should discuss the already existing equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The tetrachotomy was first discovered by Barbot [Bar95] and Fenley [Fen94] simultaneously  (corresponding to the line 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The equivalence between the lines 1 and 2 is proven in [ABM22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For the first column, Barbot [Bar01] proved that the 4th cell implies the 1st cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Barbot does  not specify a sign for α∧dα, but it can be recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We give a second proof of this implication in  Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='3, with the sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The main theorem of this article of the converse implication: the 1st  cell implies the 4th cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We additionally introduce the 3rd line which is intersecting on its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The equivalence between the 2nd and 3rd cells is more elementary and in the spirit of the work  of Ghys [Ghy09] (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For the second column, Solodov1 proved the equivalence between cell 2 and cell 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The  equivalence between the 2nd and 3rd cells follows from Schwartzman-Sullivan’s Theorem on cross-  sections [Sch57, Sul76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The equivalence with the 4th cell is well known and more elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  To prove the main theorem, we roughly follow the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We need to find an orbit  equivalent Anosov flow which preserves a smooth volume form V , where V would correspond  to α∧dα for a contact form α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Asaoka constructed an orbit equivalent, volume preserving Anosov  flow using a Gibbs measure [Asa07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Gibbs measures form an important family of invariant  probability measure, which satisfies some dynamical properties similar to the invariant volume  forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given a Gibbs measure, Asaoka builds a new C1+H differential structure on the ambient  manifold, for which the Gibbs measure is induced by a H¨older continuous volume form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then he  approximates the flow by smooth Anosov flows, smooth for the new differential structure, each  of them preserving a smooth volume form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For an Anosov flow in general, preserving a smooth volume form is not enough to be a  1V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Solodov did not publish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Tha main argument is that actions on R, for which every element  acts with at most one fixed point, are conjugated to affine actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  2  reparametrization of a Reeb-Anosov flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We need two additional conditions, which we ex-  press in terms of linking number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' To an invariant probability measure corresponds an element  in H2(M, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by M0(φ) the set of null-cohomologous φ-invariant signed measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Theorem C (Made precise in Theorem 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let φ be a transitive Anosov flow in a closed, ori-  ented, 3-dimensional manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' There exists a unique continuous bilinear map linkφ : M0(φ) ×  M0(φ) → R extending the linking number for two disjoint, null-homologous knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Addition-  ally linkφ is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The new linking number map extends previous linking numbers given by Ghys2 [Ghy09],  Arnold [Arn74], Vogel [Vog03] and Prasad [Pra22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Our definition of the linking number map  requires the existence of a Markov partition, so it cannot be extended past maybe transitive  pseudo-Anosov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' One can rephrase McDuff criterion [Duf87, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='2] for contact-type  differential 2-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Theorem D (McDuff [Duf87] - Prasad [Pra22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let φ be a smooth Anosov flow on a closed,  oriented 3-dimensional manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then φ is a smooth reparametrization of a Reeb-Anosov flow  for a contact form α with α ∧ dα > 0, if and only if it preserves a smooth probability measure ν,  which is null-cohomologous and has positive linking number with all null-cohomologous invariant  probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  McDuff’s theorem is written in the language of structural boundaries developed by Sulli-  van [Sul76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Prasad recently interpreted McDuff’s theorem in terms of linking number, for  right-handed flows on rational homology spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We give a proof of the reformulation in Sec-  tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We need to find a Gibbs measure satisfying the two hypothesis in the previous theorem:  it is null-cohomologous and has positive linking numbers with all null-cohomologous invariant  probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We first construct an invariant probability measure Lebγ supported on a  single periodic orbit γ, satisfying these two properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then we approximate Lebγ by a sequence  of null-homologous Gibbs measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map linkφ is continuous, so a sub-sequence of Gibbs  measures has positive linking numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then we adapt Asaoka’s strategy to get a smooth Anosov  flow preserving a smooth volume form, for which McDuff’s criterion is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  It remains to explain the existence of the measure Lebγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Fried [Fri83] proved the existence of  Birkhoff section for transitive Anosov flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Birkhoff section are roughly speaking transverse sur-  faces, bounded by periodic orbits, which intersect efficiently the orbits of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The Birkhoff  sections constructed by Fried bring little information on the topology of the ambient manifold  and of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In a previous work [ABM22], we characterized positively skewed R-covered  Anosov flows using the existence of Birkhoff sections with positive boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In Section 5, we  improve this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let φ be a R-covered and positively skewed Anosov flow on a closed, oriented 3-  dimensional manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then φ admits an embedded positive Birkhoff section with only one bound-  ary component, whose corresponding periodic orbit has orientable stable and unstable leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In  particular, it gives an open book decomposition of M, adapted to φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take a periodic orbit γ bounding a Birkhoff section as in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Consider the Lebesgue  measure Lebγ on γ, invariant by the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Up to rescaling, Lebγ is a null-cohomologous invari-  ant probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Its linking number with any null-cohomologous invariant probability  measure is positive, since γ is the positive boundary of a Birkhoff section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence one can apply  the above discussion on Lebγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  2Ghys proposes a general definition for the linking number, but it is not clear why it is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Florio  and Hryniewicz proposed a similar definition by replacing limits with limits inferior, for right-handed flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  3  We end the introduction by discussing two annex theorems, which we obtain with the same  techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Our smoothing strategy also works for Anosov flows said to by homologically full,  that is every element in H1(M, Z) is the homology class of a periodic orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Sharp [Sha93]  proved that an Anosov flow is homologically full if and only it admits a null-cohomologous Gibbs  measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We can smooth the Gibbs measure with Asaoka’s technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Theorem F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let φ be a homologically full Anosov flow on a closed 3-dimensional manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  There exists a smooth Anosov flow ψ and a smooth volume form V on M which satisfy the  following: ψ is orbit equivalent to φ, it preserves V , and the interior product ι  �  ∂ψt  ∂t  �  V is null-  cohomologous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  In [Ghy09, CDHR22], a positive linking number condition is used to construct a Birkhoff  section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For Anosov flows, their idea and the linking number map give a criterion for the  existence of Birkhoff section bounding a specific boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Theorem G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let φ be an Anosov flow on a closed, oriented, 3-dimensional manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let Γ  be a collection of periodic geodesic, with multiplicities in Z, supposed to be null-homologous  in H1(M, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then the following statement are equivalent:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' there exist n > 0 and a Birkhoff section bounded nΓ,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' the invariant Lebesgue measure LebΓ supported on Γ has a positive linking number with  every null-cohomologous probability measure invariant by φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Notice that the theorem holds even when there exists no null-cohomologous probability mea-  sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In which case both conditions are satisfied independently on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The author does not know if  one can always take n = 1 in the first assertion, assuming that Γ is null-homologous in H1(M, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We prove the theorem in Section 5, and deduce a new proof of Barbot’s Theorem [Bar01] on  Reeb-Anosov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  In our strategy, we change the differential structure on the ambient manifold, and the  parametrization of the flow comes as a consequence of the technique used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It comes as a sur-  prise since the parametrization of the flow seems to play a more important role than the smooth  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Barbot introduced a notion of topologically contact Anosov flow [Bar01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Being topo-  logically contact is invariant by (only continuous) conjugation, so it does not depend on the  specific choice of the smooth structure on the ambient manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Can we characterize all topologically contact parametrizations of a given Reeb-  Anosov flow?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  I am grateful to M Asaoka for the discussion and help regarding the use  of Gibbs measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' I thank the Max Plank Institute in Bonn for its financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' I thank P  Dehornoy and A Boulanger for the discussions on the linking number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' I also thank A Rechtman,  T Barbot, M Postic and V Colin for the various discussions on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Contents  1  Preliminary  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='1  Anosov flows  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
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+page_content='  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='2  Orbit space .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
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+page_content='  6  4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='3  Reeb flow of a contact form .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
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+page_content='  50  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='2  C1+H differential structure .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
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+page_content='  56  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='3  Smoothing the flow .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
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+page_content='  61  A Fried-desingularisation  67  References  70  1  Preliminary  We introduce the notions appearing in the main theorem: Anosov flows, orbits space, positively  skewed R-covered Anosov flows and Reeb flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='1  Anosov flows  Let M be a closed three-dimensional manifold and φ a C1 flow on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The flow φ is said Anosov  if there exists a φ-invariant splitting of T M into three line bundles T M = Es ⊕ X ⊕ Eu and two  real numbers A, B > 0 such that for one/any Riemannian norm ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='∥ on T M, we have: X is tangent to the flow,  for all t ≥ 0, ∥dEsφt∥ ≤ A exp−Bt,  for all t ≤ 0, ∥dEuφt∥ ≤ A exp−B|t|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  5  Here dEsφt and dEuφt correspond to the differential dφt restricted to the line bundles Es  and Eu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The flow is said transitive if there exists an orbit of φ dense inside M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We compare three  properties for Anosov flows, each of them implies that the flow is transitive: R-covered Anosov  flows [Bar95], Reeb-Anosov flows [Bar01] and Anosov flows admitting a Birkhoff section [Fri83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  In the rest of the article, M is supposed oriented and φ is supposed transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The bundles Es and Eu are integrable in two 1-foliations, which we denote respectively by Fss  and Fuu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' They are called the strong stable and unstable foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The plane bundles Es ⊕ X  and Eu ⊕ X are also integrable in two 2-foliations, invariant by φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' They are denoted by Fs  and Fu and called weak stable and unstable foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Every leaf of the foliations Fss and Fuu is homeomorphic to R and each leaf of the foli-  ations Fs and Fu is homeomorphic to either R/Z × R or an open M¨obius strip if it contains  a periodic orbit, or R2 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Through the article, we use the notation F(x) for the leaf  of the foliation F containing the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We also denote by Leaf(F) the set of leaves of the  foliation F, equipped with the quotient topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Let φ and ψ be two flows on respectively M and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' An orbit equivalence h: (M, φ) →  (N, ψ) is a homeomorphism h: M → N such that for each orbit arc γ of φ, h(γ) is an orbit arc  of ψ and h preserves the orientation by the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' When M and N are oriented, we additionally  suppose that h preserves the orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Important notions in this article are invariant by orbit  equivalence: the homeomorphism class of the bi-foliated plane, existence of a Birkhoff section  with positive boundary and linking number between invariant measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Orientations-bundle covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Let M ′ be the manifold given as the set of tuples (x, os  x, ou  x)  with x ∈ M, and os  x, ou  x are local orientations of respectively the stable and unstable foliations  at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The projection M ′ → M on the first coordinate is a degree 4 covering map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by �  M  one connected component of M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The projection �π : �  M → M is a covering map, which is of  degree one if and only φ has orientable stable and unstable foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Recall that M is supposed  orientable, so �  M → M is of degree one or two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We call �  M the orientations-bundle covering  of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We also denote by �φ the lift of φ to �  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice that �φ is Anosov, its stable and unstable  foliations are the lifts of the corresponding foliations for φ, and they are orientable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Notice that �π∗(π1(�  M)) is a subgroup of π1(�  M) of index one or two, so it is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote  by sign: π1(M) → {−1, 1} the group morphism obtained as π1(M) ։ π1(�  M)/�π∗(π1(�  M)) ֒−→  {−1, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' That is given a simple loop γ based on x, sign([γ]) = 1 if and only if the stable and  unstable foliations of M are orientable in a neighborhood of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Notice that sign is a group  morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map sign is defined so that for a closed and injective loop γ, the stable and  unstable foliations of φ are orientable on a small neighborhood of γ if and only if sign(γ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='2  Orbit space  Take an Anosov flow φ on a closed 3-manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by πM : �  M → M the universal  covering map of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The flow and the foliations lifts to a flow �φ and foliations �Fs, �Fu, �Fss  and �Fuu in �  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We call orbit space of φ, the quotient Pφ of �  M by orbits of the flow �φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Barbot and Fenley proved [Bar95, Fen94] that Pφ is homeomorphic to a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We denote  by πPφ : �  M → Pφ the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The foliations �Fs, �Fu are saturated by �φ, so they project  in Pφ to two transverse foliations by curves, which we denote by Ls and Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If the flow φ is  smooth, the set Pφ comes with a C∞ structure, for which the foliations Ls and Lu are of class C1  (see [FH19, Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We call bi-foliated plane of φ the set (Pφ, Ls, Lu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Suppose that M is oriented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We lift the orientation on M to an orientation on �  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We  equip any surface S ⊂ �  M transverse to the flow �φ, with the orientation which satisfies that S is  6  Ls  Lu  Figure 1: Classification of the bi-foliated plane of R-covered Anosov flows: suspension flows,  positively skewed and negatively skewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  positively transverse to the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The projection πPφ restricts to an immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We fix on Pφ  the orientation for which for any positively transverse surface S ⊂ �  M, the immersion S → Pφ is  orientation preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  A particular family of Anosov flow are the ones for which the space of leaves Leaf(Fs) (or  equivalently Leaf(Ls)) is homeomorphic to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' This condition is equivalent to having Leaf(Fu)  and Leaf(Lu) homeomorphic to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' These flows are said to be R-covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Theorem 2 (Barbot-Fenley [Bar95, Fen94]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let φ be a R-covered Anosov flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Then φ is  transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Additionally there is a homeomorphism, preserving the orientation and the foliations,  from the bi-foliated plane of φ to one of the three following cases (illustrated in Figure 1):  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  (R2, H, V ) where H and V are the foliations made of horizontal and vertical lines,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  (D+, H, V ) where D+ is the diagonal strip {(x, y) ∈ R2, |x − y| < 1},  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  (D−, H, V ) where D− is the anti-diagonal strip {(x, y) ∈ R2, |x + y| < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Here R2, D+, D− are equipped with the trigonometric orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  In the first case, an argument from Solodov implies that φ is topologically equivalent to a  suspension of a linear Anosov diffeomorphism on the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In the second and third case, M  is additionally orientable, and we say that φ is respectively positively skewed and negatively  skewed (they are sometime call twisted instead of skewed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Notice that these two cases are  disjoint because of the choice of the orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In particular reversing the orientation on M  switches the rolls of the cases 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The fundamental group π1(M) acts on �  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' That action preserves the foliations of �φ, so it  induces an action π1(M) ↷ Pφ which preserves Ls and Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by g·�p the action of π1(M)  on �  M, and g ∗ξ the action π1(M) ↷ Pφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' A point ξ in Pφ corresponds to the orbit πM(π−1  P ({ξ}))  of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Additionally a point ξ invariant by a non-trivial element in π1(M) corresponds to a periodic  orbit of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' More precise, let �x ∈ �  M, x = πM(�x) ∈ M and ξ = πPφ(�x) ∈ Pφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then x is φ-periodic  if and only if there exists g ∈ π1(M) \\ {0} such that g ∗ ξ = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Suppose x to be periodic,  then g can be taken as the homotopy class of the orbit of x in π1(M, x) (see [VS73] for the  non-triviality of that element).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by g ∈ π1(M, x) the homotopy class of the orbit of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Then g preserves the leaves Ls(ξ) and Lu(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  If the stable leaf Fs(x) is orientable, then g  preserves the orientation on Ls(ξ) and Lu(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Additionally g contracts Lu(ξ) and expends Ls(ξ)  (see [ABM22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' When Fs(x) is not orientable, then g2 satisfies that property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 3 (Fenley [Fen94]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let φ be a skewed R-covered Anosov flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any periodic orbit γ  of φ, there exists a periodic orbit δ and n, m ≥ 1 for which γn is anti-homotopic to δm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  7  In the lemma, γn stands for the orbit γ traveled n times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' A skewed R-covered Anosov flow is homologically full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' This is a consequence of two facts: the homology class of periodic orbits span H1(M, Z)  (see [PP86]), and every periodic orbit has a positive multiple which is anti-homotopic to a positive  multiple of another periodic orbit (see Lemma 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that there exists a finite family of  periodic orbits (γi)i, for which the convex hull of the homology class [γi] ∈ H1(M, R) contains 0 in  its interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So the flow is homologically full according to the Sharp’s characterization [Sha93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='3  Reeb flow of a contact form  A contact form on a 3-dimensional manifold M is a C1 differential form α such that α ∧ dα is  never zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In particular α ∧ dα is a volume form on M, so M is orientable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given a contact  form α, the plane distribution given by ξ = ker(α) is called a contact structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given a contact  form α, there exists a unique vector field Yα on M satisfying ι(Yα)α = 1 and ι(Yα)dα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' This  vector field is called the Reeb vector field of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' When α is of class C∞, Yα is also of class C∞, so  it generates a C∞ flow ψα, called Reeb-flow of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Flows which are both Anosov and Reeb-flows  are called Reeb-Anosov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Theorem 5 (Barbot [Bar01]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In dimension 3, any Reeb-Anosov flow is R-covered and skewed,  and in particular transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Barbot does not relate the nature positively/negatively skewed of the flow with the orientation  given by the contact form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We give a new proof of his theorem in Section 5, which additionally  gives the sign correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Given a smooth flow ψ, we say that it is Reeb-like if there exists a smooth reparametrization  of ψ which is a Reeb-flow of a contact from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It is equivalent to the existence of a contact form α  on M such that ι(X)α > 0 and ι(X)dα = 0, where X =  ∂ψt  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Indeed in this case, the flow  generated by Yα =  1  ι(X)αX is a Reeb-flow and a smooth reparametrization of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The contact structure of the Reeb flow of a smooth contact form α, that is the plane dis-  tribution given by ker(α) ⊂ T M, is smooth, transverse to the flow and invariant by the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We the flow is Anosov, only one plane distribution satisfies this property, the sum Es ⊕ Eu of  the strong stable and strong unstable directions of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In particular this sum is smooth  for Reeb-Anosov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It is known that when Es ⊕ Eu is of class C1, then either the flow is a  constant time suspension of an Anosov diffeomorphism of the torus, or Es ⊕ Eu is the kernel of  an invariant C1 contact form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In general, this plane field is only H¨older, but it can still satisfy  some topological version of being of the contact type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Barbot call these flows topologically con-  tact, and proved [Bar01] that these flows are also R-covered and skewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In Section 4, we give  a condition for an Anosov flow to be Reeb-like, in terms of linking number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='4  Invariant measures  We give our notation for invariant measures and current in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  In this article, ex-  cept when stated otherwise, the term measure refers to signed measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by Mp(φ)  and M(φ) the sets of φ-invariant probability measure and φ-invariant signed measure on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We  equip these sets with the weak topology a sequence of measures (µn)n converges toward µ∞ if  for any continuous function f : M → R, the integral  �  M fµn converges toward  �  M fµ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The  set Mp(φ) is compact for the weak topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  8  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' A consequence of Banach-Steinhaus Theorem is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given a sequence of  signed measure µn converging toward a µ∞, the total variation of µn is bounded by some constant  in dependent on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  A transverse measure µ is the data for any compact embedded surface S transverse to φ,  of a finite measure µS on S, such that for any compact embedded surfaces S1, S2 transverse to φ,  and any holonomy map h: U1 ⊂ S1 → U2 ⊂ S2 along the flow, h∗µS1 and µS2 coincide on h(U1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Given a measure µ ∈ M(φ), we denote by µ⊥ the corresponding transverse measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It is defined  by µ⊥(A) = 1  ǫµ(φ[0,ǫ](A)) for any measurable set A ⊂ S transverse to φ, and ǫ > 0 small enough  (which depend on S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by Lebφ the unique measure on orbits of φ in M, satisfying that  for every x ∈ M and ǫ > 0 small enough, φ[0,ǫ](x) = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given a transverse measure µ⊥, we  denote by µ⊥ ⊗ Lebφ the invariant measure obtained locally as a product of µ⊥ and Lebφ along  the orbits of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The maps µ �→ µ⊥ and µ⊥ �→ µ⊥ ⊗ Lebφ are continuous and inverse to each  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Any orbit equivalence h: (M, φ) → (N, ψ) induces a natural homeomorphism µ⊥ �→ h∗µ⊥  between the sets of transverse measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' There is a natural homeomorphism Θh : M(φ) → M(ψ)  between the set of invariant measures of the two flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For a measure µ ∈ M(φ), Θh(µ) is the  measure h∗(µ⊥) ⊗ψ Lebψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We introduce later the cohomology class and the linking number of  signed measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' These notions are invariant by orbit equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since they do not depend on the parametrization of the flow nor on the smooth  structure on the manifold, transverse measures is the natural notion to define the cohomology  class and the linking number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For convenience, we still use invariant measures instead, but we  keep in mind the underlining transverse measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Cohomology class of an invariant measure  Given an invariant measure, there is a natural  current associated to that measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We introduce currents and use it to define the cohomology  class of an invariant measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' See [DR84] for more background on currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by ∧ the  wedge product, by ι(X)ω the inner product of ω by X, by ·∩ the algebraic intersection and by ·  the integration of a differential k-form on a k-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by n = 3 the dimension of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' A k-current is a linear form on the set Ωk(M) of  smooth k-differential form on M, which is continuous for the C∞ topology on Ωk(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given  a k-current C and a smooth vector field Y on M, we define the currents dC and ι(Y )C to be  the k − 1 and k + 1 currents given respectively by: dC(α) = C(dα), ι(Y )C(α) = C(ι(Y )α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Given an invariant measure µ, the map Ω0(M) = C∞(M, R) → R given by f �→  �  M f(x)µ(x)  is a 0-current, which we denote by Cµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  In particular if X is the vector field generating the  flow φ, ι(X)Cµ is a 1-current, which corresponds to the transverse measure µ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For a (n − k)  differential form β (only supposed continuous), we denote by Cβ the k-current defined by Cβ(α) =  �  M α ∧ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The differential map d on currents induces a current homology class given by ker(d)/ im(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  DeRham [DR84] proved that the current homology is equivalent to the simplicial homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Additionally, if β is a continuous differential form, the homology class of Cβ is Poincar´e-dual to  the cohomology class of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For that reason for a k-current C, we denote by [C] ∈ Hn−k(M, R)  the Poincar´e-dual of the homology class of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' More precisely [C] is equal to the cohomology  class [ω] for a smooth (n−k)-differential form ω, if and only if for all smooth differential k-form α,  one has C(α) =  �  M α ∧ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  9  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any invariant measure µ ∈ M(φ), we have d(ι(X)Cµ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let µ ∈ M(φ) be an invariant measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For any smooth function f : M → R, we  have Cµ(φ∗  t f) = Cµ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that LXCµ(f) = Cµ(LXf) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Here LX is the Lie derivative  along the vector filed X = ∂φt  ∂t , that is LXf = ι(X)(df).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore we have d(ι(X)Cµ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  We denoted by [µ]φ ∈ H2(M, R) the cohomology class of ι(X)Cµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map µ ∈ M(φ) → [µ]φ ∈ H2(M, R) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Using Poincar´e duality, the cohomology class of a measure µ ∈ M(φ) is determined by the  integrals  �  M ι(X)αkdµ for a finite family (αk)k of closed 1-form for which ([αk])k spans H1(M, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Since the functions ι(X)αk are continuous, the maps µ �→ �  M ι(X)αkdµ are also continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Hence the map µ �→ [µ]φ is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let V be an invariant volume form and µV be the invariant measure induced by V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Then ι(X)V is closed and [µV ]φ = [ι(X)V ] in H2(M, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a closed 1-form α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We have 0 = ι(X)(α ∧ V ) = (ι(X)α) ∧ V − α ∧ (ι(X)V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It  follows that:  [µ]φ · [α] =  �  M  ι(X)αdµV  =  �  M  (ι(X)α) ∧ V  =  �  M  α ∧ (ι(X)V )  = [ι(X)V ] · [α]  The equality holds for all [α] ∈ H1(M, R), so [µ]φ = [ι(X)V ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Recall some homology/cohomology homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a homeomorphism h: M → N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Given a 1-chain in M, c = �  i aifi, where ai ∈ R and fi is a continuous map from [0, 1] → M, we  denote by h∗c the 1-chain in N given by h∗c = �  i aih ◦ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We also denote by h∗ : H1(M, R) →  H1(N, R) the corresponding map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We view the DeRham cohomology space as the dual of H1, in the sens that H1(M, R) is natu-  rally identified with HomR(H1(M, R), R), where the map is given by integrating a closed 1-form  on a C1 1-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by h∗ : H1(N, R) → H1(M, R) the map given by h∗(α)(c) = α(h∗(c))  for α ∈ HomR(H1(N, R), R) and c ∈ H1(M, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By Poincar´e duality, H1(M, R) and H2(M, R)  are dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So h induces a map h−∗ : H2(M, R) → H2(N, R) given by h−∗[ω] · [α] = [ω] · h∗[α].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Here h∗α is not well-defined, only h∗[α] is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proposition 11 (Invariance of the cohomology class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let h: M → N be an orbit equivalence  between two flows φ and ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then for any µ in M(φ), we have [Θh(µ)]ψ = h−∗[µ]φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Essentially, the cohomology class is a topological data which depends on the transverse mea-  sure corresponding to µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any closed 1-forms α and β on M and N with [α] = h∗[β], we have:  �  M  ι  �∂φt  ∂t  �  αdµ =  �  N  ι  �∂ψt  ∂t  �  βd(Θh(µ))  10  Proof of Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We temporarily admit the above lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take α, β as in the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Then we have:  h−∗[µ]φ · [β] = [µ]φ · h∗[β] = [µ]φ · [α]  =  �  M  ι  �∂φt  ∂t  �  αdµ  =  �  N  ι  �∂ψt  ∂t  �  βd(Θh(µ))  = [Θh(µ)]ψ · [β]  This equation is true for all [β] ∈ H1(M, R), which implies [Θh(µ)]ψ = h−∗[µ]φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Proof of Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a subset Y ⊂ M topologically transverse to φ, that is no short orbit  arc intersects Y twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take also a continuous map T : Y → R∗  +, so that the map Y ×[0, 1) → M  given by (x, s) �→ φsT (x) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by φ[0,T )(Y ) the union of φ[0,T (x))(x) for x in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take a continuous map uY : φ[0,T )(Y ) → �  M lifting the universal covering map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' There exists a finite partition of M of the form (φ[0,T i)(Y i))i of M, for some sets Y i ⊂ M  topologically transverse to φ and some continuous maps T i : Y i → R∗  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For g ∈ π1(M), and θ ∈ H1(M, R) we denote by θ(g) the integral of α on any C1 closed  loop representing g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' A continuous map f : �  M → R is said θ-equivariant if f(g · q) = θ(g) + f(q)  for all g ∈ π1(M) and q ∈ �  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Fix a point p ∈ �  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Given a closed 1-form α, we consider  the [α]-equivariant map fα uniquely defined by fα(p) = 0 and dfα = π∗  Mα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then we have:  �  φ[0,T )(Y )  ι  �∂φt  ∂t  �  αdµ =  �  x∈Y  ��  s∈[0,T (x))  ι  �∂φt  ∂t  �  α(φs(x))ds  �  dµ⊥  =  �  Y  (fα(uY (φT (Y )) − fα(uY (x)))) dµ⊥  =  �  φT (Y )  fα ◦ uY dµ⊥ −  �  Y  fα ◦ uY dµ⊥  �  M  ι  �∂φt  ∂t  �  αdµ =  �  i  �  φ[0,T )(Y )  ι  �∂φt  ∂t  �  αdµ  =  �  i  ��  φT i (Y i)  fα ◦ uY idµ⊥ −  �  Y i fα ◦ uY idµ⊥  �  =  �  i  �  φT i(Y i)−Y i fα ◦ uY idµ⊥  Here we write  �  A−B fdµ =  �  A fdµ −  �  B fdµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given a θ-equivariant map f : �  M → R, the quantity �  i  �  φT i(Y i)−Y i f ◦uY idµ⊥ depends  on θ but not on the specific choice of θ-equivariant map f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Assume the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by �h: �  M → �  N a lifting map of h: M → N, and by h∗ : π1(M) →  π1(N) the map induced by h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By assumption, we have [α] = h∗[β], so the map fβ ◦ �h is a [α]-  equivariant map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that we have the equality of the integrals:  �  M  ι  �∂φt  ∂t  �  αdµ =  �  N  ι  �∂ψt  ∂t  �  βd(Θh(µ))  11  Proof of Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take a smooth triangulation T of M for which every face of positive codi-  mension is transverse to φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We also suppose each simplex small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For a 3-dimensional simplex S of T , denote by YS the set of point x ∈ ∂S for which φ(0,ǫ)(x)  is included in S for some ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Also define the map TS : YS → R∗  + as the first return map  from YS to ∂S \\ YS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice that YS and ∂S \\ YS are transverse to the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore, TS is  well-defined and continuous, as restriction of a first return time from YS to ∂S \\YS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Additionally,  the family (φ[0,TS)(YS))S form a finite partition of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  □  Proof of Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take two θ-equivariant maps f1, f2 : �  M → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then f1 −f2 is invariant by the  action of π1(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore, there exists a continuous map f3 : M → R such that f1−f2 = f3◦πM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We have:  �  i  �  φT i (Y i)−Y i(f1 − f2) ◦ uY idµ⊥ =  �  i  �  φT i (Y i)−Y i f3dµ⊥  The latter is equal to zero thanks to a telescoping argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Indeed, the family of non-empty  intersections of the form φT j(Y j) ∩ Y i gives a partition of Y i when j varies, and a partition  of φT j(Y j) when i varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So we have:  �  i  �  φT i (Y i)−Y i f3dµ⊥ =  �  i  �  j  ��  φT i (Y i)∩Y j f3dµ⊥ −  �  Y i∩φT i (Y j)  f3dµ⊥  �  = 0  □  ■  Flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  One can interpret [µ]φ as the flux of φ for the measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' That is [µ]φ · [S] corresponds  to the quantity of flow going through a closed surface S, relatively to µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In [CHL08], the authors  establish a connection between first return maps of Reeb flows and area preserving diffeomor-  phisms having zero flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' More precisely fix a compact surface S with non-empty boundary, and a  diffeomorphism f : S → S preserving an area form ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Consider the kernel of the action of f∗ −id  on H1(S, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The flux of f is a map flux: ker(f∗ − id) → R, where fluc([c]) =  �  d ω, where d is  a 2-chain bounding f(c) − c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It is proven in the same article that if f is the first return map of  a Reeb flow, then it has zero flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Consider the suspension N = S × R/(x, t + 1) ≃ (f(x), t) of the map f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We lift ω to a  2-form in N, denoted by ωN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' One can verify that H2(N, R) ≃ ker(f∗ − id) ⊂ H1(S, R), where an  element [c] ∈ ker(f∗ − id) is in correspondence with the element [c × [0, 1] + d × 0] ∈ H2(N, R),  where d a 2-chain in S bounded f(c) − c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then the map f has zero flux if and only the integral  of ωN over all element in H2(N, R) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' That is [ωN] = 0 in H2(N, R) by Poincar´e duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For the suspension with boundary case, it follows from [CHL08] that the suspension flow on N  is a reparametrization of a Reeb-flow if and only if the preserving 2-form ωN is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='5  Invariant 1-cycle  Let γ be a periodic orbit, we denote by Lebγ the unique invariant measure on γ whose total  mass is the length of γ for φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  That is Lebγ is the Lebesgue measure on γ obtained as the  pullback f ∗  γ Leb, where fγ : R/T Z �→ γ is map given by fγ(t) = φt(x) for some x ∈ γ of period T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  A map M → R is called an invariant 1-cycle if it is invariant by φt and is zero outside  finitely many periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The support of an invariant 1-cycle is the union of these periodic  12  orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a simplicial 1-chain c = �n  i=1 λifi, where fi : [0, 1] → M is an immersion with  image in a single orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map defined by gc(x) = �  i λ(x ·∩[(fi)|[0,1)]) is a 1-cycle if and only  if ∂c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In that case we identify c and gc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Similarly given a finite union Γ of periodic orbits  of M and a: Γ → R, �  γ∈Γ a(γ)γ is identified with the invariant 1-cycle g given by g(x) = a(γ)  if x ∈ γ ∈ Γ, g(x) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' An invariant 1-cycle is said positive if the corresponding map  has non-negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given an invariant 1-cycle Γ = �  γ a(γ)γ, we define the invariant signed  measure LebΓ = �  γ a(γ) Lebγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The cohomology class [LebΓ]φ is Poincar´e dual to [Γ] ∈ H1(M, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any closed 1-form β and any periodic orbit γ of period T , we have:  [Lebγ]φ · [β] =  �  M  ι(X)β Lebγ =  � T  0  ι(X)β(φt(x))dt =  �  γ  β = [β] · [γ]  ■  Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given a measure µ ∈ M(φ) and a finite union of periodic orbit ∆ ⊂ M, there  exists a sequence of invariant 1-cycles Γn with rational coefficients, such that LebΓn converges  toward µ, and so that for all n:  the support of Γn is disjoint from ∆,  [LebΓn]φ = [µ]φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Additionally if ∆ is empty and µ is a probability measure, Γn can be taken so that LebΓn is a  probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  It is essentially a variation of a well known result on the density of finitely supported measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  This lemma follows from similar Lemmas 41 and 42, about symbolic dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  2  Markov partition  Here φ is supposed to be a transitive Anosov flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We call Markov cuboid for φ a compact  subset R ⊂ M for which there exists a homeomorphism f : [0, 1]3 → R, such that for all x, y ∈  [0, 1], we have:  f(x × [0, 1]2) is included in a weak unstable leaf,  f([0, 1] × y × [0, 1]) is included in a weak stable leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Similarly a Markov rectangle is the image of a topological embedding f : [0, 1]2 → M, topo-  logically transverse to φ, which sends the horizontal and vertical foliations in [0, 1]2 to the weak  stable and unstable foliations in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a Markov cuboid R and a map f : [0, 1]3 → R given  as above, and suppose that for every x, y ∈ [0, 1], we have t ∈ [0, 1] �→ f(x, y, t) is orientation  preserving, for the orientation given by the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We define the sets:  ∂sR = f([0, 1] × {0, 1} × [0, 1]) the stable boundary of R,  ∂uR = f({0, 1} × [0, 1] × [0, 1]) the unstable boundary of R,  ∂suR = ∂sR ∪ ∂uR,  ∂−R = f([0, 1]2 × 0), ∂−R = f([0, 1]2 × 1), which are Markov rectangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We call the  transverse boundaries of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  13  we denote Int ∂ǫR for ǫ ∈ {s, u, su, −, +}, the image by f of the sames set, where [0, 1] is  replaced by (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' That is Int ∂ǫR is the interior of the rectangles corresponding to ∂ǫR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The weak stable and unstable foliations on M induce stable and unstable foliations on R,  on ∂−R and on ∂+R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by W s  R and W u  R the stable and unstable foliations on ∂−R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given  a Markov rectangle R, a subset U ⊂ R is said to be a vertical (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' horizontal) sub-rectangle  of R if it is connected and is the union of unstable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' stable) leaves on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' A Markov  partition P is a finite family of Markov cuboids {R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' , Rn} satisfying:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' ∪iRi = M,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' for any distinct i, j, the interior of Ri and Rj are disjoint,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' for any i, j, either the intersection (Int ∂+Ri) ∩ (Int ∂−Rj) is empty, or ∂+Ri ∩ ∂−Rj is a  horizontal sub-rectangle of ∂+Ri and a vertical sub-rectangle of ∂−Rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Any transitive Anosov flows admits a Markov partition (see [Rat69, Rat73]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Our definition  of Markov partition is more restrictive than the usual definition, in the sense that a Markov  cuboid R have a nice transverse boundary ∂+R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By taking Markov sub-partitions, any Markov  partition in the usual sense can be transformed in a Markov partition in our sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We de-  fine ∂P, ∂sP, ∂uP, ∂suP, ∂−P, ∂+P as the union of the corresponding boundary of the Markov  cuboids of the Markov partition P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Symbolic dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Here we fix a Markov partition P = {R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' , Rn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We call the transition  matrix of P the matrix A ∈ Mn(Z), containing a 1 in position (i, j) if Int(∂+Ri) ∩ Int(∂−Rj)  is non-empty, and a zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We call a P-word any sequence u: I ⊂ Z → P satisfying  Aj−i  ui,uj > 0 whenever i and j are in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The set of bi-infinite P-words (that is sequences u: Z → P)  is denoted by ΣP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For two P-words u, v: I ⊂ Z → P, we denote by dP the distance between u  and v, defined by:  dP(u, v) =  �  i∈I  ui̸=vi  2−|i|  It induces on ΣP the topology for which cylinders form a basis of the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Here cylinders  are the subsets of the form {u ∈ ΣP, ui = vi for all i ∈ I}, where I is a finite subset of Z and  v: I → P is a P-word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by σ the shift map on ΣP, given by σ(u)i = ui+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice  that σ is 2-Lipchitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Similar to Anosov flows, σ admits a stable foliation and an unstable foliation, denote by W s  and W u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Here W s(v) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' W u(v)) is the set of sequence w ∈ ΣP with wi = vi for all i large  enough (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' for all i ≤ i0 for some i0 ∈ Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The shift map divides the distances by 2 in each  stable leaf, and multiplies the distances by 2 on each unstable leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We additionally call the local  stable and unstable leaves of v the sets defined by:  W s  l (v) = {w ∈ ΣP, wi = vi for all i ≥ 0}  and  W u  l (v) = {w ∈ ΣP, wi = vi for all i ≤ 0}  A cyclic P-word is a map u: Z/nZ → P such that we have Aui,ui+1 > 0 for all i ∈ Z/nZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We call n the length of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We say that u is primitive if it is not k-periodic for some k < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Given a cyclic P-word u, we denote by u: Z → P the P-word given by ui = u(i mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For two  cyclic P-words u, v of length n, m such that u0 = v0, we denote by uv or u ◦ v the concatenation  14  of u and v, which is the cyclic P-word of length n + m given for i ∈ �0, n − 1� by (uv)i = ui, and  for i ∈ �n, m + n − 1� by (uv)i = vi−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Given x ∈ M and u ∈ ΣP, we say that x admits an itinerary (un, tn)n∈Z if (tn)n is an  increasing sequence in R, such that for any n ∈ Z and s ∈ [tn, tn+1], the point φs(x) lies in the  cuboid un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We also say that x admits an itinerary along u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It implies that tn converges to −∞  and +∞ when n goes to −∞ and +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Every point x ∈ M admits an itinerary, but there is no  uniqueness in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Additionally every u in ΣP corresponds to an itinerary of a point in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Given a set I ⊂ Z and a P-word u: I → P, we say that x ∈ M admits a short itinerary along u  if x admits an itinerary of the form (vn, tn)n where v is a bi-infinite P-word extending u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Let R be a Markov cuboid of a Markov partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take a finite P-word u: �0, n� → P  satisfying u0 = R, and denote by Ru the set of points in ∂−R admitting a short itinerary  along u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by T u : Ru → R∗  + the smooth map obtained the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For z ∈ Ru,  the point z admits an itinerary of the form (wi, ti)i∈Z such that w�0,n� = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by  T u(z) = �n−1  i=0 ti, which satisfies that φT u(z)(z) is in ∂−un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The element T u(z) does not depend  on the specific choice of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We also denote by σu : Ru → un the map given by σu(x) = φT u(x)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The following holds:  Ru is a closed horizontal sub-rectangle of ∂−R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  σu(Ru) is a closed vertical sub-rectangle of ∂−un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' When n = 1, it is a consequence of the definition of Markov partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' When n ≥ 2, it  can be verified by induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  We denote by πP : ΣP → M the encoding map given by πP(u) = x, where x is the unique  point in M which admits an itinerary of the form (un, tn)n for some sequence (tn)n satisfying  t0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In particular πP(u) lies in the rectangle ∂−u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map πP is well-defined, continuous,  and sends the foliations W s  l , W u  l to the foliations W s  R, W u  R in each rectangle ∂−R for R ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='1  Suspension of a Markov partition  Fix a Markov partition P of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given u in ΣP, denote by TP(u) ∈ R+ the first return time  from πP(u) ∈ ∂−u0 to πP ◦ σ(u) ∈ ∂+u0 ∩ ∂−u1 for φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map TP : ΣP → R+ is continuous  since πP is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by (MP, φP) the suspension flow for the map σ and the  time function TP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' That is MP is the compact quotient ΣP × R/ ≃ where ≃ is the relation  equivalence induced by (u, s) ≃ (σ(u), s−TP(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The flow φP is defined by φP  t (u, s) = (u, t+s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map hP : MP → M given by hP(u, s) = φs(πP(u)) is well-defined and is a  semi-conjugation between φP and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' That is hP ◦ φP  t = φt ◦ hP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Additionally it is continuous,  and is injective on the set of point u satisfying that the orbit of hP(x) is disjoint from ∂suP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by M(σ) and M(φP) the set of signed measures on ΣP and MP which are invariant  by σ and φP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any measure µ ∈ M(σ), we denote by µ ⊗ LebP ∈ M(φP) the measure which  coincide locally with the product of µ and the Lebesgue measure on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We define ΘP : M(σ) →  M(φ), the surjective map given by ΘP(µ) = h∗  P(µ ⊗ LebP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map ΘP is used in Section 3 to  relate a linking number between φ-invariant measures with a linking number between σ-invariant  measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Given a cyclic P-word u, Denote by �γu the periodic orbit of (u, 0) for φP, and by γu the  periodic orbit of πP(u) for φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We call the realization of u the map obtained as a restriction  of πP to �γu → γu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  15  Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let u be a cyclic P-word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' When πP(u) is on the stable or unstable boundary of P  and have non-orientable stable and unstable leaves, the realization of u is a degree two covering  map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Otherwise, it is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let u, v be two cyclic P-words with u0 = v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then either u = v or πP(u) and  πP(v) are on distinct stable and unstable leaves of W s,u  R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Sketch of proof, for Lemma 17 and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The realization map is a cover since hP is a semi-  conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We give a sketch, the details are let to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take ǫ0 > 0, so that for each cuboid R, the two connected components of ∂sR are at distance  at least 2ǫ, and similarly for ∂uR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given a point x and 0 < ǫ < ǫ0, an ǫ-quadrant at x is defined  to be a close set containing x, of the form C, where C is a connected component of B(x, ǫ)\\∂suP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  An ǫ1-quadrant and an ǫ2-quadrant are said equivalent if they contain a common ǫ3-quadrant  for some ǫ3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' An equivalence class of ǫ-quadrant is called a germ of quadrant at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Given a periodic point x ∈ M and t ∈ R, we claim that φt gives a bijection between the  germs of quadrant at x and the germs of quadrant at φt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' To see that, notice that the stable  boundary ∂sP is invariant by φt for t > 0, and ∂uP is invariant by φt for t < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore  for periodic points x ∈ ∂suP of periodic T , there exists a neighborhood U ⊂ ∂suP of x which  satisfies φt(U) ⊂ ∂suP for any t ∈ [−T, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It implies the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  One can verify that the map sending (u, s) ∈ MP with s ∈ [0, TP(u)) to the germ of quadrant  at hP(u, s) which intersects Int u0 ∩ B(hP(u, s), ǫ), is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If T denote the period of x, the  map φT permutes the germ of quadrant at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' This permutation is the identity either when x  is in the interior of a Markov cuboid, or if it has orientable stable and unstable leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In the  remaining case, φT exchanges the germ of quadrant which are opposite relatively to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So each  germ of quadrant are 2T periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The injectivity of the previous map also proves the second Lemma (stable and unstable leaves  contain at most one periodic orbit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='2  Vertical and horizontal Order  Here we suppose that the foliations Fss and Fuu are oriented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Each curve c in Fs or Fu,  transverse to the flow, is endowed with an orientation, induced by the first return map from the  curve c to any strong stable or unstable leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The set of leaves Leaf(W u  R) is homeomorphic to a  segment, and is endowed with a natural orientation coming from the orientation on Fss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote  by ≤h the order on Leaf(W u  R), coming from its orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If x, y are in R, we write x ≤h y  if W u  R(x) ≤h W u  R(y), and x <h y if W u  R(x) <h W u  R(y) (<h is not the strict order on R coming  from ≤h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We similarly define the orders ≤v on Leaf(W s  R) and on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The orders ≤h and ≤v  are called the horizontal and vertical orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Here we picture the stable foliation to be made of  horizontal leaves, oriented to the right, and the unstable foliation to be made of vertical leaves  oriented upward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lexical Order in ΣP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We define in the same manner horizontal and vertical orders on ΣP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We  do not simply lift the orders on R, because the map ΣP → M is not injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given some cuboids  R0, R1, R2 in P, we say that R1 is below R2 relatively to R if Int ∂+R0 intersects both Int ∂−R1  and Int ∂−R2, and the intersection Int ∂+R0 ∩ Int ∂−R1 is inferior to Int ∂+R0 ∩ Int ∂−R2 for  the order ≤v on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by Σ+  P and Σ−  P the sets of P-word of the form u: �0, +∞� → P and respectively  u: � − ∞, 0� → P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' They correspond to the sets of leaves of the foliations W s  l and W u  l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' They  come with the distances induced by dP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We define on Σ+  P the order ≤h defined by u ≤h v if and  only if either u = v, or there exists n ≥ 0 satisfying that ui = vi for all 0 ≤ i ≤ n, and un+1 is  16  on the left of vn+1 relatively to vn = un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In particular, u and v are comparable for ≤h if and  only if u0 = v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We define similarly an order ≤h on Σ−  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by p+, p− the natural maps ΣP → Σ+  P and ΣP → Σ−  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We define the order ≤h, ≤v  and strict order <h, <v on ΣP by pulling back the respective orders by the maps p+ and p−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' One  can verify that for u ∈ ΣP, the leaf W s  R(πP(u)) ⊂ u0 depends only on p+(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by Σ+  P,R  the set of u ∈ Σ+  P satisfying u0 = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map πP induces a map πs  P : Σ+  P,R → Leaf(W s  R), given  by πs  P(u) = W s  R(πP(v)) for any v ∈ ΣP with p+(v) = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows from the construction that πs  P  is increasing for the orders ≤v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='3  Word combinatorial  We prove a lemma which is essential to build primitive periodic orbits of an Anosov flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let A  be a finite set and A∗ the free monoid generated by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' That is elements in A∗ are finite words  (possibly empty) whose letters are inside A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For u ∈ A∗, we denote by |u| the length of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We  also denote by u ≤ v if u is a prefix of w, that is v ∈ u · A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let two elements x, y ∈ A∗ which are not powers of a common element in A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Then for all n, m ≥ 1, the cyclic P-word x2ny2m is not a power of another element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  To prove the lemma, we use the following fact, which can be found in [Lot97, Corollary  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If two words x, y ∈ A∗ satisfy a non-trivial relation, then they are power of a common  word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By satisfy non-trivial relation, one should understand that the unique monoid morphism  f : {a, b}∗ → A, satisfying f(a) = x and f(b) = y, is not injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In particular the hypothesis  of the lemma implies that xk and yl are distinct for all k, l ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Assume the lemma true for n = m = 1 and use it to prove it for any n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take u = xn and  v = ym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If u and v are powers on a common a word, then uk = vl for some integers k, l ≥ 1,  then x and y satisfy a non-trivial relation, so they are powers of a common element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It is not  possible by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So one can apply the case n = m = 1 to u and v, which proves the  lemma for any other n, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We prove now the case n = m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Suppose that x2y2 = zk for some z ∈ A∗ and k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Without lost of generality, we can suppose |x| ≥ |y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We consider several cases for the lengths  of x and z, and prove that they are all impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Suppose that |x| = |z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since zk and x2y2 are equal and start with z and x, one  has z = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that some power of x is equal to some power of y, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Suppose that |x| > |z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since x2 ≤ zk, there exist u, v prefixes of z and integers  1 ≤ a, b < k such that x = zau and x2 = zbv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice that b ≥ 2a ≥ a+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We claim that xz = zx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Indeed z ≤ x, so xz ≤ x2 ≤ zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Similarly we have zx = za+1u, so zx ≤ zbu ≤ zk when b = a + 1,  and zx ≤ za+2 ≤ zb ≤ zk when b > a+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence zx and xz have the same length and are prefixes  of the same word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So xz = zx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that x = tc and z = td for some t ∈ A∗ and c, d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Then we have y2 = tkd−2c, which contradicts that x and y are not power of a common element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We suppose now that |z| > |x| Sine 4|x| ≥ 2|x| + 2|y| = k|z|, it follows that k is equal  to 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Suppose that |z| > |x| and k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' One has x ≤ z ≤ x2 ≤ z2, so there  exists u, v ∈ A∗ such that x2 = zv and z = xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that x = uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We have z2 = uvuuvu  and x2y2 = uvuvy2, so vy2 = uvu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Thus y and u have the same length, and they are both suffix  of the same word, so y = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then uvu = vy2 = vuu, so uv = vu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence u and v are power of a  common word, which contradicts the assumption on x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Suppose that |z| > |x| and k = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Similarly we can write x = uv and z = uvu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Notice that we have 2|y| = 3|z| − 2|x| = 4|u| + |v| ≤ 2|x| = 2|u| + 2|v|, so |v| ≥ 2|u|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' On has  17  x2 = uvuv ≤ z2 = uvuuvu, so v ≤ uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by w the suffix of v satisfying uv = vw = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since  le length of w is equal to |u| ≤ |v|/2, there exists an integer p ≥ 2 and t ∈ A∗ with |t| < |w| such  that v = twp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We can write:  z3 = uvuuvuuvu  = uvuvwuvwu  = x2wvw2u  Hence y2 = wvw2u = wtwp+2u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since we have |w| = |u| and |t| < |w| < (p+2)|w|, the middle  index in wtwp+2u is inside the term wp+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So there exist two words a, b ∈ A∗ and q, r ∈ N such  that w = ab, q + r + 1 = p + 2 and y = wtwqa = bwru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We can not have r = 0, otherwise  |wtwqa| ≥ (q + 1)|w| ≥ 4|w| > 2|w| ≥ |bu|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence the words wtwqa bwru are equal and start  with ab and ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So ab = ba, and they are power of a common element s ∈ A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows the  sequence of equalities:  u = ba ∈ {s}∗  wtwqa = bwru ∈ {s}∗ so t ∈ {s}∗  v = twp ∈ {s}∗  uv = vw so u ∈ {s}∗  Hence x, y are powers of s, which is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore the last case is also impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let x, y be two cyclic P-words with x0 = R = y0, such that x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then the  point xy lies horizontally and vertically strictly between the points x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We mimic a classical argument used to find invariant points in the orbit space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We  prove the result for the vertical order only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by x′ = p+(x), y′ = p+(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We suppose  that x′ <v y′, the other case is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Consider the two maps fx, fy : Σ+  P,R → Σ+  P,R satisfying that fx(w) is the element in Σ+  P  obtained by concatenating x to the left of w, and similarly for fy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' That is fx starts with the  same first |x| elements than x, and σ|x| ◦ fx = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by d+  P the distance on Σ+  P,R given  for a ̸= b by:  d+  P(a, b) = 2− inf{i≥0,ai̸=bi}  Notice that fx and fx are increasing and Lipchitz for d+  P and for the Lipchitz constants  2−|x| and 2−|y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence the map fx ◦ fy as a unique fixed point, which is p+(xy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The goal is  to prove that the fixed point is between x′ and y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given two points a, b ∈ ΣP,R, we denote  by [a, b]v the set of points between a and b for the order ≤v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any point w ∈ [a, b]v, we have  d+  P(a, w) ≤ d+  P(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Indeed if n equals inf{i ≥ 0, ai ̸= bi}, then by definition of ≤h, w have the  same (n − 1) coefficients than a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that d+  P(a, w) ≤ 2−n = d+  P(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Suppose that fx([x′, y′]v) is not included in [x′, y′]v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Then there exists w in [x′, y′]v for  which fx(w) is not in [x′, y′]v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since fx is increasing, we have x′ = fx(x′) ≤v fx(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows  that fx(w) > y′ since ≤v is a total order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So y′ lies in [x′, fx(w)v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It implies the following  inequality, which is a contradiction:  2−|x|d+  P(x′, w) ≥ d+  P(x′, fx(w)) ≥ d+  P(x′, y′) ≥ d+  P(x′, w) > 0  So [x′, y′]v is invariant by fx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Similarly it is invariant by fy and by fx ◦ fy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The set [x′, y′]v is  compact and invariant by the contracting map fx ◦ fy, so fx ◦ fy admits fixed point in [x′, y′]v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  By uniqueness of the fixed point, we have x′ <h p+(xy) <h y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  18  3  Linking number of invariant measures  We fix one transitive Anosov flow φ on a closed oriented 3-dimensional manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Except in  the following theorem, we also suppose that the stable and unstable foliations of φ are orientable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We extend the notion of linking number for invariant measures of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We construct successively  maps linkff  φ , linkf  φ, linkφ which gives the linking number of two invariant measures under the  hypothesis that respectively two, one or none of them are supported on finitely many periodic  orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Consider Γ1 and Γ2 two invariant 1-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We suppose that they are null-homologous and  supported on disjoint sets of periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by linkff  φ (LebΓ1, LebΓ2) the linking  number between the corresponding invariant 1-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' That is if S1 is a 2-chain in M with ∂S1 =  Γ1, then one has:  linkff  φ (LebΓ1, LebΓ2) = S1 ·∩Γ2  Denote by M0(φ) the set of measures ν ∈ M(φ) which are null-cohomologous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Theorem 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let φ be a transitive Anosov flow on an oriented 3-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' There exists a unique  continuous bilinear map linkφ : M0(φ) × M0(φ) → R which satisfies linkφ(ν, µ) = linkff  φ (ν, µ)  for all measures ν, µ ∈ M0(φ) which are supported on disjoint finite unions of periodic orbits,  Additionally the map linkφ is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The proof is technically simpler when the foliations of φ are orientable, and can be deduced on  the general case from the orientable case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Recall that M0  p(φ) is the set of null-cohomologous in-  variant probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' As a consequence of the continuity and the compactness of M0  p(φ),  we obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Corollary 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Suppose that M0  p(φ) is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map M0  p(φ) → R defined as follows is  well-defined and continuous:  µ �→  min  ν∈M0p(φ) linkφ(µ, ν)  The proof of the theorem follows two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' First consider a null-homologous invariant 1-  cycle Γ and a measure µ ∈ M0(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We define the linking number linkf  φ(LebΓ, µ) roughly as the  integral of the transverse measure µ⊥ on a 2-chain S bounding Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We actually lift S and µ in  the manifold obtained by blowing up M along each periodic orbit in the support of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then the  linking number linkf  φ(LebΓ, µ) is well-defined for all µ, even when µ charges the support of LebΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  In a second part, we use Markov partitions to give a precise description of the linking number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We build a map Σ2  P → R which has a similar role to the linking form Given by Gauss and  DeRham [DR84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The difficulty is to prove that this map indeed computes linking numbers, and  that its integral for measures ν ⊗ µ is continuous in (ν, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='1  Linking number linkf  φ  Fix a null-homologous invariant 1-cycle Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We define the linking number between LebΓ and  any µ ∈ M0(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Blowup flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take a set ∆ ⊂ M, obtained as union of periodic orbits, containing the support  of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We consider the blowing up M∆ of M along each orbit of ∆, constructed bellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by X = ∂φt  ∂t the vector field generating φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given a periodic orbit γ in ∆, we denote  by Nγ the normal vector bundle to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It is obtained as the set of equivalence classes of vectors v  based on a point p in γ, non-parallel to X(p), up to the equivalence relation v1 ≃ v2 if there  19  exist a > 0, b ∈ R such that v1 = av2 + bX(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The set M∆ is the union of M \\ ∆ and of the  circle bundle ∪γNγ, equipped with the atlas induced by M and by the angular sectors around ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We define the map π∆ : M∆ → M by π∆(x) = x for x ∈ M \\ ∆, and π∆(v) = x if v is a vector  based on x ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For a periodic orbit γ ⊂ ∆, we denote by Tγ = π−1  ∆ (γ) ⊂ M∆ the boundary component  corresponding to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since M is supposed orientable, Tγ is a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If the flow φ is of class Ck,  k ≥ 1, it lifts to a Ck−1 flow φ∆ on M∆, by setting φ∆  t (v) = dφt(v) for a vector v based on a  point in ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by X∆ = ∂φ∆  t  ∂t  the vector fields generating φ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Notice that on each torus Tγ, the restriction flow φ∆  |Tγ has one or two periodic orbits corre-  sponding to the stable direction, and one or two corresponding to the unstable direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We  refer to them as the stable and unstable orbits of Tγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Every other orbit converges in the past  toward one stable orbit, and in the future toward one unstable orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' From this remark follows  the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any measure µ ∈ M(φ∆), the measure µ|Tδ is supported on the stable and  unstable orbits of Tγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For a given measure µ ∈ M(φ), one can build a measure µ∆ ∈ M(φ∆) so that π∗  Pµ∆ = µ and  whose total variation is the same as the total variation of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' To do this, consider the invariant  measure ν given by ν(A) = µ(A ∩ ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any γ ⊂ ∆, take a measure νγ ∈ M(φ∆), supported  by one of the stable orbits in Tγ, so that π∗  Pνγ coincides with ν on γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then the measure µ∆ can  be obtained as the sum of the measures νγ and of (((πP)| Int M∆)−1)∗µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Linking number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Given an invariant measure µ ∈ M(φ∆), we denote by [µ]φ∆ ∈ H2(M∆, R)  the cohomology class of µ for φ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by M0(φ∆) the set of measure µ ∈ M(φ∆) satisfy-  ing [π∗  ∆µ]φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We consider the Poincar´e duality P∆ : H1(M∆, R) → H2(M∆, ∂M∆, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For a  smooth map f : M∆ → R/Z, the element P∆([df]) is the relative homology class of the surfaces  obtained as f −1(x) for any regular value x of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given a measure µ ∈ M(φ∆) with [π∗  ∆µ]φ = 0, and ω ∈ H1(M∆, R), the quan-  tity [µ]φ∆ · ω depends only on µ and on (π∆)∗(∂P∆(ω)) ∈ H1(∆, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let ω ∈ H1(M∆, R) be a 1-form which satisfies (π∆)∗(∂P∆(ω)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by i the  inclusion ∆  i∗  ֒−→ M, so that H2(M, R)  i∗  −→ H2(M, Γ, R)  ∂−→ H1(Γ, R) is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since ∂(π∆)∗(P∆(ω))  is equal to (π∆)∗(∂P∆(ω)) = 0, there exists δ in H2(M, R) satisfying (π∆)∗(P∆(ω)) = i∗(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by Q: H1(M, R) → H2(M, R) the Poincar´e duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Similarly to P∆, Q can be  described by Q([df]) = [f −1(x)] for any smooth map f : M → R/Z and any regular value x of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Using this characterization, one can verify that the following diagram commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  H1(M, R)  H2(M, R)  H2(M, ∆, R)  H1(M∆, R)  H2(M∆, ∂M∆, R)  Q  π∗  ∆  i∗  P∆  (π∆)∗  It follows that (π∆)∗ ◦ P∆(ω) = (π∆)∗ ◦ P∆ ◦ π∗  ∆ ◦ Q−1(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map (π∆)∗ so injective so we  20  have ω = π∗  ∆ ◦ Q−1(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take α a smooth closed 1-form on M with [α] = Q−1(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then we have:  [µ]φ∆ · ω =  �  M∆  ι(X∆)(π∗  ∆α)dµ  =  �  M  ι(X)αd(π∗  ∆µ)  = [π∗  ∆µ]φ · α  = 0  since [π∗  ∆µ]φ = 0  ■  Given µ ∈ M(φ∆) satisfying [π∗  ∆µ]φ = 0, we define link∆  Γ (µ) = [µ]φ∆ · [α] for any smooth  differential 1-form α with (π∆)∗(∂P∆(ω)) = [Γ] ∈ H1(Γ, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Notice that (µ ∈ M0(φ∆)) �→  link∆  Γ (µ) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let µ ∈ M(φ∆) be an invariant measure, γ be a periodic orbit in ∆ and α be a closed  1-form on Tγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then the integral  �  Tγ ι(X∆)αdµ depends only on µ(Tγ) and on [α] ∈ H1(Tγ, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Thanks to Lemma 8, the integral  �  Tγ ι(X∆)αdµ does not depend on the choice of the closed  1-form α in its cohomology class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by dt the closed 1-form on γ satisfying ι  �  ∂φt  ∂t  �  dt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We denote by β⊥ = π∗  ∆dt and by β∥ a closed 1-form which is not null-cohomologous and such  that the stable and unstable periodic orbits on Tγ are tangent to ker(β∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then ι(X∆)β∥ is zero  on the support of µ|Tγ (see Lemma 23), so one has:  �  Tγ  ι(X∆)β∥dµ = 0  and  �  Tγ  ι(X∆)β⊥dµ =  �  Tγ  π∗  ∆(ι(X)dt)dµ  =  �  Tγ  1dµ  = µ(Tγ)  The lemma follows by linearity and from the fact that [β⊥] and [β∥] form a basis of H1(Tγ, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Lemma 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given a null-homologous invariant 1-cycle Γ and a measure µ ∈ M0(φ∆), the  quantity link∆  Γ (µ) depends only on Γ and π∗  ∆(µ) ∈ M0(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In particular, it does not depend  on ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Consider a closed 1-form α and two measures µ1, µ2 ∈ M0(φ∆) satisfying π∗  ∆(µ1) =  π∗  ∆(µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since π∆ is injective outside ∆, the measure µ1 and µ2 coincide outside ∪γTγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Ad-  ditionally their restrictions to Tγ have the same weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence according to Lemma 25, the  integral of ι(X∆)α on Tγ is the same for µ1 and µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that [µ1]φ∆ · [α] = [µ2]φ∆ · [α], and  that the linking numbers are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We now prove that it does not depend on ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote ∆1 and ∆2 two choices of union of  periodic orbit containing Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by ∆3 = ∆1 ∪ ∆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take µ ∈ M0(φ), µ1 ∈ M0(φ∆1), µ3 ∈  M0(φ∆3) satisfying π∗  ∆1µ1 = µ = π∗  ∆3µ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by π: M∆3 → M∆1 the blow-down projection,  which satisfies π∆1◦π = π∆3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' According to the previous case, we have link∆1  Γ (π∗µ3) = link∆1  Γ (µ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  21  Fix a closed 1-form α on M∆1 satisfying (π∆1)∗◦(∂P∆1[α]) = [Γ] as in the definition of link∆1  Γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Notice that π∗α is a closed 1-form on M∆3 satisfying the same equality (π∆3)∗◦(∂P∆3[π∗α]) = [Γ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Hence one has:  link∆3  Γ (µ3) =  �  M∆3  ι(X∆3)π∗αdµ3  =  �  M∆1  ι(X∆1)αdπ∗µ3  = link∆1  Γ (π∗µ3)  = link∆1  Γ (µ1)  It follows that link∆1  Γ (µ1) = link∆2  Γ (µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Definition 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For a null-homologous invariant 1-cycle Γ in M, and µ ∈ M0(φ), we de-  fine linkf  φ(LebΓ, µ) as link∆  Γ (ν) for any union of periodic orbit ∆ containing the support of Γ,  and any measure ν ∈ M0(φ∆) satisfying π∗  ∆ν = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by support(µ) the support of a measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proposition 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let ∆ be a finite union of periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The map linkf  φ is continuous  on {ν ∈ M0(φ), support(µ) ⊂ ∆} × M0(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Additionally, we have linkf  φ(LebΓ1, LebΓ2) =  linkff  φ (LebΓ1, LebΓ2) for any null-homologous invariant 1-cycle Γ1, Γ2 with disjoint support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a sequence (νn, µn)n in {ν ∈ M0(φ), support(µ) ⊂ ∆} × M0(φ), converging to-  ward (ν∞, µ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by Γn and Γ∞ the invariant 1-cycle corresponding to νn and ν∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' As  explain in Remark 6, the total variations of νn and µn are bounded by some C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We take a sequence (µ∆,n)n in M0(φ∆) so that π∗  ∆µ∆,n = µn and the total variation of µ∆,n is  at most C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We can decompose µ∆,n as a difference of two invariant positive measures µ+  ∆,n, µ−  ∆,n,  so that each of them as total mass at most C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By compactness, the sequence (µ+  ∆,n, µ−  ∆,n)n  accumulates toward a pair of positive measures (µ+  ∆,∞, µ−  ∆,∞), which satisfies π∗  P(µ+  ∆,∞−µ−  ∆,∞) =  µ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So we have linkf  φ(ν∞, µ∞) = link∆  Γ∞(µ+  ∆,∞) − link∆  Γ∞(µ−  ∆,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Since ∆ is a finite union of periodic orbits, there exists a continuous map Γ �→ ωΓ, so  that ωΓ is in H1(M∆, R) and satisfies (π∆) ∗ (∂P∆(ωΓ)) = [Γ] ∈ H1(∆, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The quanti-  ties link∆  Γn(µ±  ∆,n) = [µ±  ∆,n]φ∆ · ωΓn converges toward link∆  Γ∞(µ±  ∆,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that linkf  φ(νn, µn)  converges toward linkf  φ(ν∞, µ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The last point is a consequence of Poincar´e duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take two invariant 1-cycles Γ1 and Γ2,  an invariant 1-cycle �Γ2 = �  i miγi in M∆ with π∗(�Γ2) = Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a closed 1-form α on M∆ and  a 2-chain S satisfying [S] = P∆[α] and (π∆)∗(∂S) = Γ1, then we have:  �  M∆  ι(X∆)αd Leb�Γ2 =  �  i  mi  �  γi  α  =  �  i  miγi ·∩S  by duality  = Γ2 ·∩(π∆)∗(S)  = linkff  φ (LebΓ1, LebΓ2)  ■  22  As a consequence, one obtain the self-linking number of a null-homologous invariant 1-cycle  as the linking number between a Seifert surface of the 1-cycle and the stable direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The  following lemma is used as a definition by Ghys [Ghy09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 29 (Self-linking number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let Γ = �  i miγi be a null-homologous invariant 1-cycle, S ⊂  M be a 2-chain bounding Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For each i, denote by γs  i a 1-cycle in Fs(γi) \\ γi homologous to γi  in Fs(γi), and Γs = �  i miγs  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then we have:  linkf  φ(LebΓ, LebΓ) = S ·∩Γs  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Consider ∆ the support of Γ and M∆ the blowing up manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We lift S to a 2-chain S∗ ∈  H2(M∆, ∂M∆, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We also lift γs  i to a 1-cycle γs∗  i  in M∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By hypothesis, there exists a 1-cycle δs∗  i  in the stable direction of the torus Tγi which is homologous to γs∗  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then δs∗  i  is an invariant  1-cycle and (π∆)∗δs∗  i  = γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by α a closed 1-form on M∆, dual to S∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We have:  linkf  φ (LebΓ, LebΓ) = link∆  Γ  ��  i  mi Lebδs∗  i  �  =  �  i  mi  �  M∆  ι(X∆)αd Lebδs∗  i  =  �  i  miS∗ ·∩δs∗  i  =  �  i  miS∗ ·∩γs∗  i  = S∗ ·∩  ��  i  miγs∗  i  �  = S ·∩Γs  ■  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='2  Linking function for a Markov partition  In this subsection, we prove Theorem 21, by extending the linking number for two invariant  measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' First we transfer the problem to a question on invariant measure by a sub-shift using  a Markov partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given an invariant measure µ on ΣP, we associate to µ a measure N(µ)  supported on finitely many periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then we give a measurable function L: Σ2  P → R  such that computing the linking number between µ − R(µ) and ν corresponds to integrating L  for the measure µ ⊗ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We use that relation to define the linking number for any two measures,  knowing what is the linking number between two invariant measures finitely supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then  we prove the continuity of the linking number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We suppose that φ is an Anosov flow whose stable and unstable foliations are orientable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The main theorem in the non orientable case is proven using a degree two covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We fix a  Markov partition P for the flow φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by (R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' , Rp) the family of Markov cuboids in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We fix two orientations on the stable and unstable foliations, and consider the induced vertical  and horizontal orders on ΣP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Invariant measure of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Recall that we denote by ΘP : M(σ) → M(φ) the map sending  σ-invariant measures to their corresponding φ-invariant measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For µ ∈ M(σ), we denote  by [µ]φ = [ΘP(µ)]φ ∈ H2(M, R) the cohomology class corresponding to µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by M0(σ)  the set in measure µ ∈ M(σ) satisfying [µ]φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  23  Definition 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For two measures ν, µ ∈ M0(σ) finitely supported, we define the linking number  of ν and µ by:  linkff  σ (ν, µ) = linkf  φ(ΘP(ν), ΘP(µ))  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='1  Measure zero sets  Because of the Portmanteau Lemma, it is useful to determine some sets which have zero measures  for all measures in M(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  A set X ⊂ ΣP is said to have uniformly zero measure if for  all µ ∈ M(σ), one has |µ|(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice that it is enough to satisfy the property on probability  measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 31 (Portmanteau Lemma for almost continuous functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let X be a metrizable  topological space endowed with the Borel algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let Y ⊂ X be a measurable set and f : X → R  a bounded function, which is continuous on X \\ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given a sequence (µk)k of signed measures  on X, weakly converging toward a measure ν, one has:  lim sup  k→+∞  ����  �  X  fdµk −  �  X  fdν  ���� ≤ 2|ν|(Y ) · ∥f∥∞  A particular case is when |ν|(Y ) = 0, where we have convergence of the integrals of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a distance d compatible with the topology on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For t ∈ R+, denote by h+  t : X → R  the map defined by h+  t (x) = supy∈X(f(y) − td(x, y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map h+  t is well-defined since f is  bounded, and t-Lipchitz as supremum of t-Lipchitz maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice that we have f ≤ ht ≤ sup f ≤  ∥f∥∞ for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that:  lim sup  k  �  X  fdµk ≤ lim sup  k  �  X  h+  t dµk =  �  X  h+  t dν  For any x in X \\ Y , f is continuous at x so when t goes to +∞, h+  t (x) is decreasing and  converges toward f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence for all k one has:  lim sup  t→+∞  �  X  h+  t dν =  �  X  �  lim sup  t  h+  t  �  dν  according to Fatou’s Lemma  =  �  Y  �  lim sup  t  h+  t  �  dν +  �  X\\Y  �  lim sup  t  h+  t  �  dν  ≤ |ν|(Y ) · ∥f∥∞ +  �  X\\Y  fdν  ≤ |ν|(Y ) · ∥f∥∞ +  �  X  fdν −  �  Y  fdν  ≤  �  X  fdν + 2|ν|(Y ) · ∥f∥∞  It follows that:  lim sup  k  �  X  fdµk −  �  X  fdν ≤ 2|ν|(Y ) · ∥f∥∞  Using the map h−  t (x) = infy(f(y) + td(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' y)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' the same arguments prove that:  − lim inf  k  ��  X  fdµk −  �  X  fdν  �  ≤ 2|ν|(Y ) · ∥f∥∞  ■  24  We prove in the following two lemmas that two families of sets have uniformly zero measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Define the sets:  En+  R  = {u ∈ ΣP, un = R, and for all k > n, uk ̸= R}  and  En−  R  = {u ∈ ΣP, un = R, and for all k < n, uk ̸= R}  We also define E±  R = ∪n∈Z(En−  R  ∪ En+  R ), the set of P-word containing a R at some index,  and admitting a half sub-word without R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The set E±  R has uniformly zero measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a measure ν ∈ M(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By definition, we have σm(En+  R ) = E(n+m)+  R  and En+  R  ∩  Em+  R  = ∅ when n ̸= m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since ν is σ-invariant, one has ν(ΣP) ≥ �  n∈Z ν(En+  R ) ≥ �  n ν(E0+  R ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The measure ν is finite so ν(En+  R ) = 0 for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Similarly ν(En−  R ) = 0 and ν(E±  R) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  A second family of sets with uniformly zero measures are the union of stable and unstable  leaves, after removing the periodic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a point v ∈ ΣP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Its strong stable manifold W s(v)  is the set of point w such that d(σn(v), σn(w)) −−−−−→  n→+∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice that W s(v) is the set of P-words  which coincide with v for large enough indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let ws ⊂ ΣP be a strong stable leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If ws contains a periodic point u, then it has  a unique periodic point u and ws \\ {u} has uniformly zero measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Otherwise ws has uniformly  zero measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Suppose that u, v ∈ ws are periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by n a common multiple of the periods of u  and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then the distance between σkn(u) = u and σkn(v) = v goes to zero when k goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Hence we have u = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Suppose that ws contains a periodic point u, of period N ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by ws  n = {v ∈  ΣP, vn−1 ̸= un−1, ∀k ≥ n, vk = uk}, so that ws \\ {u} = ∪nws  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The sets ws  n are pairwise disjoints  and σN(ws  n) = ws  n−N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So using the same argument as in the previous proof, the sets ws  n−N(v)  have uniformly zero measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then ws \\ {u} has uniformly zero measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Suppose now that ws contains no periodic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If there would exist two distinct integers n <  m ∈ Z and an element u in σn(ws) ∩ σm(ws), then u would be (m − n)-periodic after a certain  index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  So the limit limk σk(m−n)+n(u) would exist and be in ws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  It is not possible so for  any n ̸= m, we have σn(ws) ∩ σm(ws) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then the previous argument shows that ws has  uniformly zero measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='2  Reduction to finitely many orbits  We denote by Prime(P) the set of periodic orbits of σ intersecting every rectangle at most once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  That is u ∈ ΣP lies in Prime(P) if and only if it is periodic of period some T ≥ 1, and for  any R ∈ P there is at most one i ∈ �1, T � such that ui = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice that Prime(P) has a finite  cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Fix a cuboid R ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We call R-reduction map the map NR : ΣP → ΣP defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For n, m ∈ Z with n ≤ m, we denote Sn,m = {u ∈ ΣP, un = R = um, R ̸∈ u�n+1,m−1�}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then  we define:  NR(u) =  �  u|�n,m−1�  if u belongs in Sn,m for some n ≤ 0 and m ≥ 1  u  otherwise  25  In the first case, u|�n,m−1� is the unique periodic P-word of period m − n, which coincides  with u on the set of indexes �n, m�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice that for all u ∈ ΣP, NR(u) coincides with u for the  indexes 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' After giving a formula for N ∗  Rµ for finitely supported measures µ, we prove the  following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map N ∗  R induces a continuous map M(φ) → M(φ), which additionally preserves  the cohomology class of measures on M(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Let u be a cyclic P-word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by ηu ∈ M(σ) the measure �|u|−1  k=0 1σk(u), which is the  unique invariant measure supported on the orbit of u, normalized by ηu(u) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let u be a cyclic P-word with u(0) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' , vn the unique family of  cyclic P-word which satisfies v1v2 · · · vn = u and v−1  j (R) = {0} for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then we have:  N ∗  Rηu =  n  �  j=1  ηvj  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If k ∈ �0, |u| − 1� satisfies �j−1  i=1 |vi| ≤ k < � i = 1j|vi| for some j ∈ �1, n�, then we  have NR ◦ σk(u) = σk−�j−1  i=1 |vi|(vj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that:  N ∗  R1σi(u) = 1NR◦σk(u)  = 1  σk−�j−1  i=1 |vi|(vj)  Hence we have �|u|−1  k=0 N ∗  R1σk(u) = �  j  �|vj|−1  k=0  N ∗  R1σk(vj) = �  j N ∗  Rηvj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Lemma 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map NR : ΣP → ΣP is continuous on ΣP \\ E±  R, and is measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The topology on ΣP is generated by the family of open sets of the form Cv = {u ∈  ΣP, u|�−m,m� = v} for all m ∈ N and for P-words v: �−m, m� → P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We determine the reciprocal  image of these sets by NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  When the letter R does not appear inside v, we have N −1  R (Cv) = Cv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Suppose that v contains  the letter R in position n1 ≤ 0 and n2 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We take n1 maximal and n2 minimal with these  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' When v is the restriction to [−m, m] of the periodic P-word of period v|�n1,n2−1�,  then we have N −1  R (Cv) = Cv|�n1,n2�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' When v is not such a restriction, we have N −1  R (Cv) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lastly suppose that v contains the letter R in position n ≤ 0, but it does not contain this  letter on positive positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The last case is similar to this one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We can take n ≤ 0 maximal  with vn = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For u ∈ ΣP, the image NR(u) is in Cv if and only if one of the two following cases  are satisfied: u contains no R on positive positions, and u|�−m,m� = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The set of such word u is a  closed subset of ΣP, and is contained inside E±  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For some k > m, u contains a R on position k, contains no R on positions [1, k −  1], u|�−m,m� = v and u|�k−n−m,k� coincides with v|�−m,n�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For k fixed, the set of such  word u is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Hence in any case, N −1  R (Cv) is a union of open sets and of one closed set contained in E±  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  So it is measurable and continuous outside E±  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Lemma 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For all µ ∈ M(σ),the pushed forward measure N ∗  Rµ is invariant by σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  26  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We prove that µ((NR ◦ σ)−1(A)) = µ((σ ◦ NR)−1(A)) by considering the intersection  between (NR ◦ σ)−1(A) and two subspaces of ΣP, which are S1,1 = {u ∈ ΣP, u1 = R} and its  complementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For any element u in ΣP \\ S1,1, we have σ ◦ NR(u) = NR ◦ σ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Indeed, when u contains  no coefficient equal to R for all positive indexes or for all non-negative indexes, then NR(u) = u  and σ(u) satisfies the same condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' When there exist n ≤ 0 maximal and m ≥ 2 minimal such  that un = R = um, it follows from the definition that σ ◦ NR(u) = NR ◦ σ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence:  µ((NR ◦ σ)−1(A) \\ S1,1) = µ((σ ◦ NR)−1(A) \\ S1,1)  Consider a P-words u ∈ S1,1 with u1 = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by E±  R = ∪n∈Z(En+  R ∪ En−  R ) the set of P-  words which contains the letter R somewhere and admits an infinite half sub-word without R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  According to Lemma 32, we have µ(E±  R) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice that S1,1 \\ E±  R = (⊔n≤0(Sn,1)) \\ E±  R =  ⊔m≥2(S1,m) \\ E±  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For n ≤ 0, we define the homeomorphism ˜gn : Sn,1 �→ S1,2−n by ˜gn(u) =  σ−n−1(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice that NR ◦ σ(˜gn(u)) = σ ◦ NR(u), so ˜gn restricts to a homeomorphism gn : (σ ◦  NR)−1(A) ∩ Sn,1 \\ E±  R �→ (NR ◦ σ)−1 ∩ S1,2−n \\ E±  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that:  µ((σ ◦ NR)−1(A) ∩ S1,1) = µ((σ ◦ NR)−1(A) ∩ (S1,1 \\ E±  R))  =  �  n≤0  µ((σ ◦ NR)−1(A) ∩ (Sn,1) \\ E±  R)  =  �  n≤0  µ  �  σ−n−1 �  (σ ◦ NR)−1(A) ∩ (Sn,1) \\ E±  R  ��  since µ is σ-invariant  =  �  n≤0  µ  �  (NR ◦ σ)−1(A) ∩ (S1,2−n) \\ E±  R  �  = µ((NR ◦ σ)−1(A) ∩ S1,1 \\ E±  R)  = µ((NR ◦ σ)−1(A) ∩ S1,1)  Hence one has σ∗(N ∗  Rµ)(A) = µ(N −1  R  σ−1(A)) = µ(σ−1 ◦ N −1  R (A)) = µ(N −1  R (A)) =  (N ∗  Rµ)(A), concluding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Lemma 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map N ∗  R : M(σ) → M(σ) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Consider a sequence of measures µk ∈ M(σ) weakly converging toward the measure ν ∈  M(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take a continuous map g : ΣP → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The map g ◦ NR : ΣP → R is bounded, mea-  surable and continuous outside the set E±  R which has uniformly zero measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' According to  the Portmanteau Lemma 31, the sequence of integrals  �  gd(N ∗  Rµk) =  �  g ◦ NRdµk converges  toward  �  gd(N ∗  Rν), which proves the continuity of N ∗  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  It is not essential for the proof of the main theorem to prove now that µ and N ∗  Rµ are  cohomologous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We prove latter a stronger statement when µ is supported by finitely many  orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' To be complete, we add a sketch of proof of the cohomologous statement in the general  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proposition 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For every measure µ ∈ M(σ), we have [N ∗  Rµ]φ = [µ]φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Additionally we have  |N ∗  Rµ|(ΣP) ≤ |µ|(ΣP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The statement on the total variation is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For x, y ∈ P, denote by Ax,y the set {v ∈  ΣP, v0 = x, v1 = y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice that for all u ∈ ΣP, NR(u) and u have the same coefficients at the  indexes 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence for ν = N ∗  Rµ − µ, we have ν(Ax,y) = 0 for all x, y ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The proposition  follows from the next lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  27  Lemma 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any measure ν ∈ M(σ) satisfying ν(Ax,y) = 0 for all x, y ∈ P, we have [ν]φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a smooth closed 1-form α on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For each Markov cuboid R ∈ P, we denote  by fR : R → R a smooth function for which dfR coincides with α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by iR : ∂−R → ∂+R  the map obtained by pushing ∂−R along the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Recall that MP is the suspension set over ΣP and hP : MP → M is the corresponding semi-  conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by T : ΣP → R+ the suspension time and by M R  P = {(x, t) ∈ ΣP × R, 0 ≤  t < T (x)} the subset of MP corresponding to the cuboid R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We have:  �  M  ι(X)αdΘ(ν) =  �  MP  (ι(X)α) ◦ hPd(ν ⊗ Leb)  =  �  R∈P  �  MR  P  (dfR(X)) ◦ hPd(ν ⊗ Leb)  Using the local product of MP, the integral  �  MR  P (dfR(X)) ◦ hPd(ν ⊗ Leb) is equal to  �  ΣR(fR ◦  iR − fR) ◦ hP(x, 0)dν(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by f +, f −: ΣP → R the maps whose restrictions to ΣR are  given respectively by fR ◦ iR ◦ πP and fR ◦ πP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then:  �  M  ι(X)αdΘ(ν) =  �  R∈P  �  ΣP,R  (f +  R − f −  R )dν  =  �  ΣP  (f + − f −)dν  =  �  ΣP  (f + − f − ◦ σ)dν  Notice that in a neighborhood of a point u ∈ ΣP, f + and f − ◦ σ are obtained from integrating α  on the Markov sub-rectangle given by ∂+u0 ∩ ∂−u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that f + − f − ◦ σ is constant on  each set Ax,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore the integral �  Ax,y(f +−f −◦σ)dν is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since the non empty sets of the  form Ax,y give a partition of ΣP, we have  �  ΣP (f +−f −◦σ)dν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that  �  M ι(X)αdΘ(ν) = 0  for all α, and that [ν]φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Denote by (R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' , Rp) the cuboids of the Markov partition P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Consider the product of the  maps �p  i=1 fRi : ΣP → ΣP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any µ ∈ M(σ), the support of (�p  i=1 fRi)∗ µ is included inside Prime(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For a measure µ ∈ M(σ), denote by S(µ) ⊂ ΣP the support of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' First notice that for  any i and u ∈ ΣP, NRi(u) is either in Prime(P) or is equal to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that given a mea-  sure µ ∈ M(σ), we have S(N ∗  Riµ) ∪ Prime(P) ⊂ S(µ) ∪ Prime(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence S(  ��k  i=1 fRi  �∗  µ) ∪  Prime(P) is a decreasing sequence in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take µ ∈ M(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by AR the union of Prime(P) and of the set of P-words of the  form u: Z → P \\ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any u in ΣP, the element v = NR(u) is in one of the three cases:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  v is in Prime(P), if u is already in Prime(P), or if un = R = um for some n < 0, m ≥ 1,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  v = u contains no R if u contains no R,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  v = u if either u|�1,+∞� of u|�−∞,−1� contains no R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The set of u in the third case has uniformly zero measure according to Lemma 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  So  the support S(N ∗  Rµ) is included in AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows from the above discussion that the support  of (�p  i=1 fRi)∗ µ is included in ∩RAR, which is equal to Prime(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  28  Density of finitely supported measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given ν ∈ M(σ), there exists a sequence of measures (νn)n converging toward ν,  so that for all n: νn is finitely supported, [νn]φ = [ν]φ, |νn|(ΣP) ≤ |ν|(ΣP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Additionally if ν is positive, νn can be taken positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For n ≥ 1, we denote by Pn the set of P-words of the form u: �−n, n� → P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  An  element u ∈ Pn correspond to a sub-cuboid of u−n, consisting of points admitting a short  itinerary along σ−n(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Then Pn corresponds to the Markov partition of φ given by these  cuboids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by Σn the set of bi-infinite Pn-words, and σn the shift map on Σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We have a natural  map πn : Σn → ΣP, so that for v ∈ Σn, πn(v) and i ∈ Z, we have (πn(v))i = (vi)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice  that πn is continuous bijective, so ν ∈ M(σn) → π∗  nν ∈ M(σ) is continuous and bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For n and R ∈ Pn, we denote by N n  R : Σn → Σn the R-reduction map for the Markov  partition Pn of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Given n, we fix an order on Pn and denote by Nn the product of the  maps N n  Ri for all Ri ∈ Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Here the product is ordered using the previously fixed order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We claim that the sequence πn ◦ Nn ◦ π−1  n  converges uniformly toward the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice  that for any u ∈ Σn, we have Nn(u)0 = u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that for v ∈ ΣP, πn ◦ Nn ◦ π−1  n (v) coincide  with v on �−n, n�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence πn ◦ Nn ◦ π−1  n (v) converges uniformly toward v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take ν ∈ M(σ) and define the invariant measure νn = (πn ◦ Nn ◦ π−1  n )∗ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' From the claim  follows that the sequence νn converges toward ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The second and third points follow from  Proposition 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Lemma 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given a measure ν ∈ M(σ) and a finite set ∆ ⊂ ΣP, there exists a sequence of  measures (νn)n converging toward ν, so that for all n: νn has a finite support which is disjoint from ∆, [νn]φ = [ν]φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given a cyclic P-word w, denote by ηw the invariant measure �|w|−1  i=0  1σi(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Thanks to  Lemma 41, it is enough to prove the lemma when we have ν = ηv for some cyclic P-word v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We fix a cyclic P-word u containing all cuboids of P, so that u0 = v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We additionally suppose  that u and v are not power of a common cyclic P-word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The lemma follows from the two claims:  Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The sequence ( 1  kηvku)k≥0 converges weakly toward the measure ηv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The measure 1  k(ηvku − ηu) is cohomologous to ηv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof of Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The sequence σi(vku) converges toward σi(v) when k goes to +∞ and  √  k ≤  i ≤ k|v| −  √  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So the Dirac measures on σi(vku) converges toward the Dirac measures on σi(v)  for all sequences of pairs (k, i) with k → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence 1  kηvku converges toward ηv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  □  Proof of Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  According to Lemma 34, we have N ∗  R(ηvku − ηu) = kN ∗  Rηv, and the map N ∗  R  preserves the cohomology according to Proposition 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  □  ■  29  u  w  σ|u|(w)  w′  σ|u|(w′)  x  x′  R−  u  R+  u  Figure 2: Illustration of (w, x) ∈ R−  u and (w′, x′) ∈ R+  u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='3  Linking function  Denote by ΣR the set of u ∈ ΣP satisfying u0 = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We construct a measurable func-  tion LR : ΣR × ΣR → R, such that for all measures ν, µ ∈ M(σ) supported on finitely many  orbits, one has linkσ(ν − N ∗  Rν, µ) = �  Σ2  R LRdν ⊗ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then we prove the continuity in (ν, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by PrimeR the set of cyclic P-word u with u0 = R and ui ̸= R for all i ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For u ∈ PrimeR, We denote by Σu the set of w ∈ ΣP which satisfy wk = uk for all k ∈ �0, |u|�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Notice that for w ∈ Σu, the element σ|u|(w) is in ΣR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For u ∈ PrimeR, we define the two  following measurable sets:  R+  u = {(w, x) ∈ Σu × ΣR, such that:  u <h x and σ|u|(w) ≤v x <v w}  R−  u = {(w, x) ∈ Σu × ΣR, such that:  u <h x and w ≤v x <v σ|u|(w)}  The lemma follow from the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We call linking function the function LR : Σ2  P → R  defined by:  LR =  �  u∈P rimeR  �  1R+  u − 1R−  u  �  Here 1X is the characteristic map of the set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' From the definitions follows the next lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any u ∈ PrimeR, the following holds:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any (w, x) ∈ R+  u , we have w <v u,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any (w, x) ∈ R−  u , we have w >v u,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  R+  u ∩ R−  u = ∅,  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  LR is measurable,  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ∥LR∥∞ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The coming theorem and property are critical to define the linking number between two  invariant measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  30  Theorem 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For ν, µ ∈ M(σ) be finitely supported, with [µ]φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then we have:  linkff  σ (ν − N ∗  Rν, µ) =  �  Σ2  R  LRdν ⊗ µ  The proof of the theorem is postponed to next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proposition 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map M(σ)×M(σ) → R given by (ν, µ) �→  �  Σ2  R LRd(ν ⊗µ) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  To prove the proposition, we first estimate the integral  �  Σ2  R LRd(ν ⊗ µ) in three estimations  lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 46 (Estimation Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any ǫ > 0, there exists A ∈ N, such that for all  measures ν, µ ∈ M(σ), we have:  |ν ⊗ µ|  \uf8eb  \uf8ed  �  u∈P rimeR,|u|≥A  R+  u  \uf8f6  \uf8f8 ≤ ǫ|ν ⊗ µ|(Σ2  P)  For u in PrimeR and k ≥ 0, we denote by uk the concatenation of k copies of u, with u0 = R  by convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We define the sets:  Σu,k = Σuk \\ Σuk+1  and  R+  u,k = {(w, x) ∈ R+  u , w ∈ Σu,k}  Recall that we denote W s  l (v) = {w ∈ ΣP, for all k ≥ 0, wk = vk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice that Σu \\ W s  l (u)  is the union of Σuk \\ Σuk+1 for all k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given w in W s  l (u), we do not have σ|u|(w) <v w nor  σ|u|(w) >v w, so no (w, x) can be in R+  u ∪ R−  u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore the sets R+  u and R−  u are the unions of  the set R+  u,k and R−  u,k for all k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 47 (Estimation Lemma 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For all ǫ > 0 and u ∈ PrimeR, there exists B ∈ N such  that for all measures ν, µ ∈ M(σ), we have:  |ν ⊗ µ|  \uf8eb  \uf8ed �  k≥B  R+  u,k  \uf8f6  \uf8f8 ≤ ǫ|ν ⊗ µ|(Σ2  P)  Lemma 48 (Estimation Lemma 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given u ∈ PrimeR and k ≥ 1, the quantity (ν ⊗ µ)(R+  u,k)  is continuous in (ν, µ) ∈ M(σ) × M(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof of Proposition 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Assume the three estimation lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let (νn, µn)n be a sequence of  measures in M(σ), converging toward a measure (ν∞, µ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We should prove that �  Σ2  R LRd(νn ⊗  µn) converges toward  �  Σ2  R LRd(ν∞ ⊗ µ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  According to Remark 6, the total variations of the measures νn ⊗ µn are bounded by some  C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take ǫ > 0 and A ∈ N given by the first estimation lemma for ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by N ∈ N the  number of element u ∈ PrimeR with |u| < A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take B ∈ N the maximal integer given by the  31  second estimation lemma, applied for all u with |u| < A and for ǫ/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We have:  lim sup  n→+∞  �  νn ⊗ µn  �  �  u∈P rimeR  R+  u  ��  = lim sup  n  \uf8eb  \uf8edνn ⊗ µn  \uf8eb  \uf8ed  �  |u|<A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k<B  R+  u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k  \uf8f6  \uf8f8  + νn ⊗ µn  \uf8eb  \uf8ed �  |u|≥A  R+  u  \uf8f6  \uf8f8  +νn ⊗ µn  \uf8eb  \uf8ed  �  |u|<A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k≥B  R+  u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k  \uf8f6  \uf8f8  \uf8f6  \uf8f8  ≤ lim sup  n  \uf8eb  \uf8edνn ⊗ µn  \uf8eb  \uf8ed  �  |u|<A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k<B  R+  u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k  \uf8f6  \uf8f8  \uf8f6  \uf8f8  + Cǫ + N · Cǫ  N  ≤ ν∞ ⊗ µ∞  \uf8eb  \uf8ed  �  |u|<A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k<B  R+  u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k  \uf8f6  \uf8f8 + 2Cǫ  according to the estimation Lemma 3  ≤ ν∞ ⊗ µ∞  ��  u  R+  u  �  + 4Cǫ for all ǫ > 0  ≤ ν∞ ⊗ µ∞  ��  u  R+  u  �  lim inf  n  �  νn ⊗ µn  �  �  u∈P rimeR  R+  u  ��  ≥ ν∞ ⊗ µ∞  ��  u  R+  u  �  Hence (νn ⊗ µn)(∪uR+  u ) converges toward (ν∞ ⊗ µ∞)(∪uR+  u ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The same statement holds  for R−  u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore the integrals  �  Σ2  R LRd(νn ⊗ µn) converges toward  �  Σ2  R LRd(ν∞ ⊗ µ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Now we prove the estimation lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof of Lemma 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given A ∈ N, denote by LA the union of the sets Σu for u ∈ PrimeR  with |u| ≥ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice that if w is in LA, then w0 = R and wk ̸= R for all k ∈ �1, A − 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Hence for two distinct indexes i, j ∈ �0, A − 1�, the set σi(LA) and σj(LA) are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For a  measure ν ∈ M(σ), we have:  A|ν|(LA) =  A−1  �  i=0  |ν|(σi(LA)) ≤ |ν|(ΣP)  So |ν|(LA) ≤ |ν|(ΣP)  A  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We take A satisfying 1 ≤ Aǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then we have:  |ν ⊗ µ|  �  ∪|u|≥AR+  u  �  ≤ |ν ⊗ µ| (LA × ΣP) ≤ ǫ|ν ⊗ µ|(Σ2  P)  ■  32  Proof of Lemma 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take (w, x) in R+  u,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then both w and σ|u|(w) are in Σuk−1 \\ Σuk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By  definition, we have σ|u|(w) ≤v x <v w, so we also have x ∈ Σuk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' According to Lemma 43, we  have x <v w <v u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So if x is in Σuk+1, u is in Σuk+1, so w is also in Σuk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since w is not is  this set, x lies to Σuk−1 \\ Σuk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take B > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice that we have σ|u|(Σuk) ⊂ Σuk−1, so given i  with 0 ≤ 2i < B − 1, one has:  (id ×σ2i)(R+  u,k) ⊂ (Σuk \\ Σuk+1) × (Σuk−1−2i \\ Σuk+1−2i)  It follows that the sets (id ×σ2i)(∪k≥BR+  u,k) for 0 ≤ 2i < B − 1 are pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For  any ν, µ ∈ M(σ), we have (ν ⊗ µ)  �  (id ×σ2i)(∪k≥BR+  u,k)  �  = (ν ⊗ µ)  �  ∪k≥BR+  u,k  �  , so using a  similar argument to above, we have:  (ν ⊗ µ)  �  ∪k≥BR+  u,k  �  ≤ |ν ⊗ µ|(Σ2  P)  ⌊ B−1  2 ⌋  ■  Denote by Sk = {w ∈ Σu,k, w <v u, u <h w} for k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We have Sk <v Sk+1 <v u for all k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take w1 ∈ Sk and w2 ∈ Sk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Suppose that w2 ≤v w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So w1 lies vertically between w1  and u, which start by uk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By definition of the vertical order, w1 starts by uk+1, which is not  possible since Σu,k = Σuk \\ Σuk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  We prove the estimation Lemma 3: ν ⊗ µ(R+  u,k) is continuous in (ν, µ) ∈ M(σ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The idea  is to use the Portmanteau Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The boundary of R+  u,k can have a positive measure, so we  need a first step before using the Portmanteau Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For that we notice that a measure ν ⊗ µ  charges ∂R+  u,k only on pairs (z, z) and (z, σ|u|(z)), where z is periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We cut a neighborhood  of (z, z) in R+  u,k and glue it on (z, σ|u|(z)), to remove these points from the boundary of ∂R+  u,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Then we use the Portmanteau Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof of Lemma 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let (νn, µn)n be a sequence of measures in M(σ) converging toward a  measure (ν∞, µ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We use a Portmanteau Lemma to prove the convergence of the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take ǫ > 0 and k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We can take a subset Xǫ ⊂ Sk for which the sum of |ν∞|({x}) for x ∈  Sk \\ Xǫ atom of |ν∞|, is less that ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' That is:  �  x∈Sk\\Xǫ  |ν∞|({x}) ≤ ǫ  For each z ∈ Xǫ, we take a small neighborhood Tz ⊂ ΣP of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We can take it such that the  following are satisfied:  Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' There exist neighborhoods Tz of the points z ∈ Xǫ satisfying the following:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ∂Tz = ∅,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' the sets Tz and the sets σ|u|(Tz) all z ∈ Xǫ are pairwise disjoint,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  σ|u|(Tz) <v σ(Tz),  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' σ|u|(Tz) >h u for all z ∈ Xǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  33  Proof of Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We take Tz = {w ∈ ΣP, for all |k| ≤ N, wk = zk} for some N > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It is open  and close, so it has no boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If we prove, the points 2, 3 and 4 when replacing Tz by z, then  points 2 to 4 will be satisfied when taking N large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If z is in Xǫ ⊂ Sk, then σ|u|(z) is not  in Σuk, so it is not in Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence the set Xǫ and σ|u|(Xǫ) are pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since z is in Σu  and z <v u, we have σ|u|(z) <v z according to Lemma 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For z ∈ Xǫ, we have u <h z, and u  and z coincide on �0, |u|�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence we have u <h σ|u|(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore the points 2 to 4 are satisfied  for large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  □  We define the sets:  T =  � �  z∈Xǫ  Tz × Tz  � �  R+  u,k  and  R′ =  �  R+  u,k \\ T  �� �  id ×σ|u|�  (T )  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  R+  u,k \\ T and (id ×σ|u|)(T ) are disjoint, so ν∞ ⊗ µ∞(R+  u,k) = ν∞ ⊗ µ∞(R′)  Proof of Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  When (w, x) is in R+  u,k we have σ|u|(w) ≤v x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' When (w, x) is in (id ×σ|u|)(T ),  we have (w, σ−|u|(x)) is in T ⊂ R+  u,k, so σ−|u|(x) <v w and x <v σ|u|(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  □  We now prove that the boundary of R′ ⊂ Σ2  P has small measure for ν∞ ⊗ µ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice that  for z in Xǫ, the point (z, z) is in the interior of T , so it is not in the adherence of R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The set Tz × σ|u|(Tz) is included in R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Assume the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote x ≡v w when we have x ≤v w ≤v x, that is x ∈ W s  l (w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The  boundary ∂R′ is contained in union of the three sets:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  {(w, x) ∈ Σu,k × ΣR, x ≡h u, u <h x}  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  {(w, x) ∈ (Σu,k \\ Xǫ) × ΣR, x ≤v u, u ≤h x, x ≡v w}  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  {(w, x) ∈ (Σu,k \\ Xǫ) × ΣR, x ≤v u, u ≤h x, x ≡v σ|u|(w)}  The measure for |ν∞ ⊗ µ∞| of the first set is zero (see Lemma 33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For the two other sets,  we have:  |ν∞ ⊗ µ∞| ({(w, x) ∈ (Σu,k \\ Xǫ) × ΣR, x ≤v u, u ≤h x, x ≡v w})  =  �  w∈Σu,k\\Xǫ  ��  x∈W s  l (w)  1x≤vu,u≤hxd|µ∞|(x)  �  d|ν∞|(w)  =  �  w∈Σu,k\\Xǫ  w periodic  ��  x∈W s  l (w)  1x≤vu,u≤hxd|µ∞|(x)  �  d|ν∞|(w) according to Lemma 33  =  �  w∈Σu,k\\Xǫ  w periodic  1w≤vu,u≤hw|µ∞|({w})d|ν∞|(w)  =  �  w∈Sk\\Xǫ  |µ∞|({w})d|ν∞|(w)  ≤ ǫ|ν∞|(ΣP)  34  A similar argument shows that the third sets has measure at most ǫ|ν∞|(ΣP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows from  the Portmanteau Lemma 31 that we have:  lim sup  n  |νn ⊗ µn(R′) − ν∞ ⊗ µ∞(R′)| ≤ 2ǫ|ν∞|(ΣP)  This holds for all ǫ > 0, and ν⊗µ(R′) = ν⊗µ(R+  u,k) for all ν, µ ∈ M(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore νn⊗µn(R+  u,k)  converges in n toward ν∞ ⊗ µ∞(R+  u,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof of Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Recall that for x ∈ σ|u|(Tz), we have u <h x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If (w, x) is in Tz × σ|u|(Tz)  and satisfies w ≤v σ−|u|(x), then we have σ|u|(w) ≤v x <v w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The second equation is due  to σ|u|(Tz) <v σ(Tz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In that case, (w, x) lies in R+  u,k and not in T , so (w, x) ∈ R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  If (w, x) ∈ Tz × σ|u|(Tz) satisfies σ−|u|(x) <v w, we also have σ|u|(w) ≤v σ−|u|(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  In-  deed σ|u|(w) belongs to Sk−1, σ−|u|(x) belongs to Sk, and Sk >v Sk−1 according to Lemma 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Hence (w, σ−|u|(x)) is in both R+  u,k and T 2  z , so it lies in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence (w, x) is in R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  □  ■  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='4  Adapted cells  We prove Theorem 44, that is the linking number linkff  σ (N ∗  Rν − ν, µ) is equal to  �  Σ2  R LRdν ⊕ µ  when µ and ν are finitely supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by ∆n the n-dimensional simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We call n-chains,  the abstract finite sums of continuous maps ∆n → ∂−R, with coefficients in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We also denote  by ∂ the boundary map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' A differential embedding ∆1 → ∂−R is said adapted if it is either  transverse to the stable and the unstable foliations, or is tangent to one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' A 1-chain is  said adapted if it is the sum of adapted embeddings ∆1 → ∂−R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For λi ∈ R and fi : ∆2 → ∂−R,  the 2-chain c = �n  i=1 λifi is said adapted if the maps fi are orientation preserving immersions  and ∂fi are adapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Here ∆2 is endowed with the anti-clockwise orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Recall that W s  R and W u  R are the stable and unstable foliations on ∂−R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We identify the bi-  foliated rectangle (∂−R, W s  R, W u  R) with a sub rectangle of (R2, H, V ), with a C1 diffeomorphism  preserving the orientations of the foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Let C be an adapted 2-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We define the  support function of C has SuppC : ∂−R → R defined for x ∈ R ⊂ R2 by SuppC(x) =  limt→0+ C ·∩(x + (−t, t2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Let c be an adapted 1-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We define Suppc = SuppC for any  2-chain C satisfying ∂C = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Remark 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given an adapted 2-cell C, if either ∂C = 0 or C is contained in finitely many  stable leaves and one unstable leave, then SuppC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Reducing a cyclic P-word  We fix a primitive cyclic P-word r with r0 = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by ηr  the measure ηr = �|r|−1  k=0 1σk(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We construct an adapted 2-cell to prove the Theorem 44 for  the measure ηr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Write r = s1 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' ◦ sn where all si are cyclic words such that (si)−1(R) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  According to Lemma 34, we have N ∗  Rηr = �n  j=1 ηsj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We suppose that n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For i ∈ �0, n − 1�, we denote by ri = si ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' ◦ sn ◦ s1 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' ◦ si−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Notice that ri and si are different since n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So thanks to Lemma 18, we can find a smooth  curve ai : [0, 1] → ∂−R, from πP (ri) to πP (si), which is additionally transverse to the stable  and unstable foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since the set of periodic orbits is countable, we can take the curve ai  such that its interior is disjoint from all period orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Recall that when x ∈ ∂−R admits an itinerary starting with u0, u1, · · · , u|u| = u0, then T u(x)  is the time realizing this itinerary on x, and σu(x) = φT u(x)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by bi : [0, 1] → ∂−R the  curve given by bi(t) = σsi ◦ ai(1 − t), and define the adapted 2-chain c = �  i(ai + bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  35  Suppc = 1  Suppc = −1  a1  b1  a2  b2  a3  b3  Figure 3: An adapted cell c = �  i(ai + bi) and its support Suppc on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' On the right is  represented the corresponding 2-chain in M, bounded the invariant 1-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any cyclic P-word r and any measure µ ∈ M0(σ) finitely supported, so that  the support of ΘP(µ) is disjoint from the support of ΘP(ηr), we have:  linkff  σ (ηr − N ∗  Rηr, µ) =  �  ΣR  Suppc ◦ πPdµ  The map Suppc is used to build a 2-chain bounding the invariant 1-cycle corresponding  to ηr − N ∗  Rηr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It comes with a choice for the curves ai and bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We remove later that choice by  replacing c by another adapted 1-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by vi = ai(0) = bi−1(1), wi = ai(1) = bi(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given a periodic P-word u, we denote by γu ⊂ M the invariant 1-cycle corresponding to  the periodic orbit of πP(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' According to Lemma 17, the realization of u has degree one (here  the foliations are orientable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence we have ΘP(ηu) = Lebγu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We also write Γ = γr − �  i γsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The Lemma is clear when r does not contain R, since N ∗  Rηr = ηr in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Otherwise  write r = s1 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' ◦ sn as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' When n = 1 we also have N ∗  Rηr = ηr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We suppose now n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Consider the curves ai, bi and c = �  i(ai + bi) as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote  by C an adapted 2-chain in ∂−R, such that ∂C = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We also denote by ei ⊂ M the 2-chain  obtained as the image of the map [0, 1] × [0, 1] → M, (s, t) �→ φtT si ◦ai(s)(ai(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Its boundary  is ∂ei = ai + φ[0,T si(wi)](wi) + bi − φ[0,T si(vi)i](vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Define a 2-chain by S = �  i ei − C, illustrated  in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We have ∂S = Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take an invariant 1-cocycle Γ′ = �  γ m(γ)γ, whose support is disjoint from the support of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  By definition of the linking number linkff  φ , we have linkff  φ (LebΓ, LebΓ′) = S ·∩Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By construction  of ai, ei is disjoint from Γ′, so:  linkff  φ (LebΓ, LebΓ′) = C ·∩Γ′  =  �  γ  m(γ)  �  x∈γ∩∂−R  d ·∩x  =  �  γ  m(γ)  �  x∈γ∩∂−R  Suppc(x)  =  �  ∂−R  Suppc Leb⊥  Γ′  =  �  ΣR  Suppc ◦ πPdµ  36  The last equality is due to the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The measures Leb⊥  Γ′ and π∗  Pµ coincide on the interior  of ∂−R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' On the boundary of ∂−R, the support supc is equal to zero, maybe outside finitely  many points wi and vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' These points are in the support of ηr, so by hypothesis they are not in  the support of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefor we have  �  ∂−R Suppcdπ∗  Pµ =  �  ∂−R Suppc Leb⊥  Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof of Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Notice that vi = πP(ri), so we have:  φT si (vi)(vi) = φT si (πP(ri))  = πP(σ|si|(ri))  = πP(ri+1)  = ai+1(0)  Similarly we have φT si (wi)(wi) = wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By construction, we have:  ∂S =  �  i  ∂ei − ∂C  =  �  i  �  ai + bi + φ[0,T si(wi)](wi) − φ[0,T si (vi)i](vi)  �  − c  =  �  i  φ[0,T si(vi)i](vi) −  �  i  φ[0,T si(wi)](wi)  = φ[0,T r(v1)](v1) −  �  i  φ[0,T si(wi)](wi)  = γr −  �  i  γsi  □  ■  We denote by w−  i = W s  R(vi) ∩ W u  R(wi) and w+  i = W s  R(vi+1) ∩ W u  R(wi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We write [a, b]s the  stable segment in ∂−R between two points a, b on a same stable leaf, and similarly for [a, b]u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by □c the adapted 1-cycle given by �  i([vi, w−  i ]s + [w−  i , w+  i ]u + [w+  i , vi+1]s), which we  call the square model of c (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any µ ∈ M(σ), we have  �  ΣR(Suppc − Supp□c) ◦ πPdµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof of Lemma 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote di = [vi, w−  i ]s + [w−  i , w+  i ]u + [w+  i , vi+1]s − ai − bi, which is a closed  adapted 1-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by fi : ∆2 → ∂−R an adapted 2-simplex whose boundary is ai +  [wi, w−  i ]s + [w−  i , vi]s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The image of fi lies in the smallest horizontal rectangle of ∂−R containing  both wi = πP(si) and vi = πP(ri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence according to Lemma 15, the image of fi is inside Rsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Hence σsi ◦ fi is well-defined and ∂σsi ◦ fi = −bi + [wi, w+  i ]u + [w+  i , vi+1]s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The 1-chain ∂(σsi ◦ fi − fi) − di is equal to [wi, w+  i ]u − [wi, w−  i ]u − [w−  i , w+  i ]u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' According to  Remark 50, the support of the previous adapted 1-chain is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore, we have Suppdi =  Suppσsi◦fi − Suppfi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Let µ ∈ M(σ) be an invariant measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since σsi ◦fi is obtained from fi by pushing along the  φ, and the φ preserves the orientations of the foliations W s  R, W u  R, we have  �  Suppσsi◦fi ◦ πPdµ =  �  Suppfi ◦ πPdµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that  �  Suppdi ◦ πPdµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice that we have �  i di = □c − c, so  the integral  �  ΣR(Suppc − Supp□c) ◦ πPdµ is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  We now prove that the integral  �  Σ2  R Ld(ν ⊗ µ) computes linking numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  37  □c  c  v1  v2  w1  w−  1  w+  1  w−  1  w+  1  x1  x2  x3  {x ∈ R, v2 ≤v x <v v1, w1 <h x}  {x ∈ R, v3 ≤v x <v v1, w3 <h x}  Fu(w1)  Figure 4: Adapted cell c = �(ai + bi) and its corresponding square model □c on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  In dashed filling are two sets of equal areas, used in the proof of Lemma 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  On the right  is represented the set of points vertically between vi and vi+1, and on the right of wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' These  sets are uses to relate the support function of □c and the linking function GR, in the proof of  Theorem 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof of Theorem 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice that if ν(ΣR) = 0, then N ∗  Rν = ν and the two terms in the equality  are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By linearity, it is enough to prove the result when ν is equal to the measure ηr for some  primitive cyclic P-word r with r0 = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We use the notations si, ri used in the above discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Suppose that n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then N ∗  Rηr = ηr so linkff  σ (ηr − N ∗  Rηr, µ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Fix u ∈ PrimeR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  If u ̸= r, then ηr ⊗ µ(R±  u ) = 0 since no points w ∈ Σu are in the support of ηr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If u = r,  then we also have ηr ⊗ µ(R±  r ) = 0, since if w is in Σu and in the support of ηr, then w = u  and there is no x ∈ ΣR satisfying w ≤v x <v σ|u|(w) = w or σ|u|(w) ≤v x <v w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence we  have  �  Σ2  P LRdηr ⊗ µ = 0 = linkff  σ (ηr − N ∗  Rηr, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Now suppose that n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by Iu  + the unstable boundary component of ∂−R, which  is on the right, and by xi = W s(vi) ∩ Iu  + the projection of vi on Iu  +, in parallel to W s  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We define  the adapted 2-cycle fi by fi = [w−  i , w+  i ]u + [w+  i , xi+1]s + [xi+1, xi]u + [xi, w−  i ]s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We have �  i Suppfi = Supp□c (illustrated in Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any µ ∈ M(σ),  �  ΣR Suppfi ◦ πPdµ = (1ri ⊗ µ)(R+  si) − (1ri ⊗ µ)(R−  si)  Assume the two claims for now, and take µ ∈ M(φ) finitely supported, so that the support  of ΘP(µ) is disjoint from the support of ΘP(ηr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take u ∈ PrimeR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Recall the definition:  R+  u = {(w, x) ∈ Σu × ΣR, such that:  u <h x and σ|u|(w) ≤v x <v w}  38  Notice that (1ri ⊗ µ)(R±  u ) = 0 when u ̸= si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that:  �  ΣR  LRd(ηr ⊗ µ) =  �  i  �  ΣR  LRd(1ri ⊗ µ)  =  �  i  �  u∈P rimeR  �  (1ri ⊗ µ)(R+  u ) − (1ri ⊗ µ)(R−  u )  �  =  �  i  �  (1ri ⊗ µ)(R+  si) − (1ri ⊗ µ)(R−  si)  �  =  �  i  �  ΣR  Suppfi ◦ πPdµ  according to Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  =  �  ΣR  Supp□c ◦ πPdµ  according to Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  = linkff  σ (ηr − N ∗  Rηr, µ) according to Lemmas 51, 52  Which implies linkff  σ (ηr − N ∗  Rηr, µ) =  �  ΣR LRd(ηr ⊗ µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof of Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  One can verify that the closed 1-chain �  i fi − □c is a chain contained in the  union of the unstable leaf Iu  + and of finitely many stable leaves containing the points vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Essen-  tially the parts in [w−  i , w+  i ]u cancel each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence its support function is Supp(�  i fi−□c) = 0  (see Remark 50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  □  Proof of Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Notice that for (w, x) ∈ R+  si to be in the support of (1ri ⊗ µ), we must  have w = ri and ri+1 <v ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Similarly for (w, x) ∈ R−  si to be in the support of the same measure,  we must have ri <v ri+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence the measure of either R+  si or R−  si is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Consider the case when ri+1 <v ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  By construction, we have w+  i  <v w−  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  So fi is the  boundary of a Markov rectangle Ri ⊂ ∂−R with the induced orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence the support of  the support of fi is equal to:  Suppfi(x) =  1  if x is in the interior of the rectangle Ri  1  if x ∈ (w+  i , xi+1]s ∪ [xi+1, xi)u  0  otherwise  So the integral �  ΣR Suppfi ◦ πPdµ is equal to the measure for µ of the set of x ∈ ΣR satisfying:  πP(si) = wi <h πP(x)  πP(ri+1) = vi+1 ≤v πP(x) <v vi = πP(ri)  According to Lemma 18 this condition on x is equivalent to having:  si <h x  σ|si|(ri+1) ≤v x <v ri  Hence we have  �  ΣR Suppfi ◦ πPdµ = µ(R+  si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In the case ri+1 >v ri, the same argument shows  that  �  ΣR Suppfi ◦ πPdµ = −µ(R−  si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  □  ■  39  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='5  Linking number linkφ  We compose the reduction maps for all Markov cuboids of P to reduce an invariant measure  on one supported by a fixed finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We then prove Theorem 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by R1, · · · , Rp the  Markov cuboids in P and by N k = NRk ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' ◦ NR1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Definition 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For ν, µ ∈ M(σ) such that [ν]φ = [µ]φ = 0, we define the linking number  between ν and µ by:  linkσ(ν, µ) = linkf  φ(ΘP(N  ∗  ρν), ΘP(µ)) +  p−1  �  k=1  �  Σ2  P  LRkd(N  ∗  kν) ⊗ µ  Lemma 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map linkσ is continuous on M0(σ) × M0(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The maps ν �→ N  ∗  ρµ and ΘP are continuous (see Lemma 37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The measure N  ∗  ρν is  supported by a subset of Prime(P) (see Lemma 40), which is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by ∆ ⊂ M the  union of the periodic orbits containing πP(Prime(P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map linkf  φ is continuous on the set  of pairs of measures (νφ, µφ) ∈ M0(φ) × M0(φ) for which ∆ contains the support of νφ (see  Proposition 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore the term linkf  φ(ΘP(N  ∗  ρν), ΘP(µ)) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The second term is  continuous according to Proposition 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Lemma 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any ν, µ ∈ M0(σ) with finite supports, so that ΘP(µ) and ΘP(ν) have disjoint  support, we have:  linkσ(ν, µ) = linkff  σ (ν, µ)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Suppose first that the support of µ is also disjoint from Prime(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then for all k, the  support of ΘP(N  ∗  kν) is disjoint from the support of ΘP(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence according to Theorem 44, we  have  �  Σ2  P LRkd(N  ∗  kν) ⊗ µ = linkff  σ (N ∗  k ν − N ∗  k+1ν, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that:  linkff  σ (ν, µ) = linkff  σ (N  ∗  ρν, µ) +  p−1  �  k=1  linkff  σ (N ∗  kν − N ∗  k+1ν, µ)  = linkf  φ(ΘP(N  ∗  ρν), ΘP(µ)) +  p−1  �  k=1  �  Σ2  P  LRkd(N  ∗  kν) ⊗ µ  = linkσ(ν, µ)  The general case follows from the density of measures in the first case (see Lemma 42) and  from the continuity of linkσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Lemma 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' There exists a continuous bilinear map linkφ : M0(φ)×M0(φ) → R such that for all  measure ν, µ ∈ M0(σ), one has linkσ(ν, µ) = linkφ(ΘPν, ΘPµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Additionally linkφ is continuous  on M0(φ) × M0(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It is enough to prove that for any measures ν1, ν2, µ ∈ M0(σ) with ΘP(ν1) = ΘP(ν2), we  have linkσ(ν1, µ) = linkσ(ν2, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Once it is proven, we define the map linkφ by linkφ(νφ, µφ) =  linkσ(νσ, µσ) for any measures νσ, µσ ∈ M0(σ) in the reciprocal images of νφ, µφ by the surjec-  tion ΘP : M(σ) → M(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  To prove the claim, take ν1, ν2, µ ∈ M0(σ) with ΘP(ν1) = ΘP(ν2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by Γ the set  of periodic orbits of φ, contained in the stable/unstable boundary of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map ΘP is injec-  tive outside the reciprocal image of the set ∪γ∈Γ(Fs(γ) ∪ Fu(γ)) for the map πP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence the  measures νi coincide on this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  40  According to Lemma 33, the measure Θ(νi) charges the set π−1(∪γ∈Γ(Fs(γ) ∪ Fu(γ))) only  on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that ν1 − ν2 has a finite support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Suppose that µ is finitely supported, and  that the support of ΘP(µ) is disjoint from the support of Θ(ν1 − ν2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' According to Lemma 55,  we have linkσ(ν1 − ν2, µ) = linkff  σ (ν1 − ν2, µ) = linkf  φ(ΘP(ν1 − ν2), ΘP(µ)) = 0 since ΘP(ν1 −  ν2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By continuity of linkσ and density of the measures satisfying that assumption, we  have linkσ(ν1, µ) = linkσ(ν2, µ) for all measures µ ∈ M0(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Proof of Theorem 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Lemma 42 implies that we can approximate any pair of invariant mea-  sures (ν, µ) ∈ M0(φ) by sequence of pairs (νn, µn) ∈ M0(φ), where νn and µn are supported  by disjoint and finite unions of periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore the uniqueness and symmetry of the  map linkφ follow from the continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  When the stable and unstable foliations of φ are orientable, the theorem follows from the  above discussion (see Lemmas 55 and 56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Suppose that the stable and unstable foliations are not orientable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by �π: �  M → M  the bundle-orientations covering (which is of degree 2), and by g the non-trivial deck transforma-  tion of �  M → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Lift φ to a flow ψ on �  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by linkψ : M0 (ψ)2 → R the linking number  map obtained in the orientable case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a measure µ ∈ M0(φ) and denote by µ ◦ �π ∈ M(ψ)  the measure which coincides with ((�π|U)−1)∗µ on each set U ⊂ �  M where �π is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice  that µ ◦ �π is the unique ψ and g invariant measure lifting 2µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take a closed 1-form �α on �  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since �α + g∗�α is invariant by the action of g, there exists a  closed 1-form α on M satisfying �π∗α = �α + g∗�α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The flow ψt and the measure µ◦ �π are invariant  by g, so we have:  �  �  M  ι  �∂ψt  ∂t  �  �αdµ ◦ �π = 1  2  ��  �  M  ι  �∂ψt  ∂t  �  �αdµ ◦ �π +  �  �  M  ι  �∂ψt  ∂t  �  g∗�αdµ ◦ �π  �  = 1  2  �  �  M  ι  �∂ψt  ∂t  �  �π∗αdµ ◦ �π  =  �  M  ι  �∂φt  ∂t  �  αdµ = 0  It follows that µ◦�π is null-cohomologous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take K ⊂ �  M a measurable subset such that (K, g·K) is  a partition of �  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then the projection K → M is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given a continuous map f : �  M → R,  one has:  �  �  M  fdµ ◦ �π =  �  K  fdµ ◦ �π +  �  g·K  fdµ ◦ �π  =  �  M  f ◦ (�πK)−1dµ +  �  M  f ◦ (g · �πK)−1dµ  =  �  M  \uf8eb  \uf8ed  �  y∈�π−1(x)  f(x)  \uf8f6  \uf8f8 dµ(x)  It follows that µ �→ µ◦ �π is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For ν, µ ∈ M0 (ψ), we define linkφ(ν, µ) = 1  2 linkψ(µ◦  �π, ν ◦ �π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By construction, it is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take Γ1, Γ2 two invariant 1-cycles for φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take S1 a  2-chain on M, bounding Γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Lift Γi to an invariant 1-cycle �Γi of ψ, which satisfies g · �Γi = �Γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  41  Also lift S1 to a 2-chain �S1 in �  M, bounding �Γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' According to the orientable case, we have:  linkφ(LebΓ1, LebΓ2) = 1  2 linkψ(LebΓ1 ◦�π, LebΓ2 ◦�π)  = 1  2 linkψ(Leb�Γ1, Leb�Γ2)  = 1  2  �S1 ·∩�Γ2  = S1 ·∩Γ2  = linkff  φ (LebΓ1, LebΓ2)  ■  Corollary 57 (Linking number is invariant by orbit equivalence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any orbit equivalence  h: (M, φ) → (N, ψ) between two Anosov flows on oriented closed 3-dimensional manifolds, we  have linkφ = linkψ ◦(Θh, Θh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By continuity and density of finitely supported orbits (see Lemma 42), it is enough to  prove it for pairs of measures LebΓ1, LebΓ2 ∈ M0(φ), for two null-homologous invariant 1-  cycles Γ1, Γ2 with disjoint supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Given a periodic orbit γ ⊂ M, we have Θh(Lebγ) =  h∗ Leb⊥  γ ⊗ Lebψ = Lebh(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore we have Θh Lebγ = Lebh(γ), and Θh LebΓi = Leb h∗(Γi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The linking number of disjoint 1-chains is invariant by preserving orientation homeomorphism,  so the linking numbers linkψ(Θh LebΓ1, Θh LebΓ2) and linkφ(LebΓ1, LebΓ2) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  4  Reeb-like condition for measures  We prove Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' An invariant measure µ for φ is said to satisfy the Reeb-like condition  if the two following conditions are satisfied:  [µ]φ = 0 in H2(M, R),  minν link(µ, ν) > 0, where the minimum is taken over all invariant probability measure ν  such that [ν]φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Notice that the Reeb-like condition is invariant by orbit equivalence of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' That is if h: (M, φ) →  (N, ψ) is an orbit equivalence between two transitive Anosov flows, then a measure µ ∈ M(φ)  satisfies the Reeb-like condition for φ if and only if Θh(µ) satisfies the Reeb-like condition for ψ  (see Proposition 11 and Corollary 57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Let α be a C1 differential 1-form such that ι(X)(dα) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The 2-form dα is invariant by φ,  so it induces a transverse measure for φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by να the invariant measure whose transverse  measure ν⊥  α corresponds to dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We have [να]φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Additionally for any invariant measure µ ∈ M0(φ), we have:  linkφ(µ, να) =  �  M  ι(X)αµ  Prasad [Pra22] define the linking number with a smooth measure as the integral appearing  in the above lemma, for homology spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Theorem D is a consequence of McDuff’s Theo-  rem [Duf87, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='2], except for the signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' That is in McDuff’s theorem, there is no relation  between the sign of the linking number and the sign of the reparametrization (McDuff formula-  tion allows one to reverse the direction of the flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For completeness, we give a sketch of proof  for the reformulation, without using McDuff’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  42  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It is enough to verify the equality for measures of the form LebΓ for null-homologous  invariant 1-cycles Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The lemma follows from continuity of linkφ and from the density Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take a smooth 1-form β which satisfies ι(X)β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then the 3-form V = dα ∧ β is invariant  by the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Indeed we have:  ι(X)V = (ι(X)dα) ∧ β + dα ∧ (ι(X)β) = dα  Therefore we have LX(dα ∧ β) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If νV denote the invariant measure corresponding to the  invariant 3-form V , then ν⊥  v corresponds to ι(X)V = dα, so νV = να.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that [να]φ =  [ι(X)V ] = 0 (see Lemma 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We denote by ∆ the support of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by α∆ = π∗  ∆α and β∆ = π∗  ∆β and take a  rational 2-chain S in MΓ si that (π∆)∗(∂S) = Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We can take a closed 1-form ω on MΓ so that  P∆([ω]) = [S].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a closed 1-form ω on M∆ satisfying (π∆)∗(∂P∆[ω]) = [Γ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then we have:  0 = ι(X∆)(ω ∧ dα∆ ∧ β∆)  = (ι(X∆)ω) ∧ (dα∆ ∧ β∆) − ω ∧ (ι(X∆)dα∆) ∧ β∆ − ω ∧ dα∆ ∧ (ι(X∆)β∆)  = (ι(X∆)ω)(dα∆ ∧ β∆) − ω ∧ dα∆  From the definition link∆  Γ follows:  linkf  φ(να, LebΓ) =  �  M∆  (ι(X∆)ω)(dα∆ ∧ β∆)  =  �  M∆  ω ∧ dα∆  =  �  S  dα∆  by Poincar´e duality  =  �  (π∆)∗(S)  dα  =  �  Γ  α  =  �  M  ι(X)αd LebΓ  ■  Lemma 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let E be a finite dimensional vector space and F ⊂ E × R be a compact convex set  such that F ∩ (0 × (−∞, 0]) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then there exist η > 0 and a linear map f : E → R satisfying:  F ⊂ {(x, t) ∈ E × R, t ≥ f(x) + η}  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since F is convex and compact, there exists a hyperplane H ⊂ E × R separating F  and (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Additionally H can be chosen so that H ∩ (0 × R) = {(0, c)} for some c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then  there exists a linear map f : E → R satisfying that (x, y) ∈ E × R lies in H if and only if  y = f(x) + c, and f(0, 1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then for all (x, y) ∈ F, we have y > f(x) + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Sketch proof of Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Suppose that a smooth reparametrization ψ of φ is the Reeb of a  contact form α with α ∧ dα > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by να ∈ M(ψ) the ψ-invariant measure correspond-  ing to α ∧ dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For any measure µ ∈ M0  p(ψ), we have linkψ(να, µ) =  �  M ι  �  ∂ψt  ∂t  �  αdµ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  43  Since α ∧ dα is positive, να is a positive multiple of a probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore to να cor-  responds a smooth φ-invariant probability measure which satisfies the Reeb-like condition (since  the cohomology class and the linking numbers are invariant by topological equivalence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Suppose now that there exists a smooth probability measure νV satisfying the Reeb-like  condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Here V is a smooth a positive volume form, invariant by φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' According to Lemma 10,  the cohomology class of ι(X)V is equal to [νV ]φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  So there exists a smooth differential  1-form β on M such that dβ = ι(X)V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We transform β in a contact form in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Apply Lemma 59 to the convex F = {([µ]φ,  �  M ι(X)βdµ) ∈ H2(M, R) × R, µ ∈ Mp(φ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We  obtain that there exists an element [δ] ∈ H1(M, R), where δ is a closed 1-form, and c > 0 such  that for all µ ∈ Mp(φ), we have  �  M ι(X)(β + δ)dµ =  �  M ι(X)βdµ + [µ]φ · [δ] ≥ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then by  averaging β + δ along the flow, one obtain a differential 1-form α =  1  T  � T  0 φ∗  t (β + δ)dt, which  satisfies ι(X)α > 0 and dα = ι(X)V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that φ is Reeb-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  5  Birkhoff Section  We give a brief overview on partial sections and Birkhoff sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Here the manifold M is  supposed oriented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We give a variation of the usual definition of partial sections using the  orientation on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  A partial section of φ is a pair (S∗, f) where S∗ is a compact C∞ oriented surface and f  is a smooth immersion of S∗ in M, which satisfies that f| Int S∗ is positively transverse the flow,  and f(∂S∗) is a finite union of periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It implies that each boundary component of S∗ is  sent to a periodic orbit γ of φ, and f : γ∗ → γ is a covering map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We usually denote by S = f(S∗),  and identify the immersed surface S with (S∗, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' A Birkhoff section is a partial section (S∗, f)  embedded in its interior, such that there exists T > 0 for which any orbit arc of φ of length T  intersects S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proposition 60 (Fried-desingularisation [Fri82]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let φ be a smooth flow on a 3-dimensional  closed manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let S1 be a Birkhoff section and S2 be a partial section for φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then there  exists a Birkhoff section for φ, relatively homologous to [S1] + [S2] in H2(M, ∂S1 ∪ ∂S2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Fried introduced a desingularisation and used it (together with the Fried sections we define  below) to prove that any transitive Anosov flow admits a Birkhoff section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Fried description is  not precise, we write a precise proof of the proposition in the Appendix A, which additionally  works for non-Anosov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Sign of boundary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Fix a partial section S = (S∗, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The orientation on the  surface S∗ induces an orientation on its boundary, so that the normal direction going outside S∗  plus the positive direction tangent to the boundary, induce the orientation on S∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We orient  the orbits of the flow with the orientation given by the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' A boundary component γ∗ is said  positive if f : γ∗ → f(γ) is orientation preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' A partial section is said positive if all its  boundary components are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Theorem 61 (Asaoka-Bonatti-Marty [ABM22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' An Anosov flow is R-covered and positively  skewed if and only if it admits a positive Birkhoff section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Given a positive Birkhoff section S as in the last theorem, we show later that the probability  measure Leb∂S has positive linking numbers with all null-cohomologous probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  In the last section, we need more than what is prescribed by the theorem, namely we need a  44  u1  u2  v1v2  v2v1  γ1  γ2  γw  Figure 5: Illustration of a 2-chain modeling a Fried Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In green is represented a transverse  polygon (corresponding to f(P) ⊂ ∂−R in the proof of Lemma 62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In orange and yellow are  represented two tangent rectangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' They can be smoothed to obtain a Fried section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Birkhoff section with only one boundary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given a partial section (S∗, f), we view ∂S  as an invariant 1-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It is given by:  ∂S =  �  γ∗⊂∂S∗  deg(γ∗ → f(γ∗))f(γ∗)  Here deg(γ∗ → f(γ∗)) is the algebraic degree of the map γ∗  f−→ f(γ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Note that S is positive  if and only if ∂S is an invariant 1-cycle with positive coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='1  Fried section  Fried constructed [Fri83] a type of partial section which we call Fried section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We give a more  precise definition to additionally control the boundary orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Let P be a markov partition, u1, u2 be two distinct, primitive cyclic P-words, which start  with the same letter R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by γi the periodic orbit of φ corresponding to ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote  by Ni ∈ {1, 2} the degree of the realization of ui, given in Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Fix two integers k1, k2 ∈  N≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We write w = uk1  1 uk2  2  and denote by γw ⊂ M the periodic orbit corresponding to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If γi  has non orientable leaves, we suppose that kiNi is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Suppose that u1 is smaller than u2 for both the vertical and horizontal orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then  there exist m ≥ 1 and a partial section S satisfying ∂S = (k1N1γ1 + k2N2γ2) − mγw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Suppose that u1 <v u2 and u2 <h u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Then there exist m ≥ 1 and a partial section S  satisfying ∂S = −(k1N1γ1 + k2N2γ2) + mγw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Suppose that k1 and k2 are both even, then m = 1 holds in the two previous statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We call Fried section the partial section S given in the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It is illustrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We prove the first case, the second case is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We use the notation Ru, σu and φT u  from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' That is Ru is the set of elements x ∈ ∂−R having a short itinerary along u, σu  is the map Ru → R corresponding to pushing along the flow along the itinerary u, and T u is  the time we push along φ to go from x to σu(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We write vi = uki  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by P the 4-gon given by [0, 1]2, equip with the anti-clockwise  orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By assumption, either γi has orientable leaves, or ki is even, or Ni = 2 (and πP(ui)  lies in the boundary of the Markov partition P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So for j ∈ {1, 2}, πP(v1v2) = σv2(πP(v2v1))  and πP(v2v1) = σv1(πP(v1v2)) are on the same side relatively to πP(uj), for the horizontal  and the vertical order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In particular πP(v1v2) and πP(v2v1) are in the Markov sub-rectangle  45  of ∂−R which admits πP(u1) and πP(u2) as opposite corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore, we can find an immer-  sion f : P → ∂−R ⊂ M such that: (0, 0), (1, 0), (1, 1) and (0, 1) are respectively sent to πP(u1), πP(v1v2), πP(u2), πP(v2v1),  the restrictions of f to each edge of P is transverse to the stable and unstable foliations, σv1 ◦ f(t, 0) = f(0, t) for every t ∈ [0, 1], σv2 ◦ (t, 1) = f(1, t) for every t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  In the 3rd point, f(t, 0) lies in the smaller horizontal rectangle containing f(0, 0) = πP(v1)  and f(1, 0) = πP(v1v2), so f(t, 0) lies in the horizontal sub-rectangle Rv1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore σ|v1|◦f(t, 0)  is well-defined, and similarly for the 4th point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by Q the blow up of P along its four corners, and byπQ : Q → P be the blow-  down projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We still denote by (t, 0) and (0, t) the points in ∂Q corresponding to lifts  of (t, 0), (0, t) ∈ ∂P, for t ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by (0+, 0) and (0, 0+) the limits in ∂Q of (t, 0)  and (0, t) when t goes to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take τ : Q → R a smooth map, and consider the map g : Q → M  given by:  g(x) = φτ(x)(f ◦ πQ(x))  We choose the map τ on the edges corresponding to [0, 1] × 0 and 0 × [0, 1] so that τ(t, 0) =  T v1 ◦f(t, 0)+τ(0, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then we have g(t, 0) = g(0, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' That is the image of g is an immersed 8-gon,  whose edges corresponding to [0, 1] × 0 and 0 × [0, 1] coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Similarly we choose τ on [0, 1] × 1  and 1×[0, 1], so that the corresponding edges in g(Q) coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We additionally choose τ so that  the surface obtained by gluing the coinciding edges, is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by S that surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The surface S is constructed by pushing im f ⊂ ∂−R along the flow, so it is transverse to the  flow in its interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  It is in fact positively transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Indeed we have πP(u1) <v πP(u2),  so πP(v1v2) <v πP(v2v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Similarly we have πP(v1v2) >h πP(v2v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  It follows that the  curve f(∂[0, 1]2) is going anti-clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  So f preserves the orientation and S is positively  transverse in its interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  By construction, S has some potential boundaries only on the images by g of the blow-up  corners of Q, which are sent in the orbits going through the corners of f([0, 1]2), namely γ1, γ2  and γw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We need to determine the boundary of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Recall the definition of the realization of a primitive cyclic P-word u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We constructed in  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='1 a suspension flow φP for the homeomorphism σ: ΣP → ΣP, with the time function  corresponding to the first return time from ∂−R to ∂+R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' To a cyclic P-word u corresponds a  periodic orbit �γu of φP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The realization is the covering map, from the orbit �γu, to the φ-periodic  orbit of corresponding to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Here we additionally call the length of the realization of u, the  length of the orbit �γu (counted with multiplicity if u is a multiple of a primitive cyclic P-word).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We have τ(t, 0) = T v1 ◦ f(t, 0) + τ(0, t) for t ∈ (0, 1), so at the limit we have τ(0+, 0) =  T v1 ◦ f(0, 0) + τ(0, 0+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore T v1 ◦ f(0, 0) is the length of the boundary component of S  corresponding to the blow-up of the corner (0, 0) (view as an orbit traveled multiple times).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The time T v1 ◦ f(0, 0) is equal to k1 times the length of the realization of u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So the boundary  component of S in γ1 has multiplicity k1N1 ̸= 0 (recall that the realization of ui has degree Ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Therefore we can choose τ so that S is additionally immersed on the boundary component cor-  responding to the edge [(0, 0+), (0+, 0)] ⊂ ∂Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Similarly we can choose τ so that S is everywhere  immersed, the boundary component in γ2 has multiplicity k2N2, and the boundary component  in γw has negative multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  To be more specific, the length of the boundary of S in γw, is given by −T v1 ◦ f(1, 0) − T v2 ◦  f(0, 1) = −T v1v2 ◦ f(1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Suppose that k1 and k2 are both even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then w = uk1  1 uk2  2  is not the  46  ξ1  ξ2  ξ  L++(ξ)  L−−(ξ)  Ls  +(ξ1)  Lu  +(ξ1)  Figure 6: Lozenge with corners ξ1 and ξ2 on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The two lozenges with a corner in ξ on the  right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' On the right, the flow is R-covered and positively skewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  power of a smaller cyclic P-word (see Lemma 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Additionally γw is not in the boundary of P  (since it intersects the interior of the Markov partition ∂−R in f(1, 0)), so the realization of w is  of degree one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By definition of the realization, the length of the realization of w is T w ◦ f(1, 0),  so the multiplicity of S in γw is -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='2  Birkhoff section with one boundary  We prove Theorem E: when the flow is positively skewed, it admits a Birkhoff section with  only one boundary component, with multiplicity one and the corresponding periodic orbit lies  in orientable stable and unstable leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For that, we follow the proof of Theorem 61 given  in [ABM22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  A key notion to prove the Theorem are lozenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Recall that (Pφ, Ls, Lu) is the bi-foliated  plane of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Fix a pair of orientations on Ls and Lu so that the orientation on Ls plus the  orientation on Lu gives the orientation on Pφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For x ∈ Pφ, we denote by Ls  +(x) the open half  leaves of x (excluding x), so that the orientation goes from x to Ls  +(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We define similarly  Ls  −(x) = Ls(x) \\ (Ls  +(x) ∪ {x}), Lu  +(x) and Lu  −(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For ξ ∈ Pφ and σ, ǫ ∈ {−, +}, we define the  quadrant Qσǫ(ξ) as the set of points ζ ∈ Pφ satisfying Ls  σ(ξ) ∩ Lu(ζ) ̸= ∅ and Lu  ǫ (ξ) ∩ Ls(ζ) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  A lozenge L ⊂ Pφ is a subset of the form Q++(ξ1) = Q−−(ξ2) or Q+−(ξ1) = Q−+(ξ2), for  two points ξ1, ξ2 ∈ Pφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In this article, only the lozenges of the form Q++(ξ1) = Q−−(ξ2) are  used, which we call positive lozenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The points ξ1 and ξ2 are respectively called the corners  (−, −) and (+, +) of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We use lozenges later to construct positive Birkhoff sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  When the flow is R-covered and positively skewed, each point ξ ∈ Pφ bounds two positive  lozenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by L++(ξ) the lozenge for which ξ is a corner (−, −), and L−−(ξ) the other  lozenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Let K ⊂ M be a compact subset of the orbit space Pφ such that for each point x ∈ M, there  exists a lift �x ∈ �  M whose image πPφ(�x) is in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For example, one can take a big compact ball  in �  M and take its image in Pφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Every point in K admits an open neighborhood lying inside  one lozenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So by compactness, there exists a finite family of lozenges (Li)i∈I covering K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  In [ABM22], these lozenges are used to build a positive Birkhoff section, whose boundary is  the union of the periodic orbits corresponding to the corners of the lozenges Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We prove the  theorem above by choosing carefully the family of lozenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof of Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by K and (Li) the compact and lozenges introduced in the  previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  One can take the family (Li)i so that there exist open subsets (Ui)i∈I  47  with U i ⊂ Li such that K is included in ∪iUi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by �  M the orientation bundle covering and �φ the lifted flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We also denote  by π�  M : �  M → �  M the universal covering map of �  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take ǫ > 0 small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since the flow φ is Anosov and transitive, �φ is also Anosov and transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  So by using the closing Lemma, there exists a periodic orbit �γ of �φ such that any point �  M  is a distance at most ǫ from �γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If ǫ is chosen small enough, one can find for every i ∈ I, a  point ξi ∈ πPφ(π−1  �  M (�γ)) close to the corner −, − of the lozenges Li, so that L++(ξi) contains U i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by g1 = 1, g2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' , gn some elements in π1(�  M) satisfying gi ∗ ξ1 = ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Also denote by ξ′  i  the corner +, + of the lozenge L++(ξi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since �φ has orientable stable and unstable foliations,  the lozenge L++(ξi) is equal to gi ∗ L++(ξ1), so gi ∗ ξ′  1 = ξ′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by γ and δ the periodic orbits in M corresponding to the points ξ1 and ξ′  1 in M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The  proof of the implication ⇐ of [ABM22, Theorem 1] gives a positive Birkhoff section S1 bounded  by γ and by γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take a Markov partition P of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Recall that �γ is ǫ-dense in �  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Up to taking ǫ > 0 smaller,  we can suppose that the orbits γ and γ′ intersect a same Markov rectangle ∂−R of P, in two  points p, q satisfying p <v q and q <h p, that is p is bellow and on the right of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by n and m the multiplicity of S1 along γ and γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' According to Lemma 62, there  exist a partial section S2 and a periodic orbit δ such that ∂S2 = δ − 2nγ − 2mγ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' According to  Proposition 60, there exists a Birkhoff section S3 relatively homologous to 2[S1]+[S2] in H2(M, γ∪  γ′ ∪ δ, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The boundary of 2[S1] + [S2] is equal to δ, with multiplicity one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence S3 has only  one boundary component in δ, with multiplicity one, so it is a positive Birkhoff section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Notice that δ is homologous to 2(nγ + mγ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Recall that the map sign: π1(M) → {−1, 1} is  a group morphism, which measures the orientability of the stable and unstable foliations, on a  neighborhood of a simple closed loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since H1(M, R) is the Abelianization of π1(M) and {−1, 1}  is Abelian, it factorizes into a group morphism �  sign: H1(M, R) → {−1, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So �  sign(δ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By  definition of the function sign, the stable and unstable leaves of δ are orientable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='3  Birkhoff section with given boundary  We prove Theorem G: given Γ, there exists a Birkhoff section bounding NΓ for some N ∈ N, if  and only if LebΓ has positive linking numbers with all the measures in M0  p(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' As a corollary,  we obtain a second proof of Barbot’s Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Given an oriented compact 3-manifold N and a flow ψ on N, a global section of ψ is an  oriented compact surface S ⊂ N which is positively transverse to the flow, with ∂S ⊂ ∂N and  intersecting all orbits of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows from compactness that S intersects every orbit arc of  length T for some T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We start by constructing a Birkhoff section from a global section on a blow up manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take ∆ ⊂ M a finite union of periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by M∆ the blow up manifold of M  along ∆, by π∆ : M∆ → M the blow down projection, and by φ∆ the lifted flow on M∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let S ⊂ M∆ a smooth global section of φ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Then there exist a smooth func-  tion T : S → R satisfying that the image π∆ ◦ φT (S) is a Birkhoff section of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In this proof, all isotopies are isotopies along the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' There are two types of boundary  components for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The curves in ∂S ∩ Tγ, for some γ ⊂ ∆, which are homologous to a fiber of  Tγ → γ, and the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We call these boundary components of type 1 and type 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We need to  prove that up to an isotopy on S, the projection π∆ restricts to a nice map on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Here ’nice  map’ means that for each boundary component δ ⊂ Tγ of type 1, δ is sent to a single point in γ  and the image of S is smooth close to that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' And for each boundary component δ of type  2, π∆ restricts to an immersion close to δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then the image of S is a Birkhoff section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  48  Take a boundary component δ ⊂ ∂S∩Tγ of type 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then δ is isotopic to one fiber (see [Fri82]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  In fact there exists a global isotopy of Tγ sending ∂S ∩ Tγ to a finite union of fibers of Tγ → γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We can extend the global isotopy on Tγ to a global isotopy on M∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by S1 the image  of the surface S by the global isotopy along the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We can take the global isotopy so that  π∆(S1) is an embedded and smooth surface, close to the orbits of type 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take a boundary component δ ⊂ ∂S1 ∩ Tγ of type 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Then similarly δ is isotopic to a  smooth curve δ′ in Tγ, transverse to the flow, and transverse to the foliation on Tγ by fibers of  Tγ → γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Similarly there exists a global isotopy on M∆, whose image of S1 is a surface S2, and  which satisfies that π∆(S2) is immersed close to the orbits of type 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then π∆(S2) is a Birkhoff  section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Proof of Theorem G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let Γ be a null-homologous invariant 1-cycle with integer coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by ∆ the support of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Suppose first that there exists a Birkhoff section S bounding NΓ  with N ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We lift S to a smooth surface S∗ ⊂ M∆ whose boundary is inside ∂M∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Up to an  isotopy along the flow for S, we can suppose that there exists ǫ > 0 for which there is no arc of  length 2ǫ from S∗ to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In particular S∗ is positively transverse to φ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a non-decreasing  smooth map h: [−ǫ, ǫ] → R such that h ≡ −1/2 close to −ǫ, h ≡ 1/2 close to ǫ,  Then define a smooth map f : M∆ → R/Z by:  f(x) =  h(t)  (mod 1)  when φ∆  −t(x) lies in S∗ for some t ∈ [−ǫ, ǫ]  1/2  (mod 1)  otherwise  By construction we have (π∆)∗(∂P∆[df]) = NΓ and ι(X∆)df ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence for any positive  measure µ ∈ M(φ∆) with [π∗  ∆µ]φ = 0, we have:  linkf  φ(LebΓ, π∗  ∆µ) = 1  N  �  M∆  ι(X∆)dfdµ  = 1  N  �  φ∆  [−ǫ,ǫ](S∗)  ι(X∆)dfdµ  = 1  N  �  S∗  �� ǫ  −ǫ  ι(X∆)df ◦ φt(x)dt  �  dµ⊥(x)  = 1  N  �  S∗ (h(ǫ) − h(−ǫ)) dµ⊥(x)  = 1  N µ⊥(S∗) > 0  So LebΓ has only positive linking numbers with M0  p(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Suppose now that LebΓ has a  positive linking number with any null-cohomologous invariant probability measure of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Fix  an element ω1 ∈ H1(M∆, R) such that (π∆)∗(∂P∆(ω1)) is equal to [Γ] ∈ H1(∆, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We can  choose ω1 ∈ H2(M∆, Q) with rational coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by F ⊂ H2(M, R) × R be the set {([π∗  ∆µ]φ, [µ]φ∆ · ω1), µ ∈ Mp(φ∆)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By assump-  tion, we have [µ]φ∆ · ω1 = linkφ(LebΓ, π∗  ∆µ) > 0 for all measure µ ∈ M0  p(φ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore the  intersection F ∩ (0 × (−∞, 0]) is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' According to Lemma 59, there exists η > 0 and a linear  map f : H2(M, R) → R satisfying that for all µ ∈ Mp(φ∆), we have [µ]φ∆ · ω1 ≥ f([π∗  ∆µ]φ) + η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  By Poincar´e duality, f corresponds to an element ω2 ∈ H1(M, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since Mp(φ∆) is compact,  we can perturbate f so that ω2 has rational coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take an integer N ∈ N>0 satisfying  that N(ω1 − π∗  ∆ω2) have integer coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then for all measures µ ∈ Mp(φ∆), we have:  [µ]φ∆ · N(ω1 − π∗  ∆ω2) ≥ η > 0  49  It follows from Schwartzman-Fried-Sullivan Theory (see [Fri82, Sul76]), that φ∆ admits a  global section S whose relative homology class [S] ∈ H2(M∆, ∂M∆, R) is dual to N(ω1 − π∗  ∆ω2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  According to Lemma 63, the global section can be taken so that π∆(S) is a Birkhoff section of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Additionally the boundary of π∆(S) is homologous to (π∆)∗(∂P∆(N(ω1 − π∗  ∆ω2))) = N[Γ] ∈  H1(∆, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore NΓ bounds a Birkhoff section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  We should discuss a particular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' When there do not exist any null-cohomologous invariant  probability measure, Sullivan [Sul76] proved that the flow admits a global section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since the  flow is Anosov, the first return map for that global section is conjugated to a linear Anosov  diffeomorphism on the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In that case, every null-homologous invariant 1-cycle, with integer  coefficient, bounds a Birkhoff section (see [Mar21, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence Theorem G is a type  of generalization of this suspension case, where we have less control over the multiplicity at the  boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof of the remaining equivalences in the Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We prove the implications in the first colum  of the table: cell 4 =⇒ cell 2 ⇐⇒ cell 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Suppose that there exists ψ orbit equivalent to φ, which preserves a contact form α with  α ∧ dα > 0 (cell 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then according to Theorem D, the ψ-invariant probability measure induced  by α ∧ dα has the Reeb-like property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The Reeb-like property is invariant by orbit equivalence,  so there exists a φ-invariant probability measure with the Reeb-like property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Namely it is  null-cohomologous and has only positive linking number with null-cohomologous φ-invariant  probability measures (cell 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The implication cell 2 (there exists a positive Birkhoff section) =⇒ cell 3 follows from the  above discussion (Theorem G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Suppose that there exists a measure µ ∈ M0  p(φ) with the Reeb-like property (cell 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Accord-  ing to Lemma 14, there exists a sequence of null-homologous invariant 1-cycles Γn, with rational  coefficients, so that LebΓn converges toward µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By continuity of the linking number, for large n,  LebΓn has positive linking numbers with measures in M0  p(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows from Theorem G that  there exists a positive Birkhoff section (cell 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  6  Reeb-like properties for positively skewed Anosov flow  Gibbs measures are a family of invariant probability measures which satisfy similar properties  than invariant smooth volume forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' On one hand a Gibbs measure corresponds to a volume  form in some other differential structure on the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' On the other hand Gibbs measures  offer some control, which we use to obtain the Reeb-like property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='1  Reeb-like Gibbs measure  We introduce some thermodynamic formalism we need later, more can be found in [Bow08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' A  H¨older continuous map p: M → R is call a H¨older potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by P(M) the set of  H¨older potentials on M, which we equip with the topology given by the infinity norm ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by hµ(φ) the measured entropy of φ relatively to an invariant probability measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Given a H¨older potential p: M → R the pressure of p is the real number:  P(p) =  sup  µ∈Mp(φ)  �  hµ(φ) +  �  M  pdµ  �  Any invariant probability measure µ for which the supremum is achieved is call a Gibbs  measure of p (or the equilibrium state of p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It is known that for Anosov flows, Gibbs measures  50  exist and are unique for H¨older potentials [Bow08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Additionally a Gibbs measure has no atom  and charges every non-empty open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given a H¨older potential p, we denote by µp its Gibbs  measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Theorem 64 (Sharp [Sha93]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let φ be an Anosov flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The φ is homologically full, then φ  admits a null-cohomologous Gibbs measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Sharp actually proves that four properties on Anosov flows are equivalent, two of them are  the ones cited above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Theorem 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let φ be an Anosov flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If φ is R-covered and positively skewed, then φ admits  a Gibbs measure satisfying the Reeb-like condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  To prove the theorem, we obtain some control on a Gibbs measure, and use the continuity  of µp in p to obtain implicitly the Gibbs measure in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The maps p ∈ P(M) �→ P(p) ∈ R and p ∈ P(M) �→ µp ∈ Mp(φ) are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The pressure is continuous as supremum of 1-Lipchitz maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take (pn)n a sequence of  H¨older potentials converging toward a H¨older potential p∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  By compactness of Mp(φ), the  sequence µpn accumulates to a measure ν ∈ Mp(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then we have:  ����  �  M  pndµpn −  �  M  p∞dν  ���� ≤  �  M  |pn − p∞| dµpn +  ����  �  M  p∞dµpn −  �  M  p∞dν  ����  ≤ ∥pn − p∞∥∞ +  ����  �  M  p∞dµpn −  �  M  p∞dν  ����  −−−−−→  n→+∞ 0  Since the entropy hµ(φ) is upper semi-continuous in µ, the entropy hν(φ) is not less than the  limit inf of the entropies hµpn (φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence we have:  P(p∞) = lim inf  n  P(pn)  = lim inf  n  �  hµpn (φ) +  �  M  pndµpn  �  ≤ hν(φ) +  �  M  p∞dν ≤ P(p∞)  Hence ν is the Gibbs measure of p∞, so the map p �→ µp is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Null-cohomologous Gibbs measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We first reprove Sharp Theorem, by constructing a  null-homologous Gibbs measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Our construction allows us to have more control on the linking  numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The proof goes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Lemma 68 gives a way to control the Gibbs measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given  a H¨older potential which charges mostly one periodic orbit, its Gibbs measure charge mostly a  neighborhood of that orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We apply this lemma on enough periodic orbits, to obtain a better  control on the Gibbs measure’s cohomology class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' To go from one to several orbits, we need a  topological property, stated in Lemma 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  51  Here we suppose that the flow φ is homologically full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by n the dimension of H1(M, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We construct a map f : Rn −→ P(M)  [µ·]φ  −−−→ H1(M, R) ≃ Rn, such that for r large enough, f has  no zero on the sphere of radius r, and induces a map ∂B(0, r) → Sn−1 of degree one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Fix a family of periodic orbits γ+  1 , · · · , γ+  p , such that ([γ+  1 ], · · · , [γ+  p ]) is a basis of H1(M, R)  (see [PP86] for the existence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' According to Lemma 3, for each index i there exists a periodic  orbit γ−  i , homologous to −[γ+  i ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take two families of closed neighborhoods U +  i  and U −  i  of γ+  i  and γ−  i , which we take pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then fix H¨older potentials g+  i and g−  i such that gǫ  i ≡ 1  on γǫ  i and gǫ  i ≡ 0 outside U ǫ  i , for ǫ ∈ {+, −}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Define the map p: Rn → P(M) given by:  p(x) =  �  1≤i≤n  xi>0  xi · g+  i +  �  1≤i≤n  xi<0  |xi| · g−  i  For each x ∈ Rn, the map p(x) is a H¨older potential, so it admits a Gibbs measure µp(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take  a basis (ωi)i of H1(M, R) which is Poincar´e dual to the basis ([γ+  i ])i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' More precisely ωi · [γ+  j ]  equals 1 when i = j, and it equals 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We define the map fp: Rn → Rn defined by  fp(x) =  �  [µp(x)]φ ∧ ω1, · · · , [µp(x)]φ ∧ ωn  �  Proposition 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' There exists r > 0 such that f −1  p (0) ⊂ [−r, r]n, and such that the map  (x ∈ ∂[−r, r]n) �→  �  fp(x)  ∥fp(x)∥2 ∈ Sn−1�  is of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Additionally there exists x ∈ (−r, r)n satis-  fying fp(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Theorem 64 is a consequence of the proposition, since fp(x) = 0 if and only if [µp(x)]φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  To prove the proposition, we first need some control on the Gibbs measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by htop the  topological entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let γ be a periodic orbit of the flow, C2 > C1 > 0 be two real numbers, U ⊂ M  be a neighborhood of γ, and p be a H¨older potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Suppose that p ≤ C2 with equality on γ,  and p ≤ C1 outside U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then we have:  µp(M \\ U) ≤ htop(φ)  C2 − C1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By definition a Gibbs measure maximizes the pressure, so we have:  C2µp(M \\ U) + C2µp(U) = C2 =  �  M  pd Lebγ  ≤  �  M  pd Lebγ +hLebγ(φ)  ≤  �  M  pdµp + hµp(φ)  ≤ C1µp(M \\ U) + C2µp(U) + htop(φ)  It follows that we have (C2 − C1)µp(M \\ U) ≤ htop(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  For x ∈ Rn, we denote by  xi its ith coordinates, such that x = (x1, · · · , xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Similarly  for a map f with values in Rn, we write f = (f1, · · · , fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For C ∈ R+ and x ∈ Rn, we  define Cx = {i, |xi| > C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Also denote by sign(x) ∈ {−1, 1} the sign of a real number x ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take 0 < C < 1 and ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let f : [−1, 1]n → Rn be a continuous map such that  for all x ∈ ∂[−1, 1]n the following holds:  52 �  i∈Cx |fi(x)| > nǫ,  for all i ∈ Cx, sign(xi)fi(x) ≥ −ǫ,  for all j ̸∈ Cx, |fj(x)| ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Then the map G: ∂[−1, 1]n → ∂[−1, 1]n given by G(x) =  fp(x)  ∥fp(x)∥∞ is well-defined and of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  In particular the exists x in (−1, 1)n with fp(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For x in ∂[−1, 1]n, we have n∥fp(x)∥∞ ≥ �  i |fi(x)| > nǫ so ∥fp(x)∥∞ > ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence G(x)  is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For t ∈ R+, we define the continuous map gt : R → R by:  gt(x) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  0  |x| ≤ t  x − t  if x > t  x + t  if x < −t  and the continuous map gn  t : Rn → Rn, (x1, · · · , xn) �→ (gt(x1), · · · , gt(xn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Then for all x  in ∂[−1, 1]n, the following holds:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' for all t ∈ [0, ǫ], ∥gn  t ◦ fp(x)∥∞ > 0,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' for all i ∈ Cx, sign(xi) · gn  ǫ ◦ fi(x) ≥ 0,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' for all j ̸∈ Cx, gn  ǫ ◦ fj(x) = 0,  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' for all i, sign(xi) · gn  ǫ ◦ fi(x) ≥ 0,  The last item is a consequence of the two previous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by S the unit sphere of Rn for  the 1-norm ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='∥1, and by N1, N∞ : Rn\\{0} → S the normalization maps given by N1(x) = x/∥x∥1  and N∞(x) = x/∥x∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Consider the homotopy ht : S → S defined by:  ht(x) = N1 ◦ gn  t ◦ f ◦ N∞(x)  The sphere S admits a natural triangulation, whose maximal dimensional faces are given by  the sets {x ∈ S, ∀i, sixi ≥ 0} for some s1, · · · , sn ∈ {−1, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The points 2 and 3 imply that any  face F of S (of any dimension) is let invariant by hǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence using a convex interpolation, hǫ is  homotopic to the identity on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since the map IdS is of degree one, hǫ is also of degree 1, and  so is h0 since the degree is invariant by homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence x ∈ ∂[−1, 1]n �→ x/∥f∥2 ∈ Sn−1 ⊂ Rn  is of degree one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Proof of Proposition 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Recall that (ωi)i is the dual base to ([γ+  i ])i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence we can take a closed  1-form αi whose cohomology class is ωi and such that the following holds: ι(X)αi ≡ 0 on U +  j ∪ U −  j whenever j ̸= i, ι(X)αi is positive on U +  i  and negative on U −  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We fix D > 0 such that for all i, |ι(X)αi| is larger than D on U +  i ∪ U −  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take ǫ > 0, r > 0, and  η = htop(φ)  r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We suppose ǫ small and r large so that we have D(1−η) > 2nǫ and η∥ι(X)α∥∞ < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Fix a point x in Rn satisfying ∥x∥∞ > r and take 0 < B < ∥x∥∞ − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We start by estimating  the measure µp(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We define the sets B+ = {1 ≤ i ≤ n, |xi| > B} and the set:  Vx =  �  i∈B+  U sign(xi)  i  53  Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We have µp(x)(M \\ Vx) < η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof of Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The function p(x) = �  i |xi| · gsign(xi)  i  is bounded by ∥x∥∞ on all M, since  the maps g±  i  have disjoint supports and ∥g±  i ∥∞ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let j be such that |xj| = ∥x∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then  for z ∈ γsign(xj)  j  , we have gsign(xj)  j  = 1 and p(x)(z) = ∥x∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For z ∈ M \\ Vx, the point z is in at most one set U ±  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If z belongs to U ±  j , then j is not in B+,  so |xj| ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence we have p(x)(z) = �  i |xi| · g±  i (z) ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows from Lemma 68 that:  µp(x)(M \\ Vx) ≤  htop(φ)  ∥x∥∞ − B  < htop(φ)  r  < η  □  To estimate fp(x), we apply Lemma 69 on the map x ∈ [−1, 1]n �→ fp(rx), with the con-  stant C = B/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For that it is enough to prove the three following points for x ∈ [−r, r]n:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' for j ̸∈ B+,  ��[µp(x)]φ · [αj]  �� ≤ ǫ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' for j ∈ B+, sign(xj)[µp(x)]φ · [αj] ≥ −ǫ  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  �  i∈B+  ��[µp(x)]φ · [αj]  �� > nǫ  Take j ̸∈ B+, so that |xj| ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We have ι(X)αj ≡ 0 on Vx and µp(x)(M \\ Vx) ≤ η, so:  ��[µp(x)]φ · [αj]  �� =  ����  �  M  ι(X)αjdµp(x)  ����  ≤  ����  �  Vx  ι(X)αjdµp(x)  ���� +  �����  �  M\\Vx  ι(X)αjdµp(x)  �����  ≤ η∥ι(X)α∥∞ ≤ ǫ  Similarly for j ∈ B+, with |xj| > B, we have:  [µp(x)]φ · [αj] =  �  M  ι(X)αjdµp(x)  =  �  U  sign(xj )  j  ι(X)αjdµp(x) +  �  Vx\\U  sign(xj )  j  ι(X)αjdµp(x)  +  �  M\\Vx  ι(X)αjdµp(x)  By hypothesis, the sign of ι(X)αj on U sign(xj)  j  is sign(xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Additionally ι(X)αj is zero  on Vx \\ U sign(xj)  j  and  ���  �  M\\Vx ι(X)αjdµp(x)  ��� ≤ η∥ι(X)α∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So one has:  sign(xj) · ι(X)µp(x)(αj) ≥ −η∥ι(X)α∥∞ ≥ −ǫ  54  Finally we have:  �  i∈B+  ��ι(X)µp(x)(αj)  �� =  �  i∈B+  ����  �  M  ι(X)αjdµp(x)  ����  ≥  �  i∈B+  �����  �  Vx  ι(X)αjdµp(x)  ���� −  �����  �  M\\Vx  ι(X)αjdµp(x)  �����  �  ≥  �  i∈B+  �  Dµp(x)  �  U sign(xi)  i  �  − ∥ι(X)α∥∞µp(x)(M \\ Vx)  �  ≥ D(1 − η) − nη∥ι(X)α∥∞  > nǫ  ■  Gibbs measures with positive linking number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We deform the Gibbs measure obtained  in Proposition 67 to find another Gibbs measure which has a positive linking number with every  null-cohomologous invariant probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  According to Lemma E, there exists a Birkhoff section bounded by a single orbit δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The  measure Lebδ satisfies the Reeb-like property according to Theorem G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We extend the map p  given in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take the sets U ±  i  disjoint from δ, and take U0 to be a compact  neighborhood of δ, disjoints from each set U ±  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Also take a map g0 : M → [0, 1] such that g0 ≡ 1  on δ and g0 ≡ 0 outside U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We additionally suppose that g0 < 1 outside δ and g±  i  < 1  outside γ±  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We define the maps q: Rn × R → P(M) and fq : Rn × R → Rn by:  q(x, z) = p(x) + z · g0(x)  and  fq(x, t) = ([µq(x,t)]φ ∧ ωi)i  Lemma 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' There exists a sequence (xn, zn)n in Rn × R such that ∥(xn, zn)∥∞ goes to +∞,  and such that for all n, we have zn ≥ 0 and [µq(xn,zn)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any sequence (xn, zn)n given by the previous lemma, the sequence µq(xn,zn)  converges toward  Lebδ  Lebδ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof of Lemma 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Suppose that all points (x, z) with x ≥ 0 and with [µq(x,z)] = 0 remain in a  bounded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take r > 0 large, and define the map h: ∂[−r, r]n × [0, 1] → Sn−1 by h′(x, t) =  fq((1 − t)x, rt) and h(x, t) =  h′(x,t)  ∥h′(x,t)∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For r > 0 large enough, h is well-defined, continuous  and is a homotopy between a degree one map (see Lemma 67) and a constant map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' This is not  possible, concluding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Proof of Lemma 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Consider a sequence (xn, zn) as in Lemma 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By compactness of Mp(φ),  the sequence µq(xn,zn) accumulates to some measure ν ∈ Mp(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Up to extraction, we can  suppose that xn  i have the same sign for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Up to reversing the roles of γ+  i  and γ−  i , we can  suppose that xn  i is non-negative for all i and all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We should prove that ν is supported by δ ∪i γ+  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take ǫ > 0 smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Define the open  subset Aǫ ⊂ M as the set of point x ∈ M satisfying g0(x) < 1−ǫ and g+  i < 1−ǫ for all i ∈ �1, n�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  55  The union of ∪ǫ<1Aǫ is equal to M \\ (δ ∪i γ+  i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by Nn = ∥(xn, zn)∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' According to  Lemma 68, we have:  µq(xn,zn)(Aǫ) ≤  htop(φ)  Nn − Nn(1 − ǫ)  ≤ htop(φ)  Nnǫ  −−−−−→  n→+∞ 0  Since Aǫ is open, at the limit we have ν(Aǫ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So ν(M \\ (δ ∪i γ+  i )) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence the measure ν  is of the form a0 Lebδ + �  i ai Lebγ+  i for some non-negative numbers ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since the cohomology  class of a measure is continuous in the measure, we have [ν]φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' According to Lemma 13,  the Poincar´e dual of [ν]φ ∈ H2(M, R) is a0[δ] + �  i ai[γ+  i ] ∈ H1(M, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The orbit δ is null-  homologous and [ν]φ = 0, so one has �  i ai[γ+  i ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The family ([γ+  i ])1≤i≤n is a basis of H1(M, R),  so the numbers ai for i ≥ 1 are null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Hence ν is the unique invariant probability measure  supported on δ, that is  Lebδ  Lebδ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The sequence (µq(xn,zn))n in Mp(φ) has a unique accumulating point, which is  Lebδ  Lebδ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Therefore it converges toward that measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Proof of Theorem 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' According to Lemmas 70 and 71, there exists a sequence of null-cohomologous  Gibbs measure µq(xn,zn) which accumulates toward  Lebδ  Lebδ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By construction, Lebδ has positive  linking number with all invariant probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The linking number is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So it follows from Corollary 22, that for n large enough, the  measure µq(xn,zn) has positive linking numbers with all null-cohomologous invariant probability  measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Namely µq(xn,zn) have the Reeb-like property for n large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='2  C1+H differential structure  Fix a H¨older potential p: M → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We construct a class C1+H differential structure on M,  and a reparametrization of φ, which is of class C1+H and Anosov for that differential structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Additionally the Gibbs measure of φ, associated to p, is induced by a H¨older volume form for that  differential structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' This as already be done by E Cawley [Caw91] for Anosov diffeomorphisms  and done by Asaoka [Asa07] for codimension one Anosov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We need an additional control on the invariant volume form, so we use the local product of  Gibbs measures given by Haydn [Hay94], and follow Asaoka’s work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The end of the section gives  some control on the invariant measure, in terms of positive linking number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Theorem 72 (Haydn [Hay94]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Fix a H¨older potential p of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' There exists positive measures µs  p  and µu  p on the strong stable and unstable foliations (that is a measure on each leaf) satisfying  the properties:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The measure µp is locally equal to a positive multiple of the product measure µs  p⊗µu  p ⊗Lebφ,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  φ∗  t µs  p = e−Ftµs  p and φ∗  t µu  p = eFtµu  p, where Ft(x) = � t  0(f ◦ φs(x) − P(f))ds,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For any points x, x′ such that x′ ∈ Fss(x), denote the holonomy map h: Fss  l (x) → Fss  l (x′)  from x to x′ along the weak unstable foliation, we have µs  p = eωh∗µs  p, where ω is the  H¨older function given by ω =  � +∞  0  (f ◦ φt ◦ h − f ◦ φt) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' A similar statement holds for  the measure µu  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  56  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The measure µs  p, restricted on a strong stable leaf, is finite on compact segments and positive  on non-empty open subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Remark 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The measure µs  p is a measure on each leaves of the strong stable foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We can  also interpret it as a pseudo-transverse measure to the flow, on each weak stable leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' That is  given a point x ∈ M, I ⊂ Fss(x) a strong stable segment and a continuous function T : I → R,  the measure µs  p induces the measure (φT )∗ �  eFT µs  p  �  on the curve φT (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We equip the orbit space with a family of measures on the stable and unstable foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For p ∈ �  M, we denote ξp the point πPφ(p), and by θs  p : Fss(�π(p)) → Ls(ξp) the lift/projection  map, defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The maps �π : �  M → M and πPφ : �  M → Pφ induce two homeomor-  phisms �Fss(p) → Fss(�π(p)) and �  Fss(p) → Ls(πPφ(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We define θs  p as the product of the  corresponding homeomorphisms Fss(�π(p))  �π−1  −−→ �Fss(p) → Ls(ξp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Then define µs  p = (θs  p)∗µs  p, which is a measure on Ls(ξp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The measure µs  p induces a transverse  measure to the unstable foliation, on an open subset of Pφ obtained as the union of unstable  leaves intersecting Ls(ξp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Thanks to Theorem 72, the transverse measure induced by µs  p is  multiplied by a H¨older continuous function when p varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Fix an orientation on Ls(ξp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by  � b  a µs  p the integral of µs  p on the unstable seg-  ment [a, b] ⊂ Ls(ξp), with a negative sign if the orientation on Ls(ξp) goes from b to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 74 (Asaoka, Lemma 42 in [Asa07]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map ηs  p : Ls(Q(p)) → R, ξ �→  � ξ  ξp µs  p is a  bi-H¨older homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 75 (Asaoka, Lemma 43 in [Asa07]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given a holonomy map along the weak unstable  foliation hp,q : I ⊂ �Fss  l (p) → J ⊂ �Fss  l (q), the map ηs  q ◦ hp,q ◦ (ηs  p)−1 is of class C1+H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The two lemmas also hold for the measures on unstable leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Using these lemmas, we define  a C1+H differential structure on Pφ transverse to Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by A⊥  p the atlas on Pφ whose  charts are given by the maps (ηs  p, ηu  p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Thanks to the two lemmas, the transition maps are of  Class C1+H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So A⊥  p induces a C1+H differential structure on the surface Pφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' From the definition  follow the next lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The action π1(M) ↷ Pφ is of class C1+H for the atlas A⊥  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The Gibbs measure µp induces a transverse measure µ⊥  p on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It lifts to a measure on Pφ  which is invariant by the action π1(M) ↷ Pφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We also denote by µ⊥  p the corresponding measure  on Pφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The measure µ⊥  p on Pφ is induced by a H¨older area form on Pφ, for the differential  structure A⊥  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows from Theorem 72 that µ⊥  p is locally a product of ηs  p and ηs  q, multiplied by a  positive function f (corresponding to the functions in points 2 and 3 in the theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since  the foliation Fss and Fuu are H¨older for the initial differential structure (see  cite[Corollary  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='11]Fisher19), the function f is H¨older for the initial differential structure on Pφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' According  to Lemma 74, the function f is also H¨older for the atlas A⊥  p , so µ⊥  p is H¨older regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  From transverse atlas to atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Given a compact surface S ⊂ M (only continuously em-  bedded), topologically transverse to the flow, the atlas A⊥  p induces a C1+H differential structure  on S (using a lift of S to �  M and the projection to Pφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For that reason on view the atlas A⊥  p  on Pφ as a transverse atlas to φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  57  Proposition 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' There exist a C1+H atlas Ap on M and a continuous reparametrization ψ of  the flow φ which satisfy the following: ψ is of class C1+H for Ap,  for any C1+H compact surface S, topologically transverse to ψ, the C1+H differential struc-  tures on S induced by Ap and A⊥  p are the same,  if id: (M, φ) → (M, ψ) is seen as an orbit equivalence, the measure Θid(µp) is induced by  a H¨older volume form for Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Given a continuous and positive function f : M → R∗  +, there exists a unique flow ψf on M,  differentiable in the direction of the flow (for the initial smooth structure), and for which we  have f ∂ψf  t  ∂t = ∂φt  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The flow ψf can also be characterized by the following formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For every t ∈  R, we have:  φt(x) = ψ� t  0 f◦φs(x)ds(x)  We construct an atlas on M whose charts are given by flow boxes of the flow ψf for some func-  tion f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a compact surface S ⊂ M, supposed to be topologically embedded and transverse  to φ, and T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If the map a: S × [0, 1] → M defined by a(x, s) = ψf  T (x)(x) is an embedding,  the image of a is called a flow box for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Fix T > 0 small and take a finite family of compact surfaces (Si)1≤i≤n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We suppose that  the set φ[0,T ](Si) is a flow box for the surface Si for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We also suppose that the union of the  interior of the flow boxes φ[0,T/2](Si) is equal to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Notice that for any continuous function f : M → [1, 2], the set ψf  [0,T ](Si) is a flow box con-  taining φ[0,T/2](Si), and is included in φ[0,T ](Si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In particular these flow boxes cover M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We  denote by af  i : Si × [0, T ] → ψf  [0,T ](Si) the parametrization of the flow box for the flow ψf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' That  is af  i (x, s) = ψf  s (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The map af  i induces a C1+H differential structure on Bi, given as the image of the product  between A⊥  p and the standard smooth structure on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take two surfaces Si, Sj whose flow  boxes intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' There is a maximal set Ui,j ⊂ Si and a continuous function T f  i,j : Ui,j → R+,  so that for any point (x, s) ∈ Si × [0, 1] the point af  i (x, s) lies in Sj if and only if x is in Ui,j  and s = T f  i,j(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We call the map Ti,j a short return time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Suppose that for all i, j ∈ I, the map T f  i,j is of class C1+H for the atlas A⊥  p  on Ui,j ⊂ Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then all transition maps of the form (af  i )−1 ◦ af  j are of class C1+H where they are  well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In particular the set of charts (ai)i form a C1+H atlas on M, compatible with the  transverse atlas A⊥  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map (af  i )−1 ◦ af  j coincides with the map (x, s) �→ (ψf  Ti,j(x)(x), s + T f  i,j(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map  (x ∈ Ui,j ⊂ Si) �→ (ψf  Ti,j(x)(x) ∈ Sj) is C1+H, by definition of the C1+H differential structure  on Si and Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So the map (af  i )−1 ◦ af  j is as regular as T f  i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  We now prove the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The idea is to define locally f to make the short return times  of class C1+H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given one short return time, one can build by hand f to make that short return  time regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' To extend f globally, we build it piece by piece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' At each step, the short return  maps which are well-defined are regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' And we choose the pieces where we define f, so that it  is afterward possible to extend it by keeping short return times regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  58  Given i, j and a continuous function g : Ui,j ⊂ Si → [1, 2], consider the map f defined  on {φt(x), x ∈ Ui,j, t ∈ [0, T 1  i,j(x)] or [T 1  i,j(x), 0]} by f(φt(x)) = T 1  i,j(z)g(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For this function f,  T f  i,j is well-defined on Ui,j and coincides with g, so it is as regular as g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' To define a function f  guarantying the regularity of all maps T f  i,j, we define f inductively using this idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof of Proposition 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For all indexes i < j, up to removing a small open neighborhood  of Si ∩ Sj to Sj, we can suppose that the surface Si and Sj are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It can be done while  preserving that the interior of the flow boxes φ(0,T/2)(Int Sk) cover M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Indeed the parts of the  flow boxes we remove, are cover by either the flow boxes of Si or by the interior of the flow boxes  which cover Si ∩ Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then all surfaces Si are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For i, j ≤ n, we consider pairs of the form (x, t) ∈ Si × R satisfying t > 0, that φt(x) lies  in Sj, and that the orbit arc φ(0,t)(x) is disjoint from ∪k Int Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We say that (x, t) is an adjacent  pair for (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For m ≥ 0, we define Ai,j,m as the set of adjacent pairs (x, t) for (i, j), which  satisfy that the curve φ(0,t)(x) intersects ∪k∂Sk in exactly m points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The surfaces Sk are called  in between surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We suppose that the surfaces (Sk)k are in general positions, so that the  adherence of the sets Ai,j,m = ∅ are compact manifolds of dimension 2 − m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In particular Ai,j,m  is finite for m = 2, and empty for m > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice that all points in M \\ ∪k Int Sk is on a unique  orbit arc of the form φ(0,t)(x), for an adjacent pair (x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  To prove that all short return times are C1+H, it is enough to prove that for every adjacent  pair (x, t) for (i, j), and every in between surfaces Sk for (x, t), the short return times T f  i,j and T f  i,k  are C1+H on a neighborhood of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Indeed all short return times are locally sums of the short  return times described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Now we construct f inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For each adjacent pair (x, t) ∈ Ai,j,2, choose a small flow  box neighborhood U ⊂ M of φ[0,t](x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by Sk1, Sk2 the two in between surfaces for the pair  (x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Choose the function fU : U → [1, 2], such that the short return times T f  i,j, T f  i,k1 and T f  i,k2  are well-defined and of class C1+H close to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice that the boundary of the 1-dimensional  sub-manifold Ai,j,1 is Ai,j,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a small neighborhood V of the set:  U ∪  �  i,j  \uf8eb  \uf8ed  �  (x,t)∈Ai,j,1  φ[0,t](x)  \uf8f6  \uf8f8  Up to taking U and V smaller, we can extend fU to a function fV : V → [1, 2] satisfying the  following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For every adjacent pair (x, t) ∈ ∪i,jAi,j,1 and in between surfaces Sk for (x, t), the  short return times T f  i,j, T f  i,k are well-defined and C1+H close to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then we can extend f on  all M so that it additionally satisfies that: for every adjacent pair (x, t) ∈ ∪i,jAi,j,0, the short  return time T f  i,j is well-defined and C1+H close to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by ψ = ψf the reparametrization of φ corresponding to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows from the above  discussion that the flow boxes for ψ, and the surfaces Sk, induce a C1+H atlas Ap, compatible  with the atlas A⊥  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The atlas Ap is constructed using flow boxes for ψ, so ψ is of class C1+H  for Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For a similar reason, Θid(µp) is H¨older regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Lemma 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' There exists T > 0 for which we have  � T  0 (p ◦ φs(x) − P(p))ds < 0 for all x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  This statement is already known for symbolic dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We give a proof using only the  definition of the pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a measure µ ∈ Mp(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By definition of the pressure, we have:  �  M  (p − P(p))dµ ≤ −hµ(φ) ≤ 0  59  The equality potential holds only when µ is the Gibbs measure of φ for the H¨older potential f,  in which case we have hµ(φ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence  �  M(p − P(p))dµ < 0 holds in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Suppose that for  all N ∈ N, there exists xN ∈ M for which we have � N  0 (p◦φs(xN)−P(p))ds ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by νn the  pull back of the Lebesgue measure on [0, T ], by the map t ∈ [0, N] �→ φt(xN) ∈ M, rescaled to be  a probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By compactness of the set of probability measures on M, the sequence νn  accumulates to some probability measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By hypothesis, we have  �  M(p − P(p))dνN ≥ 0 for  all N, so at the limit  �  M(p − P(p))dµ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a continuous function f : M → R and t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For any N ∈ N we have:  ����  �  M  (f ◦ φt − f)dνN  ���� =  �����  1  N  � N  0  (f ◦ φt+s(xN) − f ◦ φt(xN))ds  �����  =  ����  1  N  � t  0  (f ◦ φN+s(xN) − f ◦ φt(xN))ds  ����  ≤ 2t  N ∥f∥∞  Hence the integral  �  M(f ◦ φt − f)dνN converges toward  �  M(p ◦ φt − p)dµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that µ  is invariant by φ, which contradicts that  �  M(p − P(p))dµ is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore for some T > 0,  the integral  � T  0 (p ◦ φs(x) − P(p))ds is negative for all x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Lemma 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The flow ψ is Anosov for the atlas Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Write by Y = ∂ψt  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take m a continuous metric on M, for the atlas Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By construction  of the product atlas Ap, the foliation Fs is of class C1 for the atlas Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by Ecs the  invariant plane field tangent to Fs, for Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by | detm(dcsψt)| the absolute value of  the determinant of the restriction of dψt to Ecs, computed with the metric m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We first prove that | detm(dcsψt)| decreases exponentially in t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Call stable curve any  curve which remains in a single leaf of Fs, and topologically transverse to the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by µs  p  the measure on the leaves of Fss coming from the decomposition of the measure µp in a local  product (see Theorem 72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' As discuss previously, µs  p induces a measure on every stable curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We define a similar measure using the metric m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For a point p and a vector v ∈ TpM  tangent to Fs, we denote by ∥v∥⊥  m the minimum of ∥v + tY (p)∥m for t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For a C1 stable  curve γ : [0, 1] → M and A ⊂ [0, 1] measurable, define the measure νs  m(A) by:  νs  m(A) =  �  A  ∥γ′(t)∥⊥  m dt  It defines a measure νs  m on C1 stable curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Additionally it is of the Lebesgue type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By  definition of the atlas Ap, the measure µs  p is also of the Lebesgue type on C1 stable curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Therefore there exists a continuous function g : M → R, which satisfies µs  p = egνs  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 80 and Theorem 72 imply that the measure ψ∗  −tµs  p decreases exponentially in t > 0,  with uniform constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By compactness, g is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So ψ∗  −tνs  m also decreases exponentially  in t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore there exists A, B > 0 for which | detm(dcsψt)| ≤ Ae−Bt for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We use a cone argument to decompose Ecs as a sum of two invariants line bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a  bi-valued continuous section U of Ecs, so that for every p ∈ M, U(p) consist of two opposites  vectors not parallel to Y (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We can change the metric m so that (Y, U) gives an orthonormal  basis of each plane of Ecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take T > 0 satisfying Ae−T B < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by a: M → R, b: M → R+  two continuous maps satisfying dψT (U) = aY + bU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then we have:  |detm(dcsψT )| =  ����det  �  1  a  0  b  ����� = b  60  It follows that b < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take λ >  max a  1−max b and consider the cone filed C ⊂ Ecs defined for p ∈ M  by C(p) = {xU(p) + yY (p), |y| ≤ λ|x|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The image cone ψT (C) is the set of points of the form:  x′b(p)U(ψT (p)) + (y′ + x′a(p))Y (ψT (p))  with |y′| ≤ λ|x′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We claim that ψT (C) contains C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Indeed given (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' y) ̸= 0 with |y| ≤ λ|x|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' we  have:  xU(ψT (p)) + yY (ψT (p)) = x′b(p)U(ψT (p)) + (y′ + x′a(p))Y (ψT (p))  for x′ =  1  b(p)x and y′ = y − a(p)  b(p) x  Then we have:  |y′|  |x′| = |b(p)y − a(p)x|  |x|  ≤ b(p)|y|  |x| + a(p)  ≤ λ max b + max a < λ  Therefore the cone field C is invariant by ψ−T for large t > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' and we have a uniform expo-  nential expansion of ψ−T on C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' so by a classical argument of invariant cone fields,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' there exists  a continuous invariant line bundle Es ⊂ C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' disjoint from the image of Y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' which is exponen-  tially contracted by ψt for t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Similarly there exists an invariant line bundle Eu which is  exponentially contracted by ψt for t < 0, and with T M = Es ⊕ Eu ⊕ RY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that ψ is  Anosov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='3  Smoothing the flow  In this section φ is supposed homologically full, p is a H¨older potential so that µp is null-  cohomologous, ψ is a reparametrization of φ, of class C1+H and Anosov for the atlas Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We also  denote by Y = ∂ψt  ∂t the vector field generation ψ, and νp = cste·Θid(µp) the ψ-invariant probabil-  ity corresponding to µp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The measure νp is also null-cohomologous (see to Proposition 11), and  is induced by a H¨older volume form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' When φ is additionally R-covered and positively skewed,  we additionally suppose that νp satisfies the Reeb-like condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Plan of proof for Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Suppose φ to be a positively skewed R-covered Anosov flow in an  oriented closed 3-manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' According to Theorems 65, φ admits a H¨older potential p: M → R  whose Gibbs measure µp satisfies the Reeb-like condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='2, we construct a C1+H  atlas Ap on M and a reparametrization ψ of φ, which is of class C1+H and Anosov for the  atlas Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Additionally, the Gibbs measure νp of ψ, corresponding to µp, is given by a H¨older  volume form V for the atlas Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We follow Asaoka’s ideas [Asa07] to smooth the flow ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The atlas Ap can be extended in  a class C∞ atlas A∞  p , for which ψ is generated by a C1+H vector field Y (Lemma 84).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then  we approximate the H¨older volume form V , corresponding to νp, by smooth volume form Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The vector field Y can be approximated in the C1 topology, be a smooth vector field Yn which  preserve Vn (see Lemma 87, where V and Vn appear as eλV and eλnV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since Anosov flows  are C1 structurally stable, the vector field Yn generates an Anosov flow ψn, orbit equivalent to ψ  and to φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Additionally it preserves a smooth volume form Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Suppose that ι(Yn)Vn is null-cohomologous for n large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since the linking number is  continuous (see Corollary 22), for n large, the probability measure induced by Vn has positive  61  linking numbers with all null-cohomologous ψn-invariant probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence it satisfies  the Reeb-like property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows from Theorem D that ψn is a reparametrization of a Reeb-  Anosov flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence φ is orbit equivalent to a Reeb-Anosov flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  In general, ι(Yn)Vn is not null-cohomologous, but its cohomology class converges toward the  cohomology class of νp, that is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We perturbate Yn so that ι(Yn)Vn is null-cohomologous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For that we remark that given a smooth volume form W, we can find smooth vector fields Z  preserving W, so that ι(Z)W represent any element ω ∈ H2(M, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Additionally Z can be taken  with a C1 norm bounded by Cste(W)∥ω∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The idea is to replace Yn by Yn + Zn in the above  argument, for a suitable vector field Zn satisfying [ι(Zn)Vn] = −[ι(Yn)Vn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We need Yn+Zn to converges toward Y in the C1 topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We do not control the dependence  of the C1 norm of Zn on W = Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For that reason, we add a step before extending the atlas Ap  is a smooth atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We fix a subset U ⊂ M which we call generating family of tori (U corresponds  to ∪iUi in the coming definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  If U is small, we can choose the atlas A∞  p  so that the  measure νp is smooth inside U (see Lemma 83).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We approximate the volume form V by smooth  volume forms Vn, so that Vn and V coincide inside U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We approximate Y by vector fields Yn,  Yn preserves Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map H2(U, R) → H2(M, R) is surjective, so we can take a vector field Zn  as above, but supported inside U (see Lemma 82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since Zn is chosen for a fixed volume form V  inside U, its C1 norm converges toward zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then Yn + Zn converges toward Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows that  for n large enough, Yn + Zn generates a smooth Anosov flow, orbit equivalent to φ, and which  satisfies the Reeb-like property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In particular, it is a Reeb-Anosov flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  The proof of Theorem F follows the same plan, except for the consideration on the linking  number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by D2 the unit disc in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' A generating family of tori is a finite family of disjoint  compact subsets (Ui)1≤i≤n of M, such that there exists some homeomorphisms fi : D2×S1 → Ui,  satisfying that the homology classes of the curves fi(0 × S1) form a basis of H1(M, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' When M  is a R-homology sphere, generating families of tori are empty families by conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Given a  generating family of tori (Ui)i, a generating sub-family of torus is a generating family of tori (Vi)i  with Vi ⊂ Ui for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The purpure of generating families of tori is to have a subset U ⊂ M for which H1(U, R) →  H1(M, R) is surjective, but which is small enough so that we can extend the C1+H structure Ap  on M, in a smooth structure for which the flow and the invariant volume form are smooth  inside U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by Λ∞(TM) the set of C∞ vector fields on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a norm ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='∥ on H1(M, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let M be a smooth closed 3-manifold with a generating family of tori (Ui)1≤i≤n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Fix a volume form V on M, supposed to be smooth on each Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then there exist A > 0 and a  continuous map v: H2(M, R) → Λ∞(TM), such that for each x ∈ H2(M, R), the following holds:  v(x) ≡ 0 outside ∪iUi,  v(x) preserves V , that is Lv(x)V = 0,  ∥v(x)∥C2 ≤ A∥x∥,  [ι(v(x))V ] = x in H2(M, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We prove a similar result for the manifold N = D2 × S1, and push it forward to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We  denote by (x, y) the coordinates on D2 ⊂ R2, by θ the coordinate on S1, and V0 = dx ∧ dy ∧ dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take a function h: D2 → R such that h ≡ 0 near ∂D2 and  �  D2 h(x, y)dx ∧ dy = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We define  on N the vector field Y = h(x, y)∂θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  62  One has LY (V0) = d(ι(Y )V0) = d(h(x, y)dx ∧ dy) = 0, so the vector field Y preserves the  volume form V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let S ⊂ N be an oriented compact surface whose boundary is inside ∂N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We claim that the integral  �  S ι(Y )V0 is equal to the algebraic intersection S ·∩(0 × S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  To  prove that, notice that ι(X)V0 is closed and equal to zero close to ∂N, so  �  S ι(X)V0 depends  only on the homology class of S relatively to ∂N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The equality �  S ι(X)V0 = S ·∩(0 × S1) holds  when S = D2 × {∗}, so it holds for any S by linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Up to taking a generating sub-family of tori of (Ui), we can suppose that there exist some  smooth parametrizations fi : N → Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Applying Moser’s lemma, we can take fi such that f ∗  i V =  λiV0 for some λi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote Yi =  1  λi dfi(Y ), which we extend to a smooth vector field on M  by setting Yi ≡ 0 outside Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We have LYiV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Additionally for any closed surface S ⊂ M,  the integral  �  S ι(Yi)V is equal to S ·∩γi, where γi is the closed curve corresponding to fi(0 × S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Therefore [ι(Yi)V ] ∈ H2(M, R) is Poincar´e-dual to [γi] ∈ H1(M, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For any x ∈ H2(M, R), we write x = �  i xi[γi] and define v(x) = �  i xiYi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Then we  have [ι(v(x))V ] = x and ∥v(x)∥C2 ≤ Cste∥x∥ maxi ∥Yi∥C2  ■  We denote by f ∗Y the vector field x �→ df(f −1(x))(Y ◦ f −1(x)), conjugated to Y by a  diffeomorphism f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' There exists a generating family of tori (Ui)i and an embedding f : ∪i Ui → R3 of  class C1+H for the atlas Ap, such that f ∗νp and f ∗Y are smooth measure on each Ui, f(Vi) is smooth,  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Construct first a generating family of tori by taking a family of closed C1+H curves γi  whose homology classes form a basis of H1(M, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' These curves can be taken embedded and  disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then fix a family of disjoint compact neighborhoods (Ui)i of the curves γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Inside Ui, thicken γi in a C1+H compact annulus Ai, transverse to the flow ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We take a C1+H  embedding gi : Ai → R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The measure νp is given by a H¨older volume form V on M, and Y is  H¨older, so i(Y )V is a H¨older differential 2-form on Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since Ai is transverse to Y , i(Y )V is  additionally an area form on Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence (g−1  i  )∗(i(Y )V ) is a H¨older area form on gi(Ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We can suppose that Ai has be taken such that gi(Ai) has a smooth boundary in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  In [DM90, Zeh77] is proved a generalization of Moser’s Lemma, which state that there exists  a C1+H diffeomorphism hi : gi(Ai) → gi(Ai) such that (hi ◦ gi)∗νp is given by a smooth volume  form on gi(Vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Now define the map fi : ψ[0,ǫ](Ai) → R3 by fi(ψt(x)) = (h ◦ g(x), t) ∈ R2 × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Since ψ is of class C1+H, fi is also of class C1+H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By construction df(Y ) = ∂z is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take  a closed simple curve δi in ψ[0,ǫ](Ai) homotopic to γi inside Ui such that f(δi) is smooth, and  take a neighborhood Vi of δi inside ψ[0,ǫ](Ai), such that fi(Vi) is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The map f defined  as f|Vi ≡ fi + ci for some constants ci ∈ R3 satisfies the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Regularity of the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' There exists a generating sub-family of tori Ui and a C∞ smooth structure A∞  p  on M such that: A∞  p extends the C1+H structure Ap, Y is of class C1+H for A∞  p , νp is smooth inside each Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  63  To prove the lemma, we adapt Hart’s technique to smooth the generator of a flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a  space E = Rm for some m > 0 and Ψ a flow of class C1+H on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by Z the generating  vector field of Ψ, which we suppose to be compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In general, Z is only of class CH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  In [Har83], Hart built a C1 diffeomorphism on E, which conjugates Z with a C1 vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' His  method, which we adapt bellow, actually give a C1+H vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by f ∗Ψt = f ◦Ψt◦f −1  the flow conjugated to Ψ by f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In particular ∂f ∗Ψt  ∂t  = f ∗ ∂Ψt  ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Let U, V, W ⊂ R3 be three open sub-sets such that W ⊂ V , V ⊂ U, V ∩ ∂U = ∅ and U  is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a C1+α flow Ψ on U, generated by a Cα vector field Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let T > 0 be such  that ΦT (V ) ⊂ U, and c: U → [0, T ] be a C2+α function such that c is constant on W, c > 0 on V  and c ≡ 0 outside V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Define the map Fc : U → R3 by:  Fc(x) =  � 1  0  Ψc(x)s(x)ds  Lemma 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' When ∥c∥C1+α is small enough, Fc is well-defined on all U and satisfies: Fc is a C1+α diffeomorphism on U, Fc ≡ id on Fc(U \\ V ), F ∗  c Z is of class C1+H on Fc(W),  if Z is of class C1+α on a neighborhood of x ∈ U, then F ∗  c Z is of class C1+α on a neigh-  borhood of Fc(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Additionally Fc convergences toward the identity for the C1+α topology, when ∥c∥C1+α goes to  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For all x ∈ U, either x is in V , or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  In both case, Ψc(x)s(x) is well-defined for  all s ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So Fc is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For all x ∈ U, we have:  dFc(x) =  � 1  0  dΨc(x)s(x)ds + dc(x)  � 1  0  Z ◦ Ψc(x)s(x)ds  It follows that ∥Fc − id ∥C1+α is finite and converges to zero when ∥c∥C1+α goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since  the set of C1 embedding is open in the set of C1 map, Fc is a C1+α diffeomorphism on its image  when T is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Notice that Fc coincides with the identity map on U \\ V ⊃ ∂U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So for T small,  we have Fc(U) = U and Fc : U → U is a C1+H diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For x in U, we have:  dFc(x)(Z(x)) =  � 1  0  Z ◦ Ψc(x)s(x)ds + (dc(x)(Z(x)))  � 1  0  Z ◦ Ψc(x)s(x)ds  = (dc(x)(Z(x)) + 1)Ψc(x)(x) − x  c(x)  when c(x) ̸= 0  Since we have c(x) ̸= 0 and dc ≡ 0 on W, dFc(Z) is of class C1+α on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' When Z is of class C1+α  on a neighborhood of x, dc(Z) is also C1+α on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So dFc(Z) is of class C1+H on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Lemma 86 (Extended Hart’s lemma [Har83]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let Ψ be a C1+α flow on a smooth compact  manifold M, and suppose that ∂Ψt  ∂t  is C1+α on a neighborhood of a compact set K ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then  there exists a C1+α diffeomorphism f : M → M such that f|K = idK and ∂f ∗Ψt  ∂t  is of class C1+α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  64  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For U ⊂ M, we denote by O(U) some open subset containing U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let (A, B1, · · · , Bn) be  an open covering of M such that K ⊂ A, K ∩Bi = ∅, Z is of class C1+α on O(A), and such that  there exists C∞ embedding from each Bi to R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We prove by induction on n that there exists  a C1+α diffeomorphisms f : M → M such that f ∗Z is of class C1+H on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If n = 0, f = id has  this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Let n ≥ 0 for which the induction hypothesis is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take C an open neighborhood of B1,  disjoint from K and g : C → R3 be a C∞ embedding with pre-compact image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We take U = im(g),  and V, W ⊂ U such that g(B1) ⊂ W, W ⊂ V and V ⊂ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by h: U → U the C1+α  diffeomorphism denoted by Fc in Lemma 85, for some small function c: U → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Define a  map l: M → M by l ≡ id on M \\ C and l ≡ g−1 ◦ h ◦ g on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The map h is equal to the identity on a neighborhood of ∂U, so l is of class C1+α on all M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We can take h arbitrarily C1 close to the identity so that h(W) contains g(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then l∗Z is of  class C1+α on a neighborhood of B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Similarly we can take h such that l∗Z is of class C1+α on  a neighborhood of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Indeed g∗Z is of class C1+α on a neighborhood of g(A ∩ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So for x in a  smaller neighborhood of g(A ∩ C), (h ◦ g)∗Z is C1+α on h(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If ∥c∥C1 is small enough, for the  previous reason h(O(g(A ∩ C))) contains g(A ∩ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence (h ◦ g)∗Z is C1+α on a neighborhood  of g(A∩C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So l∗Z is C1+α on a neighborhood of A∩B1, and also on A∩(M \\B1) by hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The map l is a C1+α diffeomorphism such that l∗Z is C1+α on a neighborhood of A ∪ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  According to the induction hypothesis, there exists a C1+α diffeomorphism j : M → M such  that j∗(l∗Z) is C1+α on M, and with j|K = idK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So f = j ◦ i satisfies the property at the rank n,  which conclude the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Proof of Lemma 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let (Vi)i be a generating family of tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a C1+H map f : ∪i Vi → R3  given by Lemma 83, so that f ∗νp is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take n ≥ 2 and consider a C1+H map g : M →  R3 × R2n which extends f × 0 on ∪iVi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' If g is chosen generic outside each Vi, then it is an  embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Denote by G2n the Grassmannian of 2n-dimensional vector sub-spaces of R2n+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take a C1+H  map p: M → G2n so that for each x ∈ M, p(x) is transverse to dg(TxM), the tangent space  of g(M) at g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by E = {(x, u) ∈ M × R2n+3, u ∈ p(x)} the vector bundle over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Define a C1+H map h: E → R2n+3 by h(x, u) = g(x)+u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since p(x) is transverse to dg(TxM), the  map h is injective on a neighborhood U of the zero section, that is on a neighborhood of {(x, 0) ∈  E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take a smooth sub-manifold N ⊂ U ⊂ R2n+3 obtained by a C1+H-small perturbation  of g(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since h is C1+H, h−1(N) is the image of a C1+H section s of the bundle E → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then  map h ◦ s: M → R2n+3 is of class C1+H and its image is a smooth sub-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For a set I ⊂ �1, 2n + 3� of cardinal 3, denote by πI : R2n+3 → R3 the forgetting map  corresponding to the indexes not in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Consider the atlas A whose charts are of the form (πI ◦  h◦s)U for all I ⊂ �1, 2n + 3� of cardinal 3, and U ⊂ M open on which πI ◦h◦s is an embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Since N is smooth, the transition maps are smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So A is a C∞ atlas on M, which additionally  extends Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since g(∪iVi) is smooth, we can smooth N only outside that set, so that h◦s coincides  with g on ∪iVi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then π�1,3� ◦ h ◦ s coincides with f on that set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  We now regularize Y and νp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take a generating sub-family of tori (Ui)i of (Vi)i, such  that Ui ⊂ Int Vi for all i, and denote by K = ∪iUi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  According to Lemma 86 there exists  a C1+H diffeomorphism l of M, such that l|K = idK and such that l∗Y is of class C1+H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since l  coincides with the identity inside each Ui, (l ◦ h)∗νp are smooth inside each Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The atlas A∞  p is  obtained by conjugating the chart of A with l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Smoothing the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For this subsection, we use the notation from Lemma 84 and  consider the atlas denoted by A∞  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' According to the lemma, there exists a generating family  of tori (Ui)i, on which νp is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take α ∈ (0, 1) such that Y is of class C1+α and such  65  that νp is induced by an α-H¨older volume form on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let V be a smooth volume form on M  and λ: M → R be the function which satisfies νp = eλV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It is α-H¨older continuous on M and  smooth on each tori Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take λk : M → R a sequence of C∞ function, which approximates λ for  the Cα topology, and which coincide with λ on each Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 87 (Asaoka 2008 [Asa07]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' There exists a sequence of C∞ vector fields Yk on M such  that: (Yk)k converges toward Y in the C1+α topology, Yk preserves eλkV for all k,  Denote by ψk the flow generated by Yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By C1 structural stability of Anosov flows, ψk is  Anosov and orbit equivalent to φ for k large enough (see [KKW91]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by hn : (M, ψk) →  (M, ψ) an orbit equivalence so that (hn)n converges toward id for the C0 topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Also denote  by µk the probability measure induced by eλkV , invariant by ψk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Lemma 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The sequence of measures Θhn(µn) ∈ Mp(ψ) converges toward νp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We prove that the sequence of transverse measures h∗  k(µ⊥  k ) converges toward ν⊥  p , which  implies the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Recall that (λk)k converges toward λ, so (µk)k converges toward νp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Let S ⊂ M be a compact surface, transverse to Y and Yk for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So there exists ǫ > 0 for  which the maps ψk are all injective on the set [0, ǫ] × S → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Then for any k, we have µ⊥  k (S) =  1  ǫµk(ψk  [0,ǫ](S)) ≤ 1  ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take a continuous function f : S → R so that f ≡ 0 on a neighborhood of ∂S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We can  extend f to a continuous function M → R, such that f ≡ 0 on a neighborhood of ∂S inside M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Fix three neighborhoods U1, U2, U3 ⊂ S of ∂S for which fUi = 0 and U i ⊂ Int Ui+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Since the sequence (hk)k converges toward the identity, there exist for k large enough an  embedding gk : h−1  k (S \\ U1) → S, obtained as a small isotopy along the flow ψk from h−1  k (S \\ U1)  to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For large enough, the image of gk contains U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Since hk converges toward id and Yk  converges toward Y , the map gk ◦ h−1  k  converges toward the inclusion S \\ U1 ֒→ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Hence for k  large we have S \\ U2 ⊂ gk ◦ h−1  k (S \\ U1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore for large k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' the following holds:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='S  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='fdh∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k(µ⊥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k ) =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='S\\U1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='fdh∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k(µ⊥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='h−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k (S\\U1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='f ◦ hkdµ⊥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='gk◦h−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k (S\\U1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='f ◦ hk ◦ g−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k dµ⊥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='S\\U2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='f ◦ hk ◦ g−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k dµ⊥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='S  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='fdν⊥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='p −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='S  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='fdh∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k(µ⊥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='���� ≤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='�����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='S  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='fdν⊥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='p −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='S\\U2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='f ◦ hk ◦ g−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k dµ⊥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='�����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='≤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='�����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='S\\U2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='fd  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='ν⊥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='p − µ⊥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='����� +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='S\\U2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='��f − f ◦ hk ◦ g−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='�� dµ⊥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='≤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='�����  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='S\\U2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='fd  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='ν⊥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='p − µ⊥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='����� + µ⊥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k (S)∥f − f ◦ hk ◦ g−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k ∥S\\U2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='−−−−−→  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='k→+∞ 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='66  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='The first term converges toward zero since µk converges toward νp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The second term converges  toward zero since we have µ⊥  k (S) ≤ 1  ǫ, and (f ◦hk◦g−1  k )k converges toward f on S\\U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Therefore  the measure h∗  k(µ⊥  k ) converges toward ν⊥  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Lemma 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' There exists a sequence Zk of C∞ vector fields on M such that: Zk converges toward zero in the C1+α topology, Zk preserves eλkV for all k, ι(Yk)eλkV and ι(Zk)eλkV are cohomologous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Let v: H2(M, R) → Λ∞(T M) be the map constructed in Lemma 82 for the volume  form eλV , which is smooth inside each Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Notice that for x ∈ H2(M, R), v(x) is a vector  field supported inside the union of the tori Ui, where λ and λk coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Hence the vector  field v(x) preserves the volume forms eλkV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Now take Zk = v([ι(Yk)eλkV ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By hypothesis, we  have [ι(Zk)eλkV ] = [ι(Yk)eλkV ], which converges toward zero according to Lemmas 88 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It  follows that Zk converges toward zero for the C2 topology, and so also for the C1+α topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  Corollary 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' For k large enough, the smooth vector field Yk − Zk generates an Anosov flow,  orbit equivalent to φ, which preserves the smooth volume form eλkV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Additionally the probability  measure induced by eλkV satisfies the Reeb-like property for k large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The corollary and Theorem D give a proof to Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by ψn the flow generated by the vector field Yn − Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The vector fields (Yk − Zk)  converges toward Y for the C1 topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So by structural stability of Anosov flows [KKW91],  for k large enough, ψn is Anosov and is orbit equivalent to φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by ηn the ψn-invariant  probability measure induced by ι(Yk − Zk)eλkV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows from the previous lemma that ηn  is null-cohomologous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take hn a sequence of orbit equivalences between ψn and φ, so that hn  converge toward the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' According to Lemma 88, the measure Θhn(ηn) converges  toward νp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So for n large enough, the minimum of linkφ(Θhn(ηn), ν) with all ν ∈ M0  p(φ), is  positive (see Corollary 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' It follows from Lemma 57 that the measure ηn satisfies the Reeb-like  property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  A  Fried-desingularisation  We prove Proposition 60: given a Birkhoff section S1 and a partial section S2, there exists a  Birkhoff section homologous to [S1] + [S2] in H2(M, ∂S1 ∪ ∂S2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take ∆ = ∂S1 ∪ ∂S2, M∆  the blow up manifold and fix a Riemannian metric m on M∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We denote by D the diame-  ter of M∆ for m, and by ∥ · ∥ one norm on H2(M∆, ∂M∆, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Define the normalization map  N : H2(M∆, ∂M∆, R) → H2(M∆, ∂M∆, R) given by N(0) = 0 and N(x) = x/∥x∥ when x ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  For x ∈ M∆ and T ≥ 0, denote by γT  x : [0, T + 1] → M∆ one map given by γT  x (t) = φ∆  t (x),  for t ∈ [0, T ], and (γT  x )|[T,T +1] is smooth path from φT (x) to x, whose length is at most D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The  choice of the curve (γT  x )|[T,T +1] has no influence on the following notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The accumulating points  of elements of the form N([γT  x ]), for T −→ +∞, are called asymptotic directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Schwartzman  and Fried [Sch57, Fri82] proved that a compact surface is relatively homologous to a global section  of flow if and only if it intersects algebraically positively every asymptotic direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Proof of Proposition 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We lift the surface Si to a surface �Si inside M∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' We write �S = �  S1 ∪ �  S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Since S1 is a Birkhoff section, there exists T0 > 0 for which every orbit arc of length T0 of φ∆  intersects �  S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Recall that Si is positively transverse to the φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Denote by C1 > 0 the supremum  67  of the algebraic intersection of �S and a smooth curve of length D, which is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Take C2 > 0  for which every closed C1 curve γ ⊂ M∆ satisfies ∥[γ]∥ ≤ C2lengthm(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Also take C3 > 0 the  maximum of  ��� ∂φ∆  t  ∂t  ���  m, so that the length for m of an orbit arc φ[0,T ](x) is at most C3T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Take x ∈ M∆ and T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' When x is not in the boundary of M∆, the algebraic intersection  between γT  x and �S is at least φ∆  [0,T ](x) ·∩([�  S1] + [�  S2]) − C1 ≥  T  T0 − 1 − C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' By continuity, it also  holds for x ∈ ∂M∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' So we have:  N  �  [γT  x ]  �  ∩[�S] ≥ [γT  x ] ·∩[�S]  ∥[γTx ]∥  ≥  T  T0 − 1 − C1  C2(C3T + D)  lim inf  T−→+∞ N  �  [γT  x ]  �  ∩[�S] ≥  1  T0C2C3  Hence [�S] intersects positively every asymptotic direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' According to Schwartzman-Fried  Theorem [Fri82], there exists a global section relatively homologous to �S in H2(M∆, ∂M∆, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  The existence of a Birkhoff section homologous to [S1] + [S2] follows from Lemma 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  ■  References  [ABM22]  Masayuki Asaoka, Christian Bonatti, and Th´eo Marty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Oriented Birkhoff sections of  Anosov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
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+page_content=' Topology, 21(4):353 – 371, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  [Fri83]  David Fried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Transitive Anosov flows and pseudo-Anosov maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Topology, 22(3):299  – 303, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  [Ghy09]  ´Etienne Ghys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Right-handed vector fields & the Lorenz attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Japanese Journal  of Mathematics, 4(1):47–61, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  [Goo83]  Sue Goodman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Dehn surgery on Anosov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Geometric Dynamics, pages 300–307,  1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  [Har83]  David Hart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' On the smoothness of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Topology, 22(3):357–363, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  [Hay94]  Nicolai TA Haydn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Canonical product structure of equilibrium states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Random and  computational dynamics, 2(1):79–96, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  [HT80]  Michael Handel and William P Thurston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Anosov flows on new three manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Inventiones mathematicae, 59(2):95–103, 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  [KKW91]  Anatole Katok, Gerhard Knieper, and Howard Weiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Formulas for the derivative and  critical points of topological entropy for Anosov and geodesic flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Communications  in Mathematical Physics, 138(1):19–31, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  [Lot97]  M Lothaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Combinatorics on words, volume 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Cambridge university press, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  [Mar21]  Theo Marty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Anosov flows and Birkhoff sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Phd manuscript, Universit´e Greno-  ble Alpes [2020-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='.], September 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  [PP86]  William Parry and Mark Pollicott.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' The Chebotarov theorem for Galois coverings of  axiom A flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Ergodic Theory and Dynamical Systems, 6(1):133–148, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  [Pra22]  Rohil Prasad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Volume-preserving right-handed vector fields are conformally Reeb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  arXiv preprint arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='12935, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  [Rat69]  M Rather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Markov partitions for C-flows on 3-dimensional manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Zametki,  6(6):693–704, 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  [Rat73]  Marina Ratner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Markov partitions for Anosov flows on n-dimensional manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  Israel Journal of Mathematics, 15(1):92–114, 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  69  [Sal22]  Federico Salmoiraghi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Goodman surgery and projectively Anosov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  [Sch57]  Sol Schwartzman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Asymptotic cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Annals of Mathematics, pages 270–284, 1957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  [Sha93]  Richard Sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Closed orbits in homology classes for Anosov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Ergodic Theory  and Dynamical Systems, 13(2):387–408, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  [Sul76]  Dennis Sullivan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Cycles for the dynamical study of foliated manifolds and complex  manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Inventiones mathematicae, 36(1):225–255, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  [Vog03]  Thomas Vogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' On the asymptotic linking number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Proceedings of the American  Mathematical Society, 131(7):2289–2297, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  [VS73]  Santiago Alberto Verjovsky-Sola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Codimension 1 Anosov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Brown University,  1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  [Zeh77]  Eduard Zehnder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Note on smoothing symplectic and volume preserving diffeomor-  phisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' In Geometry and topology, pages 828–854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content=' Springer, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
+page_content='  70' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAyT4oBgHgl3EQf7voS/content/2301.00842v1.pdf'}
diff --git a/QNFQT4oBgHgl3EQfZDbf/content/tmp_files/2301.13315v1.pdf.txt b/QNFQT4oBgHgl3EQfZDbf/content/tmp_files/2301.13315v1.pdf.txt
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index 0000000000000000000000000000000000000000..19a2390480f3a1073e9f9e4f818fba63f662a020
--- /dev/null
+++ b/QNFQT4oBgHgl3EQfZDbf/content/tmp_files/2301.13315v1.pdf.txt
@@ -0,0 +1,221 @@
+arXiv:2301.13315v1  [physics.atom-ph]  30 Jan 2023
+Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022
+1
+Probing Gravity for one Minute
+with an Optical-Lattice Atom Interferometer
+C.D. Panda, M. Tao, J. Eggelhof, M. Ceja, A. Reynoso, V. Xu, and H. M¨uller
+University of California, Berkeley, CA 94720, USA
+We have realized an atom interferometer that probes gravitational potentials
+by holding, rather than dropping, atoms.
+Up to one minute of coherence
+times are realized by suspending the spatially separated atomic wave packets
+in an optical lattice that is mode-filtered by an optical cavity. This trapped
+configuration suppresses phase variance due to vibrations by four to five orders
+of magnitude, overcoming the dominant noise source in atom-interferometric
+gravimeters.
+Recent progress in characterizing and reducing interferometer
+decoherence led to major increases in coherence and precision, paving the way
+to measurements of dark-energy candidates and probes of the quantum nature
+of gravity through measuring the gravity of source masses with record precision
+and spatial resolution.
+Light-pulse atom interferometry1 has been used to measure the fine struc-
+ture constant,2 the gravitational constant,3 to test the equivalence princi-
+ple,4 to look for laboratory signatures of dark energy,5 or to search for CPT
+or Lorentz violation.6,7 Atom interferometers are among the most accurate
+and sensitive tools for measuring gravity, in and out of the laboratory.8
+The achieved precision is limited by two main factors: coherence time and
+sensitivity to vibrations. In atom interferometers performed with atomic
+fountains, coherence is limited by the available free-fall time to below three
+seconds (for ∼ 10 m drop towers).
+Our interferometer suspends atoms against Earth’s gravity in an optical
+lattice that is formed by the mode of an optical cavity (Fig. 1). We observe
+up to 1 minute of coherence time (Fig. 2), which is by far the longest-lasting
+realization of the fundamental notion that a massive quantum particle can
+be in a superposition of being located in two different places at once and
+still three times longer than we have realized with the same experiment
+in a previous publication.9 Achieving this in a conventional free-fall atom
+interferometer would require a drop tower 4.5 km tall.
+
+Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022
+2
+Δz
+Fig. 1.
+Lattice interferometer atom trajectories.
+Ultracold Cesium atoms are
+launched against gravity.
+A pair of π/2 pulses creates a spatial superposition state
+where two atomic wavepackets are separated by a distance ∆z. After a time tA, atoms
+reach the apex of their free-fall trajectory and are loaded in a far-detuned optical lattice
+formed by the mode of an optical cavity (horizontal stripes) and remain held for a time
+τ. The two wavepackets are recombined and their interference is measured.
+Atoms held at fixed locations can measure extremely weak fields, such
+as the gravity due to small milligram source masses, for many seconds. This
+is orders of magnitude longer than possible with atomic fountains and rep-
+resents a major sensitivity boost for fundamental-physics experiments, such
+as tests of fifth forces,10 searches for dark energy in the laboratory,5 pro-
+posed measurements of the gravitational Aharonov–Bohm effect,11 or tests
+on whether gravity can mediate entanglement between quantum mechan-
+ical systems.12 The optical-lattice interferometer is also significantly less
+susceptible to noise due to acousto-mechanical vibrations than its fountain
+counterpart. Holding the atoms in an optical lattice for 60 seconds provides
+104–105 times suppression of environmental vibrations in the 1–1000 Hz vi-
+bration band, making operation near the standard quantum limit possible.
+Bringing optical-lattice technology, common to atomic-clock metrol-
+ogy,13 to atom interferometry is nontrivial, as spatial-superposition states
+are particularly susceptible to external forces. The interaction of the atoms
+with the optical lattice causes shifts of the atomic partial wavepacket phases
+that are large compared to the measured signals. This common-mode phase
+
+B
+元/2
+元/2
+元/2
+元/2
++hk.
+X
+ZProceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022
+3
+Fig. 2.
+One-minute interference fringes measured with the lattice interferometer.
+is intrinsically subtracted by the interferometer, but any imperfections in
+the optical potential can lead to phase shifts and decoherence. Precise con-
+trol of the optical field is therefore necessary, which we achieve at least
+partially by using an optical cavity as a mode-filter for the optical lattice.
+We empirically observe a robust scaling in the decoherence rate of
+our interferometer, where the contrast follows an exponential decay, C =
+C0 exp(−τ/τC) with C0 the starting contrast. τC is inversely proportional
+to the wavepacket separation ∆z and trap depth U: 1/τC = U∆z/c, where
+c = 110 ± 30 µm Er s is a measured constant and Er is the recoil energy
+of the Cs D2 transition (Fig. 3a).
+The decoherence results from resid-
+ual transverse motion of the atoms in the one-dimensional optical lattice,
+where atoms sample tiny differences in the optical potential holding the top
+and bottom partial wavepackets. We confirm this by observing increased
+contrast when only selecting cold atoms by loading atoms in the narrower
+higher-order Laguerre–Gaussian modes of the cavity (Fig. 3b). Planned
+reductions in the atom sample temperature and control of lattice imperfec-
+tions promise to increase coherence times past the one-minute mark.
+We have described an optical-lattice atom interferometer with coher-
+ence times reaching one minute, realizing the longest spatial superposi-
+tion states to date, more than an order of magnitude longer than atomic-
+fountain interferometers.
+This geometry comes along with intrinsic ad-
+vantages: vibration-noise rejection and micron-scale spatial resolution of
+the measured signals. Cognizance and reduction of decoherence and atom-
+source upgrades promise to push this technology to record precision and
+ultra-long coherence. This opens a new era in atom interferometry, with
+
+Time offset (μus)
+-20-10.0 10.20
+-20-1001020-20-1001020
+0.4
+T
+0.2
+AA
+0.0
+As
+-0.2
+-0.4
+0.2109
+1.0509
+5.1109
+15.0509
+30.0309
+60.0609
+Interferometer hold time (seconds)Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022
+4
+Contrast decay constant c (
+�
+m Er s)
+Transverse mode
+�
+�
+�
+�
+■
+■
+■
+■
+◆
+◆
+◆
+◆
+▲
+▲
+▲
+▲
+▼
+▼
+▼
+▼
+0
+2
+4
+6
+8
+10
+12
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+U (Er)
+1/τC (s)
+Δz (μm)
+▼ 1.8
+▲ 4.2
+●
+6.5
+■
+9.4
+◆ 11.3
+a
+b
+Fig. 3.
+Interferometer coherence. a.) Measured linear scaling of 1/τC with trap
+depth U and separation, ∆z.
+b.)
+The magnitude of the contrast decay constant
+c increases when the optical-lattice radius is lowered through the use of higher-order
+Laguerre–Gaussian modes of the cavity.
+applications in gravimetry, searches for fifth forces, a measurement of the
+gravitational Aharonov–Bohm effect in the absence of forces, or new tests
+of CPT and Lorentz violation.
+References
+1. A.D. Cronin, J. Schmiedmayer, and D.E. Pritchard, Rev. Mod. Phys. 81,
+1051 (2009).
+2. L. Morel, Z. Yao, P. Clade, and S. Guellati-Kh´elifa, Nature 588, 61 (2020).
+3. G. Rosi, F. Sorrentino, L. Cacciapuoti, M. Prevedelli, and G.M. Tino, Nature
+510, 518 (2014).
+4. P. Asenbaum, C. Overstreet, M. Kim, J. Curti, and M. A. Kasevich, Phys.
+Rev. Lett. 125, 191101 (2020).
+5. M. Jaffe et al., Nature Phys. 13, 938 (2017).
+6. K. Chung, S. Chiow, S. Herrmann, S. Chu, and H. M”uller, Phys. Rev. D
+80, 016002 (2009).
+7. Q.G. Bailey and V.A. Kosteleck´y, Phys. Rev. D 74, 045001 (2006).
+8. G.M. Tino, Quantum Sci. Technol. 6, 024014 (2021).
+9. V. Xu, M. Jaffe, C.D. Panda, S.L. Kristensen, L.W. Clark, and H. M¨uller,
+Science 366, 745 (2019).
+10. P. Wolf, P. Lemonde, A. Lambrecht, S. Bize, A. Landragin, and A. Clairon,
+Phys. Rev. A 75, 063608 (2007).
+11. M.A. Hohensee, B. Estey, P. Hamilton, A. Zeilinger, and H. M¨uller, Phys.
+Rev. Lett. 108, 230404 (2012).
+12. D. Carney, H. M¨uller, and J.M. Taylor, PRX Quantum 2, 030330 (2021).
+13. G.E. Marti, R.B. Hutson, A. Goban, S.L. Campbell, N. Poli, and J. Ye, Phys.
+Rev. Lett. 120, 103201 (2018).
+
+500
+400
+300
+I
+200
+I
+100
+LG 00
+LG 10
+LG 20
+0
\ No newline at end of file
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf,len=203
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='13315v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='atom-ph]  30 Jan 2023  Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022  1  Probing Gravity for one Minute  with an Optical-Lattice Atom Interferometer  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' Panda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' Tao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' Eggelhof, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' Ceja, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' Reynoso, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' Xu, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' M¨uller  University of California, Berkeley, CA 94720, USA  We have realized an atom interferometer that probes gravitational potentials  by holding, rather than dropping, atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  Up to one minute of coherence  times are realized by suspending the spatially separated atomic wave packets  in an optical lattice that is mode-filtered by an optical cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' This trapped  configuration suppresses phase variance due to vibrations by four to five orders  of magnitude, overcoming the dominant noise source in atom-interferometric  gravimeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  Recent progress in characterizing and reducing interferometer  decoherence led to major increases in coherence and precision, paving the way  to measurements of dark-energy candidates and probes of the quantum nature  of gravity through measuring the gravity of source masses with record precision  and spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  Light-pulse atom interferometry1 has been used to measure the fine struc-  ture constant,2 the gravitational constant,3 to test the equivalence princi-  ple,4 to look for laboratory signatures of dark energy,5 or to search for CPT  or Lorentz violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='6,7 Atom interferometers are among the most accurate  and sensitive tools for measuring gravity, in and out of the laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='8  The achieved precision is limited by two main factors: coherence time and  sensitivity to vibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' In atom interferometers performed with atomic  fountains, coherence is limited by the available free-fall time to below three  seconds (for ∼ 10 m drop towers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  Our interferometer suspends atoms against Earth’s gravity in an optical  lattice that is formed by the mode of an optical cavity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' We observe  up to 1 minute of coherence time (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' 2), which is by far the longest-lasting  realization of the fundamental notion that a massive quantum particle can  be in a superposition of being located in two different places at once and  still three times longer than we have realized with the same experiment  in a previous publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='9 Achieving this in a conventional free-fall atom  interferometer would require a drop tower 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='5 km tall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022  2  Δz  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  Lattice interferometer atom trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  Ultracold Cesium atoms are  launched against gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  A pair of π/2 pulses creates a spatial superposition state  where two atomic wavepackets are separated by a distance ∆z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' After a time tA, atoms  reach the apex of their free-fall trajectory and are loaded in a far-detuned optical lattice  formed by the mode of an optical cavity (horizontal stripes) and remain held for a time  τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' The two wavepackets are recombined and their interference is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  Atoms held at fixed locations can measure extremely weak fields, such  as the gravity due to small milligram source masses, for many seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' This  is orders of magnitude longer than possible with atomic fountains and rep-  resents a major sensitivity boost for fundamental-physics experiments, such  as tests of fifth forces,10 searches for dark energy in the laboratory,5 pro-  posed measurements of the gravitational Aharonov–Bohm effect,11 or tests  on whether gravity can mediate entanglement between quantum mechan-  ical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='12 The optical-lattice interferometer is also significantly less  susceptible to noise due to acousto-mechanical vibrations than its fountain  counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' Holding the atoms in an optical lattice for 60 seconds provides  104–105 times suppression of environmental vibrations in the 1–1000 Hz vi-  bration band, making operation near the standard quantum limit possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  Bringing optical-lattice technology, common to atomic-clock metrol-  ogy,13 to atom interferometry is nontrivial, as spatial-superposition states  are particularly susceptible to external forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' The interaction of the atoms  with the optical lattice causes shifts of the atomic partial wavepacket phases  that are large compared to the measured signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' This common-mode phase  B  元/2  元/2  元/2  元/2  +hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  X  ZProceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  One-minute interference fringes measured with the lattice interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  is intrinsically subtracted by the interferometer, but any imperfections in  the optical potential can lead to phase shifts and decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' Precise con-  trol of the optical field is therefore necessary, which we achieve at least  partially by using an optical cavity as a mode-filter for the optical lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  We empirically observe a robust scaling in the decoherence rate of  our interferometer, where the contrast follows an exponential decay, C =  C0 exp(−τ/τC) with C0 the starting contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' τC is inversely proportional  to the wavepacket separation ∆z and trap depth U: 1/τC = U∆z/c, where  c = 110 ± 30 µm Er s is a measured constant and Er is the recoil energy  of the Cs D2 transition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  The decoherence results from resid-  ual transverse motion of the atoms in the one-dimensional optical lattice,  where atoms sample tiny differences in the optical potential holding the top  and bottom partial wavepackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' We confirm this by observing increased  contrast when only selecting cold atoms by loading atoms in the narrower  higher-order Laguerre–Gaussian modes of the cavity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' Planned  reductions in the atom sample temperature and control of lattice imperfec-  tions promise to increase coherence times past the one-minute mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  We have described an optical-lattice atom interferometer with coher-  ence times reaching one minute, realizing the longest spatial superposi-  tion states to date, more than an order of magnitude longer than atomic-  fountain interferometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  This geometry comes along with intrinsic ad-  vantages: vibration-noise rejection and micron-scale spatial resolution of  the measured signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' Cognizance and reduction of decoherence and atom-  source upgrades promise to push this technology to record precision and  ultra-long coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' This opens a new era in atom interferometry, with  Time offset (μus)  20-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='20  20-1001020-20-1001020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='4  T  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='2  AA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='0  As  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='2109  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='0509  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='1109  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='0509  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='0309  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='0609  Interferometer hold time (seconds)Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022  4  Contrast decay constant c (  �  m Er s)  Transverse mode  �  �  �  �  ■  ■  ■  ■  ◆  ◆  ◆  ◆  ▲  ▲  ▲  ▲  ▼  ▼  ▼  ▼  0  2  4  6  8  10  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='4  U (Er)  1/τC (s)  Δz (μm)  ▼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='8  ▲ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='5  ■  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='4  ◆ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='3  a  b  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  Interferometer coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=') Measured linear scaling of 1/τC with trap  depth U and separation, ∆z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=')  The magnitude of the contrast decay constant  c increases when the optical-lattice radius is lowered through the use of higher-order  Laguerre–Gaussian modes of the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  applications in gravimetry, searches for fifth forces, a measurement of the  gravitational Aharonov–Bohm effect in the absence of forces, or new tests  of CPT and Lorentz violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
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+page_content=' Schmiedmayer, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
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+page_content=' Yao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' Clade, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
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diff --git a/RtE3T4oBgHgl3EQfywtH/content/tmp_files/2301.04722v1.pdf.txt b/RtE3T4oBgHgl3EQfywtH/content/tmp_files/2301.04722v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..9d1ebe8c3380b77cb5b7250cdf52ec0e6bf362a9
--- /dev/null
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@@ -0,0 +1,1715 @@
+arXiv:2301.04722v1  [math.PR]  11 Jan 2023
+RATE OF CONVERGENCE IN MULTIPLE SLE USING
+RANDOM MATRIX THEORY
+ANDREW CAMPBELL, KYLE LUH, AND VLAD MARGARINT
+Abstract. We provide an order of convergence for a version of the Carath´eodory
+convergence for the multiple SLE model with a Dyson Brownian motion driver
+towards its hydrodynamic limit, for β = 1 and β = 2. The result is obtained
+by combining techniques from the field of Schramm-Loewner Evolutions with
+modern techniques from random matrices. Our approach shows how one can
+apply modern tools used in the proof of universality in random matrix theory,
+in the field of Schramm-Loewner Evolutions.
+1. Introduction and main results
+Schramm-Loewner Evolution (SLE) and random matrix theory (RMT) are two
+active and well-studied fields of research within modern probability theory [2,38].
+The SLE was introduced by Oded Schramm in 2000 in his study of scaling limits
+of various discrete processes [49]. RMT appeared earlier in the statistical work of
+Wishart [61] and the pioneering physics of Wigner [60]. Both SLE and RMT have
+been thriving areas of mathematical research since their advent.
+When studying SLE theory, one introduces the notion of compact hulls, which
+are compact sets with simply connected complements in the upper half-plane. If
+Kt is a growing set of hulls parameterized by t ∈ [0, T ] and the growth is local in
+some sense, then it is known that gt := gKt obeys the Loewner differential equation
+∂tgt(z) =
+2
+gt(z) − Wt
+where Wt is referred to as the driving function and captures the local growth of Kt.
+SLE are the random curves corresponding to the gt when the driving function is a
+constant multiple of Brownian motion, that we denote by √κBt, for κ ≥ 0. With
+probability one, gt is continuous up to the boundary and the limit
+γ(t) = lim
+y→0 g−1
+t
+(√κBt + iy)
+exists and is continuous in time, by the Rohde-Schramm Theorem [46]. The curve
+γ(t) is called the SLE trace. Also, it can be shown that with probability one, gt is
+a continuous family of conformal maps from Ht to H, where Ht is the unbounded
+component of the complement in H of γ(t), for t ∈ [0, T ] [46]. Moreover, the nature
+of the curve changes as κ increases from simple a.s. when κ ∈ [0, 4], to having
+double points a.s. for κ ∈ (4, 8) and space-filling a.s., for κ ≥ 8. For different
+parameters κ, the SLE models the scaling limits of an astoundingly diverse set of
+discrete models. For instance, it was proved in [39] that the scaling limit of the loop
+erased random walk (with the loops erased in a chronological order) converges in the
+K. Luh was supported in part by the Ralph E. Powe Junior Faculty Enhancement Award.
+1
+
+2
+A. CAMPBELL, K. LUH, AND V. MARGARINT
+scaling limit to SLEκ with κ = 2. Moreover, other two dimensional discrete models
+from Statistical Mechanics including the Ising model cluster boundaries, Gaussian
+free field interfaces, percolation on the triangular lattice at critical probability, and
+Uniform spanning trees were proved to converge in the scaling limit to SLE for
+values of κ = 3, κ = 4, κ = 6 and κ = 8 respectively in the series of works [52], [50],
+[51] and [39].
+One can consider more generally the Loewner equation driven by a time-dependent
+real-valued measure µt
+∂
+∂tgt(z) =
+�
+R
+µt(dx)
+gt(z) − x,
+g0(z) = z.
+When the driving measure µt is a Dirac-delta mass at location √κBt, we recover
+the previous SLE maps. In the case µt = �N
+i=1 ωi(t)δUi(t), for some non-intersecting
+continuous functions Ui(t) ∈ R (called driving functions), and weights ωi(t) ∈ R+,
+we obtain the multi-slit Loewner equation with driving functions Ui(t), i = 1, · · · , n.
+In this work, we consider the case ωi(t) = 1/N, for all t ∈ [0, T ].
+For a real parameter β > 0, Dyson Brownian motion (DBM) is defined by the
+following system of N equations
+(1)
+dλ(i)
+t
+=
+2
+√Nβ dB(i)
+t
++ 2
+N
+�
+j̸=i
+dt
+λ(j)
+t
+− λ(i)
+t
+,
+for i = 1, 2, ..., N.
+Due to its connections with other fields, an important Loewner equation is the
+multiple SLE with DBM as a driver. The multiple SLE maps that are obtained
+when the driving measure is an empirical measure on N DBM particles are denoted
+in this paper by gN
+t (z). This model was introduced by Cardy in [10], and studied
+further by Lawler and Healey in [30], in connection with the quantum Calogero-
+Sutherland model and Conformal Field Theory.
+More works on the connection
+between Multiple SLE and CFT can be found in [40] and [48]. In the case of N = 2
+curves, perturbations of this model in the parameter β have been studied in [11].
+We note that the parameters β in the DBM model and κ in SLE theory are related
+via β = 8/κ.
+We refer to the multiple SLE model with Dyson Brownian motion as a driver
+as the simultaneously growing multiple SLE model. There is also a version of the
+multiple SLE that has non-simultaneous growth that has received a lot of attention
+in the previous years. There have been several results on the multiple SLE model
+in both the upper half-plane and the unit disk versions [5, 12, 13, 15, 31, 32, 35, 37,
+40,45,47,58,62,63].
+In [14], the authors consider the N → ∞ limit of multiple SLE driven by DBM. In
+particular they show that the empirical measure of the initial positions converges to
+a probability measure µ0, then gN
+t converges in distribution with respect to locally
+uniform convergence to g∞
+t
+solving
+(2)
+∂
+∂tg∞
+t (z) = M ∞
+t (z),
+g0(z) = z,
+Where M ∞
+t
+is a solution to the complex Burgers equation
+(3)
+� ∂M∞
+t (z)
+∂t
+= −2M ∞
+t (z) ∂M∞
+t (z)
+∂z
+, t > 0,
+M ∞
+0 (z) =
+�
+R
+2
+z−xdµ0(x).
+
+RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY
+3
+Their result serves as the multiple SLE analog of Wigner’s famous semicircle law
+in random matrix theory.
+We consider this model and we obtain more refined
+information by providing an order of convergence of this model in a weaker ver-
+sion of the Carath´eodory type convergence. We aim in future works to study the
+full Carath´eodory convergence by strengthening the estimates as we approach the
+multiple SLE hull.
+In this work, we combine elements of the proof of Local Laws in random matrix
+theory, such as resolvent techniques, with elements of the SLE theory. In other
+words, we apply modern techniques from random matrix theory to the analysis of
+SLE. Local Laws are a very important research direction in random matrix theory
+in the last years (see [9], [16], [18], [17], [19], [20], [21], [22], [25], [24], [26], [23],
+[42], [44], [54], [53], [57], [56], [55] for a non-exhaustive list). They are one of the
+fundamental ingredients in proving the universality of Wigner ensembles in random
+matrix theory (see [27]).
+Given the outstanding developments in the proof of the universality in RMT
+using the analysis of the DBM, the interaction between multiple SLE and random
+matrices will provide many avenues to explore. The approach in the current work
+represents one of the possible directions of exploration between these two major
+fields of Probability theory. In a different direction, which we aim to explore in the
+future, one can study the geometry of the multiple SLE curves using the analysis
+of the Dyson Brownian motion drivers, as well as good approximation schemes of
+the model (see, for example, [28], [59], [33] in the one SLE curve case). Yet another
+possibility is to study the continuity of the multiple SLE model in the parameter
+β, motivated by the great interest and progress throughout the years in this yet
+unresolved conjecture in the one SLE curve case (see [6], [7], [29], [34]). In addition,
+the fact that the multiple SLE curves grow from the positions of the drivers along
+with some knowledge about the structure of the drivers gives the possibility of
+defining new observables in order to study the convergence of discrete models to
+the multiple SLE. Examples of such observables include the statistics of the kth
+smallest distance between drivers, for k ≥ 1, (see [8]) or the probability of having
+no drivers in a symmetric region about the origin (see [41] for β = 2).
+Although our result can be obtained for general bounded initial conditions, we
+state it in the case in which all the Dyson Brownian motion particles start from
+the origin. We prefer this choice for the simplicity of the notation and exposition.
+Theorem 1.1. Let β = 1 or β = 2, and let us consider Dyson Brownian motion
+beginning at the origin. Let KT be the multiple SLE hull at time T > 0. Then, for
+any ε > 0, for the multiple SLE maps for N curves, we have that
+sup
+t∈[0,T ], z∈G
+|gN
+t (z) − g∞
+t (z)| = O
+�
+1
+N 1/3−ε
+�
+,
+with overwhelming probability1, for a given G ⊂ H \ KT .
+Remark 1.2. It is well-known that for the special values of the parameters β = 1,
+β = 2 and β = 4, the Dyson Brownian motion particles statistics can be understood
+using matrices as these values correspond to the well-studied models of the Gauss-
+ian Orthogonal Ensemble, Gaussian Unitary Ensemble (GUE), and the Gaussian
+1An event E holds with overwhelming probability if, for every p > 0, P(E) ≥ 1 − Op(n−p); see
+Definition 3.1 for details.
+
+4
+A. CAMPBELL, K. LUH, AND V. MARGARINT
+Symplectic Ensemble (GSE) respectively. An n × n real symmetric matrix A is
+drawn from the Gaussian Orthogonal Ensemble (GOE) if the upper-triangular en-
+tries Aij, 1 ≤ i ≤ j ≤ n are independent Guassian random variables, where Aij has
+mean zero and variance 1+δij
+n
+and δij is the Kronecker delta. The GUE and GSE
+ensembles are defined similarly with complex and quaternic Gaussian off-diagonal
+entries. We study the cases β = 1 and β = 2 respectively as they correspond to
+the critical parameters κ = 8 and κ = 4 in SLE theory. We expect that a similar
+analysis will hold for the case β = 4 that corresponds to the value κ = 2.
+We note that the N −(1/3−ǫ) order of convergence to the hydrodynamic limit
+of multiple SLE is obtained via an estimate in [44] which is, to the best of our
+knowledge, the best stability estimate in this setting available in the literature.
+Theorem 1.1 relies on the following technical result.
+Theorem 1.3. Let β = 1 or β = 2, and let us consider Dyson Brownian motion
+started from the origin
+�
+λ(1)
+t , . . . , λ(N)
+t
+�
+and M N
+t
+: C+ → C− defined by
+M N
+t (z) = 1
+N
+N
+�
+j=1
+2
+z − λ(j)
+t
+.
+Let M ∞
+t
+: C+ → C− be the solution to the complex Burgers equation
+(4)
+� ∂M∞
+t (z)
+∂t
+= −2M ∞
+t (z) ∂M∞
+t (z)
+∂z
+, t > 0,
+M ∞
+0 (z) = 2
+z.
+Then for any compact set G ⊂ C+, ε > 0, and fixed t ∈ [0, T ]
+(5)
+sup
+z∈G
+��M N
+t (z) − M ∞
+t (z)
+�� = OG,ε
+�
+t
+N
+1
+3 −ε
+�
+,
+with overwhelming probability.
+The remainder of the paper is organized into several sections. In the second
+section, we present probabilistic estimates involving the multiple SLE hull and sub-
+sets of its complement. The third section focuses on the random matrix techniques
+we use, as well as on the proof of Theorem 1.3. In subsection 3.4 we utilize a net
+argument that extends the previously obtained results for a fixed time t ∈ [0, T ], to
+all times simultaneously. In section 4, we prove Theorem 1.1 and in the Appendix
+we provide the stability part of the argument.
+2. Subset of the complement of the multiple SLE hull
+In this section, we provide probabilistic estimates for general β ≥ 1 that are
+useful in deducing the choice of the set G ⊂ H \ KT , where we establish the order
+of convergence of the family of maps. We present the estimates for general β ≥ 1,
+and specialize to the β = 1 and β = 2 cases in our application.
+Let ∂tgt(z) =
+1
+N
+�N
+i=1
+2
+gt(z)−λi
+t , where (λ(1)
+t , · · · , λ(N)
+t
+) is a Dyson Brownian
+motion (DBM) with parameter β ≥ 1. We first consider λi
+t ≡ 0, ∀t ∈ [0, T ],
+
+RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY
+5
+for all i = {1, 2, · · · , N}. Then, we have that ∂tgt(z) =
+2N
+Ngt(z) =
+2
+gt(z). Since
+gt(z) = Re(gt(z)) + iIm(gt(z)), we have that
+∂tIm(gt(z)) = −2Im(gt(z))
+|gt(z)|2
+≥
+2
+(Im(gt(z))2 .
+This allows us to conclude that
+Im (gt(z))2 ⩾ (Im(z))2 − 4t > 0,
+whenever Im(z) > 2
+√
+T.
+In order to control the real part, for a Dyson Brownian motion (λ(1)
+t , · · · , λ(N)
+t
+)
+with parameter β ≥ 1, we observe that
+∂tRe(gt(z)) = 1
+N
+N
+�
+i=1
+Re (gt(z)) − λi
+t
+��gt(z) − λi
+t
+��2
+> 0,
+whenever Re(gt(z)) > M = supt∈[0,T ] supi={1,2,...,N}
+��λi
+t
+�� . Then, combining the two
+estimates, we have that
+{z ∈ H| : |Re(z) > M or Im > 2
+√
+T} ⊂ H \ KT .
+We also note that for all t ∈ [0, T ], we have
+Kt ⊂ {z ∈ H : | Re z| ≤ M and Im z ≤ 2
+√
+T}.
+Next, we use the following probabilistic result on the behaviour of the extreme
+eigenvalues.
+Lemma 2.1 (Lemma 4.3.17 in [3]). Let λ∗
+N(t) := max1≤i≤N
+���λ(i)
+t
+��� = max
+�
+λ(N)
+t
+, −λ(1)
+t
+�
+.
+Let β ≥ 1. Then there exist finite constants α = α(β) > 0, C = C(β), and for all
+t ≥ 0 a random variable η∗
+N(t) with law independent of t, such that
+P (η∗
+N(t) ≥ x + C) ≤ e−αNx
+and, for all t ≥ 0,
+λ∗
+N(t) ≤ λ∗
+N(0) +
+√
+tη∗
+N(t).
+In the case of the DBM drivers, using Lemma 2.1, we have that for β ≥ 1 and
+for C = C(β) and α = α(β) some finite constants that
+P
+�
+sup
+t∈[0,T ]
+sup
+i={1,2,...,N}
+��λi
+t
+�� ≤ (C + x)
+√
+T
+�
+≥ 1 − e−αNx.
+For conformal maps, we have the following result.
+Lemma 2.2 (Lemma 4.5 in [36]). Let K be a hull and H = H\K. If K ⊂ B (x0, r),
+then gK maps H ∩ B (x0, 2r) into B (x0, 3r) and
+sup
+z∈H
+|gK(z) − z| ≤ 5r.
+For a box G ⊂ HT = H \ KT , we have that with overwhelming probability that
+(6)
+gN
+t (G) ⊂ {z :
+�
+Im(z0))2 − 4t ≤ Im(z) ≤ Im(z0); |Re(z)| ≤ f(N, T )},
+where f(N, T ) can be deduced from the following:
+
+6
+A. CAMPBELL, K. LUH, AND V. MARGARINT
+(7)
+|RegK(z)| ≤ |gK(z)| ≤ |z| + 5r.
+In the case of the multiple SLE hull KT , we have r =
+�
+M 2 + (2
+√
+T)2.
+3. Random Matrix Techniques
+In this section we prove some random matrix results leading to the proof of
+Theorem 1.3. It is worth noting that for β = 1 and β = 2, DBM
+�
+λ(1)
+t , . . . , λ(N)
+t
+�
+defined as the solution to (1) starting from initial positions
+�
+λ(1)
+0 , . . . , λ(N)
+0
+�
+is equal
+in distribution to the eigenvalues of D−2
+√
+tA where D is an N ×N diagonal matrix
+of the initial positions and A is a matrix drawn from the Gaussian Orthogonal
+Ensemble for β = 1 or Gaussian Unitary Ensemble (GUE) for β = 2. We establish
+the results in this section for the case when A is drawn from the GOE, since the
+adjustments to the GUE model are straightforward.
+3.1. Tools. This section introduces the tools we will use throughout. We begin
+with a definition describing high probability events.
+Definition 3.1 (High probability events). Let E be an event that depends on n.
+• E holds asymptotically almost surely if P(E) = 1 − o(1).
+• E holds with high probability if P(E) = 1−O(n−c) for some constant c > 0.
+• E holds with overwhelming probability if, for every p > 0, P(E) ≥ 1 −
+Op(n−p).
+For z = E + iη ∈ C+, n × n Hermitian matrix H, and G(z) := (H − zI)−1 the
+Ward identity states that
+(8)
+n
+�
+j=1
+|Gij(z)|2 = 1
+η Im Gii(z).
+If A and B are invertible matrices, the resolvent identity states that
+(9)
+A−1 − B−1 = A−1(B − A)B−1 = B−1(B − A)A−1.
+If ξ is a Gaussian random variable with mean zero and variance σ2 and f : R → C
+is continuously differentiable, the Gaussian integration by parts formula states that
+(10)
+E[ξf(ξ)] = σ2E[f ′(ξ)],
+provided the expectations are finite.
+The next lemma is a convenient moment
+bound for a martingale difference sequence.
+Lemma 3.2 (Lemma 2.12 from [4]). Let {Xk} be a complex martingale difference
+sequence and Fk = σ(X1, . . . , Xk) be the σ-algebra generated by X1, . . . , Xk. Then,
+for any p ≥ 2,
+E
+�����
+n
+�
+k=1
+Xk
+�����
+p
+≤ Cp
+
+E
+� n
+�
+k=1
+Ek−1|Xk|2
+�p/2
++ E
+n
+�
+k=1
+|Xk|p
+
+ .
+where Cp is a constant that only depends on p and Ek−1[·] := E[·|Fk−1].
+The next concentration lemma is helpful in controlling the deviation of a qua-
+dratic form from its expectation.
+
+RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY
+7
+Lemma 3.3 (Equation (3) from [1]). Let X be an n-vector containing iid standard
+Gaussian random variables, A a deterministic n × n matrix and ℓ ≥ 1 an integer.
+Then
+E[X∗AX − tr A|2ℓ ≤ Cℓ(tr AA∗)ℓ
+where Cℓ is a constant that only depends on ℓ.
+Finally, we will require the following algebraic identity in Section 3.2.
+Lemma 3.4 (Theorem A.5 from [4]). Let A be an n × n symmetric matrix and
+Ak be the k-th major submatrix of size (n − 1) × (n − 1). If A and Ak are both
+invertible, then
+tr(A−1) − tr(A−1
+k ) =
+1 + α∗
+kA−2
+k αk
+Akk − α∗
+kA−1
+k αk
+where αk is obtained from the k-th column of A by deleting the k-th entry.
+3.2. Concentration of the Gaussian Orthogonal Ensemble. In this section
+we show that |M N
+t (z)−EM N
+t (z)| is small for a fixed z ∈ C+. To match the random
+matrix literature we will consider for fixed t > 0, mN(z) := − 1
+2M N
+t (z). We let At
+be
+√
+tA where A is drawn from the Gaussian Orthogonal Ensemble.
+We note that mN(z) − EmN(z) can be written as the following telescopic sum
+mN(z) − EmN(z) =
+n
+�
+k=1
+(EkmN(z) − Ek−1mN(z)) :=
+N
+�
+k=1
+γk
+Observe that
+mN(z) = 1
+N tr(At − z)−1 = 1
+N tr
+1
+√
+tA − z =
+1
+N
+√
+t tr
+1
+A − z/
+√
+t =
+1
+N
+√
+t tr
+1
+A − z′
+We define E′ = E/
+√
+t and η′ = η/
+√
+t. Let Ek denote the conditional expectation
+with respect to the σ-field generated by Aij with i, j ≤ k, so that ENmN(z) =
+mN(z) and E0mN(z) = EmN(z).
+
+8
+A. CAMPBELL, K. LUH, AND V. MARGARINT
+γk =
+1
+N
+√
+t(Ek tr(A − z′)−1 − Ek−1 tr(A − z′)−1)
+=
+1
+N
+√
+t
+�
+Ek
+�
+tr(A − z′)−1 − (Ak − z′)−1�
+− Ek−1
+�
+tr(A − z′)−1 − tr(Ak − z′)−1��
+=
+1
+N
+√
+t(Ek − Ek−1)
+�
+a∗
+kG2
+kak − Eaka∗
+kG2
+kak
+Akk − z′ − a∗
+kGkak
++
+1 + Eaka∗
+kG2
+kak
+Akk − z′ − a∗
+kGkak
+−
+1 + Eaka∗
+kG2
+kak
+Akk − z′ − Eaka∗
+kGkak
+�
+=
+1
+N
+√
+t(Ek − Ek−1)
+�
+a∗
+kG2
+kak − Eaka∗
+kG2
+kak
+Akk − z′ − a∗
+kGkak
+−
+(1 + Eaka∗
+kG2
+kak)(a∗
+kGkak − Eaka∗
+kGkak)
+(Akk − z′ − a∗
+kGkak)(Akk − z′ − Eaka∗
+kGkak)
+�
+=
+1
+N
+√
+t(Ek − Ek−1)
+�
+a∗
+kG2
+kak − 1
+N tr G2
+k
+Akk − z′ − a∗
+kGkak
+−
+(1 + 1
+N tr G2
+k)(a∗
+kGkak − 1
+N tr Gk)
+(Akk − z′ − a∗
+kGkak)(Akk − z′ − 1
+N tr Gk)
+�
+where ak denotes the k-th row of A with the k-th entry removed. We define the
+following quantities,
+αk = a∗
+kG2
+kak − 1
+N tr G2
+k,
+βk =
+1
+Akk − z′ − a∗
+kGkak
+,
+¯βk =
+1
+Akk − z′ − 1
+N tr Gk
+,
+δk = a∗
+kGkak − 1
+N tr Gk,
+ǫk = 1 + 1
+N tr G2
+k,
+so that
+mN(z) − EmN(z) =
+1
+N
+√
+t
+N
+�
+k=1
+(Ek − Ek−1)αkβk −
+1
+N
+√
+t
+N
+�
+k=1
+(Ek − Ek−1)ǫkδkβk ¯βk
+:= 1
+√
+tS1 − 1
+√
+tS2.
+(11)
+For a fixed ε > 0, we will show that N 1−ε(η′)3|S1| = o(1) and N 1−ε(η′)3|S2| =
+o(1) with overwhelming probability. This will be done via the method of moments.
+We begin with S1. By Markov’s inequality, it suffices to bound E|N 1−ε(η′)3S1|2ℓ =
+E|N −ε(η′)3 �n
+k=1(Ek − Ek−1)αkβk|2ℓ for ℓ ∈ N.
+
+RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY
+9
+By Lemma 3.2, for any ℓ ≥ 1,
+E|N −ε(η′)3
+N
+�
+k=1
+(Ek − Ek−1)αkβk|2ℓ ≤ Cℓ
+�
+E
+� N
+�
+k=1
+Ek−1|N −ε(η′)3αkβk|2
+�ℓ
++
+N
+�
+k=1
+E|N −ε(η′)3αkβk|2ℓ
+�
+.
+We use Cℓ to indicate a constant that only depends on ℓ, but may change from line
+to line. Since Im a∗
+kGkak > 0,
+|βk| ≤ (η′)−1.
+Therefore,
+E
+�����N −ε(η′)3
+n
+�
+k=1
+(Ek − Ek−1)αkβk
+�����
+2ℓ
+≤ CℓN −2εℓ
+�
+E
+� N
+�
+k=1
+Ek−1|(η′)2αk|2
+�ℓ
++
+N
+�
+k=1
+E|(η′)2αk|2ℓ
+�
+.
+(12)
+By Lemma 3.3,
+E|(η′)2αk|2ℓ ≤ Cℓ(η′)4ℓN −2ℓE| tr G2
+kG∗2
+k |ℓ.
+We use the simple bound that
+tr G2
+kG∗2
+k =
+� N
+�
+i=1
+1
+((λi − E)2 + (η′)2)2
+�
+≤ N(η′)−4
+(13)
+We now have that
+E|(η′)2αk|2ℓ ≤ Cℓ(η′)4ℓN −2ℓE|N(η′)−4|ℓ
+≤ CℓN −ℓ
+Therefore, by equation (12),
+E
+�����N −ε(η′)3
+N
+�
+k=1
+(Ek − Ek−1)αkβk
+�����
+2ℓ
+≤ CℓN −2εℓ
+
+E
+� N
+�
+k=1
+Ek|(η′)2αk|2
+�ℓ
++ N −ℓ+1
+
+
+By the same reasoning as in (13), we also have that Ek|αk|2 ≤ N(η′)−4. Thus,
+Ek|(η′)2αk|2 ≤ KN −1
+so
+E
+� N
+�
+k=1
+Ek|(η′)2αk|2
+�ℓ
+≤ Cℓ.
+Finally, we can conclude that
+E
+�����N −ε(η′)3
+N
+�
+k=1
+(Ek−1 − Ek)αkβk
+�����
+2ℓ
+≤ CℓN −2εℓ.
+As ℓ is arbitrary, we have shown that |S1| = oη(t3/2/N 1−ε) with overwhelming
+probability.
+
+10
+A. CAMPBELL, K. LUH, AND V. MARGARINT
+Now we address S2. We first observe that
+����1 + 1
+N tr G2
+k
+���� ≤ 1 + 1
+N tr GkG∗
+k
+= (η′)−1 Im
+�
+−Akk + z′ + 1
+N tr Gk
+�
+Therefore,
+|ǫk ¯βk| =
+|1 + 1
+N tr G2
+k|
+|Akk − z′ − 1
+N tr Gk| ≤ (η′)−1
+Recalling that |βk| ≤ (η′)−1, we have that
+E|N 1−ε(η′)4S2|2ℓ = N −2εℓ(η′)2ℓ
+�����
+N
+�
+k=1
+(Ek − Ek−1)δk
+�����
+2ℓ
+Again, by Lemma 3.2
+E|N 1−ε(η′)4S2|2ℓ ≤ CℓN −2εℓ(η′)2ℓ
+
+E
+� N
+�
+k=1
+Ek−1|δk|2
+�ℓ
++
+N
+�
+K=1
+E|δk|2ℓ
+
+ .
+Note that by Lemma 3.3,
+E|δk|2ℓ ≤ CℓN −2ℓE| tr GkG∗
+k|ℓ.
+We have that
+tr GkG∗
+k ≤ N(η′)−2
+so
+E|δk|2ℓ ≤ CℓN −ℓ(η′)−2ℓ.
+Additionally,
+Ek−1|δk|2 ≤ N −1(η′)−2.
+Thus,
+E|N 1−ε(η′)4S2|2ℓ ≤ CℓN −2εℓ.
+We can then conclude that S2 is oη(t2/N 1−ε) with overwhelming probability. Re-
+turning to (11) we have shown that
+(14)
+|mN(z) − EmN(z)| = o
+�
+t
+N 1−ε
+�
+with overwhelming probability.
+3.3. Proof of Theorem 1.3. In this section we provide the proof of Theorem
+1.3. We will give begin the proof for generic initial starting positions of the Dyson
+Brownian motion, before specializing to the starting positions at the origin. Define
+the matrix
+(15)
+Lt = D − 2
+√
+tA
+where A is drawn from the Gaussian Orthogonal/Unitary Ensemble and D is an
+N × N deterministic diagonal matrix. Define the resolvent matrices
+Gt(z) := (Lt − zI)−1 ,
+and
+Q(z) := (D − zI)−1 .
+
+RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY
+11
+Next, we define the functions
+M N
+t (z) = − 2
+N tr Gt(z),
+and
+SN(z) = − 2
+N tr Q(z).
+Fix t, η > 0 and z such that Im(z) ≥ η. Additionally, define the matrices
+G := Gt(z),
+and
+Q := Q
+�
+z − 2tEM N
+t (z)
+�
+.
+In particular SN �
+z − 2tEM N
+t (z)
+�
+= − 2
+N tr Q. By the resolvent identity (9)
+EM N
+t (z) − SN �
+z − 2tEM N
+t (z)
+�
+= − 2
+N (tr Gt − tr Qt)
+(16)
+= −2E 1
+N tr
+�
+G ˜AQ
+�
++ 4tEM N
+t (z)E 1
+N tr (GQ)
+where ˜A = 2
+√
+tA. We now consider the term
+(17)
+− 2E 1
+N tr
+�
+G ˜AQ
+�
+= − 2
+N
+�
+i,j
+QiiE
+�
+Gij ˜Aji
+�
+.
+A computation involving the resolvent identity (9) shows that
+∂Gkl
+∂Aij
+=
+�
+GkiGji + GkjGil,
+if i ̸= j,
+GkiGjl,
+if i = j .
+Applying Gaussian integration by parts to (17) yields
+−2E 1
+N tr
+�
+G ˜AQ
+�
+= −8
+N 2 E
+�
+i,j
+QiiG2
+ij − 4t
+N EM N
+t (z) tr(QG),
+which when combined with (16) gives
+EM N
+t (z) − SN �
+z − 2tEM N
+t (z)
+�
+= −8
+N 2 E
+�
+i,j
+QiiG2
+ij − 4t
+N EM N
+t (z) tr(QG)
+(18)
++ 4tEM N
+t (z)E 1
+N tr (GQ) .
+We now fix z = E + iη ∈ C+. By the Ward identity (8)
+������
+8
+N 2 E
+�
+i,j
+QiiG2
+ij
+������
+≤ E 8
+N 2
+�
+j
+|Qii|
+�
+j
+|Gij|2
+≤ E
+8
+N 2η
+�
+i
+|Qii| Im Gii
+≤
+8
+Nη3 .
+
+12
+A. CAMPBELL, K. LUH, AND V. MARGARINT
+For the difference 4tEM N
+t (z)E 1
+N tr (GQ) − 4t
+N EM N
+t (z) tr(QG), note that
+����
+4t
+N tr (GQ)
+���� =
+�����
+4t
+N
+�
+i
+QiiGii
+�����
+≤ 4t
+η2 .
+It then follows from (14) with D equal to the zero matrix that
+E
+�����4t
+�
+EM N
+t (z)
+� 1
+N tr (GQ) − 4t
+N M N
+t (z) tr(QG)
+����
+�
+≤ E
+���M N
+t (z)E − EM N
+t (z)
+��
+����
+4t
+N tr (GQ)
+����
+�
+= o
+�4 max(t, t2)
+N 1−εη2
+�
+.
+Thus, we conclude that
+(19)
+EM N
+t (z) − SN �
+z − 2tEM N
+t (z)
+�
+= O
+�4 max(t, t2)
+N 1−εη3
+�
+,
+where SN(z) = 2
+z for all N. Let M ∞
+t
+be defined as in Theorem 1.3, then
+M ∞
+t (z) − SN (z − 2tM ∞
+t (z)) = 0.
+Note for each z ∈ C+, st = − 1
+2M ∞
+t (z), ˜st = − 1
+2EM N
+t (z), and s0(z) = − 1
+2SN(z)
+satisfy the conditions of Proposition A.1, (see Appendix) and hence it follows from
+Proposition A.1 and (44) (see Appendix) that
+(20)
+EM N
+t (z) − M ∞
+t (z) = O
+�41/3 max(t, t2)1/3
+N 1/3−εη
+�
+.
+Applying (14) to (20) completes the proof of Theorem 1.3.
+3.4. Extension to uniform bound over [0, T ]. In this section we outline how to
+extend Theorem 1.3 uniformly in t ∈ [0, T ]. This relies on the continuity of DBM.
+Without loss of generality, we work with the interval [0, 1] instead of the interval
+[0, T ]. Let us consider a partition of the time interval [0, 1] into a uniform partition
+with tk = k
+n, k = 0, 1, . . . , n. The intervals of this partition are all equally-sized and
+their lengths are equal to 1
+n.
+Let us consider t ∈ (t1, t2) an intermediate time. We have that
+sup
+z∈G
+|M ∞
+t (z) − M N
+t (z)|
+≤ sup
+z∈G
+|M ∞
+t (z) − M ∞
+t1 (z)| + sup
+z∈G
+|M ∞
+t1 (z) − M N
+t1 (z)| + sup
+z∈G
+|M N
+t1 (z) − M N
+t (z)|,
+(21)
+with G being a particular subset of the complement of the hull as in the previous
+section. The first term can be controlled from the Burgers equation as the solution
+is locally Lipschitz in time.
+For the second term of the right hand side of (21), we have that from Theorem
+1.3, for any ǫ > 0
+
+RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY
+13
+(22)
+sup
+z∈G
+|M ∞
+t1 (z) − M N
+t1 (z)| = Oε
+�
+1
+N 1/3−ǫ
+�
+with overwhelming probability, that is with probability at least 1 − e−cN, for some
+constant c.
+By a union bound for any tj, j = 1, . . . , n in the net, we have that
+P
+��
+ti
+|M ∞
+ti (z) − M N
+ti (z)| = Ω
+�
+C
+N 1/3−ǫ
+��
+≤
+n
+�
+i=1
+P
+�
+|M ∞
+ti (z) − M N
+ti (z)| = Ω
+�
+C
+N 1/3−ǫ
+��
+≤ ne−CN,
+(23)
+where g = Ω(f) means g(x)
+f(x), as x → ∞.
+For the third term of the right hand side of (21), using the notation ˜ηi
+t = z − λi
+t,
+for i = 1, 2, . . ., N, we have that
+(24)
+|M N
+t1 (z) − M N
+t (z)| ≤ 2
+N
+N
+�
+i=1
+|λi
+t − λi
+t1|
+|˜ηi
+t1 ˜ηi
+t|
+≤
+˜C|t − t1|1/2−ǫ
+Im(z0)2
+,
+where we have used the regularity of the Dyson Brownian Motion driver ( [43])
+and the bound |˜ηi
+t| ≥ | Im(z)| ≥ | Im(z0)| where z0 ∈ H such that Im(z0) ≤
+minz∈G(Im(z)).
+Using the notation ˆC =
+˜
+C
+Im(z0)2 , if we want the error to not accumulate in our
+net we need
+ˆC
+1
+n1/2−ǫ ≤
+C
+N 1/3−ǫ .
+Thus, for our partition of the time interval we have
+n >
+ˆC2(N (1/3−ǫ))2
+C2
+,
+for ˆC and C some constants. It then follows from (21), that
+(25)
+sup
+t∈[0,1], z∈G
+��M N
+t (z) − M ∞
+t (z)
+�� = O
+�
+1
+N
+1
+3 −ε
+�
+.
+4. Proof of Theorem 1.1
+In this section we will complete the proof of Theorem 1.1. Fix ε > 0. Let G be
+a suitable compact subset of C+ and let ˜G be a compact subset of C+ such that
+gN
+t (G) ⊆ ˜G with overwhelming probability (see (6) for the existence of such a ˜G).
+
+14
+A. CAMPBELL, K. LUH, AND V. MARGARINT
+Begin by defining η := minz∈ ˜
+G(Im z) > 0. Note that
+|gN
+t (z) − g∞
+t (z)| =
+����
+� t
+0
+M N
+s (gN
+s (z)) − M ∞
+s (g∞
+s (z))ds
+����
+(26)
+≤
+����
+� t
+0
+M N
+s (gN
+s (z)) − M ∞
+s (gN
+s (z))ds
+����
+(27)
++
+����
+� t
+0
+M ∞
+s (gN
+s (z)) − M ∞
+s (g∞
+s (z))ds
+���� .
+For the term M N
+s (gN
+s (z)) − M ∞
+s (gN
+s (z)), observe that from Theorem 1.3
+sup
+z∈ ˜
+G
+��M N
+s (z) − M ∞
+s (z)
+�� = O
+� 4T 2
+N
+1
+3 −ε
+�
+,
+for fixed s ∈ [0, T ] with overwhelming probability. From the argument in Section
+3.4 this can be extended to
+(28)
+sup
+s∈[0,T ], z∈ ˜
+G
+��M N
+s (z) − M ∞
+s (z)
+�� = O
+� 4T 2
+N
+1
+3 −ε
+�
+.
+For the term M ∞
+s (gN
+s (z)) − M ∞
+s (g∞
+s (z)), note that M ∞
+s
+is at most
+2
+η2 -Lipschitz on
+˜G, and hence
+(29)
+��M ∞
+s (gN
+s (z)) − M ∞
+s (g∞
+s (z))
+�� ≤ 2
+η2 |gN
+t (z) − g∞
+t (z)|.
+From (26), (28), and (29), we conclude that
+|gN
+t (z) − g∞
+t (z)| ≤ O
+� 4T 2
+N
+1
+3 −ε
+�
++
+� t
+0
+2
+η2 |gN
+s (z) − g∞
+s (z)|ds.
+Theorem 1.1 then follows from Gr¨onwall’s iequality.
+
+RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY
+15
+Appendix A. Stability
+The following is essentially a result of O’Rourke and Vu ( [44]). We provide the
+details for the time change for the convenience of the reader.
+Proposition A.1 (Stability for positive time). Let t > 0 and z, st, ˜st be elements
+of the upper half-plane such that
+(30)
+st = s0(z + 4tst),
+and
+(31)
+˜st = s0(z + 4t˜st) + O(ε),
+with s0(z) =
+�
+R
+dµ0
+x−z for some compactly supported probability measure µ0, some
+R ≥ z, and small ε > 0. Additionally assume there exists η > 0 such that Im(z) ≥ η.
+Then s, s′ = O(1) and
+(32)
+st = ˜st + O
+�
+ε1/3
+(4t)2/3η
+�
+.
+Proof. Showing st, ˜st = O(1) requires no change from O’Rourke and Vu.
+Let
+wt = z + 4tst and ˜wt = z + 4t˜st. We aim now to show |wt − ˜wt| is sufficiently
+small It follows from (30) and (31) that
+(33)
+s0(wt) − s0( ˜wt) = wt − ˜wt
+4t
++ O(ε).
+It additionally follows from the definition of s0 that
+s0(wt) − s0( ˜wt) = (wt − ˜wt)
+�
+R
+dµ0(z)
+(x − wt)(x − ˜wt),
+which when combined with (33) yields
+(34)
+�
+R
+dµ0(z)
+(x − wt)(x − ˜wt) = 1
+4t + O
+�
+ε
+|wt − ˜wt|
+�
+.
+On the other hand
+Im(st) = Im(s0(wt))
+= Im(wt)
+�
+R
+dµ0(x)
+|x − wt|2 ,
+(35)
+and rearranging yields
+(36)
+�
+R
+dµ0(x)
+|x − wt|2 =≤ 1
+4t.
+An identical argument yields
+(37)
+�
+R
+dµ0(x)
+|x − ˜wt|2 =≤ 1
+4t + O
+�ε
+η
+�
+.
+From the arithmetic mean-geometric mean inequality, we have
+����
+1
+(x − wt) (x − ˜wt)
+���� ≤ 1
+2
+1
+|x − wt|2 + 1
+2
+1
+|x − ˜wt|2 .
+
+16
+A. CAMPBELL, K. LUH, AND V. MARGARINT
+Since wt ̸= ˜wt, it follows that
+����Re
+�
+1
+(x − wt) (x − ˜wt)
+����� = (1 − δ)
+�
+1
+2
+1
+|x − wt|2 + 1
+2
+1
+|x − ˜wt|2
+�
+for some δ > 0. Then we have
+|x − wt| = (1 + O(δ)) |x − ˜wt| .
+and
+∠ (x − wt, x − ˜wt) = O
+�
+δ1/2�
+.
+Since x, wt, ˜wt = O(1), we obtain wt− ˜wt = O
+�
+δ1/2�
+. We obtain that Re
+�
+1
+(x−wt)(x− ˜
+wt)
+�
+≤
+�
+1 − C |wt − ˜wt|2� �
+1
+2
+1
+|x−wt|2 + 1
+2
+1
+|x− ˜wt|2
+�
+for some C > 0, and hence
+Re
+�
+R
+dµ(x)
+(x − wt) (x − ˜wt) ≤
+�
+1 − C |wt − ˜wt|2� � 1
+4t + O
+�ε
+η
+��
+.
+We have that
+(38)
+�
+1 − C|wt − ˜wt|2� � 1
+4t + O
+� ǫ
+η
+��
+= 1
+4t + O
+�
+ǫ
+|wt − ˜wt|
+�
+.
+Then,
+(39)
+1
+4t + O
+� ǫ
+η
+�
+− c
+4t|wt − ˜wt|2 − C|wt − ˜wt|2O
+� ǫ
+η
+�
+= 1
+4t + O
+�
+ǫ
+|wt − ˜wt|
+�
+.
+Furthermore, we obtain
+(40)
+O (ǫ) = |wt − ˜wt|O
+� ǫ
+η
+�
+− C
+4t|wt − ˜wt|3 − C|wt − ˜wt|3O
+� ǫ
+η
+�
+.
+Using that the first and the third term are bounded we obtain
+(41)
+O(ǫ) + O
+� ǫ
+η
+�
+= |wt − ˜wt|3 C
+4t.
+Thus, we have that
+(42)
+|wt − ˜wt| = O
+��4tǫ
+η
+�1/3�
+,
+and
+(43)
+|st − ˜st| = O
+�
+ǫ1/3
+(4t)2/3η1/3
+�
+.
+□
+For small t the following observation is useful. Fix η > 0, then s0 is Lipschitz
+with Lipschitz constant at most
+1
+η2 on {z : Im(z) ≥ η}. Thus for st and ˜st as in
+(30) and (31) one has
+(44)
+st − ˜st = 4t
+η2 (st − ˜st) + O (ε) .
+
+RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY
+17
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+Math., 131:73–99, 2017.
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+symmetries. Nuclear Phys. B, 867(2):429–447, 2013.
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+Ising model. Ann. of Math. (2), 172(2):1435–1467, 2010.
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+206(1):127–204, 2011.
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+ces. Electron. J. Probab., 16:no. 77, 2104–2121, 2011.
+
+RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY
+19
+[56] T. Tao and V. Vu. Random matrices: the universality phenomenon for Wigner ensembles.
+In Modern aspects of random matrix theory, volume 72 of Proc. Sympos. Appl. Math., pages
+121–172. Amer. Math. Soc., Providence, RI, 2014.
+[57] T. Tao and V. Vu. Random matrices: universality of local spectral statistics of non-Hermitian
+matrices. Ann. Probab., 43(2):782–874, 2015.
+[58] N.-G. K. Tom Alberts, Sung-Soo Byun and N. G. Makarov. Pole Dynamics and an Integral
+of Motion for Multiple SLE(0). https://arxiv.org/pdf/2011.05714/.
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+Math., 40(2):601–616, 2015.
+[60] E. P. Wigner. Characteristic vectors of bordered matrices with infinite dimensions. Annals of
+Mathematics, 62(3):548–564, 1955.
+[61] J. Wishart. The generalised product moment distribution in samples from a normal multi-
+variate population. Biometrika, pages 32–52, 1928.
+[62] D. Zhan. Two-curve Green’s function for 2-SLE: the interior case. Comm. Math. Phys.,
+375(1):1–40, 2020.
+[63] D. Zhan. Two-curve Green’s function for 2-SLE: the boundary case. Electron. J. Probab.,
+26:Paper No. 32, 58, 2021.
+Department of Mathematics, University of Colorado, Campus Box 395, Boulder, CO
+80309-0395, USA
+Email address: andrew.j.campbell@colorado.edu
+Department of Mathematics, University of Colorado, Campus Box 395, Boulder, CO
+80309-0395, USA
+Email address: kyle.luh@colorado.edu
+Department of Mathematics, University of Colorado, Campus Box 395, Boulder, CO
+80309-0395, USA
+Email address: vlad.dumitrumargarint@colorado.edu
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf,len=936
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='04722v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='PR]  11 Jan 2023  RATE OF CONVERGENCE IN MULTIPLE SLE USING  RANDOM MATRIX THEORY  ANDREW CAMPBELL, KYLE LUH, AND VLAD MARGARINT  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' We provide an order of convergence for a version of the Carath´eodory  convergence for the multiple SLE model with a Dyson Brownian motion driver  towards its hydrodynamic limit, for β = 1 and β = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' The result is obtained  by combining techniques from the field of Schramm-Loewner Evolutions with  modern techniques from random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Our approach shows how one can  apply modern tools used in the proof of universality in random matrix theory,  in the field of Schramm-Loewner Evolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Introduction and main results  Schramm-Loewner Evolution (SLE) and random matrix theory (RMT) are two  active and well-studied fields of research within modern probability theory [2,38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  The SLE was introduced by Oded Schramm in 2000 in his study of scaling limits  of various discrete processes [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' RMT appeared earlier in the statistical work of  Wishart [61] and the pioneering physics of Wigner [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Both SLE and RMT have  been thriving areas of mathematical research since their advent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  When studying SLE theory, one introduces the notion of compact hulls, which  are compact sets with simply connected complements in the upper half-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' If  Kt is a growing set of hulls parameterized by t ∈ [0, T ] and the growth is local in  some sense, then it is known that gt := gKt obeys the Loewner differential equation  ∂tgt(z) =  2  gt(z) − Wt  where Wt is referred to as the driving function and captures the local growth of Kt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  SLE are the random curves corresponding to the gt when the driving function is a  constant multiple of Brownian motion, that we denote by √κBt, for κ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' With  probability one, gt is continuous up to the boundary and the limit  γ(t) = lim  y→0 g−1  t  (√κBt + iy)  exists and is continuous in time, by the Rohde-Schramm Theorem [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' The curve  γ(t) is called the SLE trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Also, it can be shown that with probability one, gt is  a continuous family of conformal maps from Ht to H, where Ht is the unbounded  component of the complement in H of γ(t), for t ∈ [0, T ] [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Moreover, the nature  of the curve changes as κ increases from simple a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' when κ ∈ [0, 4], to having  double points a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' for κ ∈ (4, 8) and space-filling a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=', for κ ≥ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' For different  parameters κ, the SLE models the scaling limits of an astoundingly diverse set of  discrete models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' For instance, it was proved in [39] that the scaling limit of the loop  erased random walk (with the loops erased in a chronological order) converges in the  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Luh was supported in part by the Ralph E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Powe Junior Faculty Enhancement Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  1  2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' CAMPBELL, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' LUH, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' MARGARINT  scaling limit to SLEκ with κ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Moreover, other two dimensional discrete models  from Statistical Mechanics including the Ising model cluster boundaries, Gaussian  free field interfaces, percolation on the triangular lattice at critical probability, and  Uniform spanning trees were proved to converge in the scaling limit to SLE for  values of κ = 3, κ = 4, κ = 6 and κ = 8 respectively in the series of works [52], [50],  [51] and [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  One can consider more generally the Loewner equation driven by a time-dependent  real-valued measure µt  ∂  ∂tgt(z) =  �  R  µt(dx)  gt(z) − x,  g0(z) = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  When the driving measure µt is a Dirac-delta mass at location √κBt, we recover  the previous SLE maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' In the case µt = �N  i=1 ωi(t)δUi(t), for some non-intersecting  continuous functions Ui(t) ∈ R (called driving functions), and weights ωi(t) ∈ R+,  we obtain the multi-slit Loewner equation with driving functions Ui(t), i = 1, · · · , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  In this work, we consider the case ωi(t) = 1/N, for all t ∈ [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  For a real parameter β > 0, Dyson Brownian motion (DBM) is defined by the  following system of N equations  (1)  dλ(i)  t  =  2  √Nβ dB(i)  t  + 2  N  �  j̸=i  dt  λ(j)  t  − λ(i)  t  ,  for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Due to its connections with other fields, an important Loewner equation is the  multiple SLE with DBM as a driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' The multiple SLE maps that are obtained  when the driving measure is an empirical measure on N DBM particles are denoted  in this paper by gN  t (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' This model was introduced by Cardy in [10], and studied  further by Lawler and Healey in [30], in connection with the quantum Calogero-  Sutherland model and Conformal Field Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  More works on the connection  between Multiple SLE and CFT can be found in [40] and [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' In the case of N = 2  curves, perturbations of this model in the parameter β have been studied in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  We note that the parameters β in the DBM model and κ in SLE theory are related  via β = 8/κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  We refer to the multiple SLE model with Dyson Brownian motion as a driver  as the simultaneously growing multiple SLE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' There is also a version of the  multiple SLE that has non-simultaneous growth that has received a lot of attention  in the previous years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' There have been several results on the multiple SLE model  in both the upper half-plane and the unit disk versions [5, 12, 13, 15, 31, 32, 35, 37,  40,45,47,58,62,63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  In [14], the authors consider the N → ∞ limit of multiple SLE driven by DBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' In  particular they show that the empirical measure of the initial positions converges to  a probability measure µ0, then gN  t converges in distribution with respect to locally  uniform convergence to g∞  t  solving  (2)  ∂  ∂tg∞  t (z) = M ∞  t (z),  g0(z) = z,  Where M ∞  t  is a solution to the complex Burgers equation  (3)  � ∂M∞  t (z)  ∂t  = −2M ∞  t (z) ∂M∞  t (z)  ∂z  , t > 0,  M ∞  0 (z) =  �  R  2  z−xdµ0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY  3  Their result serves as the multiple SLE analog of Wigner’s famous semicircle law  in random matrix theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  We consider this model and we obtain more refined  information by providing an order of convergence of this model in a weaker ver-  sion of the Carath´eodory type convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' We aim in future works to study the  full Carath´eodory convergence by strengthening the estimates as we approach the  multiple SLE hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  In this work, we combine elements of the proof of Local Laws in random matrix  theory, such as resolvent techniques, with elements of the SLE theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' In other  words, we apply modern techniques from random matrix theory to the analysis of  SLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Local Laws are a very important research direction in random matrix theory  in the last years (see [9], [16], [18], [17], [19], [20], [21], [22], [25], [24], [26], [23],  [42], [44], [54], [53], [57], [56], [55] for a non-exhaustive list).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' They are one of the  fundamental ingredients in proving the universality of Wigner ensembles in random  matrix theory (see [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Given the outstanding developments in the proof of the universality in RMT  using the analysis of the DBM, the interaction between multiple SLE and random  matrices will provide many avenues to explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' The approach in the current work  represents one of the possible directions of exploration between these two major  fields of Probability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' In a different direction, which we aim to explore in the  future, one can study the geometry of the multiple SLE curves using the analysis  of the Dyson Brownian motion drivers, as well as good approximation schemes of  the model (see, for example, [28], [59], [33] in the one SLE curve case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Yet another  possibility is to study the continuity of the multiple SLE model in the parameter  β, motivated by the great interest and progress throughout the years in this yet  unresolved conjecture in the one SLE curve case (see [6], [7], [29], [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' In addition,  the fact that the multiple SLE curves grow from the positions of the drivers along  with some knowledge about the structure of the drivers gives the possibility of  defining new observables in order to study the convergence of discrete models to  the multiple SLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Examples of such observables include the statistics of the kth  smallest distance between drivers, for k ≥ 1, (see [8]) or the probability of having  no drivers in a symmetric region about the origin (see [41] for β = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Although our result can be obtained for general bounded initial conditions, we  state it in the case in which all the Dyson Brownian motion particles start from  the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' We prefer this choice for the simplicity of the notation and exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Let β = 1 or β = 2, and let us consider Dyson Brownian motion  beginning at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Let KT be the multiple SLE hull at time T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Then, for  any ε > 0, for the multiple SLE maps for N curves, we have that  sup  t∈[0,T ], z∈G  |gN  t (z) − g∞  t (z)| = O  �  1  N 1/3−ε  �  ,  with overwhelming probability1, for a given G ⊂ H \\ KT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' It is well-known that for the special values of the parameters β = 1,  β = 2 and β = 4, the Dyson Brownian motion particles statistics can be understood  using matrices as these values correspond to the well-studied models of the Gauss-  ian Orthogonal Ensemble, Gaussian Unitary Ensemble (GUE), and the Gaussian  1An event E holds with overwhelming probability if, for every p > 0, P(E) ≥ 1 − Op(n−p);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' see  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  4  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' CAMPBELL, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' LUH, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' MARGARINT  Symplectic Ensemble (GSE) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' An n × n real symmetric matrix A is  drawn from the Gaussian Orthogonal Ensemble (GOE) if the upper-triangular en-  tries Aij, 1 ≤ i ≤ j ≤ n are independent Guassian random variables, where Aij has  mean zero and variance 1+δij  n  and δij is the Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' The GUE and GSE  ensembles are defined similarly with complex and quaternic Gaussian off-diagonal  entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' We study the cases β = 1 and β = 2 respectively as they correspond to  the critical parameters κ = 8 and κ = 4 in SLE theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' We expect that a similar  analysis will hold for the case β = 4 that corresponds to the value κ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  We note that the N −(1/3−ǫ) order of convergence to the hydrodynamic limit  of multiple SLE is obtained via an estimate in [44] which is, to the best of our  knowledge, the best stability estimate in this setting available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='1 relies on the following technical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Let β = 1 or β = 2, and let us consider Dyson Brownian motion  started from the origin  �  λ(1)  t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' , λ(N)  t  �  and M N  t  : C+ → C− defined by  M N  t (z) = 1  N  N  �  j=1  2  z − λ(j)  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Let M ∞  t  : C+ → C− be the solution to the complex Burgers equation  (4)  � ∂M∞  t (z)  ∂t  = −2M ∞  t (z) ∂M∞  t (z)  ∂z  , t > 0,  M ∞  0 (z) = 2  z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Then for any compact set G ⊂ C+, ε > 0, and fixed t ∈ [0, T ]  (5)  sup  z∈G  ��M N  t (z) − M ∞  t (z)  �� = OG,ε  �  t  N  1  3 −ε  �  ,  with overwhelming probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  The remainder of the paper is organized into several sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' In the second  section, we present probabilistic estimates involving the multiple SLE hull and sub-  sets of its complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' The third section focuses on the random matrix techniques  we use, as well as on the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' In subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='4 we utilize a net  argument that extends the previously obtained results for a fixed time t ∈ [0, T ], to  all times simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' In section 4, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='1 and in the Appendix  we provide the stability part of the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Subset of the complement of the multiple SLE hull  In this section, we provide probabilistic estimates for general β ≥ 1 that are  useful in deducing the choice of the set G ⊂ H \\ KT , where we establish the order  of convergence of the family of maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' We present the estimates for general β ≥ 1,  and specialize to the β = 1 and β = 2 cases in our application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Let ∂tgt(z) =  1  N  �N  i=1  2  gt(z)−λi  t , where (λ(1)  t , · · · , λ(N)  t  ) is a Dyson Brownian  motion (DBM) with parameter β ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' We first consider λi  t ≡ 0, ∀t ∈ [0, T ],  RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY  5  for all i = {1, 2, · · · , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Then, we have that ∂tgt(z) =  2N  Ngt(z) =  2  gt(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Since  gt(z) = Re(gt(z)) + iIm(gt(z)), we have that  ∂tIm(gt(z)) = −2Im(gt(z))  |gt(z)|2  ≥  2  (Im(gt(z))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  This allows us to conclude that  Im (gt(z))2 ⩾ (Im(z))2 − 4t > 0,  whenever Im(z) > 2  √  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  In order to control the real part, for a Dyson Brownian motion (λ(1)  t , · · · , λ(N)  t  )  with parameter β ≥ 1, we observe that  ∂tRe(gt(z)) = 1  N  N  �  i=1  Re (gt(z)) − λi  t  ��gt(z) − λi  t  ��2  > 0,  whenever Re(gt(z)) > M = supt∈[0,T ] supi={1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=',N}  ��λi  t  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Then, combining the two  estimates, we have that  {z ∈ H| : |Re(z) > M or Im > 2  √  T} ⊂ H \\ KT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  We also note that for all t ∈ [0, T ], we have  Kt ⊂ {z ∈ H : | Re z| ≤ M and Im z ≤ 2  √  T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Next, we use the following probabilistic result on the behaviour of the extreme  eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='1 (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='17 in [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Let λ∗  N(t) := max1≤i≤N  ���λ(i)  t  ��� = max  �  λ(N)  t  , −λ(1)  t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Let β ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Then there exist finite constants α = α(β) > 0, C = C(β), and for all  t ≥ 0 a random variable η∗  N(t) with law independent of t, such that  P (η∗  N(t) ≥ x + C) ≤ e−αNx  and, for all t ≥ 0,  λ∗  N(t) ≤ λ∗  N(0) +  √  tη∗  N(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  In the case of the DBM drivers, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='1, we have that for β ≥ 1 and  for C = C(β) and α = α(β) some finite constants that  P  �  sup  t∈[0,T ]  sup  i={1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=',N}  ��λi  t  �� ≤ (C + x)  √  T  �  ≥ 1 − e−αNx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  For conformal maps, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='2 (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='5 in [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Let K be a hull and H = H\\K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' If K ⊂ B (x0, r),  then gK maps H ∩ B (x0, 2r) into B (x0, 3r) and  sup  z∈H  |gK(z) − z| ≤ 5r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  For a box G ⊂ HT = H \\ KT , we have that with overwhelming probability that  (6)  gN  t (G) ⊂ {z :  �  Im(z0))2 − 4t ≤ Im(z) ≤ Im(z0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' |Re(z)| ≤ f(N, T )},  where f(N, T ) can be deduced from the following:  6  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' CAMPBELL, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' LUH, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' MARGARINT  (7)  |RegK(z)| ≤ |gK(z)| ≤ |z| + 5r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  In the case of the multiple SLE hull KT , we have r =  �  M 2 + (2  √  T)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Random Matrix Techniques  In this section we prove some random matrix results leading to the proof of  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' It is worth noting that for β = 1 and β = 2, DBM  �  λ(1)  t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' , λ(N)  t  �  defined as the solution to (1) starting from initial positions  �  λ(1)  0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' , λ(N)  0  �  is equal  in distribution to the eigenvalues of D−2  √  tA where D is an N ×N diagonal matrix  of the initial positions and A is a matrix drawn from the Gaussian Orthogonal  Ensemble for β = 1 or Gaussian Unitary Ensemble (GUE) for β = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' We establish  the results in this section for the case when A is drawn from the GOE, since the  adjustments to the GUE model are straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' This section introduces the tools we will use throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' We begin  with a definition describing high probability events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='1 (High probability events).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Let E be an event that depends on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  E holds asymptotically almost surely if P(E) = 1 − o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  E holds with high probability if P(E) = 1−O(n−c) for some constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  E holds with overwhelming probability if, for every p > 0, P(E) ≥ 1 −  Op(n−p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  For z = E + iη ∈ C+, n × n Hermitian matrix H, and G(z) := (H − zI)−1 the  Ward identity states that  (8)  n  �  j=1  |Gij(z)|2 = 1  η Im Gii(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  If A and B are invertible matrices, the resolvent identity states that  (9)  A−1 − B−1 = A−1(B − A)B−1 = B−1(B − A)A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  If ξ is a Gaussian random variable with mean zero and variance σ2 and f : R → C  is continuously differentiable, the Gaussian integration by parts formula states that  (10)  E[ξf(ξ)] = σ2E[f ′(ξ)],  provided the expectations are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  The next lemma is a convenient moment  bound for a martingale difference sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='2 (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='12 from [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Let {Xk} be a complex martingale difference  sequence and Fk = σ(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' , Xk) be the σ-algebra generated by X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' , Xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Then,  for any p ≥ 2,  E  �����  n  �  k=1  Xk  �����  p  ≤ Cp  \uf8eb  \uf8edE  � n  �  k=1  Ek−1|Xk|2  �p/2  + E  n  �  k=1  |Xk|p  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  where Cp is a constant that only depends on p and Ek−1[·] := E[·|Fk−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  The next concentration lemma is helpful in controlling the deviation of a qua-  dratic form from its expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY  7  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='3 (Equation (3) from [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Let X be an n-vector containing iid standard  Gaussian random variables, A a deterministic n × n matrix and ℓ ≥ 1 an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Then  E[X∗AX − tr A|2ℓ ≤ Cℓ(tr AA∗)ℓ  where Cℓ is a constant that only depends on ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Finally, we will require the following algebraic identity in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='4 (Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='5 from [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Let A be an n × n symmetric matrix and  Ak be the k-th major submatrix of size (n − 1) × (n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' If A and Ak are both  invertible, then  tr(A−1) − tr(A−1  k ) =  1 + α∗  kA−2  k αk  Akk − α∗  kA−1  k αk  where αk is obtained from the k-th column of A by deleting the k-th entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Concentration of the Gaussian Orthogonal Ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' In this section  we show that |M N  t (z)−EM N  t (z)| is small for a fixed z ∈ C+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' To match the random  matrix literature we will consider for fixed t > 0, mN(z) := − 1  2M N  t (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' We let At  be  √  tA where A is drawn from the Gaussian Orthogonal Ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  We note that mN(z) − EmN(z) can be written as the following telescopic sum  mN(z) − EmN(z) =  n  �  k=1  (EkmN(z) − Ek−1mN(z)) :=  N  �  k=1  γk  Observe that  mN(z) = 1  N tr(At − z)−1 = 1  N tr  1  √  tA − z =  1  N  √  t tr  1  A − z/  √  t =  1  N  √  t tr  1  A − z′  We define E′ = E/  √  t and η′ = η/  √  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Let Ek denote the conditional expectation  with respect to the σ-field generated by Aij with i, j ≤ k, so that ENmN(z) =  mN(z) and E0mN(z) = EmN(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  8  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' CAMPBELL, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' LUH, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' MARGARINT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='γk =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='t(Ek tr(A − z′)−1 − Ek−1 tr(A − z′)−1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='Ek  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='tr(A − z′)−1 − (Ak − z′)−1�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='− Ek−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='tr(A − z′)−1 − tr(Ak − z′)−1��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='t(Ek − Ek−1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='a∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kG2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kak − Eaka∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kG2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='Akk − z′ − a∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kGkak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='1 + Eaka∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kG2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='Akk − z′ − a∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kGkak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='1 + Eaka∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kG2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='Akk − z′ − Eaka∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kGkak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='t(Ek − Ek−1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='a∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kG2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kak − Eaka∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kG2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='Akk − z′ − a∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kGkak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='(1 + Eaka∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kG2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kak)(a∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kGkak − Eaka∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kGkak)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='(Akk − z′ − a∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kGkak)(Akk − z′ − Eaka∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kGkak)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='t(Ek − Ek−1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='a∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kG2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kak − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='N tr G2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='Akk − z′ − a∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kGkak  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='(1 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='N tr G2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='k)(a∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kGkak − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='N tr Gk)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='(Akk − z′ − a∗  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='kGkak)(Akk − z′ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='N tr Gk)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='where ak denotes the k-th row of A with the k-th entry removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' We define the  following quantities,  αk = a∗  kG2  kak − 1  N tr G2  k,  βk =  1  Akk − z′ − a∗  kGkak  ,  ¯βk =  1  Akk − z′ − 1  N tr Gk  ,  δk = a∗  kGkak − 1  N tr Gk,  ǫk = 1 + 1  N tr G2  k,  so that  mN(z) − EmN(z) =  1  N  √  t  N  �  k=1  (Ek − Ek−1)αkβk −  1  N  √  t  N  �  k=1  (Ek − Ek−1)ǫkδkβk ¯βk  := 1  √  tS1 − 1  √  tS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  (11)  For a fixed ε > 0, we will show that N 1−ε(η′)3|S1| = o(1) and N 1−ε(η′)3|S2| =  o(1) with overwhelming probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' This will be done via the method of moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  We begin with S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' By Markov’s inequality, it suffices to bound E|N 1−ε(η′)3S1|2ℓ =  E|N −ε(η′)3 �n  k=1(Ek − Ek−1)αkβk|2ℓ for ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY  9  By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='2, for any ℓ ≥ 1,  E|N −ε(η′)3  N  �  k=1  (Ek − Ek−1)αkβk|2ℓ ≤ Cℓ  �  E  � N  �  k=1  Ek−1|N −ε(η′)3αkβk|2  �ℓ  +  N  �  k=1  E|N −ε(η′)3αkβk|2ℓ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  We use Cℓ to indicate a constant that only depends on ℓ, but may change from line  to line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Since Im a∗  kGkak > 0,  |βk| ≤ (η′)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Therefore,  E  �����N −ε(η′)3  n  �  k=1  (Ek − Ek−1)αkβk  �����  2ℓ  ≤ CℓN −2εℓ  �  E  � N  �  k=1  Ek−1|(η′)2αk|2  �ℓ  +  N  �  k=1  E|(η′)2αk|2ℓ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  (12)  By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='3,  E|(η′)2αk|2ℓ ≤ Cℓ(η′)4ℓN −2ℓE| tr G2  kG∗2  k |ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  We use the simple bound that  tr G2  kG∗2  k =  � N  �  i=1  1  ((λi − E)2 + (η′)2)2  �  ≤ N(η′)−4  (13)  We now have that  E|(η′)2αk|2ℓ ≤ Cℓ(η′)4ℓN −2ℓE|N(η′)−4|ℓ  ≤ CℓN −ℓ  Therefore, by equation (12),  E  �����N −ε(η′)3  N  �  k=1  (Ek − Ek−1)αkβk  �����  2ℓ  ≤ CℓN −2εℓ  \uf8eb  \uf8edE  � N  �  k=1  Ek|(η′)2αk|2  �ℓ  + N −ℓ+1  \uf8f6  \uf8f8  By the same reasoning as in (13), we also have that Ek|αk|2 ≤ N(η′)−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Thus,  Ek|(η′)2αk|2 ≤ KN −1  so  E  � N  �  k=1  Ek|(η′)2αk|2  �ℓ  ≤ Cℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Finally, we can conclude that  E  �����N −ε(η′)3  N  �  k=1  (Ek−1 − Ek)αkβk  �����  2ℓ  ≤ CℓN −2εℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  As ℓ is arbitrary, we have shown that |S1| = oη(t3/2/N 1−ε) with overwhelming  probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  10  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' CAMPBELL, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' LUH, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' MARGARINT  Now we address S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' We first observe that  ����1 + 1  N tr G2  k  ���� ≤ 1 + 1  N tr GkG∗  k  = (η′)−1 Im  �  −Akk + z′ + 1  N tr Gk  �  Therefore,  |ǫk ¯βk| =  |1 + 1  N tr G2  k|  |Akk − z′ − 1  N tr Gk| ≤ (η′)−1  Recalling that |βk| ≤ (η′)−1, we have that  E|N 1−ε(η′)4S2|2ℓ = N −2εℓ(η′)2ℓ  �����  N  �  k=1  (Ek − Ek−1)δk  �����  2ℓ  Again, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='2  E|N 1−ε(η′)4S2|2ℓ ≤ CℓN −2εℓ(η′)2ℓ  \uf8eb  \uf8edE  � N  �  k=1  Ek−1|δk|2  �ℓ  +  N  �  K=1  E|δk|2ℓ  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Note that by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='3,  E|δk|2ℓ ≤ CℓN −2ℓE| tr GkG∗  k|ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  We have that  tr GkG∗  k ≤ N(η′)−2  so  E|δk|2ℓ ≤ CℓN −ℓ(η′)−2ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Additionally,  Ek−1|δk|2 ≤ N −1(η′)−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Thus,  E|N 1−ε(η′)4S2|2ℓ ≤ CℓN −2εℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  We can then conclude that S2 is oη(t2/N 1−ε) with overwhelming probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Re-  turning to (11) we have shown that  (14)  |mN(z) − EmN(z)| = o  �  t  N 1−ε  �  with overwhelming probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' In this section we provide the proof of Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' We will give begin the proof for generic initial starting positions of the Dyson  Brownian motion, before specializing to the starting positions at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Define  the matrix  (15)  Lt = D − 2  √  tA  where A is drawn from the Gaussian Orthogonal/Unitary Ensemble and D is an  N × N deterministic diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Define the resolvent matrices  Gt(z) := (Lt − zI)−1 ,  and  Q(z) := (D − zI)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY  11  Next, we define the functions  M N  t (z) = − 2  N tr Gt(z),  and  SN(z) = − 2  N tr Q(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Fix t, η > 0 and z such that Im(z) ≥ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Additionally, define the matrices  G := Gt(z),  and  Q := Q  �  z − 2tEM N  t (z)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  In particular SN �  z − 2tEM N  t (z)  �  = − 2  N tr Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' By the resolvent identity (9)  EM N  t (z) − SN �  z − 2tEM N  t (z)  �  = − 2  N (tr Gt − tr Qt)  (16)  = −2E 1  N tr  �  G ˜AQ  �  + 4tEM N  t (z)E 1  N tr (GQ)  where ˜A = 2  √  tA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' We now consider the term  (17)  − 2E 1  N tr  �  G ˜AQ  �  = − 2  N  �  i,j  QiiE  �  Gij ˜Aji  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  A computation involving the resolvent identity (9) shows that  ∂Gkl  ∂Aij  =  �  GkiGji + GkjGil,  if i ̸= j,  GkiGjl,  if i = j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Applying Gaussian integration by parts to (17) yields  −2E 1  N tr  �  G ˜AQ  �  = −8  N 2 E  �  i,j  QiiG2  ij − 4t  N EM N  t (z) tr(QG),  which when combined with (16) gives  EM N  t (z) − SN �  z − 2tEM N  t (z)  �  = −8  N 2 E  �  i,j  QiiG2  ij − 4t  N EM N  t (z) tr(QG)  (18)  + 4tEM N  t (z)E 1  N tr (GQ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  We now fix z = E + iη ∈ C+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' By the Ward identity (8)  ������  8  N 2 E  �  i,j  QiiG2  ij  ������  ≤ E 8  N 2  �  j  |Qii|  �  j  |Gij|2  ≤ E  8  N 2η  �  i  |Qii| Im Gii  ≤  8  Nη3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  12  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' CAMPBELL, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' LUH, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' MARGARINT  For the difference 4tEM N  t (z)E 1  N tr (GQ) − 4t  N EM N  t (z) tr(QG), note that  ����  4t  N tr (GQ)  ���� =  �����  4t  N  �  i  QiiGii  �����  ≤ 4t  η2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  It then follows from (14) with D equal to the zero matrix that  E  �����4t  �  EM N  t (z)  � 1  N tr (GQ) − 4t  N M N  t (z) tr(QG)  ����  �  ≤ E  ���M N  t (z)E − EM N  t (z)  ��  ����  4t  N tr (GQ)  ����  �  = o  �4 max(t, t2)  N 1−εη2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Thus, we conclude that  (19)  EM N  t (z) − SN �  z − 2tEM N  t (z)  �  = O  �4 max(t, t2)  N 1−εη3  �  ,  where SN(z) = 2  z for all N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Let M ∞  t  be defined as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='3, then  M ∞  t (z) − SN (z − 2tM ∞  t (z)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Note for each z ∈ C+, st = − 1  2M ∞  t (z), ˜st = − 1  2EM N  t (z), and s0(z) = − 1  2SN(z)  satisfy the conditions of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='1, (see Appendix) and hence it follows from  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='1 and (44) (see Appendix) that  (20)  EM N  t (z) − M ∞  t (z) = O  �41/3 max(t, t2)1/3  N 1/3−εη  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Applying (14) to (20) completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Extension to uniform bound over [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' In this section we outline how to  extend Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='3 uniformly in t ∈ [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' This relies on the continuity of DBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Without loss of generality, we work with the interval [0, 1] instead of the interval  [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Let us consider a partition of the time interval [0, 1] into a uniform partition  with tk = k  n, k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' The intervals of this partition are all equally-sized and  their lengths are equal to 1  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Let us consider t ∈ (t1, t2) an intermediate time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' We have that  sup  z∈G  |M ∞  t (z) − M N  t (z)|  ≤ sup  z∈G  |M ∞  t (z) − M ∞  t1 (z)| + sup  z∈G  |M ∞  t1 (z) − M N  t1 (z)| + sup  z∈G  |M N  t1 (z) − M N  t (z)|,  (21)  with G being a particular subset of the complement of the hull as in the previous  section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' The first term can be controlled from the Burgers equation as the solution  is locally Lipschitz in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  For the second term of the right hand side of (21), we have that from Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='3, for any ǫ > 0  RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY  13  (22)  sup  z∈G  |M ∞  t1 (z) − M N  t1 (z)| = Oε  �  1  N 1/3−ǫ  �  with overwhelming probability, that is with probability at least 1 − e−cN, for some  constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  By a union bound for any tj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' , n in the net, we have that  P  ��  ti  |M ∞  ti (z) − M N  ti (z)| = Ω  �  C  N 1/3−ǫ  ��  ≤  n  �  i=1  P  �  |M ∞  ti (z) − M N  ti (z)| = Ω  �  C  N 1/3−ǫ  ��  ≤ ne−CN,  (23)  where g = Ω(f) means g(x)  f(x), as x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  For the third term of the right hand side of (21), using the notation ˜ηi  t = z − λi  t,  for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=', N, we have that  (24)  |M N  t1 (z) − M N  t (z)| ≤ 2  N  N  �  i=1  |λi  t − λi  t1|  |˜ηi  t1 ˜ηi  t|  ≤  ˜C|t − t1|1/2−ǫ  Im(z0)2  ,  where we have used the regularity of the Dyson Brownian Motion driver ( [43])  and the bound |˜ηi  t| ≥ | Im(z)| ≥ | Im(z0)| where z0 ∈ H such that Im(z0) ≤  minz∈G(Im(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Using the notation ˆC =  ˜  C  Im(z0)2 , if we want the error to not accumulate in our  net we need  ˆC  1  n1/2−ǫ ≤  C  N 1/3−ǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Thus, for our partition of the time interval we have  n >  ˆC2(N (1/3−ǫ))2  C2  ,  for ˆC and C some constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' It then follows from (21), that  (25)  sup  t∈[0,1], z∈G  ��M N  t (z) − M ∞  t (z)  �� = O  �  1  N  1  3 −ε  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='1  In this section we will complete the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Fix ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Let G be  a suitable compact subset of C+ and let ˜G be a compact subset of C+ such that  gN  t (G) ⊆ ˜G with overwhelming probability (see (6) for the existence of such a ˜G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  14  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' CAMPBELL, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' LUH, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' MARGARINT  Begin by defining η := minz∈ ˜  G(Im z) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Note that  |gN  t (z) − g∞  t (z)| =  ����  � t  0  M N  s (gN  s (z)) − M ∞  s (g∞  s (z))ds  ����  (26)  ≤  ����  � t  0  M N  s (gN  s (z)) − M ∞  s (gN  s (z))ds  ����  (27)  +  ����  � t  0  M ∞  s (gN  s (z)) − M ∞  s (g∞  s (z))ds  ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  For the term M N  s (gN  s (z)) − M ∞  s (gN  s (z)), observe that from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='3  sup  z∈ ˜  G  ��M N  s (z) − M ∞  s (z)  �� = O  � 4T 2  N  1  3 −ε  �  ,  for fixed s ∈ [0, T ] with overwhelming probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' From the argument in Section  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='4 this can be extended to  (28)  sup  s∈[0,T ], z∈ ˜  G  ��M N  s (z) − M ∞  s (z)  �� = O  � 4T 2  N  1  3 −ε  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  For the term M ∞  s (gN  s (z)) − M ∞  s (g∞  s (z)), note that M ∞  s  is at most  2  η2 -Lipschitz on  ˜G, and hence  (29)  ��M ∞  s (gN  s (z)) − M ∞  s (g∞  s (z))  �� ≤ 2  η2 |gN  t (z) − g∞  t (z)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  From (26), (28), and (29), we conclude that  |gN  t (z) − g∞  t (z)| ≤ O  � 4T 2  N  1  3 −ε  �  +  � t  0  2  η2 |gN  s (z) − g∞  s (z)|ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='1 then follows from Gr¨onwall’s iequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY  15  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Stability  The following is essentially a result of O’Rourke and Vu ( [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' We provide the  details for the time change for the convenience of the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='1 (Stability for positive time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Let t > 0 and z, st, ˜st be elements  of the upper half-plane such that  (30)  st = s0(z + 4tst),  and  (31)  ˜st = s0(z + 4t˜st) + O(ε),  with s0(z) =  �  R  dµ0  x−z for some compactly supported probability measure µ0, some  R ≥ z, and small ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Additionally assume there exists η > 0 such that Im(z) ≥ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Then s, s′ = O(1) and  (32)  st = ˜st + O  �  ε1/3  (4t)2/3η  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Showing st, ˜st = O(1) requires no change from O’Rourke and Vu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Let  wt = z + 4tst and ˜wt = z + 4t˜st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' We aim now to show |wt − ˜wt| is sufficiently  small It follows from (30) and (31) that  (33)  s0(wt) − s0( ˜wt) = wt − ˜wt  4t  + O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  It additionally follows from the definition of s0 that  s0(wt) − s0( ˜wt) = (wt − ˜wt)  �  R  dµ0(z)  (x − wt)(x − ˜wt),  which when combined with (33) yields  (34)  �  R  dµ0(z)  (x − wt)(x − ˜wt) = 1  4t + O  �  ε  |wt − ˜wt|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  On the other hand  Im(st) = Im(s0(wt))  = Im(wt)  �  R  dµ0(x)  |x − wt|2 ,  (35)  and rearranging yields  (36)  �  R  dµ0(x)  |x − wt|2 =≤ 1  4t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  An identical argument yields  (37)  �  R  dµ0(x)  |x − ˜wt|2 =≤ 1  4t + O  �ε  η  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  From the arithmetic mean-geometric mean inequality, we have  ����  1  (x − wt) (x − ˜wt)  ���� ≤ 1  2  1  |x − wt|2 + 1  2  1  |x − ˜wt|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  16  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' CAMPBELL, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' LUH, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' MARGARINT  Since wt ̸= ˜wt, it follows that  ����Re  �  1  (x − wt) (x − ˜wt)  ����� = (1 − δ)  �  1  2  1  |x − wt|2 + 1  2  1  |x − ˜wt|2  �  for some δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Then we have  |x − wt| = (1 + O(δ)) |x − ˜wt| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  and  ∠ (x − wt, x − ˜wt) = O  �  δ1/2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Since x, wt, ˜wt = O(1), we obtain wt− ˜wt = O  �  δ1/2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' We obtain that Re  �  1  (x−wt)(x− ˜  wt)  �  ≤  �  1 − C |wt − ˜wt|2� �  1  2  1  |x−wt|2 + 1  2  1  |x− ˜wt|2  �  for some C > 0, and hence  Re  �  R  dµ(x)  (x − wt) (x − ˜wt) ≤  �  1 − C |wt − ˜wt|2� � 1  4t + O  �ε  η  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  We have that  (38)  �  1 − C|wt − ˜wt|2� � 1  4t + O  � ǫ  η  ��  = 1  4t + O  �  ǫ  |wt − ˜wt|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Then,  (39)  1  4t + O  � ǫ  η  �  − c  4t|wt − ˜wt|2 − C|wt − ˜wt|2O  � ǫ  η  �  = 1  4t + O  �  ǫ  |wt − ˜wt|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Furthermore, we obtain  (40)  O (ǫ) = |wt − ˜wt|O  � ǫ  η  �  − C  4t|wt − ˜wt|3 − C|wt − ˜wt|3O  � ǫ  η  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Using that the first and the third term are bounded we obtain  (41)  O(ǫ) + O  � ǫ  η  �  = |wt − ˜wt|3 C  4t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Thus, we have that  (42)  |wt − ˜wt| = O  ��4tǫ  η  �1/3�  ,  and  (43)  |st − ˜st| = O  �  ǫ1/3  (4t)2/3η1/3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  □  For small t the following observation is useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Fix η > 0, then s0 is Lipschitz  with Lipschitz constant at most  1  η2 on {z : Im(z) ≥ η}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Thus for st and ˜st as in  (30) and (31) one has  (44)  st − ˜st = 4t  η2 (st − ˜st) + O (ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY  17  References  [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Adamczak, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Lata�l a, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Meller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Hanson-Wright inequality in Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
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+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
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+page_content=' Yau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Semicircle law on short scales and delocalization of  eigenvectors for Wigner random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
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+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Yau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
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+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
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+page_content=' Schlein, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
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+page_content=' Yau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
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+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
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+page_content=' Yau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Universality of local spectral statistics of random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
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+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
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+page_content=' Conformal welding problem, flow line problem, and multiple  Schramm-Loewner evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
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+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=',  375(1):1–40, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  [63] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Zhan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Two-curve Green’s function for 2-SLE: the boundary case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=',  26:Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content=' 32, 58, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='  Department of Mathematics, University of Colorado, Campus Box 395, Boulder, CO  80309-0395, USA  Email address: andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='campbell@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='edu  Department of Mathematics, University of Colorado, Campus Box 395, Boulder, CO  80309-0395, USA  Email address: kyle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='luh@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
+page_content='edu  Department of Mathematics, University of Colorado, Campus Box 395, Boulder, CO  80309-0395, USA  Email address: vlad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfywtH/content/2301.04722v1.pdf'}
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diff --git a/TtE3T4oBgHgl3EQfzwsF/content/tmp_files/2301.04730v1.pdf.txt b/TtE3T4oBgHgl3EQfzwsF/content/tmp_files/2301.04730v1.pdf.txt
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index 0000000000000000000000000000000000000000..d9b67e9752dfdd59b577ff36ba9dede037cd99eb
--- /dev/null
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+Graphene amplifier reaches the quantum-noise limit
+Kin Chung Fong1, ∗
+1Raytheon BBN Technologies, Quantum Engineering and Computing Group, Cambridge, Massachusetts 02138, USA
+(Dated: January 13, 2023)
+To make yourself heard in a noisy environment
+is no easy task.
+An amplifier, like a megaphone,
+can come to your rescue by increasing your voice’s
+volume over the background noise.
+Your speech
+can be heard clearly.
+This is analogous to mea-
+suring superconducting qubits. Since their energy
+quanta are a few orders of magnitude smaller than
+the thermal noise, amplifiers are necessary to boost
+the signal up before being registered by apparatus
+at room temperature. However, having a high gain
+from amplifiers is not nearly enough. The physical
+process of amplification is also subjected to fluctua-
+tions, resulting in added noise by the amplifier that
+can degrade the signal-to-noise ratio. For a phase-
+insensitive linear amplifier, the minimum amount of
+this added noise is half a quantum because of quan-
+tum fluctuations[1].
+Employing a parametric pro-
+cess to achieve this fundamental limit of amplifica-
+tion at radio and microwave frequencies has a long
+history: from using variable-capacitor diodes in the
+1960s to Josephson junctions in the 1980s[2]. With
+high gain and low noise, modern Josephson para-
+metric amplifiers (JPAs)[3] have quickly become a
+must-have in laboratories[4]: enabling high-fidelity
+qubit readouts, observing quantum jumps, tracking
+quantum trajectory, and even searching for the rare
+event when the axion dark matter converts into a
+microwave photon under a high magnetic field.
+Writing in Nature Nanotechnology, two indepen-
+dent reports by Guilliam Butseraen, et. al.[5] and
+Joydip Sarkar, et. al.[6], now further advance JPAs
+using a two-dimensional material—graphene.
+We can consult children on swings about the pro-
+cess of parametric amplification[7] (Fig. 1a). A child
+can amplify the pendulum oscillation by standing
+up and squatting down when the swing reaches its
+maximum and minimum height, respectively. This
+stand-and-squat action is pumping the pendulum
+motion at twice its resonant frequency, and doing
+work on the harmonic oscillator. The amplitude of
+the originally small oscillation (signal) will gradu-
+ally increase, i.e. amplify. Parametric amplification
+sets apart from other amplification mechanisms be-
+∗ fongkc@gmail.com
+cause it modulates the reactance, rather than the
+resistance of the system. As such, it minimizes the
+noise that is inevitably brought into the amplifica-
+tion process according to the fluctuation-dissipation
+theorem.
+A JPA constructed by a superconducting LC res-
+onator, schematically shown in Fig.
+1b, operates
+under the similar principle. In addition to suppress-
+ing dissipation, superconductors can form Josephson
+junctions, which can provide an inductance—from
+the inertia of the Cooper pairs tunneling through the
+junction—to the circuit. When two Josephson junc-
+tions are made in the form of a loop, their supercur-
+rents will interfere and form a device known as super-
+conducting quantum interference device (SQUID)
+(Fig. 1c-d). Due to the Aharonov-Bohm phase, the
+total critical current of the SQUID depends on the
+magnetic flux through the loop. A magnetic field
+generated by an electrical current can control the
+Josephson inductance.
+Hence, we can pump the
+JPA by modulating its resonant frequency with a
+pump current at twice the frequency of the super-
+conducting resonators. Alternatively, JPAs can be
+powered up by feeding the pump tone directly into
+the input port for parametric amplification.
+The
+second method exploits the non-linear dependence
+of Josephson inductance on the magnitude of the
+current running through the junction, which allow
+for non-degenerate parametric amplifications. JPAs
+operate as a reflection amplifier: when the minute
+signal enters the JPA via the coupling capacitor, it
+is parametrically amplified before making its way
+back to the input port.
+The two research teams now replace the insula-
+tor traditionally used as the Josephson weak link
+with a layer of graphene encapsulated by hexagonal-
+boron nitride (Fig.
+1e-f).
+By doing so, the re-
+searchers can employ the gate voltage response of
+the graphene to tune the Josephson critical current
+and thus the Josephson inductance.
+Hence, they
+can control the resonant frequency of the JPA by
+voltage, rather than current. Butseraen, et. al. at-
+tain parametric amplification using a transmission-
+line resonator at ∼5 GHz and a pump tone through
+the gate; whereas Sarkar, et. al. exploit lump com-
+ponents and a pump tone applied at the input port,
+arXiv:2301.04730v1  [cond-mat.mes-hall]  11 Jan 2023
+
+FIG. 1. (a) Parametric amplification in a mechanical oscillator. The red dot marks the center of mass of the child
+in the swing. Its oscillation periodically modulates the effective length of the pendulum, and thus, the resonant
+frequency. (b) Schematic model of the parametric amplifier at radio or microwave frequencies. The signal will enter
+the LC resonator and be amplified by the modulation of the inductance before leaving the resonator. (c) A SQUID as
+an implementation of the variable inductor. The Josephson inductance depends inversely on the total critical current,
+which is controlled by the magnetic flux through the loop. (d) Josephson junctions based on superconductor-insulator-
+superconductor (SIS) heterostructure. (e) Graphene-based Josephson junction with a gate control can operate as
+a variable inductor. (f) Josephson junctions based on superconductor-normal metal-superconductor (SNS) lateral
+junctions. The latest reports demonstrate JPA by SNS junction with graphene, encapsulated in hexagonal-boron
+nitride (blue), as the weak link. This new JPA is voltage-tunable and can operate under a high-magnetic field.
+respectively.
+They both achieve similar figures of
+merit: ∼10 MHz bandwidth, ∼500 MHz tuning of
+the resonant frequency with a gain of >20 dB, and
+1-dB compression point at about -127 dBm of in-
+put power.
+Most importantly, both teams show
+that the added noise from the graphene JPA can
+reach the quantum limit. This is a pleasant surprise
+given that graphene Josephson junctions are more
+dissipative with a lower quality-factor at microwave
+frequency[8], than the conventional superconductor-
+insulator-superconductor (SIS) junctions.
+With
+their careful designs and implementations, the two
+teams demonstrate that a superconductor-normal
+metal-superconductor (SNS) junction can overcome
+its intrinsic dissipation to achieve quantum-limited
+amplification.
+Given the ubiquity of JPAs based on SQUIDs, it
+is natural to question the utility of graphene JPA
+beyond mere scientific curiosity. One of the poten-
+tial benefits may lie in the mitigation of cross-talks
+among different tunable components in the quantum
+circuitry due to the magnetic field generated by a
+current bias. Without the SQUID loop, graphene
+JPAs can also be more densely packed and oper-
+ate under a larger in-plane magnetic field.
+More-
+over, these latest results exemplify the renewed in-
+terest in applying SNS junctions to quantum sci-
+ence. Unlike their insulating counterpart, SNS junc-
+tions possess more variety in their physical proper-
+ties, arising from the interplay of the superconduc-
+tor and the materials in the weak link.
+Not only
+would it enable voltage control of quantum ampli-
+fiers as shown here, but, by developing this mo-
+tif, we can also create new hybrid superconducting
+electronics[9], qubits[10, 11], high-sensitivity pho-
+ton detectors[12], and topological superconductivity
+which may host Majorana zero modes[13]. With the
+rise of quantum technology, we shall expect more
+innovations from SNS junctions to come!
+The author declares no conflict of interest.
+[1] C. M. Caves, Physical Review D 26, 1817 (1982).
+[2] B. Yurke, P. G. Kaminsky, R. E. Miller, E. A. Whit-
+taker, A. D. Smith, A. H. Silver, and R. W. Simon,
+Physical Review Letters 60, 764 (1987).
+[3] M. A. Castellanos-Beltran and K. W. Lehnert, Ap-
+plied Physics Letters 91, 083509 (2007).
+[4] J. Aumentado, IEEE Microwave Magazine 21, 45
+(2020).
+[5] G. Butseraen, A. Ranadive, N. Aparicio, K. R.
+Amin,
+A.
+Juyal,
+M.
+Esposito,
+K.
+Watanabe,
+2
+
+a
+a
+S
+**
+S
+OB
+e
+gateT. Taniguchi, N. Roch, F. Lefloch, et al., Nature
+Nanotechnology 17, 1153 (2022).
+[6] J. Sarkar, K. V. Salunkhe, S. Mandal, S. Ghatak,
+A.
+H.
+Marchawala,
+I.
+Das,
+K.
+Watanabe,
+T. Taniguchi, R. Vijay, and M. M. Deshmukh, Na-
+ture Nanotechnology 17, 1147 (2022).
+[7] D. Rugar and P. Grtter, Physical Review Letters
+67, 699 (1991).
+[8] R. Haller,
+G. Flp,
+D. Indolese,
+J. Ridderbos,
+R. Kraft, L. Y. Cheung, J. H. Ungerer, K. Watan-
+abe, T. Taniguchi, D. Beckmann, et al., Physical
+Review Research 4, 013198 (2022).
+[9] J. J. A. Baselmans, A. F. Morpurgo, B. J. v. Wees,
+and T. M. Klapwijk, Nature 397, 43 (1999).
+[10] J. I.-J. Wang, D. Rodan-Legrain, L. Bretheau, D. L.
+Campbell, B. Kannan, D. Kim, M. Kjaergaard,
+P. Krantz, G. O. Samach, F. Yan, et al., Nature
+Nanotechnology 14, 120 (2019).
+[11] M. Hays, V. Fatemi, D. Bouman, J. Cerrillo, S. Dia-
+mond, K. Serniak, T. Connolly, P. Krogstrup, J. Ny-
+grd, A. L. Yeyati, et al., Science 373, 430 (2021).
+[12] E. D. Walsh, W. Jung, G.-H. Lee, D. K. Efetov,
+B.-I. Wu, K. F. Huang, T. A. Ohki, T. Taniguchi,
+K. Watanabe, P. Kim, et al., Science 372, 409
+(2021).
+[13] A. Fornieri, A. M. Whiticar, F. Setiawan, E. Por-
+tols, A. C. C. Drachmann, A. Keselman, S. Gronin,
+C. Thomas, T. Wang, R. Kallaher, et al., Nature
+569, 89 (2019).
+3
+
diff --git a/TtE3T4oBgHgl3EQfzwsF/content/tmp_files/load_file.txt b/TtE3T4oBgHgl3EQfzwsF/content/tmp_files/load_file.txt
new file mode 100644
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+++ b/TtE3T4oBgHgl3EQfzwsF/content/tmp_files/load_file.txt
@@ -0,0 +1,217 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf,len=216
+page_content='Graphene amplifier reaches the quantum-noise limit  Kin Chung Fong1, ∗  1Raytheon BBN Technologies, Quantum Engineering and Computing Group, Cambridge, Massachusetts 02138, USA  (Dated: January 13, 2023)  To make yourself heard in a noisy environment  is no easy task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  An amplifier, like a megaphone,  can come to your rescue by increasing your voice’s  volume over the background noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  Your speech  can be heard clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  This is analogous to mea-  suring superconducting qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' Since their energy  quanta are a few orders of magnitude smaller than  the thermal noise, amplifiers are necessary to boost  the signal up before being registered by apparatus  at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' However, having a high gain  from amplifiers is not nearly enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' The physical  process of amplification is also subjected to fluctua-  tions, resulting in added noise by the amplifier that  can degrade the signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' For a phase-  insensitive linear amplifier, the minimum amount of  this added noise is half a quantum because of quan-  tum fluctuations[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  Employing a parametric pro-  cess to achieve this fundamental limit of amplifica-  tion at radio and microwave frequencies has a long  history: from using variable-capacitor diodes in the  1960s to Josephson junctions in the 1980s[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' With  high gain and low noise, modern Josephson para-  metric amplifiers (JPAs)[3] have quickly become a  must-have in laboratories[4]: enabling high-fidelity  qubit readouts, observing quantum jumps, tracking  quantum trajectory, and even searching for the rare  event when the axion dark matter converts into a  microwave photon under a high magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  Writing in Nature Nanotechnology, two indepen-  dent reports by Guilliam Butseraen, et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' [5] and  Joydip Sarkar, et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' [6], now further advance JPAs  using a two-dimensional material—graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  We can consult children on swings about the pro-  cess of parametric amplification[7] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' A child  can amplify the pendulum oscillation by standing  up and squatting down when the swing reaches its  maximum and minimum height, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' This  stand-and-squat action is pumping the pendulum  motion at twice its resonant frequency, and doing  work on the harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' The amplitude of  the originally small oscillation (signal) will gradu-  ally increase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' amplify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' Parametric amplification  sets apart from other amplification mechanisms be-  ∗ fongkc@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='com  cause it modulates the reactance, rather than the  resistance of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' As such, it minimizes the  noise that is inevitably brought into the amplifica-  tion process according to the fluctuation-dissipation  theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  A JPA constructed by a superconducting LC res-  onator, schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  1b, operates  under the similar principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' In addition to suppress-  ing dissipation, superconductors can form Josephson  junctions, which can provide an inductance—from  the inertia of the Cooper pairs tunneling through the  junction—to the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' When two Josephson junc-  tions are made in the form of a loop, their supercur-  rents will interfere and form a device known as super-  conducting quantum interference device (SQUID)  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' 1c-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' Due to the Aharonov-Bohm phase, the  total critical current of the SQUID depends on the  magnetic flux through the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' A magnetic field  generated by an electrical current can control the  Josephson inductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  Hence, we can pump the  JPA by modulating its resonant frequency with a  pump current at twice the frequency of the super-  conducting resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' Alternatively, JPAs can be  powered up by feeding the pump tone directly into  the input port for parametric amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  The  second method exploits the non-linear dependence  of Josephson inductance on the magnitude of the  current running through the junction, which allow  for non-degenerate parametric amplifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' JPAs  operate as a reflection amplifier: when the minute  signal enters the JPA via the coupling capacitor, it  is parametrically amplified before making its way  back to the input port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  The two research teams now replace the insula-  tor traditionally used as the Josephson weak link  with a layer of graphene encapsulated by hexagonal-  boron nitride (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  1e-f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  By doing so, the re-  searchers can employ the gate voltage response of  the graphene to tune the Josephson critical current  and thus the Josephson inductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  Hence, they  can control the resonant frequency of the JPA by  voltage, rather than current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' Butseraen, et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' at-  tain parametric amplification using a transmission-  line resonator at ∼5 GHz and a pump tone through  the gate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' whereas Sarkar, et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' exploit lump com-  ponents and a pump tone applied at the input port,  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='04730v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='mes-hall]  11 Jan 2023  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' (a) Parametric amplification in a mechanical oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' The red dot marks the center of mass of the child  in the swing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' Its oscillation periodically modulates the effective length of the pendulum, and thus, the resonant  frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' (b) Schematic model of the parametric amplifier at radio or microwave frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' The signal will enter  the LC resonator and be amplified by the modulation of the inductance before leaving the resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' (c) A SQUID as  an implementation of the variable inductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' The Josephson inductance depends inversely on the total critical current,  which is controlled by the magnetic flux through the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' (d) Josephson junctions based on superconductor-insulator-  superconductor (SIS) heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' (e) Graphene-based Josephson junction with a gate control can operate as  a variable inductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' (f) Josephson junctions based on superconductor-normal metal-superconductor (SNS) lateral  junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' The latest reports demonstrate JPA by SNS junction with graphene, encapsulated in hexagonal-boron  nitride (blue), as the weak link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' This new JPA is voltage-tunable and can operate under a high-magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  They both achieve similar figures of  merit: ∼10 MHz bandwidth, ∼500 MHz tuning of  the resonant frequency with a gain of >20 dB, and  1-dB compression point at about -127 dBm of in-  put power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  Most importantly, both teams show  that the added noise from the graphene JPA can  reach the quantum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' This is a pleasant surprise  given that graphene Josephson junctions are more  dissipative with a lower quality-factor at microwave  frequency[8], than the conventional superconductor-  insulator-superconductor (SIS) junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  With  their careful designs and implementations, the two  teams demonstrate that a superconductor-normal  metal-superconductor (SNS) junction can overcome  its intrinsic dissipation to achieve quantum-limited  amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  Given the ubiquity of JPAs based on SQUIDs, it  is natural to question the utility of graphene JPA  beyond mere scientific curiosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' One of the poten-  tial benefits may lie in the mitigation of cross-talks  among different tunable components in the quantum  circuitry due to the magnetic field generated by a  current bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' Without the SQUID loop, graphene  JPAs can also be more densely packed and oper-  ate under a larger in-plane magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  More-  over, these latest results exemplify the renewed in-  terest in applying SNS junctions to quantum sci-  ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' Unlike their insulating counterpart, SNS junc-  tions possess more variety in their physical proper-  ties, arising from the interplay of the superconduc-  tor and the materials in the weak link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  Not only  would it enable voltage control of quantum ampli-  fiers as shown here, but, by developing this mo-  tif, we can also create new hybrid superconducting  electronics[9], qubits[10, 11], high-sensitivity pho-  ton detectors[12], and topological superconductivity  which may host Majorana zero modes[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' With the  rise of quantum technology, we shall expect more  innovations from SNS junctions to come!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  The author declares no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
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diff --git a/U9AyT4oBgHgl3EQf8vpk/content/tmp_files/2301.00861v1.pdf.txt b/U9AyT4oBgHgl3EQf8vpk/content/tmp_files/2301.00861v1.pdf.txt
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--- /dev/null
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@@ -0,0 +1,916 @@
+Hardware Abstractions and Hardware Mechanisms to Support
+Multi-Task Execution on Coarse-Grained Reconfigurable Arrays
+Taeyoung Kong, Kalhan Koul, Priyanka Raina, Mark Horowitz, and Christopher Torng
+Stanford University
+{kongty,kkoul,praina,horowitz,ctorng}@stanford.edu
+Abstract
+Domain-specific accelerators are used in various com-
+puting systems ranging from edge devices to data centers.
+Coarse-grained reconfigurable arrays (CGRAs) represent
+an architectural midpoint between the flexibility of an
+FPGA and the efficiency of an ASIC and are a promising
+candidate for servicing multi-tasked workloads within an
+application domain. Unfortunately, scheduling multiple
+tasks onto a CGRA is challenging. CGRAs lack abstrac-
+tions that capture hardware resources, leaving workload
+schedulers unable to reason about performance, energy,
+and utilization for different schedules. This work first pro-
+poses a CGRA architecture that can flexibly partition key
+resources, including the global buffer memory capacity,
+the global buffer memory bandwidth, and the compute re-
+sources. Partitioned resources serve as hardware abstrac-
+tions that decouple compilation and resource allocation.
+The compiler uses these abstractions for coarse-grained
+resource mapping, and the scheduler uses them for flexi-
+ble resource allocation at run time. We then propose two
+hardware mechanisms to support multi-task execution.
+A flexible-shape execution region increases the overall
+resource utilization by mapping multiple tasks with dif-
+ferent resource requirements. Dynamic partial reconfig-
+uration (DPR) enables a CGRA to update the hardware
+configuration as the scheduler makes decisions rapidly.
+We show that our abstraction can help automatic and
+efficient scheduling of multi-tasked workloads onto our
+target CGRA with high utilization, resulting in 1.05x–
+1.24x higher throughput and a 23–28% lower latency in a
+multi-tasked cloud workload and 60.8% reduced latency
+in an autonomous system workload when compared to a
+baseline CGRA running single tasks at a time.
+1. Introduction
+Domain-specific accelerators have gained growing inter-
+est in recent years as they provide improved performance
+and energy efficiency over general-purpose processors.
+Application-specific integrated circuits (ASICs) [8, 18,
+21] show the highest performance and efficiency as they
+are specialized for target applications such as image pro-
+cessing or machine learning (ML). However, the ASIC
+design process can span multiple years, and fixed-function
+accelerators quickly become obsolete as applications con-
+tinue to evolve. Some works deploy applications on FP-
+GAs [12, 16, 17]. FPGAs enable reconfiguration of the
+underlying hardware and can accelerate diverse work-
+loads, but their bit-level flexibility incurs high area and
+energy overheads. Coarse-grained reconfigurable arrays
+(CGRAs) are promising architectures that lie between
+ASICs and FPGAs. A CGRA has arithmetic units and a
+routing system that are configurable in word-level gran-
+ularity, providing flexibility at a lower overhead than
+a FPGA. With its unique advantages, a CGRA can be
+widely adopted in domains with high performance, power,
+and flexibility requirements.
+As hardware accelerators are deployed in various sce-
+narios, the demand for multi-task execution support on
+hardware is growing. For example, many vendors [21, 13]
+offer INFerence-as-a-Service, where multiple tenants
+share the same hardware to run inference tasks. Also, an
+autonomous system handles concurrent tasks to process
+various types of data from numerous sensors. Some works
+have explored multi-task execution support in ASICs and
+FPGAs. PREMA [11] and Planaria [14] propose a sys-
+tolic array that supports multi-tenancy by temporal and
+spatial multiplexing, respectively. [35, 29, 34] propose an
+FPGA virtualization framework with multi-tenancy sup-
+port. However, multi-task execution support on CGRAs
+has not been explored much thus far. A noteworthy ex-
+ception is ChordMap [27] which schedules multiple tasks
+captured in synchronous data flow graphs onto a CGRA.
+However, it assumes that all tasks are known a priori,
+whereas in a multi-tenant cloud or multi-tasked edge work-
+load scenario, tasks may arrive dynamically and require
+schedulers to react to maximize utilization.
+Unfortunately, scheduling multiple tasks onto a CGRA
+is challenging as it lacks abstractions capturing hardware
+resources. In this paper, we propose hardware abstrac-
+tions of a CGRA by partitioning key hardware resources.
+Both compilers and schedulers can exploit the abstrac-
+arXiv:2301.00861v1  [cs.AR]  2 Jan 2023
+
+tions to reason about performance, energy, and utilization.
+We also develop hardware mechanisms that allow fast and
+flexible multi-task execution on a CGRA, which sched-
+ulers exploit to improve hardware utilization. We evaluate
+our CGRA with two different multi-tasked workload sce-
+narios to show the potential. Our key contributions are:
+• 1⃝ We propose a CGRA architecture that can flexibly
+re-partition key resources, including the Global Buffer
+(GLB) memory capacity, the GLB memory bandwidth,
+and the compute resources. Specifically, we partition
+the GLB into GLB-slices and the tile array into array-
+slices, which serve as hardware abstractions. The com-
+piler uses these abstractions for coarse-grain resource
+mapping, while the scheduler uses them for flexible
+resource allocation.
+• 2⃝ We propose two hardware mechanisms to support
+multi-task execution on the CGRA. First, the CGRA
+can form a flexible-shape execution region at run time.
+It improves resource utilization by enabling a scheduler
+to allocate GLB-slices and array-slices flexibly. Sec-
+ond, we propose a fast-DPR method to reconfigure the
+underlying hardware rapidly according to scheduler de-
+cisions. It also supports run time relocation of a task to
+any available array-slice without software intervention.
+• 3⃝ We quantify the benefits of our proposed mecha-
+nisms on two different examples. Our CGRA with
+flexible execution regions and fast-DPR shows 1.05x–
+1.24x higher throughput and 23–28% lower latency in a
+cloud system scenario and 60.8% reduced latency in an
+autonomous system scenario than the baseline CGRA.
+2. Architectural Support for Multi-Task Ex-
+ecution on a CGRA
+In this section, we explore the architectural support
+needed for multi-task execution on a CGRA. Section 2.1
+first introduces a baseline CGRA architecture with com-
+mon features present in many reconfigurable accelera-
+tors [7, 32, 15, 6, 1, 28]. Section 2.2 then introduces how
+we abstract the hardware resources in the CGRA for the
+scheduler by partitioning the global buffer (GLB) and
+the tile array into GLB-slices and array-slices, respec-
+tively. We further develop hardware mechanisms that
+enable multi-task execution on top of these abstractions
+(Section 2.3), including flexible-shape execution regions
+and dynamic partial reconfiguration (DPR).
+2.1. Baseline CGRA Architecture
+Our baseline CGRA consists of a tile array with pro-
+cessing element (PE) and memory (MEM) tiles and a
+global buffer (GLB) (Figure 1). We leverage almost the
+same hardware configuration used in the Amber SoC [7].
+The CGRA has 32x16 tiles with 384 PE tiles and 128
+MEM tiles, and tiles communicate through a statically
+Figure 1: Baseline CGRA block diagram corresponding to [23].
+App.
+Task
+Ver.
+Tpt.
+Array
+slices
+GLB
+slices
+ResNet-18
+conv2_x
+a
+64
+2
+7
+b
+256
+6
+7
+conv3_x
+a
+64
+2
+4
+b
+256
+6
+4
+conv4_x
+a
+64
+2
+6
+b
+256
+6
+6
+conv5_x
+a
+64
+2
+20
+b
+128
+6
+20
+MobileNet
+conv_dw
+_pw_2_x
+1
+a
+52
+2
+4
+b
+208
+5
+4
+conv_dw
+_pw_3_x
+a
+52
+2
+4
+b
+104
+3
+4
+conv_dw
+_pw_4_x
+a
+52
+2
+4
+b
+104
+3
+4
+Camera
+pipeline
+Camera
+pipeline
+a
+3
+4
+4
+b
+12
+6
+14
+Harris
+Harris
+a
+1
+2
+4
+b
+2
+4
+7
+c
+4
+7
+14
+Table 1: Variants of tasks with different resource usage and
+throughput. ResNet-18 and MobileNet consist of several lay-
+ers, and one or more layers form a single task. The unit of
+throughput (Tpt.) for ResNet-18 and MobileNet is MACs/cycle
+and for camera pipeline and harris it is pixels/cycle.
+configured mesh interconnect. A PE tile is extended from
+Amber version to support MAC operation. Each node
+in the interconnect has five incoming and five outgoing
+tracks in each direction, and switch boxes route data from
+incoming tracks to outgoing tracks. Connection boxes
+select data from incoming tracks and route it to the PE
+or MEM tile cores. The GLB consists of 32 banks, with
+each bank containing 128 KB of SRAM. Each GLB bank
+directly communicates with the tile array through IO tiles
+located at the top of the array.
+2.2. A Scheduler-Visible Abstraction of Hardware Re-
+sources
+We focus on three key hardware resources within the
+CGRA (Figure 1): the GLB memory capacity, the GLB
+1A conv_dw_pw refers to a merged task of a depth-wise convolu-
+tional layer and a point-wise convolutional layer.
+2
+
+CGRA
+GLB
+PE
+PE
+MEM
+Bank,
+Global Buffer (GLB)
+用
+PE
+PE
+用
+MEM
++
+CGRA Interconnect
+PE
+Tiles
+PE
+PE
+MEM
+MEM
+Tiles
+Routing
+Connection
+Switch
+Tracks
+Box
+Box(a) Baseline
+(b) Fixed-sized execution region
+(c) Variably sized execution region
+(d) Flexible-shape execution region
+Figure 2: Resource allocation in the baseline CGRA and a CGRA with three different execution regions. Resources colored grey
+represent the blocks occupied by a current-running task, and those colored red represent blocks occupied by a next-running task.
+memory bandwidth, and the compute resources within
+the tile array. When a task is compiled in the Amber
+toolchain [23], a compiler converts it into a dataflow
+graph where each node and edge represents a hardware
+resource and communication, respectively. Specifically,
+GLB banks are used for medium-sized storage and com-
+munication to the host and tile array, and PE and MEM
+tiles are used for computation and as small scratchpads.
+The dataflow graph can derive the usage of memory capac-
+ity, memory bandwidth, compute units, and throughput.
+We abstract the hardware resources by partitioning the
+GLB and tile array into homogeneous GLB-slices and
+array-slices, respectively. For example, we can abstract
+each GLB bank within our CGRA as a GLB-slice and
+every set of four columns in the tile array (48 PE tiles and
+16 MEM tiles) as an array-slice. This abstraction serves
+as a middle layer that decouples offline bitstream genera-
+tion by a compiler and run time resource allocation by a
+scheduler. During compilation, we represent the resource
+usage of each task using these abstracted GLB-slices and
+array-slices. For instance, a conv2_x layer in [19] utilizes
+750KB of GLB memory capacity, 17.3MB/s of memory
+bandwidth, 80 PE tiles, and 17 MEM tiles and achieves
+64 OPs/cycle throughput at a 500MHz clock frequency.
+The task is abstracted as seven GLB-slices and two array-
+slices in coarse-grain resource slice usage. It is possible to
+produce variants of the same task with different resource
+usage and throughput by tweaking the compiler. For ex-
+ample, increasing the unroll factor of the same task by
+four would achieve 4x throughput (256 OPs/cycle) with
+288 PE tiles, 33 MEM tiles, and the same GLB mem-
+ory capacity and bandwidth, which is abstracted as seven
+GLB-slices and six array-slices. Our approach allows
+for pre-computation of bitstreams that support different
+resource usage and throughput to be cached in on-chip
+storage to support fast dynamic partial reconfiguration, as
+discussed later. Table 1 summarizes the resource usage
+and throughput for several different variants of tasks. At
+run time, a scheduler leverages the hardware slice abstrac-
+tion to decide which variant of tasks to choose, which
+resources to allocate, and when to execute.
+2.3. Hardware Mechanisms
+Flexible-Shape Execution Regions. To manage multi-
+ple tasks that are concurrently running, we need a way
+to monitor hardware resources and the status of tasks,
+that are build upon the abstractions described above. We
+introduce an execution region, a sub-region of the CGRA
+on which a single task is mapped and executed. An ex-
+ecution region consists of one or more GLB-slices and
+array-slices. The flexibility to form different sizes and
+shapes of execution regions gives the scheduler a sim-
+plified and quantized view of hardware resources while
+providing enough information to allocate resources to
+each task to maximize resource utilization in multi-tasked
+workloads.
+Figure 2 compares different mechanisms to form an
+execution region and how they affect resource allocation.
+3
+
+CGRA
+GlobalBuffer（GLB)
+GLB-slice
+Array-slice
+Tile-ArrayCGRA
+GlobalBuffer (GLB)
+1111111111
+Tile-ArrayCGRA
+GlobalBuffer(GLB)
+Tile-ArrayCGRA
+Global Buffer (GLB)
+GLB-slice
+[available
+Multi-stageNetwork
+Array-slice
+Tile-Array
+[available]The blocks colored in gray represent resources occupied
+by the currently running task, and those colored in red rep-
+resent resources allocated to the next-running task. The
+baseline CGRA (Figure 2a) is unaware of our hardware
+slice abstraction, and the entire CGRA serves as a single
+large execution region. Since an existing task is already
+mapped onto the CGRA, subsequent tasks are always
+forced to wait until the previous tasks finish and release
+the single execution region.
+The simplest mechanism to form an execution region
+is only to support fixed-sized regions. For example, all
+execution regions in Figure 2b consist of two GLB-slices
+and one array-slice. Fixed-sized regions are not optimal.
+Since each task must fit within the fixed-sized execution
+region, the largest task with the highest resource usage
+determines the size. On the other hand, when there are
+several available execution regions, a task can be unrolled
+and mapped in parallel to achieve higher throughput (e.g.,
+the next-running task is unrolled by three in Figure 2b).
+This method does not require much architectural change,
+and the implementation of a scheduling algorithm can
+be straightforward given the assumption that all target
+tasks fit within an execution region. However, although
+unrolling increases throughput, optimization across the
+unrolled dimension can be challenging to support.
+Another method is to support variably sized execution
+regions by merging multiple fixed-sized regions. We de-
+fine the unit size of a region as in the fixed-sized region
+case, but we can merge multiple unit regions to form a
+larger execution region. For example, in Figure 2c, three
+unit-sized regions are merged to execute the next-running
+task (colored in red). The benefit of variably sized execu-
+tion regions is to allow compilation optimization across
+the unrolled dimension. For example, a camera pipeline
+task with 3 pixels/cycle throughput uses four array-slices
+(Table 1). Naively unrolling it by four achieves 12 pix-
+els/cycle throughput using 16 array-slices. However, the
+compiler can optimize to time-multiplex PE tiles and
+achieve 12 pixels/cycle throughput with only six array-
+slices. Support for a variably sized region still allows for
+the pre-computation of bitstreams for multiple variants of
+tasks with different resource usage and throughput. How-
+ever, this approach may still suffer from low resource
+utilization since the ratio of GLB-slices and array-slices
+within an execution region always remains the same.
+Therefore, we propose flexible-shape execution regions
+in which GLB-slices and array-slices are no longer cou-
+pled. Decoupling of GLB-slices and array-slices enables
+finer-grained resource allocation. For example, Figure 2d
+shows how an execution region can be allocated any
+number of GLB-slices and array-slices, forming a non-
+rectangular shape, with remaining array-slices and GLB-
+slices available to be used by other tasks. The support
+for flexible-shape execution regions improves resource
+utilization, especially for multi-tasked workloads where
+memory-intensive and compute-intensive tasks are mixed.
+However, it may require additional communication be-
+tween the GLB-slices and the array-slices. In this work,
+we limit the placement of GLB-slices and array-slices
+within an execution region to be contiguous to simplify
+our study. Design space exploration on flexible placement
+support and the required network remains as future work.
+Section 3.1 describes the benefits of these mechanisms in
+more detail with a cloud system example.
+Dynamic Partial Reconfiguration. Dynamic partial re-
+configuration (DPR) is a mechanism to update the hard-
+ware configuration in reconfigurable architectures. We
+propose fast-DPR following the DPR mechanism pro-
+posed in Amber SoC [7], but with added features to
+exploit hardware abstractions. In Amber, every other
+GLB bank stores the configuration bitstreams and inde-
+pendently streams configuration into two columns of the
+tile array. Also, clocks and configuration signals are dis-
+tributed down each column together, enabling reconfigur-
+ing the tile array at high clock frequency without pipeline
+stages. In our CGRA, we also reuse GLB blocks to store
+and stream bitstreams to the tile array and follow the same
+clock distribution network. Unlike Amber, however, one
+GLB bank streams configuration into one array-slice (in
+turn, four columns of the tile array) as an array-slice is
+the minimum unit of execution regions.
+We added a feature to relocate bitstreams at run time to
+exploit hardware abstractions further. In Amber, the com-
+piler generates region-aware bitstreams; the bitstreams
+for one region cannot be reused in different regions even
+though the two regions are homogeneous. This limitation
+comes from the fact that the address of each configuration
+register in different columns has a distinct column #id. On
+the other hand, our compiler generates region-agnostic
+bitstreams by assuming that the task is always mapped to
+the leftmost region. We also added a register indicating
+the destination region of DPR to GLB banks. When the
+host processor triggers DPR, GLB banks read the register
+and stream bitstreams to the target region via the network
+between the GLB and the tile array. With this bitstream
+relocation feature, a user can pre-load bitstreams of the
+next task to the GLB in advance and rapidly map it to any
+next available region just by writing to a single register.
+3. Evaluation
+We evaluate the benefits of multi-task execution support
+under two different workload scenarios. In a cloud sys-
+tem example scenario (Section 3.1), our CGRA with
+flexible-shape execution regions enables 1.05x-1.24x
+higher throughput and 23-28% lower normalized turn-
+around time (NTAT) over the baseline CGRA. In an au-
+tonomous system example scenario (Section 3.2), our
+CGRA enables 60.8% reduced total latency.
+4
+
+(a) Cloud system example
+(b) Autonomous system example
+Figure 3: (a) Cloud system example scenario with four tenants
+submitting requests to the CGRA. Each tenant is assigned with
+a task from MobileNet, ResNet-18, camera pipeline, and Harris,
+respectively. (b) Autonomous system example with tasks that
+may be triggered under conditions.
+3.1. Example 1: Cloud System
+Overview. In this example, we construct a synthetic
+cloud computing scenario that models real-world exam-
+ples in which the CGRA serves application requests from
+multiple users (Figure 3a). We construct the multi-tasked
+workload using kernels from machine learning (ML) and
+image processing domains, including ResNet-18 [19]
+and MobileNet [20] from the ML domain, and camera
+pipeline and Harris corner detector from the image pro-
+cessing domain. Table 1 summarizes the benchmark tasks
+and their resource requirements.
+To generate the multi-tasked workload, we assume four
+tenants share the CGRA and are assigned one of the four
+target applications. Each tenant sends a request to the
+CGRA following a Poisson distribution. Whenever a new
+task arrives, or an existing task finishes, the scheduler
+is triggered and runs a greedy algorithm to schedule the
+next available task. The scheduler checks if dependencies
+are met before scheduling the task (e.g., in ResNet-18,
+conv2_x depends on conv1_x). If there is more than one
+version of a task that can be mapped onto the available
+resources, the greedy scheduler always chooses the one
+with the highest throughput.
+Metrics. We measure Normalized Turn-Around Time and
+throughput to compare the baseline CGRA and the three
+partitioning mechanisms described in Section 2.3. Turn-
+(a) NTAT
+(b) Throughput
+Figure 4: Evaluation in a cloud system example. (a) NTAT and
+(b) throughput for each task with fixed-sized, variably sized, and
+flexible-shape resource partitioning, normalized to the baseline
+CGRA. Flexible-shape partitioning decreases NTAT by 23-28%
+and increases throughput by 1.05x-1.24x.
+Around Time (TAT) is the interval from the time of request
+to submit a task to the time of task completion. Normal-
+ized Turn-Around Time (NTAT) is the ratio of the TAT to
+the execution time, which represents the relative delay of
+a task (Equation (1) - (2)). We calculate NTAT for each re-
+quest and the arithmetic average for each application. We
+also measure the average throughput for each application
+to demonstrate the performance benefit.
+TAT = wait_time + execution_time
+(1)
+NTAT = TAT / execution_time
+(2)
+Results. Figure 4 illustrates the relative improvements
+in NTAT and throughput for flexible-shape execution re-
+gions compared to fixed- and variably-sized execution
+regions. Even with a simple greedy scheduling algo-
+rithm, we achieve 23–28% decreased NTAT and 1.05x–
+1.24x higher throughput. Note that we only pre-compile
+each task to two different variants in this case study (Ta-
+ble 1), and a scheduler greedily selects the one with higher
+throughput if resources are available. Co-optimizing com-
+pilation and scheduling policy may improve NTAT and
+throughput further, which remains future work.
+3.2. Example 2: Autonomous System
+Overview. In this case study, we construct a synthetic
+edge system scenario modeling the real world in which
+multiple tasks from image processing and ML domains ex-
+ecute in parallel and can dynamically trigger. Specifically,
+we develop an autonomous system scenario as described
+in Figure 3b following a methodology used in [30]. 2 The
+system takes a RAW image in Bayer encoding format
+(RGGB) from sensors at 30 fps and first runs a camera
+2We also changed the tasks to simplify the example.
+5
+
+User 1
+User 2
+MobileNet
+ResNet-18
+CGRA
+User 3
+User 4
+Camera
+Harris Corner
+Pipeline
+DetectorBackground detected
+Senddatatothecloud
+Depth estimation
+Image compression
+(stereo)
+(gaussian)
+CGRA
+Foreveryframe
+Sign detected
+Camera pipeline
+Sign classifier
+(camerapipeline)
+(ResNet)baseline
+Fixed
+Variable
+Flexible
+1.250
+1.000
+NTAT
+hh
+0.750
+0.500
+ResNet-18
+MobileNet
+Camera
+Harrisbaseline
+Fixed
+Variable
+Flexible
+1.25
+Throughput
+1.00
+川
+0.75
+0.50
+ResNet-18
+MobileNet
+Camera
+HarrisFigure 5: The average latency of an autonomous system
+example with different execution regions. The values are nor-
+malized to the result of the baseline. A red bar indicates the
+time spent for reconfiguration, and a blue bar indicates the
+sum of wait time and execution time. To show the benefit of
+fast-DPR (Section 2.3), we assume the baseline CGRA uses
+AXI4-Lite interface for DPR, while others use fast-DPR.
+pipeline task on the CGRA to convert to an RGB image.
+Once the CGRA generates an RGB image, the system
+runs object detection and dynamically decides on the next
+tasks. 3 When an event happens (e.g., detection of a spe-
+cific background), it processes the event and executes the
+corresponding tasks (e.g., depth estimation). Except for a
+camera pipeline that runs every frame, we set the period
+from one event to the next same event to follow a uniform
+random distribution between 3–7 frames.
+Results. We evaluate the benefit of hardware resource
+partitioning and fast DPR by comparing our proposed
+CGRA to the baseline CGRA with AXI4-Lite-based DPR.
+Specifically, the baseline CGRA maps only one task at
+a time. When more than one event occurs, the base-
+line handles each task one by one and reconfigures using
+sequential AXI4-Lite configuration transactions. In the
+proposed CGRA with multi-task execution support, we ex-
+ploit flexible-shape resource partitioning to concurrently
+run more than one task on the CGRA when possible. Also,
+we use the parallel and high-frequency DPR mechanisms
+in Section 2.3 to configure bitstreams. We compute the
+arithmetic average of the latency over all frames. As de-
+scribed in Figure 5, our techniques enable a 60.8% latency
+reduction compared to the baseline. With fast DPR, re-
+configuration takes less than 5% of the total latency, an
+appreciable reduction from 14.4% in the baseline.
+4. Related Work
+As Deep Neural Networks (DNNs) are widely used in vari-
+ous domains, DNN accelerators [18, 17, 8, 9, 10, 25] have
+emerged and been deployed in the cloud system [21, 13].
+To that end, many prior works have explored multi-
+tenancy support on DNN accelerators in cloud systems.
+3This work assumes that object detection is executed in another
+hardware in the system (e.g. GPU or ASIC).
+Multi-task execution support is also studied in FPGAs
+targeting both cloud and edge computing. However, a non-
+negligible portion of FPGA resources is typically reserved
+for controlling multi-task execution, ultimately decreas-
+ing the available computing resources. ChordMap [27]
+explores the automated mapping of multi-tasked applica-
+tions onto a CGRA, but it is limited to mapping multiple
+tasks within streaming applications with all tasks known
+a priori. Our work proposes hardware abstractions and
+mechanisms, which both compilers and schedulers can
+exploit and co-optimize to improve resource utilization in
+both cloud and edge systems.
+Multi-Task Execution on DNN Accelerators. Some
+DNN accelerators service multi-DNN tasks at the soft-
+ware level. AI-MT [2] and Layerweaver [31] propose a
+scheduling policy to mix compute- and memory-intensive
+tasks to increase hardware utilization. PREMA [11] im-
+plements preemptible NPUs to support multi-tenancy
+via temporal multiplexing.
+Many works add flexibil-
+ity to an accelerator to accommodate multiple DNN
+tasks. Planaria [14] introduces a flexible systolic array
+with dynamic architecture fission to map multiple DNN
+tasks. [26] suggests a multi-directional network to sup-
+port up to four DNN tasks with different dataflow. Other
+works [24, 3] explore a computing system with multiple
+DNN accelerators with different hardware characteristics.
+While these works only support DNN workloads, our
+work can support any applications that can be mapped
+onto a CGRA.
+Multi-Task Execution on FPGAs. In FPGAs, multi-
+task execution support has been explored in the context of
+virtualization. Some works divide an FPGA into a static
+region, a shell, which serves as glue logic between the
+host and the FPGA, and a dynamic region, a role, which
+handles the computation of tasks. [4, 5, 33] partition
+a physical FPGA into several fixed-size virtual blocks
+and share them across multiple tasks. AmorphOS [22]
+presents a hardware abstraction of an FPGA, Morphlet,
+which dynamically alters its size based on resource re-
+quirements. ViTAL [35] provides a full-stack framework
+to run multiple tasks with different sizes on homogeneous
+regions. [34] supports running multi-DNN tasks on an
+FPGA by dividing hardware resources into multiple PE
+cores and spatially multiplexing them, while [30] eval-
+uates the benefits of temporal multiplexing of FPGAs
+using DPR for vision applications on embedded devices.
+While these works only target scenarios where underlying
+applications change infrequently because of long reconfig-
+uration time of FPGAs, our work can support both cloud
+systems and real-time edge systems due to rapid DPR.
+5. Conclusion
+Multi-task execution support on accelerators is becoming
+increasingly relevant in both cloud and edge systems and
+6
+
+DPR
+Waiting + Execution
+1.000
+1.0.0.0
+0.750
+latency
+0.500
+0.447
+0.445
+0.392
+0.250
+0.000
+baseline
+fixed
+variable
+flexiblehas the potential to improve performance through bet-
+ter hardware utilization. This work proposes abstracting
+hardware resources within a CGRA into coarser-grained
+units with which a workload scheduler can quickly make
+decisions. Based on the proposed abstraction, we develop
+hardware mechanisms to support multi-task execution
+through flexible-shape hardware partitioning and high-
+throughput dynamic partial reconfiguration. Our evalua-
+tions modeling both a cloud and an edge system scenario
+suggest that the abstraction and hardware mechanisms can
+enable automatic schedulers to achieve high performance
+in multi-tasked workloads on future CGRAs.
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diff --git a/U9AyT4oBgHgl3EQf8vpk/content/tmp_files/load_file.txt b/U9AyT4oBgHgl3EQf8vpk/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf,len=753
+page_content='Hardware Abstractions and Hardware Mechanisms to Support  Multi-Task Execution on Coarse-Grained Reconfigurable Arrays  Taeyoung Kong, Kalhan Koul, Priyanka Raina, Mark Horowitz, and Christopher Torng  Stanford University  {kongty,kkoul,praina,horowitz,ctorng}@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='edu  Abstract  Domain-specific accelerators are used in various com-  puting systems ranging from edge devices to data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  Coarse-grained reconfigurable arrays (CGRAs) represent  an architectural midpoint between the flexibility of an  FPGA and the efficiency of an ASIC and are a promising  candidate for servicing multi-tasked workloads within an  application domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Unfortunately, scheduling multiple  tasks onto a CGRA is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' CGRAs lack abstrac-  tions that capture hardware resources, leaving workload  schedulers unable to reason about performance, energy,  and utilization for different schedules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' This work first pro-  poses a CGRA architecture that can flexibly partition key  resources, including the global buffer memory capacity,  the global buffer memory bandwidth, and the compute re-  sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Partitioned resources serve as hardware abstrac-  tions that decouple compilation and resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  The compiler uses these abstractions for coarse-grained  resource mapping, and the scheduler uses them for flexi-  ble resource allocation at run time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' We then propose two  hardware mechanisms to support multi-task execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  A flexible-shape execution region increases the overall  resource utilization by mapping multiple tasks with dif-  ferent resource requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Dynamic partial reconfig-  uration (DPR) enables a CGRA to update the hardware  configuration as the scheduler makes decisions rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  We show that our abstraction can help automatic and  efficient scheduling of multi-tasked workloads onto our  target CGRA with high utilization, resulting in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='05x–  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='24x higher throughput and a 23–28% lower latency in a  multi-tasked cloud workload and 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='8% reduced latency  in an autonomous system workload when compared to a  baseline CGRA running single tasks at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Introduction  Domain-specific accelerators have gained growing inter-  est in recent years as they provide improved performance  and energy efficiency over general-purpose processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  Application-specific integrated circuits (ASICs) [8, 18,  21] show the highest performance and efficiency as they  are specialized for target applications such as image pro-  cessing or machine learning (ML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' However, the ASIC  design process can span multiple years, and fixed-function  accelerators quickly become obsolete as applications con-  tinue to evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Some works deploy applications on FP-  GAs [12, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' FPGAs enable reconfiguration of the  underlying hardware and can accelerate diverse work-  loads, but their bit-level flexibility incurs high area and  energy overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Coarse-grained reconfigurable arrays  (CGRAs) are promising architectures that lie between  ASICs and FPGAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' A CGRA has arithmetic units and a  routing system that are configurable in word-level gran-  ularity, providing flexibility at a lower overhead than  a FPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' With its unique advantages, a CGRA can be  widely adopted in domains with high performance, power,  and flexibility requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  As hardware accelerators are deployed in various sce-  narios, the demand for multi-task execution support on  hardware is growing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' For example, many vendors [21, 13]  offer INFerence-as-a-Service, where multiple tenants  share the same hardware to run inference tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Also, an  autonomous system handles concurrent tasks to process  various types of data from numerous sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Some works  have explored multi-task execution support in ASICs and  FPGAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' PREMA [11] and Planaria [14] propose a sys-  tolic array that supports multi-tenancy by temporal and  spatial multiplexing, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' [35, 29, 34] propose an  FPGA virtualization framework with multi-tenancy sup-  port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' However, multi-task execution support on CGRAs  has not been explored much thus far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' A noteworthy ex-  ception is ChordMap [27] which schedules multiple tasks  captured in synchronous data flow graphs onto a CGRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  However, it assumes that all tasks are known a priori,  whereas in a multi-tenant cloud or multi-tasked edge work-  load scenario, tasks may arrive dynamically and require  schedulers to react to maximize utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  Unfortunately, scheduling multiple tasks onto a CGRA  is challenging as it lacks abstractions capturing hardware  resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' In this paper, we propose hardware abstrac-  tions of a CGRA by partitioning key hardware resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  Both compilers and schedulers can exploit the abstrac-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='00861v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='AR]  2 Jan 2023  tions to reason about performance, energy, and utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  We also develop hardware mechanisms that allow fast and  flexible multi-task execution on a CGRA, which sched-  ulers exploit to improve hardware utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' We evaluate  our CGRA with two different multi-tasked workload sce-  narios to show the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Our key contributions are:  1⃝ We propose a CGRA architecture that can flexibly  re-partition key resources, including the Global Buffer  (GLB) memory capacity, the GLB memory bandwidth,  and the compute resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Specifically, we partition  the GLB into GLB-slices and the tile array into array-  slices, which serve as hardware abstractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' The com-  piler uses these abstractions for coarse-grain resource  mapping, while the scheduler uses them for flexible  resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  2⃝ We propose two hardware mechanisms to support  multi-task execution on the CGRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' First, the CGRA  can form a flexible-shape execution region at run time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  It improves resource utilization by enabling a scheduler  to allocate GLB-slices and array-slices flexibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Sec-  ond, we propose a fast-DPR method to reconfigure the  underlying hardware rapidly according to scheduler de-  cisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' It also supports run time relocation of a task to  any available array-slice without software intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  3⃝ We quantify the benefits of our proposed mecha-  nisms on two different examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Our CGRA with  flexible execution regions and fast-DPR shows 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='05x–  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='24x higher throughput and 23–28% lower latency in a  cloud system scenario and 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='8% reduced latency in an  autonomous system scenario than the baseline CGRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Architectural Support for Multi-Task Ex-  ecution on a CGRA  In this section, we explore the architectural support  needed for multi-task execution on a CGRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='1  first introduces a baseline CGRA architecture with com-  mon features present in many reconfigurable accelera-  tors [7, 32, 15, 6, 1, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='2 then introduces how  we abstract the hardware resources in the CGRA for the  scheduler by partitioning the global buffer (GLB) and  the tile array into GLB-slices and array-slices, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' We further develop hardware mechanisms that  enable multi-task execution on top of these abstractions  (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='3), including flexible-shape execution regions  and dynamic partial reconfiguration (DPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Baseline CGRA Architecture  Our baseline CGRA consists of a tile array with pro-  cessing element (PE) and memory (MEM) tiles and a  global buffer (GLB) (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' We leverage almost the  same hardware configuration used in the Amber SoC [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  The CGRA has 32x16 tiles with 384 PE tiles and 128  MEM tiles, and tiles communicate through a statically  Figure 1: Baseline CGRA block diagram corresponding to [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  Task  Ver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  Tpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='Array  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='slices  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='GLB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='slices  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='ResNet-18  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
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+page_content='Table 1: Variants of tasks with different resource usage and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' ResNet-18 and MobileNet consist of several lay-  ers, and one or more layers form a single task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' The unit of  throughput (Tpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=') for ResNet-18 and MobileNet is MACs/cycle  and for camera pipeline and harris it is pixels/cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  configured mesh interconnect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' A PE tile is extended from  Amber version to support MAC operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Each node  in the interconnect has five incoming and five outgoing  tracks in each direction, and switch boxes route data from  incoming tracks to outgoing tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Connection boxes  select data from incoming tracks and route it to the PE  or MEM tile cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' The GLB consists of 32 banks, with  each bank containing 128 KB of SRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Each GLB bank  directly communicates with the tile array through IO tiles  located at the top of the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' A Scheduler-Visible Abstraction of Hardware Re-  sources  We focus on three key hardware resources within the  CGRA (Figure 1): the GLB memory capacity, the GLB  1A conv_dw_pw refers to a merged task of a depth-wise convolu-  tional layer and a point-wise convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  2  CGRA  GLB  PE  PE  MEM  Bank,  Global Buffer (GLB)  用  PE  PE  用  MEM  +  CGRA Interconnect  PE  Tiles  PE  PE  MEM  MEM  Tiles  Routing  Connection  Switch  Tracks  Box  Box(a) Baseline  (b) Fixed-sized execution region  (c) Variably sized execution region  (d) Flexible-shape execution region  Figure 2: Resource allocation in the baseline CGRA and a CGRA with three different execution regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Resources colored grey  represent the blocks occupied by a current-running task, and those colored red represent blocks occupied by a next-running task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  memory bandwidth, and the compute resources within  the tile array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' When a task is compiled in the Amber  toolchain [23], a compiler converts it into a dataflow  graph where each node and edge represents a hardware  resource and communication, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Specifically,  GLB banks are used for medium-sized storage and com-  munication to the host and tile array, and PE and MEM  tiles are used for computation and as small scratchpads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  The dataflow graph can derive the usage of memory capac-  ity, memory bandwidth, compute units, and throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  We abstract the hardware resources by partitioning the  GLB and tile array into homogeneous GLB-slices and  array-slices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' For example, we can abstract  each GLB bank within our CGRA as a GLB-slice and  every set of four columns in the tile array (48 PE tiles and  16 MEM tiles) as an array-slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' This abstraction serves  as a middle layer that decouples offline bitstream genera-  tion by a compiler and run time resource allocation by a  scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' During compilation, we represent the resource  usage of each task using these abstracted GLB-slices and  array-slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' For instance, a conv2_x layer in [19] utilizes  750KB of GLB memory capacity, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='3MB/s of memory  bandwidth, 80 PE tiles, and 17 MEM tiles and achieves  64 OPs/cycle throughput at a 500MHz clock frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  The task is abstracted as seven GLB-slices and two array-  slices in coarse-grain resource slice usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' It is possible to  produce variants of the same task with different resource  usage and throughput by tweaking the compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' For ex-  ample, increasing the unroll factor of the same task by  four would achieve 4x throughput (256 OPs/cycle) with  288 PE tiles, 33 MEM tiles, and the same GLB mem-  ory capacity and bandwidth, which is abstracted as seven  GLB-slices and six array-slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Our approach allows  for pre-computation of bitstreams that support different  resource usage and throughput to be cached in on-chip  storage to support fast dynamic partial reconfiguration, as  discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Table 1 summarizes the resource usage  and throughput for several different variants of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' At  run time, a scheduler leverages the hardware slice abstrac-  tion to decide which variant of tasks to choose, which  resources to allocate, and when to execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Hardware Mechanisms  Flexible-Shape Execution Regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' To manage multi-  ple tasks that are concurrently running, we need a way  to monitor hardware resources and the status of tasks,  that are build upon the abstractions described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' We  introduce an execution region, a sub-region of the CGRA  on which a single task is mapped and executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' An ex-  ecution region consists of one or more GLB-slices and  array-slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' The flexibility to form different sizes and  shapes of execution regions gives the scheduler a sim-  plified and quantized view of hardware resources while  providing enough information to allocate resources to  each task to maximize resource utilization in multi-tasked  workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  Figure 2 compares different mechanisms to form an  execution region and how they affect resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  3  CGRA  GlobalBuffer（GLB)  GLB-slice  Array-slice  Tile-ArrayCGRA  GlobalBuffer (GLB)  1111111111  Tile-ArrayCGRA  GlobalBuffer(GLB)  Tile-ArrayCGRA  Global Buffer (GLB)  GLB-slice  [available  Multi-stageNetwork  Array-slice  Tile-Array  [available]The blocks colored in gray represent resources occupied  by the currently running task, and those colored in red rep-  resent resources allocated to the next-running task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' The  baseline CGRA (Figure 2a) is unaware of our hardware  slice abstraction, and the entire CGRA serves as a single  large execution region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Since an existing task is already  mapped onto the CGRA, subsequent tasks are always  forced to wait until the previous tasks finish and release  the single execution region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  The simplest mechanism to form an execution region  is only to support fixed-sized regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' For example, all  execution regions in Figure 2b consist of two GLB-slices  and one array-slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Fixed-sized regions are not optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  Since each task must fit within the fixed-sized execution  region, the largest task with the highest resource usage  determines the size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' On the other hand, when there are  several available execution regions, a task can be unrolled  and mapped in parallel to achieve higher throughput (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=',  the next-running task is unrolled by three in Figure 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  This method does not require much architectural change,  and the implementation of a scheduling algorithm can  be straightforward given the assumption that all target  tasks fit within an execution region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' However, although  unrolling increases throughput, optimization across the  unrolled dimension can be challenging to support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  Another method is to support variably sized execution  regions by merging multiple fixed-sized regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' We de-  fine the unit size of a region as in the fixed-sized region  case, but we can merge multiple unit regions to form a  larger execution region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' For example, in Figure 2c, three  unit-sized regions are merged to execute the next-running  task (colored in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' The benefit of variably sized execu-  tion regions is to allow compilation optimization across  the unrolled dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' For example, a camera pipeline  task with 3 pixels/cycle throughput uses four array-slices  (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Naively unrolling it by four achieves 12 pix-  els/cycle throughput using 16 array-slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' However, the  compiler can optimize to time-multiplex PE tiles and  achieve 12 pixels/cycle throughput with only six array-  slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Support for a variably sized region still allows for  the pre-computation of bitstreams for multiple variants of  tasks with different resource usage and throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' How-  ever, this approach may still suffer from low resource  utilization since the ratio of GLB-slices and array-slices  within an execution region always remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  Therefore, we propose flexible-shape execution regions  in which GLB-slices and array-slices are no longer cou-  pled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Decoupling of GLB-slices and array-slices enables  finer-grained resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' For example, Figure 2d  shows how an execution region can be allocated any  number of GLB-slices and array-slices, forming a non-  rectangular shape, with remaining array-slices and GLB-  slices available to be used by other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' The support  for flexible-shape execution regions improves resource  utilization, especially for multi-tasked workloads where  memory-intensive and compute-intensive tasks are mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  However, it may require additional communication be-  tween the GLB-slices and the array-slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' In this work,  we limit the placement of GLB-slices and array-slices  within an execution region to be contiguous to simplify  our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Design space exploration on flexible placement  support and the required network remains as future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='1 describes the benefits of these mechanisms in  more detail with a cloud system example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  Dynamic Partial Reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Dynamic partial re-  configuration (DPR) is a mechanism to update the hard-  ware configuration in reconfigurable architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' We  propose fast-DPR following the DPR mechanism pro-  posed in Amber SoC [7], but with added features to  exploit hardware abstractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' In Amber, every other  GLB bank stores the configuration bitstreams and inde-  pendently streams configuration into two columns of the  tile array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Also, clocks and configuration signals are dis-  tributed down each column together, enabling reconfigur-  ing the tile array at high clock frequency without pipeline  stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' In our CGRA, we also reuse GLB blocks to store  and stream bitstreams to the tile array and follow the same  clock distribution network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Unlike Amber, however, one  GLB bank streams configuration into one array-slice (in  turn, four columns of the tile array) as an array-slice is  the minimum unit of execution regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  We added a feature to relocate bitstreams at run time to  exploit hardware abstractions further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' In Amber, the com-  piler generates region-aware bitstreams;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' the bitstreams  for one region cannot be reused in different regions even  though the two regions are homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' This limitation  comes from the fact that the address of each configuration  register in different columns has a distinct column #id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' On  the other hand, our compiler generates region-agnostic  bitstreams by assuming that the task is always mapped to  the leftmost region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' We also added a register indicating  the destination region of DPR to GLB banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' When the  host processor triggers DPR, GLB banks read the register  and stream bitstreams to the target region via the network  between the GLB and the tile array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' With this bitstream  relocation feature, a user can pre-load bitstreams of the  next task to the GLB in advance and rapidly map it to any  next available region just by writing to a single register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Evaluation  We evaluate the benefits of multi-task execution support  under two different workload scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' In a cloud sys-  tem example scenario (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='1), our CGRA with  flexible-shape execution regions enables 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='05x-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='24x  higher throughput and 23-28% lower normalized turn-  around time (NTAT) over the baseline CGRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' In an au-  tonomous system example scenario (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='2), our  CGRA enables 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='8% reduced total latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  4  (a) Cloud system example  (b) Autonomous system example  Figure 3: (a) Cloud system example scenario with four tenants  submitting requests to the CGRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Each tenant is assigned with  a task from MobileNet, ResNet-18, camera pipeline, and Harris,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' (b) Autonomous system example with tasks that  may be triggered under conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Example 1: Cloud System  Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' In this example, we construct a synthetic  cloud computing scenario that models real-world exam-  ples in which the CGRA serves application requests from  multiple users (Figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' We construct the multi-tasked  workload using kernels from machine learning (ML) and  image processing domains, including ResNet-18 [19]  and MobileNet [20] from the ML domain, and camera  pipeline and Harris corner detector from the image pro-  cessing domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Table 1 summarizes the benchmark tasks  and their resource requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  To generate the multi-tasked workload, we assume four  tenants share the CGRA and are assigned one of the four  target applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Each tenant sends a request to the  CGRA following a Poisson distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Whenever a new  task arrives, or an existing task finishes, the scheduler  is triggered and runs a greedy algorithm to schedule the  next available task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' The scheduler checks if dependencies  are met before scheduling the task (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=', in ResNet-18,  conv2_x depends on conv1_x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' If there is more than one  version of a task that can be mapped onto the available  resources, the greedy scheduler always chooses the one  with the highest throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' We measure Normalized Turn-Around Time and  throughput to compare the baseline CGRA and the three  partitioning mechanisms described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Turn-  (a) NTAT  (b) Throughput  Figure 4: Evaluation in a cloud system example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' (a) NTAT and  (b) throughput for each task with fixed-sized, variably sized, and  flexible-shape resource partitioning, normalized to the baseline  CGRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Flexible-shape partitioning decreases NTAT by 23-28%  and increases throughput by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='05x-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='24x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  Around Time (TAT) is the interval from the time of request  to submit a task to the time of task completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Normal-  ized Turn-Around Time (NTAT) is the ratio of the TAT to  the execution time, which represents the relative delay of  a task (Equation (1) - (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' We calculate NTAT for each re-  quest and the arithmetic average for each application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' We  also measure the average throughput for each application  to demonstrate the performance benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  TAT = wait_time + execution_time  (1)  NTAT = TAT / execution_time  (2)  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Figure 4 illustrates the relative improvements  in NTAT and throughput for flexible-shape execution re-  gions compared to fixed- and variably-sized execution  regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Even with a simple greedy scheduling algo-  rithm, we achieve 23–28% decreased NTAT and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='05x–  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='24x higher throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Note that we only pre-compile  each task to two different variants in this case study (Ta-  ble 1), and a scheduler greedily selects the one with higher  throughput if resources are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Co-optimizing com-  pilation and scheduling policy may improve NTAT and  throughput further, which remains future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Example 2: Autonomous System  Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' In this case study, we construct a synthetic  edge system scenario modeling the real world in which  multiple tasks from image processing and ML domains ex-  ecute in parallel and can dynamically trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Specifically,  we develop an autonomous system scenario as described  in Figure 3b following a methodology used in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' 2 The  system takes a RAW image in Bayer encoding format  (RGGB) from sensors at 30 fps and first runs a camera  2We also changed the tasks to simplify the example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  5  User 1  User 2  MobileNet  ResNet-18  CGRA  User 3  User 4  Camera  Harris Corner  Pipeline  DetectorBackground detected  Senddatatothecloud  Depth estimation  Image compression  (stereo)  (gaussian)  CGRA  Foreveryframe  Sign detected  Camera pipeline  Sign classifier  (camerapipeline)  (ResNet)baseline  Fixed  Variable  Flexible  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='250  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='000  NTAT  hh  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='750  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='500  ResNet-18  MobileNet  Camera  Harrisbaseline  Fixed  Variable  Flexible  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='25  Throughput  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='00  川  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='50  ResNet-18  MobileNet  Camera  HarrisFigure 5: The average latency of an autonomous system  example with different execution regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' The values are nor-  malized to the result of the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' A red bar indicates the  time spent for reconfiguration, and a blue bar indicates the  sum of wait time and execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' To show the benefit of  fast-DPR (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='3), we assume the baseline CGRA uses  AXI4-Lite interface for DPR, while others use fast-DPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  pipeline task on the CGRA to convert to an RGB image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  Once the CGRA generates an RGB image, the system  runs object detection and dynamically decides on the next  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' 3 When an event happens (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=', detection of a spe-  cific background), it processes the event and executes the  corresponding tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=', depth estimation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Except for a  camera pipeline that runs every frame, we set the period  from one event to the next same event to follow a uniform  random distribution between 3–7 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' We evaluate the benefit of hardware resource  partitioning and fast DPR by comparing our proposed  CGRA to the baseline CGRA with AXI4-Lite-based DPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  Specifically, the baseline CGRA maps only one task at  a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' When more than one event occurs, the base-  line handles each task one by one and reconfigures using  sequential AXI4-Lite configuration transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' In the  proposed CGRA with multi-task execution support, we ex-  ploit flexible-shape resource partitioning to concurrently  run more than one task on the CGRA when possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Also,  we use the parallel and high-frequency DPR mechanisms  in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='3 to configure bitstreams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' We compute the  arithmetic average of the latency over all frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' As de-  scribed in Figure 5, our techniques enable a 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='8% latency  reduction compared to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' With fast DPR, re-  configuration takes less than 5% of the total latency, an  appreciable reduction from 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='4% in the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Related Work  As Deep Neural Networks (DNNs) are widely used in vari-  ous domains, DNN accelerators [18, 17, 8, 9, 10, 25] have  emerged and been deployed in the cloud system [21, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  To that end, many prior works have explored multi-  tenancy support on DNN accelerators in cloud systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  3This work assumes that object detection is executed in another  hardware in the system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' GPU or ASIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  Multi-task execution support is also studied in FPGAs  targeting both cloud and edge computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' However, a non-  negligible portion of FPGA resources is typically reserved  for controlling multi-task execution, ultimately decreas-  ing the available computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' ChordMap [27]  explores the automated mapping of multi-tasked applica-  tions onto a CGRA, but it is limited to mapping multiple  tasks within streaming applications with all tasks known  a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Our work proposes hardware abstractions and  mechanisms, which both compilers and schedulers can  exploit and co-optimize to improve resource utilization in  both cloud and edge systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  Multi-Task Execution on DNN Accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Some  DNN accelerators service multi-DNN tasks at the soft-  ware level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' AI-MT [2] and Layerweaver [31] propose a  scheduling policy to mix compute- and memory-intensive  tasks to increase hardware utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' PREMA [11] im-  plements preemptible NPUs to support multi-tenancy  via temporal multiplexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  Many works add flexibil-  ity to an accelerator to accommodate multiple DNN  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Planaria [14] introduces a flexible systolic array  with dynamic architecture fission to map multiple DNN  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' [26] suggests a multi-directional network to sup-  port up to four DNN tasks with different dataflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Other  works [24, 3] explore a computing system with multiple  DNN accelerators with different hardware characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  While these works only support DNN workloads, our  work can support any applications that can be mapped  onto a CGRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  Multi-Task Execution on FPGAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' In FPGAs, multi-  task execution support has been explored in the context of  virtualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Some works divide an FPGA into a static  region, a shell, which serves as glue logic between the  host and the FPGA, and a dynamic region, a role, which  handles the computation of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' [4, 5, 33] partition  a physical FPGA into several fixed-size virtual blocks  and share them across multiple tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' AmorphOS [22]  presents a hardware abstraction of an FPGA, Morphlet,  which dynamically alters its size based on resource re-  quirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' ViTAL [35] provides a full-stack framework  to run multiple tasks with different sizes on homogeneous  regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' [34] supports running multi-DNN tasks on an  FPGA by dividing hardware resources into multiple PE  cores and spatially multiplexing them, while [30] eval-  uates the benefits of temporal multiplexing of FPGAs  using DPR for vision applications on embedded devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  While these works only target scenarios where underlying  applications change infrequently because of long reconfig-  uration time of FPGAs, our work can support both cloud  systems and real-time edge systems due to rapid DPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Conclusion  Multi-task execution support on accelerators is becoming  increasingly relevant in both cloud and edge systems and  6  DPR  Waiting + Execution  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='750  latency  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='447  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='445  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='392  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='250  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='000  baseline  fixed  variable  flexiblehas the potential to improve performance through bet-  ter hardware utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' This work proposes abstracting  hardware resources within a CGRA into coarser-grained  units with which a workload scheduler can quickly make  decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Based on the proposed abstraction, we develop  hardware mechanisms to support multi-task execution  through flexible-shape hardware partitioning and high-  throughput dynamic partial reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Our evalua-  tions modeling both a cloud and an edge system scenario  suggest that the abstraction and hardware mechanisms can  enable automatic schedulers to achieve high performance  in multi-tasked workloads on future CGRAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  References  [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
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+page_content=' Bonzini, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Pozzi, “Egra: A coarse grained  reconfigurable architectural template,” IEEE Transactions on  Very Large Scale Integration (VLSI) Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
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+page_content=' Kim, “A multi-neural network accelera-  tion architecture,” in 2020 ACM/IEEE 47th Annual International  Symposium on Computer Architecture (ISCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  IEEE, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
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+page_content=' Boroumand, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
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+page_content='  [35] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Zha and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' Li, “Virtualizing fpgas in the cloud,” in Proceedings  of the Twenty-Fifth International Conference on Architectural  Support for Programming Languages and Operating Systems,  2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content=' 845–858.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
+page_content='  8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQf8vpk/content/2301.00861v1.pdf'}
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+arXiv:2301.01092v1  [math.AP]  3 Jan 2023
+Vanishing viscosity limit for axisymmetric vortex rings
+Thierry Gallay and Vladim´ır ˇSver´ak
+January 4, 2023
+Abstract
+For the incompressible Navier-Stokes equations in R3 with low viscosity ν > 0, we con-
+sider initial data ω0 that represent an infinitely thin vortex filament of arbitrary strength Γ
+supported on a closed circle. The vorticity field ω(x, t) of the fluid is smooth at any positive
+time and corresponds to a vortex ring with thickness of order
+√
+νt that is translated along its
+symmetry axis due to self-induction, an effect anticipated by Helmholtz in 1858 and quanti-
+fied by Kelvin in 1867. For small viscosities, we show that ω(x, t) is well approximated on a
+large time interval by ωlin(x−a(t), t), where ωlin(·, t) = exp(νt∆)ω0 is the solution of the heat
+equation with initial data ω0, and ˙a(t) is the instantaneous velocity of a vortex ring of thick-
+ness
+√
+νt given by Kelvin’s formula. The proof relies on the construction of a fairly precise
+approximate solution, using a perturbative expansion in self-similar variables. To verify the
+stability of this approximation, one needs to rule out potential instabilities coming from very
+large convection terms in the linearized operator. This is done by adapting V. I. Arnold’s
+geometric stability methods developed in the inviscid case ν = 0 to the slightly viscous situ-
+ation. It turns out that although the geometric structures behind Arnold’s approach are no
+longer preserved by the equation for ν > 0, the relevant quadratic forms behave nicely on
+larger subspaces than those originally used in Arnold’s theory and interact favorably with
+the viscous terms.
+1
+Introduction and main result
+We consider the Cauchy problem for the 3d incompressible Navier-Stokes equations in the vor-
+ticity form
+∂tω + u · ∇ω − ω · ∇u = ν∆ω
+in R3 × (0, ∞) ,
+(1.1)
+ω|t=0 = ω0
+in R3 ,
+(1.2)
+where we use the familiar notation ω(x, t) for the vorticity of the fluid, and the velocity u(x, t)
+is given by the Biot-Savart law u(x, t) = (4π)−1 �
+R3 ω(y, t) ∧ (x − y) |x − y|−3 dy . We focus on
+the special case where the initial vorticity ω0 is an idealized vortex filament given by a current1
+of strength Γ concentrated on an oriented circle C ⊂ R3. More precisely, ω0 is the vector-valued
+measure on R3 defined by the identity
+⟨ω0 , ϕ⟩ = Γ
+3
+�
+i=1
+�
+C
+ϕi dxi ,
+(1.3)
+which is assumed to hold for any continuous test function ϕ = (ϕ1, ϕ2, ϕ3). We will use the
+notation ω0 = ΓδC for this initial vorticity current. In the well-known analogy between fluid
+1Here the term current can be understood in its heuristic meaning but also in the technical meaning of the
+geometric measure theory, such as in [21].
+1
+
+mechanics and electromagnetism, ω0 can be thought of as an electric current of intensity Γ flowing
+through an infinitely thin wire represented by the circle C. The direction of the current is given
+by the orientation of the circle and the sign of Γ. Vortex filaments were already considered in
+the classical 1858 paper of Helmholtz [36] that deals with the inviscid case ν = 0 corresponding
+to the Euler equation. Helmholtz argued that a circular vortex filament of zero thickness would
+move with infinite speed along its symmetry axis. In 1867 Kelvin [40] established the following
+formula for the speed of vortex rings of small but finite thickness d > 0 and radius r0 ≫ d :
+V ≈
+Γ
+4πr0
+�
+log 8r0
+d − C
+�
+,
+(1.4)
+where C ∈ R is a dimensionless constant that depends on the distribution of vorticity inside
+a cross section of the ring. If the vorticity is proportional to the distance from the symmetry
+axis, which is the assumption made by Kelvin (although the details of the computations are not
+given in [40]), the relevant value is C = 1
+4.
+V
+r0
+thickness ≈
+√
+νt
+t = 0
+t > 0
+ω0 = ΓδC
+=⇒
+Figure 1: An illustration our main result. Starting from a vortex filament supported by an oriented circle
+C, the solution of the Navier-Stokes equation evolves into a viscous vortex ring of thickness proportional
+to
+√
+νt which moves along the symmetry axis at a speed V given by Kelvin’s formula (1.5). In the right
+picture, the vortex lines are circles that fill the solid torus depicted in blue, whereas the black arrows
+denote the trajectories of the fluid particles.
+In the viscous case ν > 0, the solution originating from the singular filament ω0 = ΓδC
+becomes smooth for any positive time t > 0 and is expected to represent a viscous vortex ring
+of thickness
+√
+νt, as long as that quantity is small compared to the radius r0 of the ring. Based
+on Kelvin’s formula one anticipates that the vortex ring will move with speed
+V (t) ≈
+Γ
+4πr0
+�
+log 8r0
+√
+νt − C
+�
+,
+(1.5)
+where the constant C now corresponds to a Gaussian distribution of vorticity inside the core.
+A direct calculation shows that C = 3
+2 log(2)+ 1
+2(1−γE), where γE ≈ 0.5772... is Euler’s constant,
+see for example [52,23,51]. When the initial circle C is parametrized by (r0 cos θ, r0 sin θ, 0) for
+θ ∈ [0, 2π], with the orientation in the direction of increasing θ, the translational motion will be
+in the positive direction along the x3-axis if Γ > 0.
+It is proved in [30] that the Cauchy problem (1.1), (1.2) with ω0 = ΓδC has a unique solution
+in natural classes of axisymmetric fields. The main result of the present paper, Theorem 1.1
+below, describes the precise behavior of that solution in the low viscosity regime where the
+circulation Reynolds number Re := Γ/ν is large. Our description is valid on a time interval whose
+2
+
+length is intermediate between the advection time and the diffusion time, defined respectively
+as
+Tadv = r2
+0
+Γ ,
+Tdif = r2
+0
+ν .
+(1.6)
+Note that Tadv ≪ Tdif when Re ≫ 1. The leading term in our approximation is exactly the
+one suggested by Kelvin’s formula together with the simplest diffusion heuristics: The ring
+diffuses according to the linear heat equation, and translates with speed (1.5) along its symmetry
+axis. Denoting by ωlin(x, t) the solution of the heat equation ωt = ν∆ω with initial condition
+ω|t=0 = ω0 = ΓδC, and defining ||η|| = ||η/r||L1(R3), where r = r(x) is the distance from x to the
+symmetry axis, we can state our main result as follows.
+Theorem 1.1. There exist dimensionless constants K > 0, R0 > 0, and σ > 0 such that, for
+all Γ > 0, all r0 > 0, and all ν > 0 satisfying Re := Γ/ν ≥ R0, the following holds. If ω0 = Γ δC
+where C is an oriented circle of radius r0, the unique axisymmetric solution ω of the Cauchy
+problem (1.1), (1.2) established in [30] can be expressed for t ∈ (0, Tadv Reσ) as
+ω(x, t) = ωlin(x − a(t), t) + ωcorr(x, t) ,
+with
+||ωcorr(· , t)|| ≤ K Γ
+�√
+νt
+r0
+�1−3σ
+,
+(1.7)
+where a(t) describes the translation of the ring along its symmetry axis according to Kelvin’s
+formula (1.5). Specifically, if C = {(r0 cos θ, r0 sin θ, 0) ; θ ∈ [0, 2π]} one has a(t) = (0, 0, a3(t))
+with a3(0) = 0 and
+˙a3(t) =
+Γ
+4πr0
+�
+log 8r0
+√
+νt − 3
+2 log(2) + 1
+2(γE − 1)
+�
+.
+(1.8)
+An extended version of our result, including a more precise approximate solution and a much
+stronger control of the correction term, is formulated as Theorem 2.6 below, after the necessary
+notation has been introduced in Section 2.
+In Theorem 1.1, the constants K and R0 are large, whereas the exponent σ > 0 is taken
+small. We conjecture that an approximation result of the form (1.7) remains valid on longer
+time scales, corresponding to arbitrary values of σ ∈ (0, 1), but currently our proof requires that
+σ ≪ 1. In view of (1.4), the advection time Tadv can be interpreted as the time needed for
+a vortex ring of circulation Γ and small but not infinitesimal aspect ratio d/r0 to travel over
+a distance comparable to its radius r0. In contrast, the diffusion time Tdif = Tadv · Re is the
+time at which the diffusion length
+√
+νt becomes equal to the radius r0, so that the vortex ring
+structure is essentially lost. The assumption that Re ≫ 1 means that the vortex ring can travel
+over a very long distance along its symmetry axis before being destroyed by diffusion. Already
+on the intermediate time scale T = Tadv Reσ where Theorem 1.1 provides a rigorous control we
+find, using (1.6) and (1.8),
+|a(T)| =
+� T
+0
+˙a3(t) dt = r0
+4π Reσ�
+log
+�
+Re
+1−σ
+2 �
++ C′�
+,
+for some constant C′. Obviously the right-hand side grows boundlessly as Re → +∞, even in
+the limiting case where σ = 0 and T = Tadv.
+It is instructive to compare the situation for vortex rings with the case of a rectilinear
+filament, where the vorticity is initially concentrated on a straight line ℓ in R3. Let us denote
+this initial vorticity field by ω0 = Γδℓ. In that case the solution of the full vorticity equation is
+given by ω( · , t) = Γeνt∆δℓ, because the nonlinear terms vanish identically due to symmetries
+when evaluated on the solution of the heat equation ωt = ν∆ω. Although the evolution of the
+3
+
+velocity and the vorticity fields does not look very dramatic, the fluid particles in the vicinity of
+ℓ do move at very large speeds when νt is small, and the inertial forces in the fluid are therefore
+significant. However, these forces are exactly balanced by the incompressibility constraint.
+When the rectilinear filament is bent into a vortex ring (as already considered in Helmholtz’s
+1858 paper), the inertial forces are no longer in balance and the ring has to move. To achieve a
+relatively smooth motion, the bent vortex has to be “well-prepared” so that the inertial forces
+generated by the fast-moving fluid particles are still mostly canceled and do not generate fast
+oscillations. The initial condition ω0 = ΓδC has the advantage of letting the equation to adjust
+the field into a well-prepared state without trying to achieve this “by hand”. Quite remarkably,
+the equation adjusts the vorticity field in exactly such a way that the oscillations are avoided.2
+The largest inertial forces still cancel and the situation remains somewhat close to the rectilinear
+case with only two significant differences: (a) some motion of the ring along its axis of rotational
+symmetry is needed to balance the forces, but the speed of this motion is much lower than the
+speed of the fast particles in the fluid; (b) once the thickness of the ring becomes comparable to
+its radius, new effects (not discussed in this work) appear.
+1.1
+Main ideas of the proof of Theorem 1.1
+Our analysis starts with a construction of a precise approximation of the solution ω(x, t). This is
+achieved by writing the solution in suitable self-similar coordinates that capture well the singular
+behavior of the solution at t = 0 through explicit rescalings of a smooth “profile” η that can be
+thought of as a perturbation of a suitable Gaussian η0. The perturbed profile η is expressed as
+an asymptotic series in the time-dependent parameter ǫ =
+√
+νt/¯r, with ¯r = ¯r(t) denoting the
+instantaneous radius of the ring. To achieve a precision that is sufficient for our purposes, we need
+an expansion up to the fourth order: η = η0+ǫη1+ǫ2η2+ǫ3η3+ǫ4η4+ηcorr. The “coefficients” ηj
+with j ≥ 1 are obtained by inverting operators containing the small parameter δ = 1/Re = ν/Γ,
+and in that sense we really deal with a two-parameter expansion. As far as we know, this is
+somewhat different from other expansions in the literature, such as [12,51,27]. A one-parameter
+formal expansion in ǫ would treat δ as ∼ ǫ2, in view of the relation ¯r2ǫ2 = δ Γ t. Keeping both
+parameters makes it easier to cover the regimes when ǫ2 and δ are not really comparable, as is
+the case for very small and very large times. Strictly speaking, the “coefficients” ηj for j ≥ 1
+can also depend on log ǫ. That feature is well-known, and the leading term in the speed of the
+ring is related to choosing a moving coordinate system in which the terms with log ǫ in η1 are
+eliminated.
+The main difficulty in the proof of Theorem 1.1, however, is not in the computation of an
+approximate solution, but in showing that the true solution remains close to this approximation
+on a large time interval. This requires fairly strong stability properties for the linearization of
+the vorticity equation at the approximate solution, which is very singular in the low viscosity
+regime. When the initial condition corresponds to a finite number of parallel rectilinear vortices,
+a stability analysis was carried on in [28] by using suitable weighted L2 spaces adapted to the
+specific features of the rectilinear vortices with Gaussian profiles. In the vortex ring case the
+nonlinearity of the equations starts affecting the formal expansions earlier and it is unclear
+whether the setup in [28] can be used to show that the vortex ring will not disintegrate on
+time-scales approaching zero as ν → 0. A recent important work [6] extends some of the 2d
+methods for proving stability to a relevant 3d situation, but the length of the time interval over
+which the solution is under control may approach 0 as ν tends to 0.
+In physical flows and numerical experiments one observes a remarkable degree of stability
+2In the related situation of interacting vortices in R2, this was already observed in [28].
+4
+
+of vortex rings as well as signs of instabilities with respect to non-axisymmetric perturbations,
+see for example [56,48]. At a rigorous mathematical level the stability issues have not been well
+understood. In fact, when Γ/ν is not small, not only the stability, but even the uniqueness of
+the solutions of the Cauchy problem above with ω0 = Γ δC (and also with ω0 = Γ δℓ) is open in
+classes of solutions that do not share the symmetry of the initial data.
+In the 1960s, V. I. Arnold suggested a variational method for proving stability of steady
+solutions to Euler’s equation based on a geometric insight that can be summarized as follows,
+using the Hamiltonian setup of [45]:
+(a) The incompressible Euler equation can be viewed as a Poisson system with a Hamiltonian
+function given by the usual kinetic energy.
+(b) The steady states are critical points of the energy on the symplectic leaves.
+The latter
+coincide with the coadjoint orbits, called just orbits in what follows, of the group of the volume-
+preserving diffeomorphisms of the fluid domain acting by push-forward on the vorticity fields.
+(c) When the critical point is a local maximum or a local minimum on an orbit, the corresponding
+steady-state should be stable.
+These ideas fit into a broader family of methods used for proving stability of solutions of
+Hamiltonian systems by invoking extremality properties of a conserved quantity under con-
+straints given by other conserved quantities.
+For example, a circular planetary orbit in the
+three-dimensional Kepler problem is stable because it minimizes energy for a given angular
+momentum.3 In the applications to vortex rings, it is natural to restrict the analysis to axisym-
+metric flows with no swirl, which means that the velocity field is invariant under rotations about
+a symmetry axis and under reflection across any plane containing that axis.
+Arnold’s method has found many applications to Euler flows in 2d (see, for example, [4]),
+and has also been invoked in the work of Benjamin [8] on inviscid vortex rings that is directly
+relevant for our purposes here. Although some arguments in [8] may not be fully rigorous, they
+provide important suggestions for investigating stability of inviscid vortex rings in the class of
+axisymmetric solutions. In a different direction, the conservation of energy, impulse, and vortex
+strength has been used to control the evolution of a general class of concentrated solutions of
+the Euler equations describing vortex rings, see for example [7].
+There is voluminous literature on the stationary vortex ring solutions of the Euler equation,
+starting with the explicit solution of Hill [38], see e.g. [1,2,5,10,13–16,23–26,49,50,54]. Many
+of these works rely in one way or another on variational aspects of the underlying PDEs that
+have connections to the work of Arnold and Benjamin, albeit a somewhat indirect one. Roughly
+speaking, if we compare Arnold’s setup to the maximization of a function f(x) under constraints
+gj(x) = cj, one can compare some of the variational approaches in the references above to
+searching for critical points of f(x) − λ1g1(x) − · · · − λmgm(x) when the Lagrange multipliers
+λ1, . . . , λm are given. Readers interested in related links can find more details in [31].
+The works [9,11,7] use very effectively some of the variational principles inherent in Arnold’s
+and Benjamin’s approach, essentially at the inviscid level, assuming that the viscosity is suffi-
+ciently small for the viscous term to be treated as a lower-order perturbation. In our asymptotic
+expansions of the solutions of (1.1), (1.2) inviscid vortex ring solutions can also be discerned.
+For each fixed time t > 0 the third-order expansion in our parameter ǫ =
+√
+νt/¯r is a good ap-
+proximation of an inviscid vortex ring, at least in the limiting case where our second parameter
+δ = ν/Γ is taken equal to zero. A part of our stability analysis can be thus understood in terms
+of the stability properties of this ring, see Remark 2.3 and Section 3.8 for more details.
+3It is well-known that this is no longer the case in dimension four and higher [35].
+5
+
+If one wishes to apply Arnold’s ideas to the solutions of (1.1), (1.2), there appears to be
+a non-trivial obstacle: The viscous flows do not preserve the geometric structures that are at
+the basis of Arnold’s considerations and the influence of the viscosity is too large to treat the
+viscous terms perturbatively. At first this may seem to be serious problem: If the preservation
+of the orbits and the Hamiltonian nature of the equations are violated beyond the reach of the
+perturbative approach (such as [9,11,7]), can the geometric structure relying on maximization
+of the energy on symplectic leaves be helpful? In our previous work [31] we showed, in a much
+simpler situation, that the answer to this question can be positive. It turns out that the quadratic
+forms coming up in Arnold’s stability analysis, although originally envisaged as quadratic forms
+on the tangent spaces to the orbits, are often well-behaved on much larger subspaces. This point
+can still be conceptually explained by the geometry of the Euler equation. What we find more
+surprising is that Arnold’s forms also have favorable behavior with respect to the dissipative
+term generated by the viscosity. We can show this by direct calculation, but we do not have
+a good conceptual explanation of this fortuitous circumstance. In the paper [31] we showed
+that the above ideas can be used to prove the stability of the rectilinear vortex solution (in
+self-similar variables) with respect to perturbations for which the vorticity field stays parallel to
+the original vortex line. This result has been established previously by a different method [33].
+The new proof in [31] can be thought of as a proof of concept that the ideas of Arnold can be
+applied even in the presence of viscosity. The application to vortex rings presented here is more
+complicated, but we are not aware of any simpler approach in that case.
+1.2
+Comments on the local induction approximation for general filaments
+The problem studied in this paper can be considered as a special case of the viscous version of
+the local induction approximation conjecture. In the setup considered here the conjecture could
+be formulated as follows: if we replace the circle C be a general smooth closed curve and consider
+the Cauchy problem (1.1), (1.2) with ω0 = ΓδC, the motion of the filament C should still be
+determined essentially by two effects: the diffusion, which transforms the filament into a vortex
+tube of thickness d(t) ≈
+√
+νt at time t, and the advection by the self-induced velocity field. The
+latter is described by a geometric equation that represents an extension of Kelvin’s formula to
+general smooth curves, and was derived by Da Rios [18] in 1906:
+V ≈
+� Γ
+4πr log 8r
+d
+�
+b .
+(1.9)
+Here V is the vector representing the local velocity of the filament, b denotes the local binormal
+vector, r is the local radius of the curvature, and d denotes the local thickness of the filament. (All
+these quantities may be time- and position-dependent.) In the limit ν → 0 the approximation
+should be valid until the geometric evolution of the curve by the binormal flow leads to a self-
+intersection. For general initial curves C the time of the first self-intersection may be approaching
+zero as ν approaches zero. The first important step towards this general case, a local-in time
+well-posedness result for a fixed ν > 0, was obtained in [6]. Some formal computations related
+to the conjecture are presented in [12] and we also refer the reader to the important conditional
+result in [39].
+Our result can be viewed as a proof of the viscous formulation of the conjecture in the special
+case where the curve C is a circle. Important previous works for the axisymmetric case with
+very small viscosity include [9,11], where the case of several vortex rings is also considered. The
+setup in these works is somewhat different, in that the initial vortex rings have finite thickness
+d and certain relations between d, the vortex strength Γ, the viscosity ν and the maximum of
+initial vorticity are assumed so that (among other things) the limiting motion of the rings has
+6
+
+uniformly bounded velocity. In particular, the vortex strength Γ is assumed to approach 0 as
+d tends to zero and the viscosity ν has to satisfy smallness conditions related to d, so that the
+viscous terms can be treated as a small perturbation. Yet another angle on vortex rings is taken
+in the recent work [19] that deals with “leapfrogging” of inviscid vortex rings.
+The general case of the local induction approximation conjecture for the setup considered
+in this paper seems to be difficult. In fact, it is unclear whether the strongest version of the
+conjecture is valid even for small perturbations of the circle, as the perturbed filaments may
+perhaps become unstable to general 3d perturbations before possible self-intersections.
+For
+example, the instabilities studied in [56,48] may be relevant.
+2
+Preliminaries and sketch of the proof
+In this section we introduce the notation that is necessary to formulate our result in its stronger
+form, and we give a pretty detailed sketch of the overall proof. The construction of the approx-
+imate solution will be performed in Section 3, and the stability analysis in Section 4. Technical
+calculations are postponed to Appendix A and B.
+2.1
+Formulation of the problem in cylindrical coordinates
+In a suitable Cartesian coordinate system, the circle of radius r0 > 0 which represents the
+support of the initial vorticity (1.3) is given by C = {(r0 cos θ, r0 sin θ, 0) ; θ ∈ [0, 2π]}. Due to
+the symmetries of the problem, it is natural to introduce the standard cylindrical coordinates
+(r, θ, z) defined by x1 = r cos θ, x2 = r sin θ, x3 = z and to restrict our attention to velocity and
+vorticity fields of the form
+u(x, t) = ur(r, z, t)er + uz(r, z, t)ez ,
+ω(x, t) = ωθ(r, z, t)eθ ,
+(2.1)
+where er, eθ, ez denote unit vectors in the radial, azimuthal, and vertical directions, respectively.
+In the usual terminology, we thus consider axisymmetric flows with no swirl, see [44]. Due to
+the incompressibility condition div u = r−1∂r(rur) + ∂z(uz) = 0, the velocity components ur, uz
+can be expressed in terms of the Stokes stream function ψ :
+ur = −1
+r ∂zψ ,
+uz = 1
+r ∂rψ .
+(2.2)
+With this notation the vorticity formulation of the Navier-Stokes equation (1.1) becomes
+∂tωθ +
+�
+ψ, ωθ
+r
+�
+= ν
+��
+∂2
+r + ∂2
+z)ωθ + ∂r
+ωθ
+r
+�
+,
+(2.3)
+where {·, ·} is the Poisson bracket defined by {ψ, φ} = ∂rψ ∂zφ − ∂zψ ∂rφ. Eq. (2.3) is to be
+solved in the domain Ω = {(r, z) ∈ R2 | r > 0}. The smoothness of the fields in the original
+variables imposes the “boundary conditions” ωθ(r, z, t) = rζ(r, z, t) and ψ(r, z, t) = r2Ψ(r, z, t)
+near r = 0, where ζ and Ψ can be extended to smooth functions on R2 × R+ that are even
+functions of r.
+The Biot-Savart law can be expressed in this setup as
+ψ(r, z) =
+1
+2π
+�
+Ω
+√
+r¯r F
+�(r − ¯r)2 + (z − ¯z)2
+r¯r
+�
+ωθ(¯r, ¯z) d¯r d¯z ,
+(2.4)
+where F : (0, ∞) → (0, ∞) is defined by
+F(s) =
+� π/2
+0
+1 − 2 sin2 ψ
+�
+sin2 ψ + s/4
+dψ ,
+s > 0 .
+(2.5)
+7
+
+Formula (2.4) also provides a solution to the equation
+curl curl
+�ψ
+r eθ
+�
+= ωθ eθ ,
+(2.6)
+which is familiar in magnetostatics, see for example [47, §701]. The same expression can also be
+found in the classical book [42, §161]. It is well-known (and not hard to check) that
+F(s) =
+�
+log
+8
+√s − 2 + O(s log s)
+as s → 0 ,
+π
+2s3/2 + O(s−5/2)
+as s → ∞ .
+(2.7)
+Since we wish to solve the Cauchy problem (1.1), (1.2) with initial data ω0 = ΓδC, we impose
+to the scalar function ωθ defined in (2.1) the initial condition
+ωθ
+���
+t=0 = Γδ(r0,0) ,
+(2.8)
+where we use δ(r0,z0) to denote the Dirac mass at the location (r0, z0) ∈ Ω. Our starting point
+is the following global well-posedness result for the vorticity equation (2.3) with concentrated
+initial data.
+Theorem 2.1. [30] For any Γ > 0, any ν > 0, and any (r0, z0) ∈ Ω, the axisymmetric vorticity
+equation (2.3) has a unique global mild solution ωθ ∈ C0((0, ∞), L1(Ω) ∩ L∞(Ω)) such that
+sup
+t>0
+∥ωθ(t)∥L1(Ω) < ∞ ,
+and
+ωθ(t) dr dz ⇀ Γ δ(r0,z0)
+as t → 0 .
+(2.9)
+Moreover there exists a constant C > 0, depending only on the ratio Γ/ν, such that
+�
+Ω
+���ωθ(r, z, t) −
+Γ
+4πνt e− (r−r0)2+(z−z0)2
+4νt
+��� dr dz ≤ C Γ
+√
+νt
+r0
+log
+� r0
+√
+νt + 1
+�
+,
+(2.10)
+whenever t ∈ (0, Tdif), where Tdif = r2
+0/ν.
+Here and in what follows, it is understood that L1(Ω) = L1(Ω, dr dz), and similarly for the
+other Lebesgue spaces. Theorem 2.1 establishes the existence of a four-dimensional family of
+vortex ring solutions to (2.3) parametrized by the circulation Γ > 0, the viscosity ν > 0, the
+initial radius r0 > 0, and the initial vertical position z0 ∈ R. Due to translation invariance in
+the vertical direction, we may assume without loss of generality that z0 = 0, and we can also
+take r0 = 1 by rescaling the space variables. Then a rescaling of time allows us to change the
+values of both ν and Γ, while keeping the ratio Γ/ν fixed. Hence up to symmetries, the viscous
+vortex ring solutions we consider here form a one-parameter family indexed by the circulation
+Reynolds number Re := Γ/ν.
+The uniqueness of the vortex ring solution under the minimal assumptions (2.9) is discussed
+in some detail in [30], so we concentrate here on the short-time estimate (2.10), which is of
+limited use despite appearances. For a fixed value of the Reynolds number, the right-hand side
+of (2.10) is small whenever t ≪ Tdif, which means that the solution of (2.3) with initial data
+(2.8) is well approximated by a Gaussian vortex ring of thickness proportional to
+√
+νt, located
+a the initial position (r0, z0) ∈ Ω. However, since the constant C depends (very badly) on the
+Reynolds number, estimate (2.10) gives no information on the solution at a fixed time t > 0
+in the low viscosity regime ν → 0. This limitation is not surprising: due to the translational
+motion along the vertical axis predicted by Kelvin’s formula (1.5), the vortex ring at time t > 0
+is actually located at a new position which is rather far from the initial one if ν is small.
+8
+
+Our goal in this paper is to replace (2.10) by an improved estimate of the form
+�
+Ω
+���ωθ(r, z, t) −
+Γ
+4πνt e− (r−¯r(t))2+(z−¯z(t))2
+4νt
+��� dr dz ≤ K Γ
+√
+νt
+r0
+,
+t ∈ (0, Tadv Reσ) ,
+(2.11)
+where the constant K is now independent of the Reynolds number, if Re ≫ 1. Comparing with
+(2.10), we observe that (2.11) is valid up to the intermediate time Tadv Reσ, for some σ ∈ (0, 1),
+which is shorter than Tdif ≡ Tadv Re.
+But the main difference is that (2.11) compares the
+solution ωθ(r, z, t) to a vortex ring located at a time-dependent position (¯r(t), ¯z(t)), which has to
+be determined. As we shall see, we can take ¯r(t), ¯z(t) to be smooth functions of time satisfying
+¯r(0) = r0, ¯z(0) = z0, and
+˙¯r(t) = O
+� ν
+r0
+�
+,
+˙¯z(t) =
+Γ
+4πr0
+�
+log
+1
+ǫ(t) + V
+��
+1 + O
+�
+ǫ(t)2��
+,
+(2.12)
+where ǫ(t) =
+√
+νt/¯r(t) and V = 3
+2 log(2) + 1
+2(γE − 1). The first relation in (2.12) implies that
+¯r(t) = r0
+�
+1 + O(ǫ(t)2)
+�
+, which means that the change in the radius of the vortex ring over the
+time interval under consideration is much smaller than the diffusion length
+√
+νt. The second
+equality coincides with the viscous form (1.5) of Kelvin’s formula, up to higher order corrections.
+2.2
+Self-similar variables
+From now on, we fix the circulation Γ > 0 and the position (r0, 0) ∈ Ω of the initial filament,
+and we consider the vortex ring solution given by Theorem 2.1, in the regime where the viscosity
+ν > 0 is small. In view of the approximation formula (2.11), which is our objective, it is natural
+to make the following self-similar Ansatz for the axisymmetric vorticity and the associated Stokes
+stream function :
+ωθ(r, z, t) = Γ
+νt η
+�r − ¯r(t)
+√
+νt
+, z − ¯z(t)
+√
+νt
+, t
+�
+,
+ψ(r, z, t) = Γ ¯r(t) φ
+�r − ¯r(t)
+√
+νt
+, z − ¯z(t)
+√
+νt
+, t
+�
+,
+(2.13)
+where the time-dependent position (¯r(t), ¯z(t)) ∈ Ω has to be determined. We introduce the
+important notation
+δ = ν
+Γ ,
+ǫ =
+√
+νt
+¯r(t) ,
+R = r − ¯r(t)
+√
+νt
+,
+Z = z − ¯z(t)
+√
+νt
+.
+(2.14)
+The evolution equation for the rescaled vorticity η(R, Z, t) is found to be
+t∂tη + Γ
+ν
+�
+φ ,
+η
+1 + ǫR
+�
+−
+�
+t
+ν
+�
+˙¯r ∂Rη + ˙¯z ∂Zη
+�
+= Lη + ∂R
+�
+ǫη
+1 + ǫR
+�
+,
+(2.15)
+where
+�
+φ, χ
+�
+= ∂Rφ ∂Zχ − ∂Zφ ∂Rχ is the Poisson bracket in the new variables (R, Z), and L is
+the Fokker-Planck operator
+L = ∂2
+R + ∂2
+Z + 1
+2
+�
+R∂R + Z∂Z
+�
++ 1 .
+(2.16)
+Eq. (2.15) is to be solved in the time-dependent domain
+Ωǫ =
+�
+(R, Z) ∈ R2 �� 1 + ǫR > 0
+�
+,
+(2.17)
+9
+
+with the Dirichlet boundary condition η(−1/ǫ, Z, t) = 0 for all (Z, t) ∈ R × R+.
+As in [30], it is useful to introduce the velocity field U = (UR, UZ) defined by
+UR = − ∂Zφ
+1 + ǫR ,
+UZ =
+∂Rφ
+1 + ǫR ,
+(2.18)
+in terms of which the nonlinearity in (2.15) reads
+�
+φ ,
+η
+1+ǫR
+�
+= ∂R
+�
+UR η) + ∂Z(UZ η). The
+stream function φ in (2.15) satisfies the elliptic equation
+η = ∂ZUR − ∂RUZ ≡ −∂R
+� ∂Rφ
+1 + ǫR
+�
+−
+∂2
+Zφ
+1 + ǫR ,
+(R, Z) ∈ Ωǫ ,
+(2.19)
+with boundary conditions φ(−1/ǫ, Z, t) = ∂Rφ(−1/ǫ, Z, t) = 0 for all (Z, t) ∈ R × R+. Using
+(2.4), we easily obtain the representation formula [30]
+φ(R, Z) =
+1
+2π
+�
+Ωǫ
+�
+(1+ǫR)(1+ǫR′) F
+�
+ǫ2 (R−R′)2 + (Z−Z′)2
+(1+ǫR)(1+ǫR′)
+�
+η(R′, Z′) dR′ dZ′ ,
+(2.20)
+where F is as in (2.5). In what follows we write φ = BSǫ[η] when (2.20) holds.
+The quantities introduced in (2.14) are all dimensionless. The first one is the inverse Reynolds
+number δ > 0, a fixed parameter that is assumed to be small. The second one is the time-
+dependent aspect ratio ǫ > 0, which appears in the evolution equation (2.15), in the definition
+of the domain (2.17), and in the Biot-Savart formula (2.20). Finally, the variables R, Z are self-
+similar coordinates centered at the time-dependent location (¯r(t), ¯z(t)) and normalized according
+to the size
+√
+νt of the vortex core. Note that the rescaled functions η, φ defined in (2.13) are
+also dimensionless.
+Remark 2.2. Recalling that δ = ν/Γ and Tadv = r2
+0/Γ, we observe that
+ǫ2 = νt
+r2
+0
+r2
+0
+¯r(t)2 =
+δt
+Tadv
+r2
+0
+¯r(t)2 ≈
+δt
+Tadv
+,
+(2.21)
+as long as the ratio r0/¯r(t) remains close to unity, which will always be the case thanks to (2.12).
+It follows in particular that ǫ2 is comparable to δ whenever t is comparable to Tadv. Our goal
+is to control the solution of (2.3) for t ≤ Tadvδ−σ for some σ ∈ (0, 1), and on that interval it
+follows from (2.21) that ǫ2 ≲ δ1−σ.
+2.3
+Approximate solution
+The first important step in our analysis is the construction of an approximate solution of (2.15)
+with initial data
+η0(R, Z) =
+1
+4π e−(R2+Z2)/4 ,
+(R, Z) ∈ Ω0 = R2 .
+(2.22)
+The associated stream function satisfies −∆0φ0 = η0, where ∆0 = ∂2
+R + ∂2
+Z. As η0, φ0 are both
+radially symmetric, it is clear that {φ0, η0} = 0, and the Gaussian profile (2.22) has the property
+that Lη0 = 0. Since ǫ = 0 when t = 0 in view of (2.14), we conclude that equation (2.15) is
+indeed satisfied at initial time.
+For t > 0, we construct our approximate solution as a power series in the time-dependent
+parameter ǫ =
+√
+νt/¯r, the coefficients of which depend on the small parameter δ. To this end,
+we multiply both sides of (2.15) by δ and rewrite the equation in the equivalent form
+δ t∂tη +
+�
+φ ,
+η
+1 + ǫR
+�
+− ǫ¯r
+Γ
+�
+˙¯r ∂Rη + ˙¯z ∂Zη
+�
+= δ
+�
+Lη + ∂R
+�
+ǫη
+1 + ǫR
+��
+.
+(2.23)
+10
+
+This equation is defined on the time-dependent domain Ωǫ, but expanding the factors (1+ǫR)−1
+in powers of ǫ we get at each order a relation that can be solved in the whole plane Ω0 = R2.
+The corresponding approximation for the stream function φ is obtained in a self-consistent way
+by expanding the integrand in (2.20) in powers of ǫ, and integrating order by order over the
+whole plane R2. It turns out one can work with an asymptotic expansion of the form
+ηapp(R, Z, t) =
+M
+�
+m=0
+ǫm ηm(R, Z, βǫ) ,
+φapp(R, Z, t) =
+M
+�
+m=0
+ǫm φm(R, Z, βǫ) ,
+(2.24)
+where (R, Z) ∈ R2, 0 < ǫ ≪ 1, with the dependence of the profiles ηm and φm on βǫ := log(1/ǫ)
+being polynomial. The profiles also depend on the small parameter δ, but to make the notation
+lighter this dependence is not indicated explicitly. The velocity of the vortex center is not known
+a priori, but can be expressed in a similar way as a power series in ǫ :
+˙¯r(t) =
+M−1
+�
+m=0
+ǫm ˙¯rm(βǫ) ,
+˙¯z(t) =
+M−1
+�
+m=0
+ǫm ˙¯zm(βǫ) ,
+(2.25)
+where the quantities ˙¯rm(βǫ), ˙¯zm(βǫ) depend on δ and are polynomials in βǫ.
+The outcome of the analysis carried out in Section 3 below is that, if we want our expansions
+(2.24), (2.25) to hold uniformly with respect to the parameter δ in the limit where δ → 0, there
+is a unique choice of the profiles ηm, φm and of the velocities ˙¯rm, ˙¯zm such that :
+a) Both members of Eq. (2.23) agree up to order O(ǫM+1), modulo powers of βǫ;
+b) The point (¯r(t), ¯z(t)) ∈ Ω is the center of the vorticity distribution defined by ηapp(R, Z, t).
+The integer M in (2.24), (2.25) determines the accuracy of our approximate solution. It turns
+out that M = 4 will be sufficient for our purposes. The velocities ˙¯r(t), ˙¯z(t) given by (2.25) are
+found to satisfy (2.12).
+Remark 2.3. If we set δ = ˙¯r = 0, equation (2.23) reduces to
+�
+φ ,
+η
+1 + ǫR
+�
+− ǫ¯r
+Γ
+˙¯z ∂Zη ≡
+�
+φ − ¯r ˙¯z
+2Γ (1 + ǫR)2 ,
+η
+1 + ǫR
+�
+= 0 ,
+(2.26)
+which is exactly the relation satisfied by the vorticity η and the stream function φ of a stationary
+solution of the Euler equations in a frame moving with speed ˙¯z ez. In that situation the aspect
+ratio ǫ > 0 is fixed and, as in (2.14), the dimensionless variables (R, Z) are defined so that
+(r, z) = (¯r, ¯z) + ǫ¯r (R, Z). An approximate solution of (2.26) can be constructed in the form of
+a power series in ǫ, as in (2.24), where all profiles ηm, φm are even functions of the variable
+Z ∈ R, since this is the case for the coefficients of (2.26) and for the initial approximation
+(2.22). Returning to the approximate solution (2.24), we deduce by uniqueness that ηapp, φapp
+are even functions of Z in the limit δ → 0, and that ˙¯r = Γ
+r0 O(δ) as δ → 0.
+Remark 2.4. In view of (2.14) and (2.25), the function ǫ(t) is implicitly defined by the relation
+√
+νt
+ǫ(t) = ¯r(t) = r0 +
+M−1
+�
+m=0
+� t
+0
+ǫ(s)m ˙¯rm
+�
+βǫ(s)
+�
+ds ,
+(2.27)
+which should hold when 0 < t ≪ Tdif. As was mentioned in the previous remark, the radial
+velocities ˙¯ri are small when δ ≪ 1, so that Eq. (2.27) will be easy to solve, see Section 3.6.
+11
+
+The asymptotic approximation ηapp(R, Z, t) is defined on the whole plane and does not vanish
+on the boundary ∂Ωǫ. To obtain a valid approximate solution of (2.15), we fix σ0 ∈ (0, 1) and
+we truncate ηapp outside a large ball of radius ǫ−σ0 by setting
+η∗(R, Z, t) = χ0
+�
+ǫσ0(R2+Z2)1/2�
+ηapp(R, Z, t) ,
+φ∗(·, t) = BSǫ[η∗(·, t)] ,
+(2.28)
+where χ0 : R+ → [0, 1] is a smooth function such that χ0(r) = 1 for r ≤ 1 and χ0(r) = 0 for
+r ≥ 2. The remainder of that approximation is defined as
+Rem(R, Z, t) = Lη∗ + ∂R
+� ǫη∗
+1+ǫR
+�
+− t∂tη∗ − 1
+δ
+�
+φ∗ ,
+η∗
+1+ǫR
+�
++ ǫ¯r
+δΓ
+�
+˙¯r ∂Rη∗ + ˙¯z ∂Zη∗
+�
+.
+(2.29)
+By construction, this quantity depends on time only through the parameter ǫ =
+√
+νt/¯r(t).
+The accuracy of our approximate solution is quantified by the following result, which is
+established in Section 3.7 below :
+Proposition 2.5. Given any γ0 < 1 and any γ5 < 5, there exist a constant C > 0 such that the
+remainder (2.29) satisfies
+sup
+(R,Z)∈Ωǫ
+eγ0(R2+Z2)/4 |Rem(R, Z, t)| ≤ C
+�
+ǫδ + ǫγ5δ−1�
+,
+(2.30)
+whenever the parameters ǫ, δ are small enough.
+2.4
+Stability estimates
+In our previous work [30], the evolution equation (2.15) was carefully studied in the particular
+case where ¯r(t) = r0 and ¯z(t) = z0. This does not make any substantial difference for the initial
+value problem at fixed viscosity, and we can thus infer from the results of [30] that Eq. (2.15) has
+a unique solution η(R, Z, t) with initial data η0 given by (2.22). Our purpose is to show that, if
+the inverse Reynolds number δ = ν/Γ is sufficiently small, the solution η(R, Z, t) remains close
+to the approximation (2.28) on a long time interval of the form (0, Tadvδ−σ), for some small
+σ > 0. We use the following decomposition :
+η(R, Z, t) = η∗(R, Z, t) + δ ˜η(R, Z, t) ,
+φ(R, Z, t) = φ∗(R, Z, t) + δ ˜φ(R, Z, t) ,
+(2.31)
+where ˜φ = BSǫ[˜η] in the sense of (2.20). The equation satisfied by the perturbation ˜η reads
+t∂t˜η + 1
+δ
+�
+φ∗ ,
+˜η
+1 + ǫR
+�
++ 1
+δ
+�
+˜φ ,
+η∗
+1 + ǫR
+�
++
+�
+˜φ ,
+˜η
+1 + ǫR
+�
+− ǫ¯r
+δΓ
+�
+˙¯r ∂R˜η + ˙¯z ∂Z ˜η
+�
+= L˜η + ∂R
+�
+ǫ˜η
+1 + ǫR
+�
++ 1
+δ Rem(R, Z, t) .
+(2.32)
+Since η∗(R, Z, 0) = η0(R, Z), the nonlinear evolution equation (2.32) is to be solved with
+zero initial data. The solution is therefore driven by the source term δ−1Rem(R, Z, t), which is
+small in view of Proposition 2.5 and Remark 2.2. As long as ˜η stays small, the nonlinear term
+{˜φ, (1+ǫR)−1˜η} is of course harmless. The most serious difficulty in controlling ˜η using (2.32)
+comes from the linear terms with a large prefactor δ−1 = Γ/ν. These terms could conceivably
+trigger violent instabilities that might lead to strong amplification of ˜η in a short time. Our goal
+is to show that this scenario does not occur, due to the special structure of the advection terms
+in (2.32). A similar strategy was applied in the previous work [28] devoted to the vanishing
+viscosity limit of interacting vortices in the plane, but the specific estimates used there do not
+seem to be easily adaptable to the present situation.
+12
+
+To control the time evolution of the solution of (2.32), we use the energy functional
+Eǫ(t) = 1
+2
+�
+Ωǫ
+Wǫ ˜η2 dR dZ − 1
+2
+�
+Ωǫ
+˜φ ˜η dR dZ ,
+(2.33)
+where Wǫ : Ωǫ → (0, +∞) is a weight function that will be described below. The first term in
+the right-hand side of (2.33) is a weighted L2 integral of the vorticity ˜η, similar to weighted
+enstrophies that were used for the same purposes in [33,28,30], for instance. The second term is
+just the kinetic energy associated with the vorticity perturbation ˜η, as can be seen by invoking
+(2.18), (2.19) and integrating by parts :
+1
+2
+�
+Ωǫ
+˜φ ˜η dR dZ = 1
+2
+�
+Ωǫ
+|∂R ˜φ|2 + |∂Z ˜φ|2
+1 + ǫR
+dR dZ = 1
+2
+�
+Ωǫ
+�
+| ˜UR|2 + | ˜UZ|2�
+(1 + ǫR) dR dZ .
+To construct the weight Wǫ in (2.33), we consider three different regions :
+1) The inner region where ρ := (R2+Z2)1/2 ≲ ǫ−σ1, for some small σ1 > 0. Here we choose
+Wǫ =
+1
+1 + ǫR Φ′
+ǫ
+�
+η∗
+1 + ǫR
+�
+,
+(2.34)
+where η∗ is the approximate solution (2.28) and Φǫ : (0, +∞) → R is a smooth function with
+the property that, in the region under consideration,
+φ∗ − ¯r ˙¯z
+2Γ (1 + ǫR)2 = Φǫ
+�
+η∗
+1 + ǫR
+�
++ O(ǫδ + ǫγ3) ,
+(2.35)
+for some γ3 < 3 that can be arbitrarily close to 3. It is not difficult to understand intuitively why
+such a function should exist. Indeed, in the dimensionless variables (2.14), the left-hand side
+of (2.35) is nothing but the stream function of the approximate solution φ∗ in a frame moving
+with constant speed ˙¯z in the vertical direction, see Remark 2.3. If we drop the remainder term
+O(ǫδ + ǫγ3) and consider ǫ > 0 as a fixed parameter, Eq. (2.35) expresses a functional relation
+between the potential vorticity ζ∗ := (1+ǫR)−1η∗ and the stream function, which implies that η∗
+represents a stationary solution of the Euler equation in the moving frame. This is not exactly
+true, of course, but the estimate on the remainder Rem(R, Z, t) in Proposition 2.5 ensures that
+the approximate solution η∗ (for a fixed value of ǫ > 0) is not far from a stationary solution of
+Euler, and in Section 3.8 we verify that this implies the existence of a function Φǫ satisfying
+(2.35). Moreover, an easy calculation shows that
+1
+1 + ǫR Φ′
+ǫ
+�
+η∗
+1 + ǫR
+�
+= 4
+ρ2
+�
+eρ2/4 − 1
+�
++ O(ǫ) ,
+ρ =
+�
+R2 + Z2 ≤ ǫ−σ1 .
+2) The intermediate region where ǫ−σ1 ≲ ρ ≤ ǫ−σ2, for some σ2 > 1. In this area we assume
+that the weight is approximately constant in space, with value Wǫ ≈ exp(ǫ−2σ1/4).
+3) The far field region where ρ ≥ ǫ−σ2. Here we take Wǫ ≈ exp(ρ2γ/4), where γ = σ1/σ2.
+The actual construction of the weight is more complicated, and ensures that Wǫ is Lipschitz
+continuous at the boundaries of the three regions under consideration, see Section 4 below
+for details. For the moment, we just mention that our choice of the energy functional in the
+inner region is related to Arnold’s variational characterization of the steady states of the Euler
+equation, as discussed in our previous work [31]. In fact, if we suppose that ζ∗ is a stationary
+solution of the axisymmetric Euler equation in a moving frame (which not exactly true), then
+the functional (2.33) with the weight (2.34) corresponds, up to a constant factor, to the second
+13
+
+≈ ǫ−σ1
+ǫ−σ0
+ǫ−1
+ǫ−σ2
+ρ
+1
+exp(ǫ−2σ1/4)
+Wǫ(R, Z)
+A(ρ)
+exp(ρ2γ/4)
+Figure 2: When ǫ > 0 is small, the weight Wǫ(R, Z) entering the energy functional (2.33) is close to a
+piecewise smooth radially symmetric function. In the inner region where ρ := (R2+Z2)1/2 ≲ ǫ−σ1, we
+have Wǫ ≈ A(ρ) := (4/ρ2)
+�
+eρ2/4 − 1
+�
+. When Wǫ reaches the threshold value exp(ǫ−2σ1/4), it remains
+approximately constant until ρ = ǫ−σ2, and outside that region we have Wǫ ≈ exp(ρ2γ/4) with γ = σ1/σ2.
+The dashed lines reflect the fact that exp(ρ2γ/4) ≲ Wǫ ≲ A(ρ) where the implicit constants do not depend
+on the parameter ǫ. The intermediate scales ǫ−σ0, where the truncation (2.28) occurs, and ǫ−1, which is
+the distance from the origin to the boundary ∂Ωǫ, are indicated for completeness.
+variation of the kinetic energy on the isovortical surface, which is the set of (potential) vorticities
+ζ := (1 + ǫR)−1η that are measure-preserving rearrangements of ζ∗ [3, 31]. Since the kinetic
+energy is conserved under the inviscid dynamics, the advection terms involving δ−1 in (2.32),
+which originate from the linearization of Euler’s equation at the “steady state” ζ∗, do not
+contribute to the time evolution of the functional Eǫ.
+In reality ζ∗ is only an approximate
+steady state of Euler, and the cancellations alluded to above only occur up to correction terms
+of order O(ǫδ + ǫγ3), but this is sufficient to cancel the dangerous factors δ−1 in (2.32). On the
+other hand, away from the inner region, the last term in (2.33) is extremely small, so that our
+functional Eǫ reduces to a weighted enstrophy. We assume that the weight Wǫ is approximately
+constant in the intermediate region, so that the advection terms in (2.32) do not contribute
+to the evolution of Eǫ, and in the far field region the dynamics is dominated by the diffusion
+operator L in (2.32) so that we can just take any radially symmetric weight with sufficiently
+fast growth at infinity.
+A technical difficulty inherent to our approach is the fact that the functional Eǫ is not
+coercive, unless the perturbed vorticity ˜η satisfies some moment conditions. The problem comes
+from the inner region, where the last term in (2.33) plays an important role.
+The results
+established in [31, Section 2] indicate that Eǫ is positive definite provided ˜η has zero mean and
+vanishing first order moments with respect to the space variables R, Z. In practice this means
+that, in addition to the information provided by the energy Eǫ, we must control the integral
+and the first order moments of the perturbed vorticity ˜η. It turns out that
+�
+˜η dR dZ is always
+extremely small, of the order of O(exp(−c/ǫ2)) for some c > 0. The radial moment
+�
+R ˜η dR dZ
+may take larger values, but can be controlled using the conservation of the total impulse of the
+vortex ring. Finally, to get rid of the vertical moment
+�
+Z ˜η dR dZ, we exploit the translation
+invariance of the system in the vertical direction, and we introduce a “modulation parameter”
+in the same spirit as for the stability analysis of solitary waves [55,46]. A precise description of
+our approach to control the moments of the vorticity ˜η can be found in Section 4.1 below.
+14
+
+Disregarding these technical questions for the moment, we briefly indicate how the argument
+is concluded. If we differentiate Eǫ with respect to time, and use the evolution equation (2.32)
+together with the estimate (2.30) on the source term, we obtain after lengthy calculations a
+differential inequality of the form
+tE′
+ǫ(t) ≤ −c1Eǫ(t) + c2
+�
+ǫ2 + ǫ2γ3
+δ2
+�
+,
+t ∈ (0, Tadvδ−σ) ,
+(2.36)
+for some positive constants c1, c2. Integrating (2.36) with initial condition Eǫ(0) = 0, we find
+Eǫ(t) ≤ c3
+�
+ǫ2 + ǫ2γ3
+δ2
+�
+,
+t ∈ (0, Tadvδ−σ) ,
+(2.37)
+and using in addition the bounds on the moments of ˜η that are obtained by a different argument
+we arrive at an estimate of the form δ∥˜η(t)∥Xǫ ≤ c(ǫδ + ǫγ3), where Xǫ is the weighted L2 space
+equipped with the norm
+∥˜η∥Xǫ =
+��
+Ωǫ
+Wǫ(R, Z) |˜η(R, Z)|2 dR dZ
+�1/2
+.
+(2.38)
+This space depends on time through the parameter ǫ > 0, but we recall that the weight function
+satisfies a uniform lower bound of the form Wǫ(R, Z) ≳ exp(ρ2γ/4), see Figure 2.
+The main result of Section 4 can now be formulated as follows :
+Theorem 2.6. For any γ3 ∈ (2, 3), there exist constants K > 0, δ0 > 0, and σ ∈ (0, 1) such
+that, for all Γ > 0, all r0 > 0, and all ν > 0 satisfying δ := ν/Γ ≤ δ0, the unique solution η of
+(2.15) with initial data (2.22) satisfies
+∥η(t) − η∗(t)∥Xǫ ≤ K
+�
+ǫδ + ǫγ3�
+,
+t ∈ (0, Tadvδ−σ) ,
+(2.39)
+where ǫ(t) =
+√
+νt/¯r(t) and η∗ is the approximate solution defined by (2.24), (2.28).
+In the statement above, it is understood that the location (¯r(t), ¯z(t)) of the vortex ring is
+determined by the expansion (2.25). In particular the velocities ˙¯r(t), ˙¯z(t) satisfy (2.12). It is
+not difficult to verify that Theorem 2.6 implies Theorem 1.1, see Section 4.9 below for some
+details. Here we just show how to derive estimate (2.11), which is essentially a reformulation of
+(1.7). By construction we have ∥η∗(t) − η0∥Xǫ = O(ǫ), where η0 is the Gaussian function (2.22),
+and the lower bound Wǫ(R, Z) ≳ exp(ρ2γ/4) implies that Xǫ ֒→ L1(Ωǫ) uniformly in ǫ. It thus
+follows from (2.39) that
+∥η(t) − η0∥L1(Ωǫ) ≤ C1
+�
+∥η(t) − η∗(t)∥Xǫ + ∥η∗(t) − η0∥Xǫ
+�
+≤ C2ǫ ,
+for any t ∈ (0, Tadvδ−σ), and returning to the original variables (2.13) we arrive at (2.11).
+Remark 2.7. It follows from (2.24) and (2.39) that the solution of (2.15) satisfies
+η(R, Z, t) = η0(R, Z) + ǫη1(R, Z) + ǫ2η2(R, Z, βǫ) + O
+�
+δǫ + ǫγ3�
+,
+(2.40)
+where the remainder term is understood in the topology of Xǫ as ǫ → 0. Here η0 is the Gaussian
+function (2.22), and the vorticity profiles η1, η2 are explicitly constructed in Section 3. Since
+δ ≲ ǫ2 except for very small times, see Remark 2.2, we see that (2.40) determines the deformation
+of the vortex core due to self-induction up to third order in ǫ.
+15
+
+3
+Construction of the approximate solution
+In this section we construct perturbatively an approximate solution of (2.23) such that the
+corresponding remainder satisfies (2.30). Approximations of vortex rings with varying degrees of
+accuracy were obtained by many authors, and typically rely on matched asymptotics expansions
+where the inner core of the vortex and the outer region are considered separately, see [40,37,20,
+23,24] in the inviscid case and [52,12,27] in the viscous case. Here we rather follow the direct
+approach introduced in [28] for interacting vortices in the plane, which does not rely on matched
+asymptotics techniques.
+3.1
+Expansion of the Biot-Savart formula
+Our first task is to compute an accurate asymptotic expansion of the function F(s) defined by
+(2.5) in the limit where s → 0. This can be done by expressing F in terms of elliptic integrals,
+a procedure initiated in the early references [36,47].
+Lemma 3.1. For 0 < s < 4 we have the power series representation
+F(s) = log
+� 8
+√s
+�
+∞
+�
+m=0
+Amsm +
+∞
+�
+m=0
+Bmsm ,
+(3.1)
+where Am, Bm are real numbers. Moreover
+A0 = 1 ,
+A1 = 3
+16 ,
+A2 = − 15
+1024 ,
+B0 = −2 ,
+B1 = − 1
+16 ,
+B2 =
+31
+2048 .
+(3.2)
+Proof. If s > 0 and k = 2/√s + 4 ∈ (0, 1), it is straightforward to verify that
+F(s) =
+� π/2
+0
+1 − 2 sin2 ψ
+�
+sin2 ψ + s/4
+dψ = 2 − k2
+k
+K(k) − 2
+k E(k) ,
+(3.3)
+where K(k), E(k) are the complete elliptic integrals with modulus k :
+K(k) =
+� π/2
+0
+1
+�
+1 − k2 sin2 θ
+dθ ,
+E(k) =
+� π/2
+0
+�
+1 − k2 sin2 θ dθ .
+We are interested in the limit where s → 0, namely k → 1. Introducing the complementary
+modulus κ =
+√
+1 − k2, we have the power series expansions (see [17])
+K(k) =
+∞
+�
+m=0
+a2
+m κ2m�
+log 1
+κ + 2bm
+�
+,
+E(k) = 1 +
+∞
+�
+m=0
+2m + 1
+2m + 2 a2
+m κ2m+2�
+log 1
+κ + bm + bm+1
+�
+,
+(3.4)
+where a0 = 1, b0 = log(2), and
+am = 1
+2 · 3
+4 · . . . · 2m−1
+2m
+,
+bm = log(2) +
+2m
+�
+ℓ=1
+(−1)ℓ
+ℓ
+,
+m ∈ N∗ .
+Combining (3.3), (3.4), we obtain a representation of the form
+F(s) =
+1 + κ2
+√
+1 − κ2 K(k) −
+2
+√
+1 − κ2 E(k) = log
+� 4
+κ
+�
+∞
+�
+m=0
+Cmκ2m +
+∞
+�
+m=0
+Dmκ2m ,
+(3.5)
+16
+
+which converges for 0 < κ < 1. Moreover, a direct calculation shows that
+C0 = 1 ,
+C1 = 3
+4 ,
+C2 = 33
+64 ,
+D0 = −2 ,
+D1 = −3
+4 ,
+D2 = − 81
+128 .
+(3.6)
+As κ2 = s/(s + 4), the right-hand side of (3.5) can be written in the form (3.1), and using (3.6)
+we see that the first coefficients satisfy (3.2).
+Remark 3.2. Various asymptotic expansions of the stream function given by the Biot-Savart law
+(2.4) can be found in the literature [37,20,42,52,27], and are easily recovered using Lemma 3.1.
+We next consider the rescaled Biot-Savart formula (2.20), which can be written in the equiv-
+alent form
+φ(R, Z) =
+1
+2π
+�
+Ωǫ
+Kǫ(R, Z; R′, Z′) η(R′, Z′) dR′ dZ′ ,
+(3.7)
+where
+Kǫ =
+�
+(1+ǫR)(1+ǫR′) F
+�
+ǫ2D2
+(1+ǫR)(1+ǫR′)
+�
+,
+D2 = (R−R′)2 + (Z−Z′)2 .
+(3.8)
+To simplify the notations below, we define
+βǫ = log 1
+ǫ ,
+L(R, Z; R′, Z′) = log
+� 8
+D
+�
+.
+(3.9)
+Lemma 3.3. For any (R, Z), (R′, Z′) ∈ R2 with (R, Z) ̸= (R′, Z′) and any sufficiently small
+ǫ > 0, we have the expansion
+Kǫ = (βǫ + L)
+∞
+�
+m=0
+ǫmPm +
+∞
+�
+m=0
+ǫmQm ,
+(3.10)
+where Pm(R, Z; R′, Z′), Qm(R, Z; R′, Z′) are homogeneous polynomials of degree m in the three
+variables R, R′, and Z − Z′. Moreover
+P0
+= 1
+P1
+=
+1
+2(R + R′)
+P2
+=
+1
+16(R − R′)2 + 3
+16(Z − Z′)2
+Q0 = −2
+Q1 = − 1
+2(R + R′)
+Q2 =
+1
+4(R2 + R′2) − 1
+16 D2 .
+(3.11)
+Proof. If (R, Z), (R′, Z′) are as in the statement, we take ǫ > 0 small enough so that
+max
+�
+|R|, |R′|
+�
+< 1
+ǫ ,
+and
+s :=
+ǫ2D2
+(1+ǫR)(1+ǫR′) < 4 .
+(3.12)
+As D ̸= 0 by assumption, we have 0 < s < 4, so that we can apply expansion (3.1) to the
+quantity F(s) in (3.8). In view of definitions (3.9) we have
+log
+� 8
+√s
+�
+= βǫ + L + 1
+2 log(1 + ǫR) + 1
+2 log(1 + ǫR′) .
+(3.13)
+We observe that the last two terms in (3.13), as well as the prefactor
+�
+(1+ǫR)(1+ǫR′) in (3.8)
+and each monomial sm in the series (3.1), can be expanded into a power series in the three
+variables ǫR, ǫR′, and ǫ(Z − Z′). Thus, combining (3.1) and (3.8), we obtain a representation
+of the form (3.10), where the first homogeneous polynomials Pm, Qm are easily computed using
+the explicit values (3.2).
+17
+
+Remark 3.4. In what follows, with a slight abuse of notation, we denote by L the integral
+operator on R2 given by the kernel (3.9). For any continuous and rapidly decreasing function
+η : R2 → R, we thus have
+�
+Lη
+�
+(R, Z) =
+�
+R2 log
+�
+8
+�
+(R−R′)2 + (Z−Z′)2
+�
+η(R′, Z′) dR′ dZ′ .
+(3.14)
+Similarly, we associate integral operators to the homogeneous polynomials Pm, Qm in (3.10), as
+well as to the functions LPm for all m ∈ N∗.
+Definition 3.5. Using the notation introduced in Remark 3.4, we define the linear operators
+BS0 =
+1
+2π L ,
+and
+BSm =
+1
+2π
+�
+βǫPm + LPm + Qm
+�
+,
+for all m ∈ N∗ .
+(3.15)
+Note that, for m ≥ 1, the linear operator BSm depends on the parameter ǫ through the
+constant factor βǫ = log(1/ǫ), but for simplicity this mild dependence is not indicated explicitly.
+For convenience, we do not include the constant term βǫP0 + Q0 ≡ βǫ − 2 in the definition of
+BS0, because the stream function is only defined up to an additive constant. It is important
+to observe that, in (3.14) and in the corresponding definition of the integral operators Pm, Qm,
+the integration is performed on the whole plane R2, rather than on the half-plane Ωǫ. This
+is justified because these operators will always be applied to functions that decay rapidly at
+infinity, so that the integration on R2 \ Ωǫ and gives a contribution of order O(ǫ∞) as ǫ → 0,
+which can be neglected in our perturbative expansion. If η : R2 → R is compactly supported,
+then according to Lemma 3.3 the following equality holds in any bounded region of R2:
+BSǫ[η] = βǫ − 2
+2π
+�
+R2 η(R′, Z′) dR′ dZ′ +
+∞
+�
+m=0
+ǫm BSm[η] ,
+(3.16)
+provided ǫ > 0 is sufficiently small. As was already mentioned, the first term in the right-hand
+side is a constant that can be omitted.
+3.2
+Function spaces and linear operators
+We next introduce the function spaces in which we shall construct our approximate solution of
+(2.23). These spaces consist of functions that are defined on the whole space R2, and not just
+on the half-plane Ωǫ. Indeed, at each step of the approximation, the vorticity profile ηj(R, Z, βǫ)
+and the stream function φj(R, Z, βǫ) in (2.24) are defined for all (R, Z) ∈ R2. To simplify the
+writing we often denote X = (R, Z), and we use polar coordinates (ρ, ϑ) in R2 defined by the
+relations R = ρ cos ϑ, Z = ρ sin ϑ.
+Following [32,33] we introduce the weighted L2 space
+Y =
+�
+η ∈ L2(R2)
+���
+�
+R2 |η(X)|2 e|X|2/4 dX < ∞
+�
+,
+(3.17)
+equipped with the scalar product (η1, η2)Y =
+�
+R2 η1(X)η2(X) e|X|2/4 dX and the associated
+norm.
+We also introduce the differential operator L : D(L) → Y corresponding to (2.16),
+namely
+Lη = ∆η + 1
+2 X · ∇η + η ,
+η ∈ D(L) =
+�
+η ∈ Y
+��� ∆η ∈ Y , X · ∇η ∈ Y
+�
+,
+(3.18)
+18
+
+as well as the integro-differential operator Λ : D(Λ) → Y defined by
+Λη =
+1
+2π
+��
+Lη0 , η
+�
++
+�
+Lη , η0
+��
+,
+η ∈ D(Λ) =
+�
+η ∈ Y
+���
+�
+Lη0 , η
+�
+∈ Y
+�
+,
+(3.19)
+where η0 is the Gaussian function (2.22) and L denotes the integral operator (3.14). Here and
+in what follows the Poisson bracket is understood with respect to the rescaled variables (R, Z),
+so that {φ, η} = ∂Rφ ∂Zη − ∂Zφ ∂Rη. We recall the following well-known properties :
+Proposition 3.6. [32, 33, 43]
+1) The linear operator L is self-adjoint in Y, with purely discrete spectrum
+σ(L) =
+�
+−n
+2
+��� n = 0, 1, 2, . . .
+�
+.
+The kernel of L is one-dimensional and spanned by the Gaussian function η0. More generally,
+for any n ∈ N, the eigenspace corresponding to the eigenvalue λn = −n/2 is spanned by the n+1
+Hermite functions ∂αη0 where α = (α1, α2) ∈ N2 and α1 + α2 = n.
+2) The linear operator Λ is skew-adjoint in Y, so that Λ∗ = −Λ. Moreover,
+Ker(Λ) = Y0 ⊕
+�
+β1∂Rη0 + β2∂Zη0
+�� β1, β2 ∈ R
+�
+,
+(3.20)
+where Y0 ⊂ Y is the subspace of all radially symmetric elements of Y.
+A crucial observation is that both operators L, Λ are invariant under rotations about the
+origin in R2. It is therefore advantageous to decompose the space Y into a direct sum
+Y =
+∞⊕
+n=0 Yn ,
+(3.21)
+where Y0 ⊂ Y is as in Proposition 3.6 and, for all n ≥ 1, the subspace Yn ⊂ Y consists
+of all functions η ∈ Y such that η(ρ cos ϑ, ρ sin ϑ) = a1(ρ) cos(nϑ) + a2(ρ) sin(nϑ) for some
+a1, a2 : R+ → R. It is clear that Yn ⊥ Yn′ if n ̸= n′. In particular, in view of (3.20), we have
+Yn ∈ Ker(Λ)⊥ for all n ≥ 2. When n = 1, the functions ∂Rη0, ∂Zη0 belong to Y1 ∩ Ker(Λ), and
+we define
+Y′
+1 = Y1 ∩ Ker(Λ)⊥ =
+�
+η ∈ Y1
+����
+�
+R2 η(R, Z)R dR dZ =
+�
+R2 η(R, Z)Z dR dZ = 0
+�
+.
+(3.22)
+Since Λ is skew-adjoint, we have Ker(Λ)⊥ = Ran(Λ), but the image of Λ is not dense in
+Y and, therefore, we cannot solve the equation Λη = f for any f ∈ Ker(Λ)⊥. As is shown
+in [34,28], the problem disappears if one assumes in addition that f belongs to a smaller space
+such as
+Z =
+�
+η : R2 → R
+��� e|X|2/4η ∈ S∗(R2)
+�
+⊂ Y ,
+(3.23)
+where S∗(R2) denotes the space of all smooth functions which are slowly growing at infinity.
+More precisely, a smooth function w : R2 → R belongs to S∗(R2) if, for any α = (α1, α2) ∈ N2,
+there exist C > 0 and N ∈ N such that |∂αw(X)| ≤ C(1 + |X|)N for all X ∈ R2.
+To formulate the main technical result of this section, we introduce the notation
+ϕ(ρ) =
+1
+2πρ2
+�
+1 − e−ρ2/4�
+,
+h(ρ) =
+ρ2/4
+eρ2/4 − 1 ,
+ρ > 0 .
+(3.24)
+The following statement is a slight extension of [28, Lemma 4]. For the reader’s convenience, we
+give a short proof of it in Section A.1, emphasizing the case n = 1 which was not treated in [28].
+19
+
+Proposition 3.7. If n ≥ 2 and f ∈ Yn ∩ Z, or if n = 1 and f ∈ Y′
+1 ∩ Z, there exists a
+unique η ∈ Yn ∩ Z (respectively, a unique η ∈ Y′
+1 ∩ Z) such that Λη = f. In particular, if
+f = a(ρ) sin(nϑ), then η = ω(ρ) cos(nϑ), where
+ω(ρ) = h(ρ)Ω(ρ) + a(ρ)
+nϕ(ρ) ,
+ρ > 0 ,
+(3.25)
+and where Ω : (0, ∞) → R is the unique solution of the differential equation
+− Ω′′(ρ) − 1
+ρ Ω′(ρ) +
+�n2
+ρ2 − h(ρ)
+�
+Ω(ρ) =
+a(ρ)
+nϕ(ρ) ,
+ρ > 0 ,
+(3.26)
+such that Ω(ρ) = O(ρn) as ρ → 0 and Ω(ρ) = O(ρ−n) as ρ → ∞.
+Remark 3.8. As was observed in [28], if f = a(ρ) cos(nϑ), then η = −ω(ρ) cos(nϑ), where ω
+is still given by (3.25), (3.26). The general case where f = a1(ρ) sin(nϑ) + a2(ρ) cos(nϑ) follows
+by linearity.
+In the construction of an approximate solution of (2.23), we shall encounter linear equations
+of the form δ(κ− L)η + Λη = f, where κ > 0 is fixed and the parameter δ > 0 can be arbitrarily
+small. Proposition 3.6 implies that the linear operator δ(κ − L) + Λ is invertible in Y for any
+δ > 0, so that our problem has a unique solution η for any f ∈ Y. However, the norm of
+the inverse is O(δ−1), and it is intuitively clear that η has a regular limit when δ → 0 only if
+f ∈ Ran(Λ). Under similar assumptions as in Proposition 3.7, we have the following result:
+Proposition 3.9. Assume that n ≥ 2 and f ∈ Yn ∩ Z, or that n = 1 and f ∈ Y′
+1 ∩ Z. For any
+fixed κ > 0 and all δ ≥ 0, the equation
+δ
+�
+κ − L
+�
+ηδ + Ληδ = f
+(3.27)
+has a unique solution ηδ ∈ Yn (respectively, ηδ ∈ Y′
+1). Moreover there exists a constant C > 0,
+depending only on f, such that
+∥ηδ − η∥Y ≤ Cδ
+and
+∥ηδ − η − δˆη∥Y ≤ Cδ2 ,
+for all δ ≥ 0 ,
+(3.28)
+where
+1) η ∈ Yn ∩ Z (respectively, η ∈ Y′
+1 ∩ Z) is the unique solution of Λη = f;
+2) ˆη ∈ Yn ∩ Z (respectively, ˆη ∈ Y′
+1 ∩ Z) is the unique solution of Λˆη = (L − κ)η.
+The proof of Proposition 3.9 is similar to the one of Lemma 5 in [28], and can therefore be
+omitted. We just add here the following comments. First, since f ∈ Yn ∩ Z or f ∈ Y′
+1 ∩ Z,
+Proposition 3.7 shows that there is a unique η in the same class such that Λη = f. It is then
+easily verified that (κ−L)η still belongs to Yn ∩Z (respectively, to Y′
+1 ∩Z), so that the equation
+Λˆη = −(κ − L)η also has a unique solution ˆη. This shows that the approximations (3.28) of
+ηδ are well defined. On the other hand, proceeding as in the proof of Proposition 3.7, it is
+possible to verify that the solution ηδ of (3.27) belongs to the space Z, and that the expansions
+(3.28) also hold in the (Fr´echet) topology of Z. But for our purposes it will be sufficient to use
+η + δˆη ∈ Z as an approximation of ηδ, observing that
+�
+δ
+�
+κ − L
+�
++ Λ
+�
+(η + δˆη) − f = δ2(κ − L)ˆη = OZ(δ2) ,
+as δ → 0 .
+(3.29)
+20
+
+3.3
+First order approximation
+We now begin the construction of an approximate solution of (2.23) in the form (2.24). We recall
+that, for an exact solution, the stream function is determined by the relation (2.20), which we
+write in the compact form φ = BSǫ[η]. For our approximate solution, we expand the Biot-Savart
+operator as in (3.16), omitting the constant term in the right-hand side. We thus obtain the
+formal relation
+�
+∞
+�
+m=0
+ǫm BSm
+��
+M
+�
+m=0
+ǫmηm
+�
+=
+M
+�
+m=0
+ǫmφm + O
+�
+ǫM+1�
+,
+which we assume to be satisfied order by order in ǫ, up to order M. This leads to the relations
+φ0 = BS0[η0], φ1 = BS0[η1] + BS1[η0], and more generally
+φm = BS0[ηm] + BS1[ηm−1] + · · · + BSm−1[η1] + BSm[η0] .
+(3.30)
+In particular, in view of (2.22) and (3.15), the leading order of our approximation is
+η0(R, Z) =
+1
+4π e−(R2+Z2)/4 ,
+φ0(R, Z) =
+1
+2π
+�
+Lη0
+�
+(R, Z) ,
+(3.31)
+where L is the integral operator (3.14). The stream function φ0 has the expression
+φ0(R, Z) = φ0(0) − 1
+4πEin
+�R2+Z2
+4
+�
+,
+where
+Ein(x) =
+� x
+0
+1 − e−t
+t
+dt ,
+(3.32)
+so that φ0 is radially symmetric and φ0(R, Z) ∼ −(2π)−1 log ρ as ρ := (X2 + Z2)1/2 → +∞.
+The value at the origin does not play a big role in our analysis, but can be computed too, see
+Section A.2 :
+φ0(0) = log(2)
+π
++ γE
+4π ,
+where γE is Euler’s constant.
+Before proceeding further, we estimate the time derivative of the quantity ǫ =
+√
+νt/¯r(t)
+introduced in (2.14). In view of (2.25), we have
+t˙ǫ = ǫ
+2 − ǫt ˙¯r
+¯r
+= ǫ
+2 − ǫt
+¯r
+�
+˙¯r0 + ǫ ˙¯r1 + ǫ2 ˙¯r2 + . . .
+�
+.
+(3.33)
+At this stage the radial velocity profiles ˙¯rj are not determined yet, but in view of Remark 2.3 we
+can anticipate the fact that | ˙¯r| = (Γ/r0)·O(δ) as δ → 0. Since δt = (r2
+0/Γ)·O(ǫ2) by Remark 2.2,
+it follows that ¯r(t) = r0
+�
+1 + O(ǫ2)
+�
+and that t˙ǫ = ǫ/2 + O(ǫ3) as ǫ → 0.
+With that observation in mind, we substitute the expansions (2.24), (2.25) into the evolution
+equation (2.23), keeping only the terms that are exactly of order ǫ or ǫβǫ. This gives the relation
+�
+φ1 , η0
+�
++
+�
+φ0 , η1
+�
++ η0∂Zφ0 − r0
+Γ
+�
+˙¯r0 ∂Rη0 + ˙¯z0 ∂Zη0
+�
+= δ
+�
+∂Rη0 +
+�
+L − 1
+2
+�
+η1 − t∂tη1
+�
+. (3.34)
+To solve (3.34) we first impose the relation
+˙¯r0 = −Γδ
+r0
+,
+(3.35)
+which ensures that the terms involving ∂Rη0 cancel exactly. We also assume that η1 does not
+depend on βǫ, so that ∂tη1 = 0 (this property will be verified later). On the other hand, using
+21
+
+(3.30) with m = 1, we deduce that {φ1 , η0} = {BS0[η1] , η0}+{BS1[η0] , η0}, where BS0, BS1 are
+defined in (3.15). Using (3.31) and the definition (3.19) of the linear operator Λ, we thus find
+�
+φ1 , η0
+�
++
+�
+φ0 , η1
+�
+=
+1
+2π
+��
+Lη1 , η0
+�
++
+�
+Lη0 , η1
+��
++
+�
+BS1[η0] , η0
+�
+= Λη1 + βǫ − 1
+2π
+�
+P1η0 , η0
+�
++ 1
+2π
+�
+LP1η0 , η0
+�
+,
+where in the second line we used the definition (3.15) of BS1 and the fact that Q1 = −P1 in
+view of (3.11). Now, elementary calculations that are reproduced in Section A.2 show that
+�
+P1η0 , η0
+�
+= 1
+2 ∂Zη0 ,
+and
+1
+2π
+�
+LP1η0 , η0
+�
+= 1
+2 ∂Z
+�
+φ0η0
+�
+.
+(3.36)
+It follows that we can write (3.34) in the equivalent form
+Λη1 + δ
+� 1
+2 − L
+�
+η1 =
+�r0
+Γ
+˙¯z0 − βǫ − 1
+4π
+�
+∂Zη0 − 3
+2 (∂Zφ0)η0 − 1
+2 φ0∂Zη0 .
+(3.37)
+Using the explicit expressions (3.31), (3.32) of the profiles η0, φ0, it is straightforward to
+verify that the right-hand side of (3.37), which we denote by −R1, belongs to Y1 ∩ Z, where
+Y1, Z are the function spaces defined in (3.21), (3.23). Therefore, according to Proposition 3.9,
+the linear equation (3.37) has a unique solution in Y1 for all δ > 0, and that solution has a
+well-defined limit as δ → 0 if and only if R1 ∈ (ker Λ)⊥, namely if R1 ∈ Y′
+1. In view of (3.22),
+this gives the solvability condition
+�
+R2 R1Z dR dZ = 0, which determines uniquely the value of
+the constant ˙¯z0 in (3.37). The calculations are presented in Section A.2, and yield the following
+expression of the vertical velocity to leading order :
+˙¯z0 =
+Γ
+4πr0
+�
+βǫ − 1 + 2v
+�
+,
+where
+v = 3
+4 log(2) + 1
+4 γE + 1
+4 .
+(3.38)
+Here again γE = 0, 5772 . . . denotes Euler’s constant.
+Remark 3.10. The formula (3.38), including the leading term βǫ = log(1/ǫ) and the correct
+value of the constant 2v − 1, was established by Saffman [51], see also Fukumoto & Moffatt [27].
+We assume henceforth that ˙¯z0 is given by (3.38), so that (3.37) reduces to
+Λη1 + δ
+� 1
+2 − L
+�
+η1 =
+v
+2π ∂Zη0 − 3
+2 (∂Zφ0)η0 − 1
+2 φ0∂Zη0 ,
+(3.39)
+where the right-hand side −R1 now belongs to Y′
+1 ∩ Z and is independent of ǫ. The solution of
+(3.39) given by Proposition 3.9 can be expanded as in (3.28), and if is sufficient for our purposes
+to consider only the approximate solution (3.29), which reads
+η1(R, Z) = R η10(ρ) + δZ η11(ρ) ,
+ρ =
+�
+R2 + Z2 ,
+(3.40)
+where Λ(R η10) = −R1 and Λ(Z η11) = (L − 1
+2)(R η10). Note that η1 ∈ Y′ ∩ Z, which implies
+in particular that the functions η10, η11 are smooth and have a Gaussian decay at infinity. The
+corresponding stream function φ1 = BS0[η1] + BS1[η0] has a similar expression :
+φ1(R, Z, βǫ) = βǫ − 1
+4π
+R + R
+2 φ0 − ∂Rφ0 + R φ10(ρ) + δZ φ11(ρ) ,
+(3.41)
+which can be established using the calculations in Section A.2. Here the functions φ10, φ11 are
+smooth and decay at least like 1/ρ2 as ρ → +∞. Note that φ1 involves the time-dependent term
+βǫ = log(1/ǫ), so that ∂tφ1 ̸= 0. With the choices (3.35), (3.38), (3.40), and (3.41), the relation
+(3.34) is not satisfied exactly, but the difference of both members is O(δ2) in the topology of Z,
+which is all we need.
+22
+
+3.4
+Second order approximation
+We next compute the second order terms in the asymptotic expansion (2.24). As we shall see,
+it is consistent at this stage to take
+˙¯r1 = ˙¯z1 = 0 ,
+(3.42)
+so we make that assumption from now on. As before, we deduce from (3.33), (3.35), (3.42) that
+¯r(t) = r0
+�
+1 + O(ǫ2)
+�
+and t˙ǫ = ǫ/2 + O(ǫ3) as ǫ → 0. Substituting (2.24), (2.25) into (2.23) and
+keeping only the terms involving ǫ2 or ǫ2βǫ, we obtain the relation
+�
+φ2 , η0
+�
++
+�
+φ1 , η1 − Rη0
+�
++
+�
+φ0 , η2 − Rη1 + R2η0
+�
+− r0
+Γ
+�
+˙¯r0 ∂Rη1 + ˙¯z0 ∂Zη1
+�
+= δ
+��
+L − 1
+�
+η2 + ∂R(η1 − Rη0) − t∂tη2
+�
+.
+(3.43)
+In view of (3.35), the terms involving ∂Rη1 cancel exactly. Moreover, we know from (3.15),
+(3.30) that
+φ2 =
+1
+2π
+�
+Lη2 +
+�
+βǫP1 + LP1 + Q1
+�
+η1 +
+�
+βǫP2 + LP2 + Q2
+�
+η0
+�
+,
+(3.44)
+where the notations are introduced in Lemma 3.3. Recalling the definition (3.19) of the operator
+Λ, we can thus write (3.43) in the equivalent form
+Λη2 + δ
+�
+t∂tη2 +
+�
+1 − L
+�
+η2
+�
++ R2 = 0 ,
+(3.45)
+where
+R2 = 1
+2π
+�
+(βǫ − 1)P1η1 + LP1η1 , η0
+�
++ 1
+2π
+�
+βǫP2η0 + LP2η0 + Q2η0 , η0
+�
++
+�
+φ1 , η1
+�
++ (∂Zφ1)η0 + (∂Zφ0)η1 − R
+��
+φ1 , η0
+�
++
+�
+φ0 , η1
+�
++ 2(∂Zφ0)η0
+�
++ δ∂R(Rη0) − r0 ˙¯z0
+Γ
+∂Zη1 .
+(3.46)
+We have the following result, whose proof is postponed to Section A.3 :
+Lemma 3.11. The function R2 defined in (3.46) belongs to (δY0 + Y2) ∩ Z and satisfies
+R2 = 3βǫ
+16π RZη0 + RZχ20 + δ
+�
+χ21 + (R2 − Z2)χ22
+�
++ δ2RZχ23 ,
+(3.47)
+for some (time-independent) radially symmetric functions χ20, χ21, χ22, χ23 ∈ Y0 ∩ Z.
+In view of (3.47), we look for a solution of (3.45) in the form η2 = βǫˆη20 + ˆη21 + ˆη22, where
+ˆη20, ˆη21 ∈ Y2 and ˆη22 ∈ Y0 do not depend on βǫ. Inserting this ansatz into (3.45) and using the
+fact that t∂tβǫ = −1/2 + O(ǫ2), we obtain the system
+Λˆη20 + δ
+�
+1 − L
+�
+ˆη20 +
+3
+16π RZη0 = 0 ,
+Λˆη21 + δ
+�
+1 − L
+�
+ˆη21 − δ
+2 ˆη20 + P2
+�
+R2 − 3βǫ
+16π RZη0
+�
+= 0 ,
+δ
+�
+1 − L
+�
+ˆη22 + P0R2 = 0 ,
+(3.48)
+where Pn denotes the orthogonal projection in Y onto the subspace Yn. The first two equations
+in (3.48) have a unique solution by Proposition 3.9, and as in the previous section we are satisfied
+23
+
+with the approximate solutions (3.28) which are valid up to O(δ2). Since P0R2 = δχ21 by (3.47),
+the third equation reduces to (1 − L)ˆη22 + χ21 = 0, which also has a unique solution due to
+Proposition 3.6. We conclude that we can choose η2 in the form
+η2(R, Z, βǫ) = βǫ
+�
+(R2−Z2)η20 + δRZη21
+�
++ (R2−Z2)η22 + δRZη23 + η24 ,
+(3.49)
+where all functions η2j belong to Y0 ∩ Z. Using (3.44) and the calculations at the beginning of
+Section A.3, we obtain a similar expression for the corresponding stream function
+φ2(R, Z, βǫ) = βǫ
+�
+(R2−Z2)φ20 + δRZφ21
+�
++ (R2−Z2)φ22 + δRZφ23 + βǫφ24 + φ25 ,
+(3.50)
+where the functions φ2j are radially symmetric and belong to S∗(R2). With these choices, the
+difference of both members in (3.43) is O(δ2 + ǫ2δ) in the topology of Z.
+3.5
+Third order approximation
+The third order in the asymptotic expansion (2.24) can be computed in a similar way. According
+to (3.35), (3.42) and Remark 2.3, we have ¯r(t) = r0
+�
+1 − ǫ2 + O(ǫ4−)
+�
+as ǫ → 0, and using (3.33)
+we deduce that t˙ǫ = ǫ/2 + ǫ3 + O(ǫ5−). So, if we substitute (2.24), (2.25) into (2.23) and keep
+only the terms involving ǫ3 or ǫ3βǫ, we find
+�
+φ3 , η0
+�
++
+�
+φ2 , η1 − Rη0
+�
++
+�
+φ1 , η2 − Rη1 + R2η0
+�
++
+�
+φ0 , η3 − Rη2 + R2η1 − R3η0
+�
+− r0
+Γ
+�
+˙¯r0 ∂Rη2 +
+� ˙¯r2− ˙¯r0
+�
+∂Rη0 + ˙¯z0 ∂Zη2 +
+� ˙¯z2− ˙¯z0
+�
+∂Zη0
+�
+= δ
+��
+L − 3
+2
+�
+η3 + ∂R(η2 − Rη1 + R2η0) − t∂tη3 − η1
+�
+.
+(3.51)
+On the other hand, using (3.30) with m = 3 and (3.15), we obtain
+φ3 =
+3
+�
+m=0
+BSm[η3−m] =
+1
+2π Lη3 + 1
+2π
+3
+�
+m=1
+�
+(βǫ + L)Pm + Qm
+�
+η3−m ,
+(3.52)
+where the polynomials Pm, Qm are defined in (3.11) for m ≤ 2 and in (A.19) for m = 3. We can
+thus write (3.51) in the form
+Λη3 + δ
+�
+t∂tη3 +
+� 3
+2 − L
+�
+η3
+�
++ R3 = 0 ,
+(3.53)
+where
+R3 =
+1
+2π
+�
+3
+�
+m=1
+�
+(βǫ + L)Pm + Qm
+�
+η3−m , η0
+�
++
+�
+φ2 , η1 − Rη0
+�
++
+�
+φ1 , η2 − Rη1 + R2η0
+�
+−
+�
+φ0 , Rη2 − R2η1 + R3η0
+�
+− r0
+Γ
+�� ˙¯r2 − ˙¯r0
+�
+∂Rη0 +
+� ˙¯z2 − ˙¯z0
+�
+∂Zη0 + ˙¯z0∂Zη2
+�
++ δ∂R
+�
+Rη1 − R2η0
+�
++ δη1 .
+(3.54)
+Lemma 3.12. The function R3 defined in (3.54) belongs to (Y1 + Y3) ∩ Z and satisfies
+R3 = βǫ
+�
+R2Zχ30 + Zχ31
+�
++ R2Zχ32 + Zχ33 + O(δ) ,
+(3.55)
+for some (time-independent) radially symmetric functions χ30, χ31, χ32, χ33 ∈ Y0 ∩ Z.
+24
+
+The proof of Lemma 3.12 is a direct calculation that is briefly outlined in Section A.4. In
+particular we verify there that the quantity R3 does not contain any factor β2
+ǫ , which is rather
+surprising since φ1, φ2, and η2 all contain at least one term multiplied by βǫ. We do not need
+the expression of the O(δ) terms in (3.55), but they can be computed too and are found to be
+of the form δβǫ
+�
+R3 ˜χ30 + R˜χ31
+�
++ δ
+�
+R3 ˜χ32 + R˜χ33
+�
+, where ˜χ3j are radially symmetric functions.
+Finally we mention that R3 also includes terms of the form (3.55) that are multiplied by δ2.
+As can be seen from the last line of (3.54), there is a unique way to choose the quantities
+˙¯r2 and ˙¯z2 so that R3 ∈ Y′
+1 + Y3, where Y′
+1 is the subspace defined in (3.22). In view of (3.55),
+(3.35), (3.38), the velocities obtained in this way satisfy
+r0
+Γ
+˙¯r2 =
+�
+c1βǫ + c2
+�
+δ ,
+r0
+Γ
+˙¯z2 = c3βǫ + c4 + O(δ2) ,
+(3.56)
+for some constants c1, c2, c3, c4.
+Now, decomposing R3 = βǫR31 + R32 where R31, R32 are
+independent of βǫ, we look for a solution of (3.53) in the form η3 = βǫˆη31 + ˆη32 where
+Λˆη31 + δ
+� 3
+2 − L
+�
+ˆη31 + R31 = 0 ,
+Λˆη32 + δ
+� 3
+2 − L
+�
+ˆη32 − δ
+2 ˆη31 + R32 = 0 .
+(3.57)
+Since R31, R32 ∈ Y′
+1+Y3, both equations in (3.57) can be solved using Proposition 3.9. However,
+at this stage, we are satisfied with the crude approximation which consists in neglecting the terms
+proportional to δ, so we really solve Λˆη3j + R3j = 0 for j = 1, 2 using Proposition 3.7. We thus
+obtain an approximate solution of (3.53) of the form
+η3(R, Z, βǫ) = βǫ
+�
+R3η30 + Rη31
+�
++ R3η32 + Rη33 ,
+(3.58)
+where all functions η3j belong to Y0 ∩ Z. Using (3.52) we deduce the corresponding expression
+of the stream function
+φ3(R, Z, βǫ) = βǫ
+�
+R3φ30 + Rφ31
+�
++ R3φ32 + Rφ33 ,
+(3.59)
+where the functions φ3j are radially symmetric and belong to S∗(R2). Note that (3.59) does not
+contain any factor β2
+ǫ . With the choices (3.58), (3.59), the difference of both members in (3.53)
+is O(δ) in the topology of Z.
+3.6
+Fourth order approximation
+Finally we compute the fourth order approximation, which is the final step in our construction.
+No modification of the vortex speed is needed at this stage, so we can take
+˙¯r3 = ˙¯z3 = 0 .
+(3.60)
+The full expansion of the vortex speed is therefore
+˙¯r(t) = ˙¯r0(βǫ) + ǫ2 ˙¯r2(βǫ) ,
+˙¯z(t) = ˙¯z0(βǫ) + ǫ2 ˙¯z2(βǫ) ,
+(3.61)
+where ˙¯r0, ˙¯z0 are defined in (3.35), (3.38) and ˙¯r2, ˙¯z2 satisfy (3.56). As is easily verified, this implies
+that ¯r(t) = r0
+�
+1 − ǫ2 + O(ǫ4βǫ)
+�
+and t˙ǫ = ǫ/2 + ǫ3 + O(ǫ5βǫ) as ǫ → 0.
+We look for an approximate solution of (2.23) of the form
+ηapp(R, Z, t) =
+4
+�
+m=0
+ǫmηm(R, Z, βǫ) ,
+φapp(R, Z, t) =
+4
+�
+m=0
+ǫmφm(R, Z, βǫ) ,
+(3.62)
+25
+
+where the profiles ηm, φm for m ≤ 3 have been constructed in the previous steps, and η0, η1, φ0
+are actually independent of βǫ. In analogy with (3.52), we have
+φ4 =
+1
+2π Lη4 + 1
+2π
+4
+�
+m=1
+��
+βǫ + L
+�
+Pm + Qm
+�
+η4−m ,
+(3.63)
+where the operators Pm, Qm are defined in (3.11) for m ≤ 2, in (A.19) for m = 3, and in (A.20)
+for m = 4. Replacing (3.61), (3.62), (3.63) into (2.23) and proceeding as in the previous sections,
+we obtain the following equation for the profile η4 :
+Λη4 + δ
+�
+t∂tη4 +
+�
+2 − L
+�
+η4
+�
++ R4 = 0 ,
+(3.64)
+where
+R4 =
+1
+2π
+�
+4
+�
+m=1
+�
+(βǫ + L)Pm + Qm
+�
+η4−m , η0
+�
++
+�
+φ3 , η1 − Rη0
+�
++
+�
+φ2 , η2 − Rη1 + R2η0
+�
++
+�
+φ1 , η3 − Rη2 + R2η1 − R3η0
+�
+−
+�
+φ0 , Rη3 − R2η2 + R3η1 − R4η0
+�
+(3.65)
+− r0
+Γ
+�� ˙¯r2 − ˙¯r0
+�
+∂Rη1 +
+� ˙¯z2 − ˙¯z0
+�
+∂Zη1 + ˙¯z0∂Zη3
+�
++ δ∂R
+�
+Rη2 − R2η1 + R3η0
+�
++ 2δη2 .
+Lemma 3.13. The function R4 defined in (3.65) belongs to (δY0 + Y2 + Y4) ∩ Z and satisfies
+R4 =
+2
+�
+k=0
+βk
+ǫ
+�
+R3Zχ4k + RZχ5k
+�
++ O(δ) ,
+(3.66)
+for some (time-independent) radially symmetric functions χ4k, χ5k ∈ Y0 ∩ Z.
+The proof of Lemma 3.13 is the same as that of Lemma 3.12, and can therefore be omitted.
+The only important observation is that the projection of R4 onto the subspace Y0 is of order
+O(δ). This can be seen as a consequence of Remark 2.3 : when δ = ˙¯r = 0, all profiles ηm, φm
+are even functions of Z, so that the quantities Rm are odd in Z.
+We now project Eq. (3.64) on the subspace Ym for m = 0, 2, 4, and compute an (approximate)
+solution Pmη4 invoking either Proposition 3.7 (for m = 2, 4) or Proposition 3.6 (for m = 0). In
+the latter case, we use the observation that P0R4 = O(δ) to show that P0η4 is regular in the
+limit δ → 0. Altogether, we obtain an approximate solution of (3.64) in the form
+η4(R, Z, βǫ) =
+2
+�
+k=0
+βk
+ǫ
+�
+R2Z2η4k +
+�
+R2 − Z2�
+η5k + η6k
+�
+,
+(3.67)
+where the functions ηjk ∈ Y0 ∩ Z are radially symmetric and time-independent. Using (3.63)
+we deduce a similar expression for the stream function
+φ4(R, Z, βǫ) =
+2
+�
+k=0
+βk
+ǫ
+�
+R2Z2φ4k +
+�
+R2 − Z2�
+φ5k + φ6k
+�
+,
+(3.68)
+and with these choices the difference of both members in (3.64) is O(δ) in the topology of Z.
+Since we have now completed the construction of our approximate solution, we explain pre-
+cisely how to define the vortex radius ¯r(t) and the time-dependent aspect ratio ǫ(t) =
+√
+νt/¯r(t).
+In view of (3.35), (3.56), and (3.61), the function ¯r(t) satisfies the differential equation
+˙¯r(t) = −Γδ
+r0
+�
+1 − ǫ(t)2�
+c1βǫ(t) + c2
+��
+= −Γδ
+r0
+�
+1 −
+νt
+¯r(t)2
+�
+c1 log ¯r(t)
+√
+νt + c2
+��
+,
+(3.69)
+26
+
+with initial condition ¯r(0) = r0. The right-hand side of (3.69) a smooth function of ¯r > 0,
+uniformly in t ∈ (0, Tdif), and also a C1,α function of time for any α < 1. Applying the Cauchy-
+Lipschitz theorem, we obtain a unique local solution of (3.69), which can be extended as long
+as ¯r(t) > 0. Now, if we define ǫ(t) =
+√
+νt/¯r(t), it follows that ¯r(t) = r0
+�
+1 − ǫ(t)2 + O(ǫ4βǫ)
+�
+,
+which implies that the solution of (3.69) is well defined at least as long as ǫ(t) ≪ 1, namely
+when t ≪ Tdif.
+Remark 3.14. It is useful to notice that the approximate solution ηapp given by (3.62) satisfies,
+for all t > 0,
+�
+R2 ηapp(R, Z, t) dR dZ = 1 ,
+(3.70)
+�
+R2 R ηapp(R, Z, t) dR dZ =
+�
+R2 Z ηapp(R, Z, t) dR dZ = 0 .
+(3.71)
+Indeed, at each step m ≥ 1, the vorticity profile ηm is constructed by solving equations of the form
+Ληm +
+�m
+2 − L
+�
+ηm + Rm = 0, where the source term Rm has vanishing integral (by definition)
+and zero first order moments (due to the choice of the speeds ˙¯rm−1, ˙¯zm−1). These properties are
+inherited by the profile ηm, due to Proposition 3.6, and in view of (3.31) this leads to (3.70),
+(3.71).
+3.7
+Estimate of the remainder
+This section is devoted to the proof of Proposition 2.5. Our task is to estimate the remainder
+(2.29), where η∗, φ∗ are defined in (2.28), and for this we need bounds on the derivatives of
+the stream function in terms of the vorticity. If φ = BSǫ[η], where the Biot-Savart operator is
+defined in (2.20), we have the formulas
+∂Zφ(R, Z) = − 1
+2π
+�
+Ωǫ
+�
+(1+ǫR)(1+ǫR′) ˜F(s)
+(Z−Z′) η(R′, Z′)
+(R−R′)2 + (Z−Z′)2 dR′ dZ′ ,
+∂Rφ(R, Z) = − 1
+2π
+�
+Ωǫ
+�
+(1+ǫR)(1+ǫR′) ˜F(s)
+(R−R′) η(R′, Z′)
+(R−R′)2 + (Z−Z′)2 dR′ dZ′
+(3.72)
++ ǫ
+4π
+�
+Ωǫ
+√
+1+ǫR′
+√
+1 + ǫR
+�
+F(s) + ˜F(s)
+�
+η(R′, Z′) dR′ dZ′ ,
+where ˜F(s) = −2sF ′(s), see [30, Section 4.2]. Here, as in (3.12), we use the shorthand notation
+s =
+ǫ2D2
+(1+ǫR)(1+ǫR′) ≡ ǫ2 (R−R′)2 + (Z−Z′)2
+(1+ǫR)(1+ǫR′)
+.
+(3.73)
+In view of (2.7), we have ˜F(s) → 1 as s → 0 and ˜F(s) = O(s−3/2) as s → +∞.
+Throughout the proof, we fix t > 0 and we assume that the parameters ǫ =
+√
+νt/¯r(t) and
+δ = ν/Γ are small enough. By construction the vorticity η∗(R, Z, t) defined by (2.28) vanishes
+identically when ρ := (R2+Z2)1/2 ≥ 2ǫ−σ0, so we can assume henceforth that ρ ≤ 2ǫ−σ0. In
+that region, we have for any γ ∈ (0, 1) the a priori bounds
+�
+|α|≤2
+|∂αη∗(R, Z, t)| ≤ C e−γρ2/4 ,
+�
+|α|=1
+|∂αφ∗(R, Z, t)| ≤ C ,
+(3.74)
+where α = (α1, α2) ∈ N2 and ∂α = ∂α1
+R ∂α2
+Z . Indeed, the first estimate in (3.74) holds because η∗
+is obtained by truncating the asymptotic approximation ηapp(R, Z, t) which belongs to the space
+27
+
+Z defined in (3.23). The second estimate can then be obtained using the expressions (3.72) with
+φ = φ∗ and η = η∗. To see this, we first observe that 1 + ǫR ≈ 1 and 1 + ǫR′ ≈ 1 in (3.72),
+because both quantities ρ and ρ′ := (R′2+Z′2)1/2 are smaller than 2ǫ−σ0 ≪ ǫ−1. If we use the
+estimates | ˜F(s)| ≤ C in the first two lines of (3.72) and |F(s) + ˜F(s)| ≤ Cs−1/2 in the third
+line, we thus obtain
+|∂Rφ∗(R, Z, t)| + |∂Zφ∗(R, Z, t)| ≤ C
+�
+R2
+|η∗(R′, Z′, t)|
+�
+(R−R′)2 + (Z−Z′)2 dR′ dZ′ ≤ C ,
+which concludes the proof of (3.74). Finally, since
+t∂tη∗(R, Z, t) = χ0
+�
+ǫσ0ρ
+�
+t∂tηapp(R, Z, t) + σ0 ǫσ0ρχ′
+0
+�
+ǫσ0ρ
+�
+ηapp(R, Z, t) t∂t log(ǫ) ,
+it follows from the expressions given in Sections 3.3–3.6 that t∂tη∗ satisfies the same bound as
+η∗ in (3.74). Summarizing, in view of (3.74), the remainder Rem(R, Z, t) satisfies
+eγ0ρ2/4 |Rem(R, Z, t)| ≤ C δ−1(1 + ρ) e−(γ−γ0)ρ2/4 ,
+when ρ ≤ 2ǫ−σ0 ,
+(3.75)
+for any γ0 ∈ (0, 1). If we assume that γ ∈ (γ0, 1), we conclude that the right-hand side of (3.75)
+is O(δ−1ǫ∞) if ρ ≥ ǫ−σ0. So from now on we may concentrate on the inner region ρ ≤ ǫ−σ0,
+where η∗ = ηapp is given by (3.62).
+In that region we decompose the stream function as φ∗ = BSǫ[χ0 ηapp] = φ0
+∗ − φ1
+∗ + φ2
+∗, where
+φ0
+∗ =
+4
+�
+m=0
+ǫm BSm[ηapp] ,
+φ1
+∗ =
+4
+�
+m=0
+ǫm BSm[(1−χ0) ηapp] ,
+φ2
+∗ =
+∞
+�
+m=5
+ǫm BSm[χ0 ηapp] .
+Here χ0 is a shorthand notation for χ0(ǫσ0ρ). The convergence of the series defining φ2
+∗ is easily
+justified using Lemmas 3.1 and 3.3, if we observe that both inequalities in (3.12) are satisfied
+since ρ, ρ′ ≪ ǫ−1. In fact, proceeding as in the proof of (3.74), one can verify that
+|∂Rφ2
+∗(R, Z, t)| + |∂Zφ2
+∗(R, Z, t)| ≤ Cǫ5βǫ (1 + ρ)5 ,
+ρ ≤ ǫ−σ0 ,
+where βǫ = log(1/ǫ). Moreover we have |∂Rφ1
+∗|+|∂Zφ1
+∗| = O(ǫ∞) because (1−χ0)ηapp = O(ǫ∞).
+Finally, in view of (3.30), (3.62), it is clear that
+|∂R
+�
+φ0
+∗ − φapp
+�
+| + |∂Z
+�
+φ0
+∗ − φapp
+�
+| ≤ Cǫ5β3
+ǫ (1 + ρ)5 .
+Summarizing, we have shown that the remainder (2.29) satisfies
+��Rem(R, Z, t) − �
+Rem(R, Z, t)
+�� ≤ Cδ−1ǫ5β3
+ǫ (1 + ρ)5 e−γρ2/4 ,
+ρ ≤ ǫ−σ0 ,
+(3.76)
+where �
+Rem(R, Z, t) is the quantity defined for all (R, Z) ∈ R2 by the formula
+Lηapp + ǫ∂R
+�
+S4ηapp
+�
+− t∂tηapp − 1
+δ
+�
+φapp , S4ηapp
+�
++ ǫ¯r
+δΓ
+�
+˙¯r ∂Rηapp + ˙¯z ∂Zηapp
+�
+,
+(3.77)
+with S4 = 1 − ǫR + (ǫR)2 − (ǫR)3 + (ǫR)4.
+Now the crucial observation is that the asymptotic approximation ηapp was constructed
+precisely so as to make the quantity (3.77) small in the topology of Z. More precisely, the
+results of Sections 3.3–3.6 can be rephrased as follows:
+δ �
+Rem(R, Z, t) = OZ
+�
+ǫδ2 + ǫ2βǫδ2 + ǫ3βǫδ + ǫ4β2
+ǫ δ + ǫ5β3
+ǫ
+�
+.
+(3.78)
+28
+
+Inside the parenthesis in the right-hand side, the first four terms represent what remains from
+the quantities ǫm�
+Ληm + δ
+�
+t∂t + m
+2 − L
+�
+ηm + Rm) for m = 1, 2, 3, 4 after the profiles ηm have
+been determined, and the last one corresponds to those terms in (3.77) which are of order O(ǫ5)
+or higher and were not considered in the construction of ηapp. Combining (3.76), (3.78) and
+using Young’s inequality, we obtain
+sup
+ρ≤ǫ−σ0
+eγ0ρ2/4 |Rem(R, Z, t)| ≤ C
+δ
+�
+ǫδ2 + ǫ3βǫδ + ǫ5β3
+ǫ
+�
+≤ C
+�
+ǫδ + ǫγ5δ−1�
+,
+for any γ5 < 5. This concludes the proof of (2.30).
+□
+3.8
+The Eulerian approximation
+As was already observed in Remark 2.3, if we set δ = ˙¯r = 0 in the expansion (2.24), we obtain
+an approximate solution ηE
+app, φE
+app of equation (2.26), which is nothing but the stationary Euler
+equation in a frame moving with (constant) velocity ˙¯z ez. As is well known [3], steady states of
+the Euler system are often characterized by a global functional relation between the vorticity and
+the stream function. In our case, in view of (2.26), we expect finding a function Φǫ : R+ → R
+such that
+φE
+app(R, Z) − r0 ˙¯z
+2Γ (1 + ǫR)2 = Φǫ
+�ηE
+app(R, Z)
+1 + ǫR
+�
++ O
+�
+ǫM+1−�
+,
+(3.79)
+for all (R, Z) ∈ R2 such that ρ :=
+√
+R2 + Z2 ≪ ǫ−1.
+In this section, we first verify that a relation of the form (3.79) holds to second order, namely
+with M = 2. Using the expressions (3.40), (3.41), (3.49), (3.50) with δ = 0 and simplifying
+somehow the notation, we can write our approximate solution in the form
+ηE
+app(R, Z) = η0 + ǫRη1 + ǫ2(R2−Z2)η2 + ǫ2η3 ,
+φE
+app(R, Z) = φ0 + ǫRφ1 + ǫ2(R2−Z2)φ2 + ǫ2φ3 ,
+(3.80)
+where η0, φ0 are given by (3.31), and the profiles η1, η2, η3 ∈ Z and φ1, φ2, φ3 ∈ S∗(R2) are all
+radially symmetric. Note that ηm, φm may include factors of βǫ = log(1/ǫ) when m ≥ 1, but
+this dependence is not explicitly indicated. We also expand the unknown function Φǫ in (3.79)
+in powers of ǫ :
+Φǫ(s) = Φ0(s) + ǫΦ1(s) + ǫ2Φ2(s) .
+(3.81)
+Finally, to simplify the writing, we denote
+ω =
+1
+4π
+�
+βǫ − 1 + 2v
+�
+= r0 ˙¯z
+Γ + O(ǫ2−) ,
+where the last equality follows from (3.38), (3.42).
+If we consider equality (3.79) to leading order in ǫ, thus neglecting all terms that are O(ǫ)
+or O(ǫβǫ), we obtain the relation φ0 − ω/2 = Φ0(η0), which determines the principal term Φ0 in
+the expansion (3.81). In view of (3.31), (3.32) we thus have
+Φ0(s) = φ0(0) − ω
+2 − 1
+4π Ein
+�
+log
+1
+4πs
+�
+,
+s > 0 .
+(3.82)
+The constant in (3.82) has no relevance, but it is important to note that Φ0(s) ∼ − 1
+4π log log 1
+s
+as s → 0. For later use we define
+A(ρ) = Φ′
+0
+�
+η0(ρ)
+�
+= ∂Rφ0
+∂Rη0
+= ∂Zφ0
+∂Zη0
+= 4
+ρ2
+�
+eρ2/4 − 1
+�
+,
+ρ > 0 .
+(3.83)
+29
+
+Incidentally we observe that A(ρ) = 1/h(ρ) where h is defined in (3.24).
+To the next order in ǫ, we deduce from (3.79) the relation
+(φ1 − ω)R = Φ′
+0(η0)(η1 − η0)R + Φ1(η0) ,
+(3.84)
+which can be satisfied only if Φ1 = 0, because Φ1(η0) is the only radially symmetric term in
+(3.84). Dividing by R, we obtain the equality φ1 −ω = A(η1 −η0), which happens to be satisfied
+in view of our definitions of the profiles η1, φ1. This fact can be verified by following carefully
+the calculations in Section 3.3.
+Finally we exploit (3.79) to order ǫ2, keeping in mind that Φ1 = 0. We obtain the more
+complex relation
+(R2−Z2)φ2 + φ3 − ω
+2 R2 = Φ′
+0(η0)
+�
+(R2−Z2)η2 + η3 + (η0 − η1)R2�
++ 1
+2Φ′′
+0(η0)(η0 − η1)2R2 + Φ2(η0) ,
+where it is useful to substitute R2 = 1
+2(R2+Z2) + 1
+2(R2−Z2). The terms containing R2−Z2
+cancel exactly due to the identity
+φ2 − 1
+2Ψ − Aη2 = 0 ,
+where
+Ψ = ω
+2 + Φ′
+0(η0)(η0 − η1) + 1
+2Φ′′
+0(η0)(η0 − η1)2 ,
+which is satisfied by definition of the profiles φ2, η2, as can be verified by following the calculations
+in Section 3.4. We are thus left with a relation involving only radially symmetric terms
+φ3 − 1
+2(R2+Z2)Ψ − Aη3 = Φ2(η0) ,
+(3.85)
+which provides the definition of the second order correction Φ2 in (3.81). Summarizing, if Φǫ is
+defined by (3.81) with Φ1 = 0, Φ0 given by (3.82) and Φ2 by (3.85), we have shown that (3.79)
+holds with M = 2.
+We now come back to the approximate solution η∗, φ∗ of (2.23) constructed in Sections 3.3–
+3.6, and we show that it also satisfies a relation of the form (3.79), in a sufficiently small region
+near the origin. To formulate that result, we denote
+Θ(R, Z, t) = φ∗(R, Z, t) − ¯r ˙¯z
+2Γ (1 + ǫR)2 − Φǫ
+�η∗(R, Z, t)
+1 + ǫR
+�
+,
+(R, Z) ∈ Ωǫ .
+(3.86)
+Proposition 3.15. There exist σ1 ∈ (0, σ0) and N ∈ N such that, for any γ3 < 3, the quantity
+Θ defined by (3.86) satisfies, for some C > 0,
+|∂RΘ(R, Z, t)| + |∂ZΘ(R, Z, t)| ≤ C(ǫδ + ǫγ3)(1 + ρ)N ,
+ρ ≤ ǫ−σ1 ,
+(3.87)
+whenever ǫ and δ are small enough.
+Proof. The idea is to compare Θ with the second order Eulerian approximation
+ΘE
+app(R, Z, t) = φE
+app(R, Z, t) − r0 ˙¯zE
+2Γ (1 + ǫR)2 − Φǫ
+�ηE
+app(R, Z, t)
+1 + ǫR
+�
+,
+(3.88)
+which is of size O(ǫ3−) in view of (3.79). Here we consider both quantities ηE
+app, φE
+app as time-
+dependent, because we deal with the viscous case where ǫ =
+√
+νt/¯r(t). We already estimated
+30
+
+the difference φ∗ − φapp (assuming M = 4) in the proof of Proposition 2.5, and by construction
+we know that φapp = φE
+app + O(ǫδ + ǫ3β2
+ǫ ). These arguments lead to the bound
+|∂R
+�
+φ∗ − φE
+app
+�
+| + |∂Z
+�
+φ∗ − φE
+app
+�
+| ≤ C
+�
+ǫδ + ǫ3β2
+ǫ
+�
+(1 + ρ)3 ,
+ρ ≤ ǫ−σ0 .
+(3.89)
+On the other hand, we have already observed that ¯r(t) = r0(1 + O(ǫ2)), and in view of (3.42),
+(3.56) the difference between the vertical speed ˙¯z and its second order approximation ˙¯zE is of
+size (Γ/r0) · O(ǫ2βǫ). We thus find
+��� ¯r ˙¯z
+2Γ − r0 ˙¯zE
+2Γ
+���
+��∂R(1 + ǫR)2�� ≤ Cǫ3βǫ ,
+ρ ≤ ǫ−σ0 .
+(3.90)
+Finally η∗ is just a truncation of ηapp and by definition ηapp−ηE
+app = O(ǫδ+ǫ3βǫ) in the topology
+of Z. This gives the following bound
+�
+|α|≤1
+��∂α�
+η∗ − ηE
+app
+�
+(R, Z, t)
+�� ≤ C
+�
+ǫδ + ǫ3βǫ
+�
+(1 + ρ)3e−ρ2/4 ,
+ρ ≤ ǫ−σ0 .
+(3.91)
+At this point we observe that, since η∗ and ηE
+app are O(ǫ) perturbations of η0 in the topology
+of Z, one can choose σ1 > 0 small enough so that
+1
+2 η0(ρ) ≤ η∗(R, Z, t)
+1 + ǫR
+≤ 2 η0(ρ) ,
+ρ ≤ ǫ−σ1 ,
+(3.92)
+and similarly for ηE
+app. To conclude the proof of Proposition 3.15, we need estimates on the
+derivatives of the function Φǫ defined in (3.81). We begin with the leading order term Φ0 which
+is given by the explicit formula (3.82). We have
+Φ′
+0
+� s
+4π
+�
+=
+1 − s
+s log(1/s) ,
+1
+4π Φ′′
+0
+� s
+4π
+�
+= −s − 1 + log(1/s)
+s2�
+log(1/s)
+�2
+,
+s > 0 .
+Thanks to (3.92) we only need to evaluate these expressions when the argument s/(4π) takes
+its value in the interval
+� 1
+2η0(ρ), 2η0(ρ)
+�
+. In view of Lemma 3.16 below, there exists C > 1 such
+that, for all λ ∈ [1/2, 2] and all ρ > 0,
+A(ρ)
+C
+≤ Φ′
+0
+�
+λη0(ρ)
+�
+≤ CA(ρ) ,
+��Φ′′
+0
+�
+λη0(ρ)
+��� ≤ CB(ρ) ,
+(3.93)
+where A(ρ) is defined in (3.83) and
+B(ρ) = −Φ′′
+0(η0(ρ)) = 16π
+ρ4
+�
+(ρ2 − 4)eρ2/2 + 4eρ2/4�
+,
+ρ > 0 .
+(3.94)
+The second order contribution Φ2 is not known explicitly, but from the definition (3.85), where
+the left-hand side belongs to S∗(R2), we deduce that there exist C > 0 and N ∈ N such that
+��Φ′
+2
+�
+λη0(ρ)
+��� ≤ CA(ρ)(1 + ρ)N ,
+��Φ′′
+2
+�
+λη0(ρ)
+��� ≤ CB(ρ)(1 + ρ)N ,
+(3.95)
+for all ρ > 0 and all λ ∈ [1/2, 2].
+Now, if ∂α = ∂R or ∂Z, we decompose
+∂αΦǫ
+�
+η∗
+1+ǫR
+�
+− ∂αΦǫ
+� ηE
+app
+1+ǫR
+�
+= Φ′
+ǫ
+�
+η∗
+1+ǫR
+��
+∂α�
+η∗
+1+ǫR
+�
+− ∂α� ηE
+app
+1+ǫR
+��
++
+�
+Φ′
+ǫ
+�
+η∗
+1+ǫR
+�
+− Φ′
+ǫ
+� ηE
+app
+1+ǫR
+��
+∂α� ηE
+app
+1+ǫR
+�
+,
+31
+
+and we estimate the right-hand side using (3.91), (3.93), and (3.95). Taking into account the
+preliminary bounds (3.89), (3.90), we arrive at an estimate of the form
+�
+|α|=1
+��∂α�
+Θ(R, Z, t) − ΘE
+app(R, Z, t)
+��� ≤ C(ǫδ + ǫγ3)(1 + ρ)N ,
+ρ ≤ ǫ−σ1 .
+As was already mentioned, the approximation ΘE
+app(R, Z, t) is O(ǫ3−) in the topology of S∗(R2),
+so altogether we arrive at (3.87).
+In the argument above we used the following elementary result, whose proof can be omitted.
+Lemma 3.16. Let f, g : (0, +∞) → (0, +∞) be defined by
+f(s) =
+1 − s
+s log(1/s) ,
+g(s) = s − 1 + log(1/s)
+s2�
+log(1/s)
+�2
+= −f ′(s) ,
+s > 0 .
+Then given any Λ > 1 there exists C > 1 such that, for any λ ∈ [Λ−1, Λ] and any s > 0,
+1
+C ≤ f(λs)
+f(s)
+≤ C ,
+1
+C ≤ g(λs)
+g(s)
+≤ C .
+4
+Energy estimates and stability proof
+In the previous section we constructed an approximate solution η∗(R, Z, t) of the rescaled vortic-
+ity equation (2.15) which corresponds, in the original variables, to a sharply concentrated vortex
+ring of radius ¯r(t) located at the vertical position ¯z(t). Our goal is now to control the difference
+between this approximation and the actual solution of (2.15) with initial data η0. This will
+conclude the proof of our main results, Theorems 1.1 and 2.6.
+For technical reasons that were mentioned in the introduction, it is convenient for the stability
+analysis to center the vertical coordinate Z not at the point ¯z(t), which is associated with
+the approximate solution η∗, but at a point ¯z(t) + ˜z(t) where ˜z(t) is a small correction to be
+determined later. This can be viewed as the analogue of the modulation technique that is used,
+for instance, in the stability analysis of solitary waves [55, 46]. Thus, instead of the variables
+(R, Z) defined in (2.14), we use henceforth the slightly modified coordinates
+R = r − ¯r(t)
+√
+νt
+,
+Z = z − ¯z(t) − ˜z(t)
+√
+νt
+,
+(4.1)
+where the velocities ˙¯r(t), ˙¯z(t) are given by (3.61). As is easily verified, the evolution equation
+(2.15) still holds if ˙¯z is replaced by ˙¯z + ˙˜z. As a consequence, if the solution η is decomposed as
+in (2.31), the perturbation ˜η(R, Z, t) satisfies the equation
+t∂t˜η + 1
+δ
+�
+φ∗ , ˜ζ
+�
++ 1
+δ
+�˜φ , ζ∗
+�
++
+�˜φ , ˜ζ
+�
+− ǫ¯r
+δΓ
+�
+˙¯r ∂R˜η + ˙¯z ∂Z ˜η
+�
+= L˜η + ǫ∂R˜ζ + 1
+δ Rem(R, Z, t) + ǫ¯r ˙˜z
+δ2Γ
+�
+∂Zη∗ + δ∂Z ˜η
+�
+,
+(4.2)
+where to simplify the writing we use the letter ζ to denote the potential vorticity, namely
+˜ζ(R, Z, t) = ˜η(R, Z, t)
+1 + ǫR
+,
+ζ∗(R, Z, t) = η∗(R, Z, t)
+1 + ǫR
+.
+(4.3)
+32
+
+Clearly, if ˙˜z = 0, the last term in the right-hand side of (4.2) disappears, and we recover the
+perturbation equation (2.32).
+From our previous work [30] we know that Eq. (4.2) has a unique solution ˜η, in an appropriate
+weighted L2 space, with zero initial data. Our goal is to control the evolution of that solution
+on a large time interval, uniformly with respect to the viscosity in the limit ν → 0. This is not
+an easy task, because several terms in (4.2) are multiplied by the Reynolds number δ−1 = Γ/ν,
+which becomes arbitrarily large in the regime we consider. As was explained in the introduction,
+we shall use energy estimates to control the solution of (4.2), but a few preliminary steps are
+necessary before starting the actual calculations.
+4.1
+Control of the lowest order moments
+To implement our strategy based on energy estimates, we need a precise information on the
+lowest order moments of the solution of (4.2). We first define, for all t > 0,
+µ0(t) =
+�
+Ωǫ
+˜η(R, Z, t) dX ,
+µ1(t) =
+�
+Ωǫ
+�
+R + ǫR2/2
+�
+˜η(R, Z, t) dX ,
+(4.4)
+where dX = dR dZ denotes the Lebesgue measure in R2.
+Lemma 4.1. The moments defined in (4.4) satisfy µ0(t) = O(ǫ∞δ−1) and µ1(t) = O(ǫ+ǫγ5δ−2)
+for any γ5 < 1, whenever ǫ and δ are small enough.
+Proof. The conclusion can be obtained by direct calculations, but we find it more illuminating
+to use the conserved quantities of the original equation (2.3). The first one is the total circulation
+M(t) =
+�
+Ω
+ωθ(r, z, t) dr dz = Γ
+�
+Ωǫ
+�
+η∗ + δ˜η
+�
+(R, Z, t) dX = Γ
+�
+Ωǫ
+η∗ dX + Γδµ0(t) ,
+(4.5)
+which satisfies M(0) = Γ and is almost constant in time. In fact it is proved in [30, Section 4.4]
+that 0 ≤ 1 − M(t)/Γ ≤ C exp(−c/ǫ2) for some positive constants C and c. Moreover, since the
+approximate solution ηapp lies in the space Z defined by (3.23), it follows from (2.28) and (3.70)
+that
+�
+Ωǫ η∗ dX = 1 + O(exp(−c/ǫ2)). Therefore µ0(t) = O(exp(−c/ǫ2) δ−1) by (4.5).
+We next consider the total impulse in the vertical direction
+I =
+�
+Ω
+r2ωθ(r, z, t) dr dz = Γ¯r(t)2
+�
+Ωǫ
+(1 + ǫR)2�
+η∗ + δ˜η
+�
+(R, Z, t) dX ,
+(4.6)
+which is known to be exactly conserved [44, 29], so that I = Γr2
+0 for all times. Equality (4.6)
+can be rephrased as I/Γ = I∗(t) + δ¯r(t)2µ(t), where
+I∗(t) = ¯r(t)2
+�
+Ωǫ
+(1 + ǫR)2η∗(R, Z, t) dX ,
+µ(t) = µ0(t) + 2ǫµ1(t) .
+(4.7)
+It is not difficult to show that
+tI′
+∗(t) = −¯r(t)2
+�
+Ωǫ
+(1 + ǫR)2 Rem(R, Z, t) dX .
+(4.8)
+The easiest way to establish (4.8) is to observe that the impulse I∗(t) would be conserved if
+η∗ was an exact solution of (2.15), so that the remainder Rem(R, Z, t) defined in (2.29) is
+the only term that contributes to the evolution of I∗(t). However equality (4.8) can also be
+33
+
+verified by a direct calculation. In any case, since Rem(R, Z, t) satisfies estimate (2.30) and
+�
+Ωǫ Rem(R, Z, t) dx = O(ǫ∞), we deduce from (4.8) that |tI′
+∗(t)| ≤ Cr2
+0
+�
+ǫ2δ + ǫγ5+1δ−1�
+, hence
+|I∗(t) − r2
+0| ≤
+� t
+0
+|I′
+∗(s)| ds ≤ Cr2
+0
+� t
+0
+ǫ(s)2δ + ǫ(s)γ5+1δ−1
+s
+ds ≤ Cr2
+0
+�
+ǫ2δ + ǫγ5+1δ−1�
+.
+As r2
+0 − I∗(t) = δ¯r(t)2µ(t), we conclude that µ(t) = O
+�
+ǫ2 + ǫγ5+1δ−2�
+, which gives the desired
+estimate for µ1(t).
+It is not clear if the strategy above can be applied to control the first order moment of the
+perturbation ˜η with respect to the vertical variable Z. In particular, we are not aware of any
+(approximately) conserved quantity that we could invoke as in the proof of Lemma 4.1. Instead
+we use the modulation parameter ˜z(t) introduced in (4.1) to impose that the vertical moment
+vanishes identically :
+µ2(t) :=
+�
+Ωǫ
+Z ˜η(R, Z, t) dX = 0 .
+(4.9)
+Differentiating (4.9) with respect to time and using (4.2), we obtain the relation
+˙˜z(t)
+�
+Ωǫ
+Z
+�
+∂Zη∗ + δ∂Z ˜η
+�
+dX = δ2Γ
+ǫ¯r
+�
+Ωǫ
+Z R(R, Z, t) dX ,
+(4.10)
+where
+R = 1
+δ
+�
+φ∗ , ˜ζ
+�
++ 1
+δ
+�˜φ , ζ∗
+�
++
+�˜φ , ˜ζ
+�
+− ǫ¯r
+δΓ
+�
+˙¯r ∂R˜η + ˙¯z ∂Z ˜η
+�
+− L˜η − ǫ∂R˜ζ − 1
+δ Rem(R, Z, t) .
+(4.11)
+In view of Lemma 4.1 the integral in the left-hand side of (4.10) is equal to −1 + O(ǫ∞), and is
+therefore bounded away from zero if ǫ is small enough. The integral in the right-hand side is a
+priori of size O(δ−1), but we observe that R = δ−1Λ˜η + O(ǫδ−1), where Λ is the linear operator
+defined in (3.19). Using the properties established in Proposition 3.6, we see that the leading
+term gives no contribution :
+1
+4π
+�
+R2 Z Λ˜η dX =
+�
+Zη0 , Λ˜η
+�
+Y = −
+�
+Λ(Zη0) , ˜η
+�
+Y = 0 ,
+since Zη0 = −2∂Zη0 is in the kernel of Λ. These considerations, which will be made rigorous
+in Section 4.8 below, show that the modulation speed ˙˜z is uniquely determined by (4.10) and
+satisfies ˙˜z(t) = O(δ) as long as ˜η remains O(1). In particular ˜z(t) is indeed a small correction
+to the vertical position of the vortex ring.
+4.2
+Definition and properties of the weight function
+We now provide the precise definition of the weight function Wǫ : Ωǫ → (0, +∞) which appears
+in the energy functional (2.33). We give ourselves three positive numbers σ1, σ2, γ such that
+0 < σ1 < σ0 < 1 < σ2 ,
+γ = σ1/σ2 ,
+(4.12)
+where σ0 ∈ (0, 1) is the cut-off exponent already introduced in (2.28). As we shall see σ2 > 1
+can be chosen arbitrarily, but σ1 > 0 has to be taken sufficiently small. In particular σ1 should
+be small enough so that Proposition 3.15 holds.
+34
+
+As in (4.3), if ǫ > 0 and δ > 0 are sufficiently small, we denote ζ∗ = η∗/(1 + ǫR), where
+η∗ is the approximate solution of (2.15) given by (2.28). We recall that ζ∗ and φ∗ := BSǫ[η∗]
+satisfy the relation (2.35), where Φǫ : R+ → R is the function constructed in Section 3.8. We
+decompose the domain Ωǫ =
+�
+(R, Z) ; 1 + ǫR > 0
+�
+into a disjoint union Ω′
+ǫ ∪ Ω′′
+ǫ ∪ Ω′′′
+ǫ , where
+Ω′
+ǫ =
+�
+(R, Z) ∈ Ωǫ ; Φ′
+ǫ(ζ∗(R, Z)) < exp
+�
+ǫ−2σ1/4
+��
+,
+Ω′′
+ǫ =
+�
+(R, Z) ∈ Ωǫ \ Ω′
+ǫ ; ρ ≤ ǫ−σ2
+�
+,
+Ω′′′
+ǫ =
+�
+(R, Z) ∈ Ωǫ ; ρ > ǫ−σ2�
+.
+(4.13)
+Here and in what follows, if (R, Z) ∈ R2, we denote ρ = (R2+Z2)1/2. The domains Ω′
+ǫ, Ω′′
+ǫ also
+depend (mildly) on δ, but for simplicity this dependence is not indicated explicitly.
+Lemma 4.2. If ǫ > 0 is small enough, the inner region Ω′
+ǫ defined in (4.13) is diffeomorphic to
+a open disk. Moreover, if κ is any fixed positive number, one has
+�
+(R, Z) ; ρ ≤ ǫ−σ1�
+⊂ Ω′
+ǫ ⊂
+�
+(R, Z) ; ρ ≤ ǫ−σ1 + κ log 1
+ǫ
+�
+.
+(4.14)
+Proof. The main properties of the function Φǫ are established in the proof of Proposition 3.15.
+In particular, using estimates (3.92), (3.93), (3.95), it is easy to verify that
+1
+2 A(ρ) ≤ Φ′
+ǫ
+�
+ζ∗(R, Z)
+�
+≤ 2A(ρ) ,
+when ρ ≤ 2ǫ−σ1 .
+(4.15)
+Here A(ρ) = (4/ρ2)
+�
+eρ2/4 − 1
+�
+, see (3.83). Since 2A(ǫ−σ1) < exp(ǫ−2σ1/4) as soon as ǫ−σ1 ≥ 3,
+we deduce that (R, Z) ∈ Ω′
+ǫ if ρ ≤ ǫ−σ1. Similarly, using the lower bound in (4.15), it is easy
+to verify that, for any κ > 0, the inner region Ω′
+ǫ is contained in the disk ρ ≤ ǫ−σ1 + κ log 1
+ǫ if
+ǫ > 0 is small enough. Finally Ω′
+ǫ is diffeomorphic to a disk because Φ′
+ǫ(ζ∗) is close to a strictly
+increasing radially symmetric function when ǫ > 0 is small, see (3.81).
+We next choose a smooth cut-off function χ1 : R → [1
+2, 3] such that
+χ1(x) =
+1
+1 + x
+for |x| ≤ 1
+2 ,
+χ′
+1(x) = 0
+for |x| ≥ 3
+4 .
+(4.16)
+The weight Wǫ : Ωǫ → (0, +∞) is defined by
+Wǫ(R, Z) = χ1(ǫR) ×
+
+
+
+
+
+
+
+Φ′
+ǫ
+�
+ζ∗(R, Z)
+�
+in Ω′
+ǫ ,
+exp
+�
+ǫ−2σ1/4
+�
+in Ω′′
+ǫ ,
+exp
+�
+ρ2γ/4
+�
+in Ω′′′
+ǫ ,
+(4.17)
+where γ = σ1/σ2 < 1 and Ω′
+ǫ, Ω′′
+ǫ , Ω′′′
+ǫ are the regions defined in (4.13). In other words, we assume
+that Wǫ = Φ′
+ǫ(ζ∗)/(1+ǫR) as long as the numerator remains smaller than the threshold value
+exp(ǫ−2σ1/4). Outside this inner region, the weight is radially symmetric except for the geometric
+factor χ1(ǫR), and the radial profile remains constant as long as ρ ≤ ǫ−σ2 before increasing again
+like exp(ρ2γ/4) when ρ > ǫ−σ2. By construction the function Wǫ is locally Lipschitz continuous
+in Ωǫ, and smooth in the interior of all three regions (4.13). The (mild) dependence of Wǫ upon
+the parameter δ > 0 is not indicated explicitly. A schematic representation of the graph of Wǫ
+is given in Figure 2.
+Further properties of the weight Wǫ are collected in the following lemma.
+35
+
+Lemma 4.3. There exist positive constants C1, C2 such that, if ǫ, δ, and σ1 are small enough,
+the weight Wǫ satisfies the uniform bounds
+C1 exp
+�
+ρ2γ/4
+�
+≤ Wǫ(R, Z) ≤ C2A(ρ) ,
+(R, Z) ∈ Ωǫ ,
+(4.18)
+where ρ = (R2+Z2)1/2 and A(ρ) is defined in (3.83). Moreover, given γ1 < 1 there exists C3 > 0
+such that
+��Wǫ(R, Z) − A(ρ)
+�� +
+��∇Wǫ(R, Z) − ∇A(ρ)
+�� ≤ C3 ǫγ1A(ρ) ,
+when ρ ≤ ǫ−σ1 .
+(4.19)
+Proof. Since 1
+2 ≤ χ1(ǫR) ≤ 3 and exp(ρ2γ/4) ≤ CA(ρ), we deduce from (4.15) that the bounds
+(4.18) hold in the inner region Ω′
+ǫ, and the same inequalities obviously hold in the far field region
+Ω′′′
+ǫ too. In the intermediate region Ω′′
+ǫ , we know that ρ ≤ ǫ−σ2 so that exp(ρ2γ/4) ≤ exp(ǫ−2σ1/4)
+because γ = σ1/σ2. If ρ ≥ 2ǫ−σ1, it is clear that exp(ǫ−2σ1/4) ≤ A(ρ), which gives the desired
+upper bound in that case. Finally if (R, Z) ∈ Ω′′
+ǫ and ρ ≤ 2ǫ−σ1, we deduce from (4.15) that
+exp(ǫ−2σ1/4) ≤ Φ′
+ǫ
+�
+ζ∗(R, Z)
+�
+≤ 2A(ρ), which concludes the proof of the upper bound in (4.18).
+To prove (4.19), we start from the expression (4.17) of the weight Wǫ in the inner region Ω′
+ǫ.
+We know from (3.83) that A(ρ) = Φ′
+0(η0), where η0 is defined in (3.31). We thus find
+|Wǫ(R, Z) − A(ρ)
+�� ≤
+��χ1(ǫR) − 1
+��Φ′
+ǫ(ζ∗) +
+��Φ′
+ǫ(ζ∗) − Φ′
+ǫ(η0)
+�� +
+��Φ′
+ǫ(η0) − Φ′
+0(η0)
+�� .
+(4.20)
+Since χ1(ǫR) = (1 + ǫR)−1 when ρ ≤ ǫ−σ1, the first term in the right-hand of (4.20) is smaller
+than Cǫ|R| Φ′
+ǫ(ζ∗) ≤ Cǫ1−σ1A(ρ). For the second term, we use the bounds (3.92), (3.93), and
+(3.95) to obtain
+��Φ′
+ǫ(ζ∗) − Φ′
+ǫ(η0)
+�� ≤
+sup
+1
+2≤λ≤2
+��Φ′′
+ǫ(λη0)
+�� |ζ∗ − η0| ≤ CB(ρ)(1 + ρ)Nǫη0 ≤ Cǫγ1A(ρ) ,
+where in the last inequality we assumed that σ1 > 0 is small enough so that Nσ1 ≤ 1 − γ1. The
+last term in (4.20) is bounded by ǫ2|Φ′
+2(η0)| ≤ Cǫγ1A(ρ) in view of (3.95). Altogether we arrive
+at the estimate |Wǫ(R, Z) − A(ρ)
+�� ≤ Cǫγ1A(ρ). The corresponding inequality for the first order
+derivatives can be obtained in a similar way, and we omit the details
+4.3
+Coercivity of the energy functional
+For ǫ ≥ 0 small enough, we introduce the weighted L2 space Xǫ =
+�
+η ∈ L2(Ωǫ) ; ∥η∥Xǫ < ∞
+�
+defined by the norm
+∥η∥2
+Xǫ =
+�
+Ωǫ
+Wǫ(R, Z) |η(R, Z)|2 dR dZ .
+(4.21)
+In the limiting case ǫ = 0, it is understood that Ω0 = R2 and W0(R, Z) = A(ρ), in agreement
+with (4.19). Assuming that ǫ > 0, we consider the energy functional (2.33), namely
+Eǫ[η] = 1
+2 ∥η∥2
+Xǫ − Ekin
+ǫ
+[η] ,
+η ∈ Xǫ ,
+(4.22)
+where Ekin
+ǫ
+is the kinetic energy defined by
+Ekin
+ǫ
+[η] = 1
+2
+�
+Ωǫ
+φ η dR dZ = 1
+2
+�
+Ωǫ
+|∇φ|2
+1 + ǫR dR dZ ,
+φ = BSǫ[η] .
+(4.23)
+Since we are interested in the regime where ǫ is small, it is important to observe that Ekin
+ǫ
+[η]
+becomes singular in the limit ǫ → 0, if the vorticity η has nonzero mean. This divergence is
+36
+
+related to the well-known fact that any (nontrivial) nonnegative vorticity distribution in R2
+has infinite kinetic energy. The regular part of Ekin
+ǫ
+[η] is given, to leading order, by the two-
+dimensional energy
+Ekin
+0 [η] =
+1
+4π
+�
+R2
+�
+Lη)η dX =
+1
+4π
+�
+R2
+�
+R2 log
+� 8
+D
+�
+η(R, Z)η(R′, Z′) dX dX′ ,
+(4.24)
+where L is the integral operator (3.14) and D2 = (R−R′)2 + (Z−Z′)2. More precisely, we have
+the following statement, whose proof is postponed to Section B.1.
+Lemma 4.4. If ǫ > 0 is small and η ∈ Xǫ satisfies supp(η) ⊂ Bǫ := {(R, Z) ∈ Ωǫ ; ρ ≤ ǫ−σ1},
+we have the expansion
+Ekin
+ǫ
+[η] = βǫ − 2
+4π
+µ2
+0 + Ekin
+0 [η] + O
+�
+ǫβǫ∥η∥2
+Xǫ
+�
+,
+as ǫ → 0 ,
+(4.25)
+where βǫ = log(1/ǫ) and µ0 =
+�
+Ωǫ η dR dZ.
+We now consider the (formal) limit of the functional Eǫ[η] as ǫ → 0, assuming that η has
+zero mean to avoid the logarithmic divergence in the right-hand side of (4.25). In view of (4.19)
+and Lemma 4.4, we obtain the limiting functional
+E0[η] = 1
+2
+�
+R2 A(ρ) η(R, Z)2 dR dZ − Ekin
+0 [η] = 1
+2 ∥η∥2
+X0 − Ekin
+0 [η] ,
+(4.26)
+which is studied in detail in our previous work [31]. In particular, we have the following property :
+Proposition 4.5. There exists constants C4 > 2 and C5 > 0 such that, for all η ∈ X0,
+∥η∥2
+X0 ≤ C4E0[η] + C5
+�
+µ2
+0 + µ2
+1 + µ2
+2
+�
+,
+(4.27)
+where µ0 =
+�
+R2 η dX, µ1 =
+�
+R2 Rη dX, µ2 =
+�
+R2 Zη dX.
+Proof. The results of [31, Section 2] show that (4.27) holds when µ0 = µ1 = µ2 = 0, and the
+general case is easily deduced by the following argument. Given η ∈ X0 we define
+ˆη = η − µ0η0 + µ1∂Rη0 + µ2∂Zη0 ,
+ˆφ = φ − µ0φ0 + µ1∂Rφ0 + µ2∂Zφ0 ,
+where φ = (2π)−1Lη and η0, φ0 are as in (3.31). By construction the integral and the first order
+moments of the new function ˆη ∈ X0 vanish, so that we can apply the results of [31] which give
+the bound ∥ˆη∥2
+X0 ≤ C4E0[ˆη]. On the other hand, expanding the quadratic expressions ∥ˆη∥2
+X0
+and E0[ˆη] and using H¨older’s inequality, it is straightforward to verify that
+∥ˆη∥2
+X0 ≥ 1
+2 ∥η∥2
+X0 − C
+�
+µ2
+0 + µ2
+1 + µ2
+2) ,
+E0[ˆη] ≤ E0[η] +
+1
+4C4
+∥η∥2
+X0 + C
+�
+µ2
+0 + µ2
+1 + µ2
+2) ,
+for some C > 0. If we combine these estimates, we arrive at the bound (4.27) with a deteriorated
+constant C4.
+Using Proposition 4.5, we now establish a similar coercivity property for the functional Eǫ
+when ǫ > 0 is small. The proof of the following proposition is again postponed to Section B.1.
+Proposition 4.6. If the weight Wǫ satisfies (4.18) and (4.19), there exist constants C6 > 0 and
+C7 > 0 such that, for all sufficiently small ǫ > 0 and all η ∈ Xǫ, we have the estimate
+∥η∥2
+Xǫ ≤ C6Eǫ[η] + C7
+�
+βǫµ2
+0 + µ2
+1 + µ2
+2
+�
+,
+(4.28)
+where βǫ = log(1/ǫ) and µ0 =
+�
+Ωǫ η dX, µ1 =
+�
+Ωǫ Rη dX, µ2 =
+�
+Ωǫ Zη dX.
+In what follows we use the bound (4.28) to estimate the vorticity perturbation ˜η introduced
+in (2.31). The corresponding moments µ0, µ1 are under control thanks to Lemma 4.1, and µ2 = 0
+according to (4.9). So it remains to bound the energy functional Eǫ[˜η], which is the purpose of
+the remaining sections.
+37
+
+4.4
+Time evolution of the energy
+Let ˜η be the solution of (4.2) with zero initial data.
+Assuming that δ > 0 and σ > 0 are
+sufficiently small, we consider for t ∈ (0, Tadvδ−σ) the energy function
+Eǫ(t) = 1
+2
+�
+Ωǫ
+Wǫ(R, Z) ˜η(R, Z, t)2 dX − 1
+2
+�
+Ωǫ
+˜φ(R, Z, t) ˜η(R, Z, t) dX ,
+(4.29)
+where ǫ =
+√
+νt/¯r(t) and Wǫ is the weight function defined by (4.17). The first term in the
+right-hand side of (4.29) is equal to 1
+2∥˜η∥2
+Xǫ, and the second one is the kinetic energy Ekin
+ǫ
+[˜η],
+which satisfies (4.23) and involves the stream function ˜φ = BSǫ[˜η] defined by the Biot-Savart
+formula (2.20). Differentiating (4.29) with respect to time and using in particular the relations
+(3.33), (4.2), and (4.23), we obtain by direct calculation
+t∂tEǫ =
+�
+Ωǫ
+�
+Wǫ˜η t∂t˜η + 1
+2t(∂tWǫ)˜η2�
+dX −
+�
+Ωǫ
+�
+˜φ t∂t˜η + t˙ǫ
+2
+R|∇˜φ|2
+(1 + ǫR)2
+�
+dX
+= I1 + I2 + I3 + I4 + I5 + I6 ,
+where the quantities I1, . . . , I6 gather the following terms.
+1. Local advection terms :
+I1 = −1
+δ
+�
+Ωǫ
+Wǫ˜η
+�
+φ∗ , ˜ζ
+�
+dX + ǫ¯r ˙¯z
+δΓ
+�
+Ωǫ
+Wǫ˜η ∂Z ˜η dX
+= −1
+δ
+�
+Ωǫ
+Wǫ˜η
+�
+φ∗ − ¯r ˙¯z
+2Γ (1 + ǫR)2 , ˜ζ
+�
+dX
+= − 1
+2δ
+�
+Ωǫ
+�
+Wǫ(1 + ǫR) , φ∗ − ¯r ˙¯z
+2Γ (1 + ǫR)2�
+˜ζ2 dX .
+(4.30)
+2. Nonlocal advection terms :
+I2 = 1
+δ
+�
+Ωǫ
+˜φ
+�
+φ∗ , ˜ζ
+�
+dX − ǫ¯r ˙¯z
+δΓ
+�
+Ωǫ
+˜φ ∂Z ˜η dX − 1
+δ
+�
+Ωǫ
+�
+Wǫ˜η − ˜φ
+��˜φ , ζ∗
+�
+dX
+= 1
+δ
+�
+Ωǫ
+˜φ
+�
+φ∗ − ¯r ˙¯z
+2Γ (1 + ǫR)2 , ˜ζ
+�
+dX − 1
+δ
+�
+Ωǫ
+Wǫ˜η
+�˜φ , ζ∗
+�
+dX
+= 1
+δ
+�
+Ωǫ
+�
+˜φ , φ∗ − ¯r ˙¯z
+2Γ (1 + ǫR)2�
+˜ζ dX − 1
+δ
+�
+Ωǫ
+Wǫ(1 + ǫR)
+�˜φ , ζ∗
+�˜ζ dX .
+(4.31)
+3. Nonlinear terms :
+I3 = −
+�
+Ωǫ
+�
+Wǫ˜η − ˜φ
+��˜φ , ˜ζ
+�
+dX = −
+�
+Ωǫ
+�
+Wǫ˜η , ˜φ
+�˜ζ dX .
+(4.32)
+4. Diffusive terms :
+I4 =
+�
+Ωǫ
+�
+Wǫ˜η − ˜φ
+��
+L˜η + ǫ∂R˜ζ
+�
+dX .
+Integrating by parts as indicated in Section B.2, we obtain the equivalent expression
+I4 = −
+�
+Ωǫ
+Wǫ|∇˜η|2 dX −
+�
+Ωǫ
+(∇Wǫ · ∇˜η)˜η dX −
+�
+Ωǫ
+Vǫ˜η2 dX
+− ǫ
+2
+�
+Ωǫ
+∂R
+�
+Wǫ(1 + ǫR)
+�˜ζ2 dX + ǫ
+4
+�
+Ωǫ
+R|∇˜φ|2
+(1 + ǫR)2 dX ,
+(4.33)
+38
+
+where
+Vǫ = 1
+4(R∂R + Z∂Z)Wǫ − 1
+2Wǫ − (1 + ǫR) .
+(4.34)
+5. Remainder term :
+I5 = 1
+δ
+�
+Ωǫ
+�
+Wǫ˜η − ˜φ
+�
+Rem(R, Z, t) dX .
+(4.35)
+6. Additional terms :
+I6 = 1
+2
+�
+Ωǫ
+t(∂tWǫ)˜η2 dX + ǫ¯r ˙¯r
+δΓ
+�
+Ωǫ
+�
+Wǫ˜η − ˜φ
+�
+∂R˜η dX
+− t˙ǫ
+2
+�
+Ωǫ
+R|∇˜φ|2
+(1 + ǫR)2 dX + ǫ¯r ˙˜z
+δ2Γ
+�
+Ωǫ
+�
+Wǫ˜η − ˜φ
+� �
+∂Zη∗ + δ∂Z ˜η
+�
+dX .
+(4.36)
+For the purposes of our analysis, it is useful to reorganize some terms appearing in the
+quantities I4 and I6. First, using (2.19) and integrating by parts, it is easy to verify that
+−
+�
+Ωǫ
+˜φ ∂R˜η dX =
+�
+Ωǫ
+˜η ∂R ˜φ dX = ǫ
+2
+�
+Ωǫ
+|∇˜φ|2
+(1 + ǫR)2 dX .
+(4.37)
+So, if we collect all terms involving |∇˜φ|2 in (4.33), (4.36), and (4.37), we obtain the quantity
+� ǫ
+4 − t˙ǫ
+2
+� �
+Ωǫ
+R|∇˜φ|2
+(1 + ǫR)2 dX + ǫ2¯r ˙¯r
+2δΓ
+�
+Ωǫ
+|∇˜φ|2
+(1 + ǫR)2 dX = t ˙¯r
+2¯r
+�
+Ωǫ
+|∇˜φ|2
+1 + ǫR dX ,
+where we used the expression (3.33) of t˙ǫ. Next, we prefer including the term involving t∂tWǫ in
+I4 rather than I6, because it will be combined with the diffusive terms in I4 to obtain negative
+quantities that will allow us to control the evolution of the energy. Summarizing, if we define
+ˆI4 = −
+�
+Ωǫ
+Wǫ|∇˜η|2 dX −
+�
+Ωǫ
+(∇Wǫ · ∇˜η)˜η dX −
+�
+Ωǫ
+Vǫ˜η2 dX
+− ǫ
+2
+�
+Ωǫ
+∂R
+�
+Wǫ(1 + ǫR)
+�˜ζ2 dX + 1
+2
+�
+Ωǫ
+t(∂tWǫ)˜η2 dX ,
+(4.38)
+and
+ˆI6 = ǫ¯r ˙¯r
+δΓ
+�
+Ωǫ
+Wǫ˜η∂R˜η dX + t ˙¯r
+¯r Ekin
+ǫ
+[˜η] + ǫ¯r ˙˜z
+δ2Γ
+�
+Ωǫ
+�
+Wǫ˜η − ˜φ
+� �
+∂Zη∗ + δ∂Z ˜η
+�
+dX ,
+(4.39)
+we obtain the identity t∂tEǫ = I1 + I2 + I3 + ˆI4 + I5 + ˆI6, which we exploit in Sections 4.6–4.9.
+4.5
+Bounds on the stream function
+In this section we collect a few estimates on the stream function φ = BSǫ[η], where BSǫ is the
+ǫ-dependent Biot-Savart operator (2.20). We are especially interested in bounds on the velocity
+field U = (UR, UZ) defined by (2.18).
+Lemma 4.7. There exists a constant C > 0 such that, for all ǫ ∈ (0, 1),
+��� ∂Rφ
+1 + ǫR
+��� +
+��� ∂Zφ
+1 + ǫR
+��� ≤
+�
+Ωǫ
+C
+�
+(R−R′)2 + (Z−Z′)2 |η(R′, Z′)| dX′ .
+(4.40)
+In particular, for any q > 2, we have ∥U∥Lq ≤ Cq∥η∥Xǫ where U is the velocity field (2.18).
+39
+
+Proof. Estimate (4.40) is established in [30, Lemma 4.1], which in turns relies on [29, Proposi-
+tion 2.3]. Using the Hardy-Littlewood-Sobolev inequality, we deduce from (4.40) that ∥U∥Lq ≤
+Cq∥η∥Lp if q > 2 and p ∈ (1, 2) satisfy the relation 1/p = 1/q + 1/2. Finally, the lower bound
+on Wǫ in (4.18) implies that ∥η∥Lp ≤ C∥η∥Xǫ for any p ∈ [1, 2].
+The particular case where η = η∗ is the approximate solution (2.28) plays an important role.
+Lemma 4.8. The following estimates hold for the stream function φ∗ = BSǫ[η∗] :
+��� ∂Rφ∗
+1 + ǫR
+��� +
+��� ∂Zφ∗
+1 + ǫR
+��� ≤
+C
+1 + ρ + ǫ2ρ3 ,
+���
+∂Zφ∗
+(1 + ǫR)2
+��� ≤
+C
+1 + ρ + ǫ3ρ4 ,
+(4.41)
+where ρ = (R2+Z2)1/2.
+Proof. In the region where ρ ≤ 1/(2ǫ), we can use estimate (4.40) with η = η∗. Since η∗ satisfies
+the Gaussian bound (3.74), we easily deduce that |U| ≤ C(1 + ρ)−1, which gives estimate (4.41)
+in that case.
+We now concentrate on the region ρ ≥ 1/(2ǫ), where a more careful analysis
+is needed. We start from the formulas (3.72) with η = η∗, and we first estimate the vertical
+derivative ∂Zφ∗. Since | ˜F(s)| ≤ Cs−3/2 for all s > 0, we see that
+���
+∂Zφ∗
+(1 + ǫR)2
+��� ≤ C
+ǫ3
+�
+Ωǫ
+(1+ǫR′)2 |η∗(R′, Z′)|
+�
+(R−R′)2 + (Z−Z′)2�2 dR′ dZ′ .
+(4.42)
+Note that the integral is, in fact, taken over the support of η∗, which is included in the ball
+ρ′ := (R′2+Z′2)1/2 ≤ 2ǫ−σ0 where σ0 < 1. In particular we can disregard the factor (1+ǫR′)2 in
+the numerator, and the denominator is always larger that ρ4/2 if ǫ is sufficiently small. So the
+right-hand side of (4.42) is bounded by Cǫ−3ρ−4 when ρ ≥ 1/(2ǫ), which concludes the proof of
+the second inequality in (4.41). Since 1 + ǫR ≤ 1 + ǫρ, the estimate on ∂Zφ∗/(1+ǫR) in (4.41)
+follows immediately.
+To conclude the proof of the first inequality in (4.41), we must estimate the quantity ∂Rφ∗
+which contains an additional term given by the last line in (3.72). In the region where ρ ≥ 1/(2ǫ),
+using the fact that |F(s)| + | ˜F(s)| ≤ Cs−3/2, we see that the contribution of that term to the
+vertical speed UZ = ∂Rφ∗/(1+ǫR) is bounded by
+C
+ǫ2
+�
+Ωǫ
+(1+ǫR′)2 |η∗(R′, Z′)|
+�
+(R−R′)2 + (Z−Z′)2�3/2 dR′ dZ′ ≤
+C
+ǫ2ρ3 .
+The proof of (4.41) is thus complete.
+4.6
+Control of the advection terms
+In what follows we always assume that δ > 0 is sufficiently small and that ǫ2 ≲ δ1−σ for some
+small σ > 0, see Remark 2.2. We first estimate the advection terms I1, I2 defined in (4.30),
+(4.31). These terms are potentially dangerous because they include a factor 1/δ which is very
+large in the vanishing viscosity limit, but the energy functional (2.33) was designed precisely so
+that these contributions can be controlled.
+Lemma 4.9. There exist γ1 > 0 and C > 0 such that
+|I1| ≤ Cǫγ1 ∥˜η∥2
+Xǫ + Cǫ2
+δ
+�
+Ω′′ǫ
+Wǫ|˜η|2 dX .
+(4.43)
+40
+
+Proof. To exploit the properties of the weight Wǫ, we decompose the integral (4.30) defining
+I1 in three pieces, which correspond to the subdomains (4.13). If (R, Z) ∈ Ω′
+ǫ, we know from
+(4.17), (3.86) that
+Wǫ = Φ′
+ǫ(ζ∗)
+1 + ǫR ,
+φ∗ − ¯r ˙¯z
+2Γ (1 + ǫR)2 = Φǫ(ζ∗) + Θ ,
+(4.44)
+where Θ is a remainder term that is studied in Proposition 3.15. It follows that
+�
+Wǫ(1 + ǫR) , φ∗ − ¯r ˙¯z
+2Γ (1 + ǫR)2�
+=
+�
+Φ′
+ǫ(ζ∗) , Φǫ(ζ∗) + Θ
+�
+=
+�
+Φ′
+ǫ(ζ∗) , Θ
+�
+,
+where the right-hand side can be controlled using the bounds (3.87) on Θ and the estimates
+(3.93), (3.95) on Φǫ. This gives, for some N ∈ N and any γ3 ∈ (2, 3),
+���
+Φ′
+ǫ(ζ∗) , Θ
+��� ≤ C
+�
+ǫδ + ǫγ3�
+(1 + ρ)N Wǫ ≤ C
+�
+ǫδ + ǫγ3�
+ǫ−Nσ1 Wǫ ,
+(4.45)
+where we used the fact that 1 + ρ ≤ 2ǫ−σ1 when (R, Z) ∈ Ω′
+ǫ. Since ˜ζ ≈ ˜η in Ω′
+ǫ and since
+δ−1 ≲ ǫ−2/(1−σ) in the regime we consider, it follows from (4.45) that
+1
+δ
+�
+Ω′ǫ
+���
+Φ′
+ǫ(ζ∗) , Θ
+��� ˜ζ2 dX ≤ C
+�
+ǫ + ǫγ3
+δ
+�
+ǫ−Nσ1
+�
+Ω′ǫ
+Wǫ˜η2 dX ≤ Cǫγ1 ∥˜η∥2
+Xǫ ,
+(4.46)
+where γ1 is taken so that 0 < γ1 < γ3 − 2/(1−σ) − Nσ1. As γ3 < 3 is arbitrary, such a choice is
+always possible if we assume that σ > 0 and σ1 > 0 are small enough.
+We next consider the intermediate region Ω′′
+ǫ in which Wǫ(1 + ǫR) = χ2(ǫR) exp
+�
+ǫ−2σ1/4
+�
+,
+where χ2(x) = (1 + x)χ1(x). In that region, we thus have
+Jǫ :=
+�
+Wǫ(1 + ǫR) , φ∗ − ¯r ˙¯z
+2Γ (1 + ǫR)2�
+= ǫχ′
+2(ǫR) exp
+�
+ǫ−2σ1/4
+�
+∂Zφ∗ .
+Since χ2(x) = 1 when |x| ≤ 1
+2, the quantity Jǫ vanishes when ρ := (R2+Z2)1/2 ≤ 1/(2ǫ). In the
+region where 1/(2ǫ) ≤ ρ ≤ ǫ−σ2, we know from (4.41) that |∂Zφ∗/(1+ǫR)2| ≤ Cǫ−3ρ−4 ≤ Cǫ,
+and that Wǫ ≈ exp
+�
+ǫ−2σ1/4
+�
+. Since χ′
+2 is a bounded function, we deduce
+1
+δ
+�
+Ω′′ǫ
+|Jǫ| ˜ζ2 dX = 1
+δ
+�
+Ω′′ǫ
+|Jǫ| ˜η2
+(1+ǫR)2 dX ≤ Cǫ2
+δ
+�
+Ω′′ǫ
+Wǫ|˜η|2 dX .
+(4.47)
+Finally, in Ω′′′
+ǫ we have Wǫ(1 + ǫR) = χ2(ǫR) ˆWǫ where ˆWǫ = exp(ρ2γ/4), so that
+Jǫ = ǫχ′
+2(ǫR) ˆWǫ ∂Zφ∗ + ǫ¯r ˙¯z
+Γ χ1(ǫR)(1 + ǫR)2∂Z ˆWǫ + χ2(ǫR)
+� ˆWǫ , φ∗
+�
+.
+The first term in the right-hand side is estimated as above, with the difference that we now have
+the improved bound |∂Zφ∗/(1+ǫR)2| ≤ Cǫ−3ρ−4 ≤ Cǫ4σ2−3. For the second one we observe that
+��∂R ˆWǫ
+�� +
+��∂Z ˆWǫ
+�� ≤ γ ρ2γ−1 ˆWǫ ≤ γ ǫσ2−2σ1 ˆWǫ ,
+since ρ ≥ ǫ−σ2 ,
+(4.48)
+and the last term is estimated using (4.48) and the first bound in (4.41). Altogether we find
+1
+δ
+�
+Ω′′′
+ǫ
+|Jǫ| ˜ζ2 dX ≤ C
+δ
+�
+Ω′′′
+ǫ
+� 1
+ǫ2ρ4 + ¯r| ˙¯z|
+Γ
+ǫ
+ρ1−2γ +
+1
+ǫ2ρ4−2γ
+�
+Wǫ|˜η|2 dX ≤ Cǫγ1 ∥˜η∥2
+Xǫ ,
+(4.49)
+provided 0 < γ1 < σ2 + 1 − 2σ1 − 2/(1−σ). Since σ2 > 1, such a choice is again possible if σ > 0
+and σ1 > 0 are small enough. Combining (4.46), (4.47), (4.49), we arrive at (4.43).
+41
+
+Lemma 4.10. There exist γ1 > 0 and C > 0 such that
+|I2| ≤ Cǫγ1 ∥˜η∥2
+Xǫ .
+(4.50)
+Proof. In Ω′
+ǫ we have Wǫ(1 + ǫR) = Φ′
+ǫ(ζ∗) by (4.44), hence Wǫ(1 + ǫR)
+�˜φ , ζ∗
+�
+=
+�˜φ , Φǫ(ζ∗)
+�
+.
+Using the second relation in (4.44), we deduce that
+�
+˜φ , φ∗ − ¯r ˙¯z
+2Γ (1 + ǫR)2�
+− Wǫ(1 + ǫR)
+�˜φ , ζ∗
+�
+=
+�˜φ , Θ
+�
+.
+(4.51)
+The first-order derivatives of Θ are estimated in Proposition 3.15. Proceeding as in the previous
+lemma, we thus obtain
+1
+δ
+�
+Ω′ǫ
+���˜φ , Θ
+��� |˜ζ| dX ≤ C
+�
+ǫ + ǫγ3
+δ
+�
+ǫ−Nσ1
+�
+Ω′ǫ
+|∇˜φ|
+1+ǫR |˜η| dX ≤ Cǫγ1 ∥˜η∥2
+Xǫ ,
+(4.52)
+where 0 < γ1 < γ3 − 2/(1−σ) − Nσ1. In the last inequality, we used estimate (4.40) for ∇˜φ and
+we applied H¨older’s inequality with exponents 3 and 3/2 (for instance).
+In Dǫ := Ωǫ \ Ω′
+ǫ, we consider both terms in the left-hand side of (4.51) separately. The
+contribution of the first one to I2 is estimated by
+1
+δ
+�
+Dǫ
+|∇˜φ| |∇φ∗|
+1 + ǫR
+| ˜η| dX + ǫ¯r| ˙¯z|
+δΓ
+�
+Dǫ
+|∂Z ˜φ| |˜η| dX = O
+�
+ǫ∞∥˜η∥2
+Xǫ
+�
+,
+(4.53)
+because |∇φ∗| ≤ C by (4.41), ∥∇˜φ/(1+ǫR)∥L3 ≤ C∥˜η∥Xǫ by Lemma 4.7, and
+∥˜η∥L3/2(Dǫ) ≤
+��
+Dǫ
+Wǫ|˜η|2 dX
+�1/2��
+Dǫ
+W −3
+ǫ
+dX
+�1/6
+= O
+�
+ǫ∞∥˜η∥Xǫ
+�
+.
+The second term in the left-hand side of (4.51) is nonzero only if ρ ≤ 2ǫ−σ0, in view of (2.28),
+and we know from (3.74) and (4.18) that Wǫ|∇ζ∗| is a bounded function in this region. The
+contribution of that term to I2 can therefore be estimated in the same way as above:
+1
+δ
+�
+Dǫ
+Wǫ |
+�˜φ, ζ∗
+�
+| |˜ζ| dX ≤ C
+δ
+�
+Dǫ
+|∇˜φ| |˜η|
+1 + ǫR dX = O
+�
+ǫ∞∥˜η∥2
+Xǫ
+�
+.
+(4.54)
+Combining (4.52), (4.53), (4.54), we obtain (4.50).
+4.7
+Control of the diffusive terms
+Our next task is to estimate the diffusive terms collected in (4.38). To formulate the result, we
+introduce the continuous function ργ : R2 × R+ → R+ defined by
+ργ(R, Z, ǫ) =
+
+
+
+
+
+ρ
+if
+ρ ≤ ǫ−σ1 ,
+ǫ−σ1
+if
+ǫ−σ1 < ρ < ǫ−σ2 ,
+ργ
+if
+ρ ≥ ǫ−σ2 ,
+(4.55)
+where as usual ρ = (R2 + Z2)1/2. The goal of this section is to obtain the following result.
+Proposition 4.11. There exist κ > 0 and C > 0 such that
+ˆI4 ≤ −κ
+�
+Ωǫ
+Wǫ
+�
+|∇˜η|2 + ρ2
+γ ˜η2 + ˜η2�
+dX + C
+�
+µ2
+0 + µ2
+1 + µ2
+2
+�
+,
+(4.56)
+where µ0, µ1, µ2 are defined in (4.4), (4.9).
+42
+
+The proof of Proposition 4.11 requires several steps. We first control the term in ˆI4 that
+involves the time derivative of the weight function Wǫ.
+Lemma 4.12. There exist C > 0 and γ1 > 0 such that
+�
+Ωǫ
+t(∂tWǫ)˜η2 dX ≤ −σ1
+5
+�
+Ω′′ǫ
+Wǫρ2
+γ ˜η2 dX + C
+�
+Ω′′′
+ǫ
+Wǫ˜η2 dX + Cǫγ1∥˜η∥2
+Xǫ .
+(4.57)
+Proof. Following (4.17) we decompose Wǫ(R, Z) = χ1(ǫR) ˆWǫ(R, Z), so that
+t∂tWǫ = χ1(ǫR) t∂t ˆWǫ(R, Z) + t˙ǫRχ′
+1(ǫR) ˆWǫ(R, Z) .
+(4.58)
+We first estimate the right-hand side in the region Ω′
+ǫ defined by (4.13), where ˆWǫ = Φ′
+ǫ(ζ∗). As
+Φǫ = Φ0 + ǫ2Φ2 according to (3.81), we have t∂t ˆWǫ = Φ′′
+ǫ(ζ∗) t∂tζ∗ + 2tǫ˙ǫ Φ′
+2(ζ∗) in that region.
+We recall that 2t˙ǫ = ǫ(1 + O(ǫ2)) by (3.33), and that the functions Φ0, Φ2 satisfy the estimates
+(3.93), (3.95). It follows immediately that |tǫ˙ǫ Φ′
+2(ζ∗)| ≤ Cǫ2−Nσ1 ˆWǫ ≤ CǫWǫ. Moreover, since
+ζ∗ = η∗/(1+ǫR) with η∗ given by (2.28), we also have |Φ′′
+ǫ(ζ∗)t∂tζ∗| ≤ Cǫ(1 + ρ)N ˆWǫ ≤ Cǫγ1Wǫ,
+for some γ1 > 0. Finally, the last term in (4.58) is bounded by CǫρWǫ ≤ Cǫ1−σ1Wǫ. Altogether
+we have shown that |t∂tWǫ| ≤ Cǫγ1Wǫ in Ω′
+ǫ.
+In the intermediate region Ω′′
+ǫ we have ˆWǫ = exp
+�
+ǫ−2σ1/4
+�
+and ργ = ǫ−σ1, so that
+t∂t ˆWǫ = −σ1
+2 exp
+�
+ǫ−2σ1/4
+�
+t˙ǫ
+ǫ2σ1+1 = −σ1
+2
+ˆWǫ ρ2
+γ
+t˙ǫ
+ǫ ≈ −σ1
+4
+ˆWǫ ρ2
+γ .
+Since |t˙ǫRχ′
+1(ǫR)| ≤ |ǫRχ′
+1(ǫR)| ≤ C, it follows that t∂tWǫ ≤ −(σ1/5)Wǫρ2
+γ in Ω′′
+ǫ. Finally, in
+the exterior region Ω′′′
+ǫ , the weight function ˆWǫ = exp(ρ2γ/4) does not depend on time, and we
+deduce from (4.58) that |t∂tWǫ| ≤ CWǫ. Collecting all these estimates, we arrive at (4.57).
+We next consider the term involving ˜ζ in (4.38).
+Lemma 4.13. There exist C > 0 and γ1 > 0 such that
+− ǫ
+2
+�
+Ωǫ
+∂R
+�
+Wǫ(1 + ǫR)
+�˜ζ2 dX ≤ −ǫ2
+4
+�
+Ωǫ
+Wǫ˜ζ2 dX + Cǫγ1∥˜η∥2
+Xǫ .
+(4.59)
+Proof. If Dǫ denotes one of the three regions defined in (4.13), we have
+− ǫ
+2
+�
+Dǫ
+∂R
+�
+Wǫ(1 + ǫR)
+�˜ζ2 dX = −ǫ2
+2
+�
+Dǫ
+Wǫ˜ζ2 dX − ǫ
+2
+�
+Dǫ
+�
+∂RWǫ
+�˜ζ˜η dX
+(4.60)
+≤ −ǫ2
+4
+�
+Dǫ
+Wǫ˜ζ2 dX + 1
+4
+�
+Dǫ
+(∂RWǫ)2
+Wǫ
+˜η2 dX ,
+(4.61)
+where in the second line we used Young’s inequality. In the interior region Ω′
+ǫ we observe that
+˜ζ ≈ ˜η, because |ǫR| ≤ 2ǫ1−σ1 ≪ 1. Moreover we have ǫ|∂RWǫ| ≤ Cǫγ1Wǫ for some γ1 > 0, so
+taking Dǫ = Ω′
+ǫ and using (4.60) we obtain the analogue of (4.59) in that region. Outside Ω′
+ǫ,
+we cannot directly compare ˜ζ and ˜η, so we prefer using inequality (4.61). In the intermediate
+region Ω′′
+ǫ, we have |∂RWǫ| ≤ CǫWǫ by (4.17), and (4.59) easily follows. Finally, in the exterior
+region Ω′′′
+ǫ , we observe that
+∂RWǫ =
+�ǫχ′
+1(ǫR)
+χ1(ǫR) + γR
+2 ρ2γ−2
+�
+Wǫ .
+As γ < 1/2 and ρ ≥ ǫ−σ2 in that region, we deduce that |∂RWǫ| ≤ Cǫγ1Wǫ for some γ1 > 0, and
+this leads to (4.59). The proof is thus complete.
+43
+
+To conclude the proof of Proposition 4.11, we consider the quadratic form given by the first
+line of (4.38), namely
+Qǫ[η] =
+�
+Ωǫ
+Wǫ|∇η|2 dX +
+�
+Ωǫ
+(∇Wǫ · ∇η)η dX +
+�
+Ωǫ
+Vǫη2 dX ,
+(4.62)
+where Vǫ is defined in (4.34). Taking formally the limit ǫ → 0 in (4.62), we obtain
+Q0[η] =
+�
+R2 A|∇η|2 dX +
+�
+R2(∇A · ∇η)η dX +
+�
+R2 V η2 dX ,
+(4.63)
+where A is defined by (3.83) and V = 1
+4(R∂R + Z∂Z)A − 1
+2A − 1. The limiting quadratic form
+(4.63) is carefully studied in our previous work [31], and we have the following result :
+Proposition 4.14. There exists constants C8 > 2 and C9 > 0 such that, for all η ∈ X0 with
+ρη ∈ X0 and ∇η ∈ X 2
+0 , we have
+∥∇η∥2
+X0 + ∥ρη∥2
+X0 + ∥η∥2
+X0 ≤ C8Q0[η] + C9
+�
+µ2
+0 + µ2
+1 + µ2
+2
+�
+,
+(4.64)
+where µ0 =
+�
+R2 η dX, µ1 =
+�
+R2 Rη dX, µ2 =
+�
+R2 Zη dX.
+Proof. In [31, Theorem 4.2] we prove that there exists δ0 > 0 such that Q0[η] ≥ δ0∥η∥2
+X0 for
+any η ∈ X0 such that µ0 = µ1 = µ2 = 0. On the other hand, if we apply Young’s inequality to
+the middle term in the right-hand side of (4.63), we obtain the lower bound
+Q0[η] ≥ 1
+4
+�
+R2 A|∇η|2 dX +
+�
+R2
+�
+V − |∇A|2
+3A
+�
+η2 dX ≥ 1
+4∥∇η∥2
+X0 + 1
+24∥ρη∥2
+X0 − C∥η∥2
+X0 ,
+because a direct calculation reveals that V/A − |∇A|2/(3A2) ≥ ρ2/(24) − C for some constant
+C > 0. Taking a convex combination of both estimates, we see that there exists C8 > 0 such
+that
+∥∇η∥2
+X0 + ∥ρη∥2
+X0 + ∥η∥2
+X0 ≤ C8Q0[η] ,
+(4.65)
+whenever η ∈ X0 satisfies µ0 = µ1 = µ2 = 0. It remains to deduce (4.64) from (4.65), which is
+easily done using exactly the same arguments as in the proof of Proposition 4.5.
+The analogue of Proposition 4.14 for the full quadratic form (4.62) is the following statement,
+whose proof is postponed to Section B.3.
+Proposition 4.15. There exists constants C10 > 2 and C11 > 0 such that, for all sufficiently
+small ǫ > 0 and all η ∈ Xǫ with ργη ∈ Xǫ and ∇η ∈ X 2
+ǫ , we have
+∥∇η∥2
+Xǫ + ∥η∥2
+Xǫ +
+�
+Ω′ǫ∪Ω′′′
+ǫ
+Wǫρ2
+γη2 dX ≤ C10Qǫ[η] + C11
+�
+µ2 +
+�
+Ω′′ǫ
+Wǫη2�
+,
+(4.66)
+where µ2 = µ2
+0 + µ2
+1 + µ2
+2 and µ0 =
+�
+Ωǫ η dX, µ1 =
+�
+Ωǫ Rη dX, µ2 =
+�
+Ωǫ Zη dX.
+End of the proof of Proposition 4.11. In view of (4.38) and (4.62) we have
+ˆI4 = −Qǫ[˜η] − ǫ
+2
+�
+Ωǫ
+∂R
+�
+Wǫ(1 + ǫR)
+�˜ζ2 dX + 1
+2
+�
+Ωǫ
+t(∂tWǫ)˜η2 dX .
+The three terms in the right-hand side are estimated using (4.66), (4.59), and (4.57), respectively.
+Taking ǫ > 0 sufficiently small and recalling that ργ ≥ ǫ−σ1 ≫ 1 outside the interior region
+Ω′
+ǫ, we arrive at (4.56). The slight discrepancy between the definitions of µ1 in (4.4) and in
+Proposition 4.15 is completely harmless.
+44
+
+4.8
+Control of the remaining terms
+In this section, we estimate the remaining terms I3, I5, and ˆI6 defined in (4.32), (4.35), and
+(4.39), respectively.
+Control of I3. We deduce from (4.32) that
+|I3| ≤
+�
+Ωǫ
+|∇˜φ|
+1+ǫR |˜η|
+��∇(Wǫ˜η)
+�� dX ≤
+�
+Ωǫ
+|∇˜φ|
+1+ǫR |˜η|
+�
+|˜η||∇Wǫ| + Wǫ|∇˜η|
+�
+dX .
+(4.67)
+To estimate the right-hand side, we use (4.40) to obtain the uniform bound
+��� |∇˜φ|
+1+ǫR
+���
+L∞ ≤ C∥˜η∥1/2
+L4/3∥˜η∥1/2
+L4 ≤ C∥˜η∥1/2
+Xǫ
+�
+∥˜η∥1/2
+Xǫ + ∥∇˜η∥1/2
+Xǫ
+�
+.
+On the other hand it is easy to verify that |∇Wǫ| ≤ C(1 + ργ)Wǫ where ργ is defined in (4.55).
+It follows that
+|I3| ≤ C∥˜η∥3/2
+Xǫ
+�
+∥˜η∥1/2
+Xǫ + ∥∇˜η∥1/2
+Xǫ
+��
+∥˜η∥Xǫ + ∥ργ ˜η∥Xǫ + ∥∇˜η∥Xǫ
+�
+≤ C∥˜η∥XǫDǫ[˜η] ,
+(4.68)
+where for convenience we denote
+Dǫ[˜η] = ∥∇˜η∥2
+Xǫ + ∥ργ ˜η∥2
+Xǫ + ∥˜η∥2
+Xǫ .
+(4.69)
+Control of I5. Proposition 2.5 asserts that the remainder Rem(R, Z, t) satisfies the pointwise
+estimate (2.30), which implies in particular that Rem ∈ Xǫ. In view of (4.35), we thus find
+|I5| ≤ 1
+δ ∥Rem∥Xǫ
+�
+∥˜η∥Xǫ + ∥W −1
+ǫ
+˜φ∥Xǫ
+�
+≤ C
+�
+ǫ + ǫγ5
+δ2
+��
+∥˜η∥Xǫ + ∥W −1
+ǫ
+˜φ∥Xǫ
+�
+.
+It remains to estimate the norm of W −1
+ǫ
+˜φ in the space Xǫ. This can be done by decomposing the
+Biot-Savart kernel as in the proof of Lemma 4.4, see in particular Eq. (B.3) below. Neglecting
+contributions of order O(ǫ∞), we can restrict the integrals to the region where R2 + Z2 ≤ ǫ−2σ1
+and R′2 + Z′2 ≤ ǫ−2σ1. Invoking (B.4) and recalling that µ0(t) = O(ǫ∞) by Lemma 4.1, we find
+that ∥W −1
+ǫ
+˜φ∥Xǫ = ∥W −1/2
+ǫ
+˜φ∥L2(Ωǫ) ≤ C∥˜η∥Xǫ. We conclude that
+|I5| ≤ C
+�
+ǫ + ǫγ5
+δ2
+�
+∥˜η∥Xǫ ,
+(4.70)
+Control of ˆI6. The first two terms in (4.39) are easily estimated, because ˙¯r = O(δ) by (3.69).
+Proceeding as in Lemma 4.4 to control the kinetic energy, and recalling that µ0(t) = O(ǫ∞), we
+find
+I0 :=
+����
+ǫ¯r ˙¯r
+δΓ
+�
+Ωǫ
+Wǫ˜η∂R˜η dX + t ˙¯r
+¯r Ekin
+ǫ
+[˜η]
+���� ≤ Cǫ∥˜η∥Xǫ∥∇˜η∥Xǫ + Cǫ2∥˜η∥2
+Xǫ .
+So it remains to estimate the last term in (4.39), which involves the correction ˙˜z(t) to the vertical
+speed introduced in (4.10). Using (2.19) and integrating by parts we first observe that
+I1 : =
+�
+Ωǫ
+�
+Wǫ˜η − ˜φ
+�
+∂Zη∗ dX =
+�
+Ωǫ
+�
+Wǫ∂Zη∗ − ∂Zφ∗
+�
+˜η dX
+= −
+�
+Ω′ǫ
+(∂ZΘ)˜η dX +
+�
+Ω′′ǫ ∪Ω′′′
+ǫ
+�
+Wǫ∂Zη∗ − ∂Zφ∗
+�
+˜η dX ,
+(4.71)
+45
+
+where Θ is defined in (3.86). In the second line, we used the expression (4.17) of Wǫ in the
+inner region Ω′
+ǫ to obtain the identity Wǫ∂Zη∗ − ∂Zφ∗ = Φ′
+ǫ(ζ∗)∂Zζ∗ − ∂Zφ∗ = −∂ZΘ. The
+last integral in (4.71) is of order O(ǫ∞∥˜η∥Xǫ), and the integral over Ω′
+ǫ can be controlled using
+Proposition 3.15. We thus obtain |I1| ≤ C(ǫδ + ǫγ3)∥˜η∥Xǫ. Moreover, we obviously have
+I2 :=
+����
+�
+Ωǫ
+�
+Wǫ˜η − ˜φ
+�
+∂Z ˜η dX
+���� ≤ C∥˜η∥Xǫ∥∇˜η∥Xǫ + C∥˜η∥2
+Xǫ .
+Finally, to control the velocity ˙˜z(t), we need the following lemma:
+Lemma 4.16. Let J(t) =
+�
+Ωǫ ZR(R, Z, t) dX where R is defined in (4.11). Then there exists a
+constant C > 0 such that
+|J| ≤ Cǫβǫ
+δ
+�
+∥˜η∥Xǫ + δ∥˜η∥2
+Xǫ
+�
++ C
+�
+ǫ + ǫγ5
+δ2
+�
+.
+(4.72)
+Proof. We consider separately the various terms in the right-hand side of (4.11). Integrating
+by parts, we find
+J1 := 1
+δ
+�
+Ωǫ
+Z
+��
+φ∗ , ˜ζ
+�
++
+�˜φ , ζ∗
+��
+dX = −1
+δ
+�
+Ωǫ
+� ˜η∂Rφ∗
+1+ǫR + η∗∂R ˜φ
+1+ǫR
+�
+dX .
+In the right-hand side, we can restrict the integration to the region where ρ ≤ ǫ−σ0, because the
+integral on the complement is of order O
+�
+ǫ∞∥˜η∥Xǫ
+�
+. Thus, expanding the Biot-Savart formula
+as in Section 3.1, we obtain
+− δJ1 =
+1
+2π
+�
+R2
+�
+˜η∂R(Lη∗) + η∗∂R(L˜η)
+�
+dX + O
+�
+ǫβǫ∥˜η∥Xǫ
+�
+,
+(4.73)
+where L is the convolution operator (3.14). Since L is symmetric in L2(R2) and commutes with
+∂R, the integral in (4.73) vanishes and we conclude that |J1| ≤ δ−1ǫβǫ∥˜η∥Xǫ.
+Similarly, we have
+J2 :=
+�
+Ωǫ
+Z
+�˜φ , ˜ζ
+�
+dX =
+�
+Ωǫ
+�
+Z , ˜φ
+�˜ζ dX = −
+�
+Ωǫ
+˜η∂R ˜φ
+1+ǫR dX .
+Here again, up to a negligible error, we can assume that ˜η is supported in the ball ρ ≤ ǫ−σ0.
+Proceeding as before, we thus find
+J2 = − 1
+2π
+�
+R2 ˜η∂R(L˜η) dX + O
+�
+ǫβǫ∥˜η∥2
+Xǫ
+�
+= O
+�
+ǫβǫ∥˜η∥2
+Xǫ
+�
+.
+(4.74)
+The remaining terms in (4.11) are easier to treat. In view of (4.9) we have
+�
+Ωǫ
+Z
+�
+L˜η + ǫ∂R˜ζ
+�
+dx = 0 ,
+and
+�
+Ωǫ
+Z
+�
+˙¯r ∂R˜η + ˙¯z ∂Z ˜η
+�
+dX = − ˙¯zµ0 ,
+where µ0(t) = O(ǫ∞) by Lemma 4.1. Finally, using estimate (2.30), we obtain
+1
+δ
+�
+Ωǫ
+|Z| |Rem(R, Z, t)| dX ≤ C
+�
+ǫ + ǫγ5
+δ2
+�
+.
+(4.75)
+Combining (4.73), (4.74), and (4.75), we arrive at (4.72).
+46
+
+Corollary 4.17. There exists a constant C > 0 such that the velocity ˙˜z defined by (4.10) satisfies
+¯r| ˙˜z|
+Γ
+≤ Cδβǫ
+�
+∥˜η∥Xǫ + δ∥˜η∥2
+Xǫ
+�
++ C
+�
+δ2 + ǫγ5−1�
+.
+(4.76)
+We now conclude the estimate of the term ˆI6.
+To simplify the writing, we assume that
+∥˜η∥Xǫ ≤ 1 and we use the shorthand notation (4.69). Also, since ǫ2 ≲ δ1−σ we observe that
+ǫ + ǫγ5
+δ2 ≲ Rǫ(t) ,
+where
+Rǫ(t) := ǫ + ǫγ3
+δ .
+(4.77)
+Here γ3 = γ5 − 2/(1−σ) < 3, so that γ3 can be chosen arbitrary close to γ5 − 2 if σ > 0 is small
+enough. In view of (4.10) and (4.39) we have |ˆI6| ≤ I0 + |J|
+�
+|I1| + δI2
+�
+, so that
+|ˆI6| ≤ Cǫ∥˜η∥XǫD1/2
+ǫ
++ C
+�ǫβǫ
+δ ∥˜η∥Xǫ + Rǫ
+��
+δRǫ∥˜η∥Xǫ + δ∥˜η∥XǫD1/2
+ǫ
+�
+≤ C∥˜η∥Xǫ
+�
+D1/2
+ǫ
++ Rǫ
+��
+ǫβǫ + δRǫ
+�
+≤ Cǫβǫ∥˜η∥Xǫ
+�
+D1/2
+ǫ
++ Rǫ
+�
+.
+(4.78)
+4.9
+Conclusion of the proof
+We are now in position to conclude the proof of Theorem 2.6, hence also of Theorem 1.1. Let
+˜η be the unique solution of (2.32) with zero initial data. The associated energy (2.33) satisfies
+the evolution equation
+t∂tEǫ(t) = I1 + I2 + I3 + ˆI4 + I5 + ˆI6 ,
+(4.79)
+where the various terms in the right-hand side are defined in Section 4.4 and estimated in
+Sections 4.6–4.8. Using (4.43), (4.50), (4.68), (4.56), (4.70), and (4.78), we find that, as long as
+t ≤ Tadvδ−σ and ∥˜η∥Xǫ ≤ 1, there exist positive constants C, C∗ such that
+t∂tEǫ(t) ≤ −κDǫ + C∗∥˜η∥XǫDǫ + C∥˜η∥Xǫ
+�
+Rǫ + ǫβǫD1/2
+ǫ
+�
++ Cǫ2
+δ
+�
+Ω′′ǫ
+Wǫ|˜η|2 dX + Cµ2 ,
+where Dǫ is defined in (4.69), Rǫ in (4.77), and µ2 := µ2
+0 + µ2
+1 + µ2
+2 ≤ C R2
+ǫ by Lemma 4.1. Since
+ργ ≥ ǫ−σ1 in the region Ω′′
+ǫ, the integral term can be estimated as follows
+ǫ2
+δ
+�
+Ω′′ǫ
+Wǫ|˜η|2 dX ≤ ǫ2+2σ1
+δ
+�
+Ω′′ǫ
+Wǫρ2
+γ|˜η|2 dX ≲ ǫγ∗Dǫ ,
+where γ∗ = 2+2σ1−2/(1−σ) > 0 if σ > 0 is small enough. So, if we assume that C∗∥˜η∥Xǫ ≤ κ/4
+and that ǫ is sufficiently small, we obtain by Young’s inequality
+t∂tEǫ(t) ≤ −κ
+2 Dǫ + C Rǫ∥˜η∥Xǫ + Cµ2 ≤ −κ
+4 Dǫ + C R2
+ǫ .
+Integrating that differential inequality over the time interval (0, t) and recalling that Eǫ(0) = 0,
+we arrive at
+Eǫ(t) ≤ C
+� t
+0
+Rǫ(s)2
+s
+ds ≤ C Rǫ(t)2 .
+Finally we invoke (4.28) as well as Lemma 4.1 and (4.9) to conclude that
+∥˜η(t)∥2
+Xǫ ≤ C6Eǫ(t) + C7
+�
+βǫµ0(t)2 + µ1(t)2�
+≤ C Rǫ(t)2 .
+(4.80)
+Inequality (4.80) holds as long as ∥˜η(t)∥Xǫ ≤ 1 and t < Tadvδ−σ. But on that time interval we
+know that Rǫ ≲ ǫγ3−2/(1−σ) ≪ 1, so (4.80) is actually valid for all t ∈ (0, Tadvδ−σ). Returning
+to the the solution (2.15) with initial data (2.22), we obtain in view of (2.31), (4.80)
+∥η(t) − η∗(t)∥Xǫ = δ∥˜η(t)∥Xǫ ≤ Cδ Rǫ(t) = C
+�
+ǫδ + ǫγ3�
+,
+t ∈ (0, Tadvδ−σ) ,
+which gives (2.39). This concludes the proof of Theorem 2.6.
+□
+47
+
+Remark 4.18. Strictly speaking, Theorem 2.6 deals with the rescaled vorticity η defined by
+(2.13), (2.14), whereas the proof above uses the modified self-similar variables defined by (4.1).
+However, the shift in the vertical position induced by the nonlinear correction ˜z(t) is very small,
+and produces negligible corrections. Indeed, it follows from (4.76) and (4.80) that
+¯r| ˙˜z(t)|
+Γ
+≲
+�
+δβǫ Rǫ + δ2 + ǫγ5−1�
+,
+hence
+|˜z(t)| ≲ ǫ2¯r(t)
+�
+δ + βǫ Rǫ
+�
+.
+When rephrased in self-similar coordinates, Theorem 1.1 compares the solution η of (2.15)
+to the solution ˆη of the linear heat equation
+t∂tˆη −
+�
+t
+ν
+�
+˙¯r ∂Rˆη + ˙¯z ∂Z ˆη
+�
+= Lˆη + ∂R
+�
+ǫˆη
+1 + ǫR
+�
+,
+with the same initial data η0. It is straightforward to verify that ∥ˆη − η0∥Xǫ = O(ǫ) as ǫ → 0, so
+that ∥ˆη − η∥Xǫ = O(ǫ) too. The discrepancy between the velocities ˙¯r, ˙¯z constructed in Section 3
+and the approximate values corresponding to (1.8) is responsible for the deterioration of the
+error term in (1.7), which is of order ǫ + ǫβǫδ−σ ≲ ǫ1−3σ.
+A
+Appendix to Section 3
+A.1
+Inverting the operator Λ
+Following [28], we give here a short proof of Proposition 3.7. Assume that n ≥ 2 and f ∈ Yn∩Z,
+or that n = 1 and f ∈ Y′
+1 ∩ Z. In both cases, we have f ∈ Ker(Λ)⊥. We want to show that
+there exists a unique η ∈ Yn ∩ Z (or η ∈ Y′
+1 ∩ Z if n = 1) such that Λη = f.
+To make things concrete, we assume without loss of generality that f = a(ρ) sin(nϑ), for
+some function a : R+ → R. Our hypotheses imply that a is smooth, that a(ρ) = O(ρn) as ρ → 0,
+and that eρ2/4a(ρ) grows at most polynomially as ρ → ∞. We look for a solution of the form
+η = ω(ρ) cos(nϑ), where ω : R+ → R has to be determined. By (3.19), we have
+Λη =
+�
+φ0 , η
+�
++ {Ψ , η0
+�
+,
+where
+φ0 =
+1
+2π Lη0 ,
+Ψ =
+1
+2π Lη .
+(A.1)
+The function φ0 is radially symmetric and satisfies ∂ρφ0 = −ρϕ, see (3.24) and (A.12) below. It
+follows that
+�
+φ0 , η
+�
+= ∂ρφ0
+1
+ρ ∂ϑη = nϕ(ρ)ω(ρ) sin(nϑ) .
+(A.2)
+On the other hand, as −∆Ψ = η, we have Ψ = Ω(ρ) cos(nϑ), where Ω is the unique regular
+solution of the differential equation
+− Ω′′(ρ) − 1
+ρ Ω′(ρ) + n2
+ρ2 Ω(ρ) = ω(ρ) ,
+ρ > 0 .
+(A.3)
+Since η0 is radially symmetric and ∂ρη0 = −(ρ/2)η0 = −ρϕh, see (3.24), we deduce
+{Ψ , η0
+�
+= −∂ρη0
+1
+ρ ∂ϑΨ = −nϕ(ρ)h(ρ)Ω(ρ) sin(nϑ) .
+(A.4)
+In view of (A.1), (A.2), (A.4), the equation Λη = f is equivalent to the relation (3.25), and
+using in addition (A.3) we obtain the differential equation (3.26) for the unknown function Ω.
+48
+
+The main step in the proof is to show that (3.26) has a unique solution that is regular at
+the origin and decays to zero at infinity. Here we distinguish two cases according to the value
+of the angular Fourier mode n.
+1. If n ≥ 2, the homogeneous equation (3.26) with a ≡ 0 has two linearly independent solutions
+ψ+, ψ− which satisfy
+ψ−(ρ) ∼
+�
+ρn
+as ρ → 0 ,
+κρn
+as ρ → ∞ ,
+ψ+(ρ) ∼
+�
+κρ−n
+as ρ → 0 ,
+ρ−n
+as ρ → ∞ ,
+(A.5)
+for some κ > 0, see [28]. Here we use the crucial observation that (n2/ρ2)−h(ρ) > 0 when n ≥ 2,
+so that the differential operator in the left-hand side of (3.26) satisfies the Maximum Principle.
+We deduce the following representation formula for the solution of the inhomogeneous equation :
+Ω(ρ) = ψ+(ρ)
+� ρ
+0
+r
+w0
+ψ−(r) a(r)
+nϕ(r) dr + ψ−(ρ)
+� ∞
+ρ
+r
+w0
+ψ+(r) a(r)
+nϕ(r) dr ,
+(A.6)
+where w0 = 2nκ. It is then straightforward to verify that Ω(ρ) = O(ρn) as ρ → 0 and Ω(ρ) =
+O(ρ−n) as ρ → ∞. Moreover, if ω is defined by (3.25), the function η = ω(ρ) cos(nϑ) lies in
+Yn ∩ Z and satisfies Λη = f by construction. The details can be found in [28, Lemma 4].
+2. The situation is quite different when n = 1, because the lower order term 1/ρ2 −h(ρ) in (A.3)
+is no longer positive. In that case, it happens that the homogeneous equation (3.26) with a ≡ 0
+has a solution ψ(ρ) = ρϕ(ρ) which satisfies ψ(ρ) ∼ ρ/(8π) as ρ → 0 and ψ(ρ) ∼ 1/(2πρ) as
+ρ → ∞. In other words, the linear operator in the left-hand side of (3.26) has a one-dimensional
+kernel, and for that reason we have to impose the solvability condition
+f ∈ Y′
+1 ⊂ Ker(Λ)⊥ ,
+or equivalently
+� ∞
+0
+a(ρ)ρ2 dρ = 0 .
+(A.7)
+To solve (3.26) for n = 1, we look for a solution of the form Ω(ρ) = b(ρ)ψ(ρ), which leads to a
+first-order differential equation for b(ρ). In view of (A.7), we thus find
+b′(ρ) = −
+1
+ρψ(ρ)2
+� ρ
+0
+a(r)r2 dr =
+1
+ρψ(ρ)2
+� ∞
+ρ
+a(r)r2 dr .
+(A.8)
+Integrating (A.8) gives the representation formula
+b(ρ) = b0 −
+� ρ
+0
+a(r)r2�
+F(ρ) − F(r)
+�
+dr ,
+for some b0 ∈ R ,
+where
+F(ρ) = 8π2
+�
+log
+�
+eρ2/4 − 1
+�
+−
+1
+eρ2/4 − 1
+�
+,
+F′(ρ) =
+1
+ρψ(ρ)2 .
+We now substitute Ω(ρ) = b(ρ)ψ(ρ) into (3.25) with n = 1, and we choose the constant b0 so
+that
+� ∞
+0 ω(ρ)ρ2 dρ = 0. This is always possible in a unique way, since
+� ∞
+0
+ρ2h(ρ)ψ(ρ) dρ =
+� ∞
+0
+ρ3h(ρ)ϕ(ρ) dρ =
+1
+8π
+� ∞
+0
+ρ3 e−ρ2/4 dρ = 1
+π ̸= 0 .
+To conclude the proof, it remains to verify that the function η = ω(ρ) sin(ϑ) constructed above
+belongs to Y′
+1 ∩ Z and satisfies Λη = f. These are straightforward calculations, which can be
+omitted.
+□
+49
+
+A.2
+First order calculations
+We first establish the relations (3.36). As η0 ∈ Y0 has unit mass we find, using (3.11),
+�
+P1η0
+�
+(R, Z) =
+�
+R2
+R+R′
+2
+η0(R′, Z′) dR′ dZ′ = R
+2 ,
+(A.9)
+hence {P1η0 , η0} = 1
+2 ∂Zη0. On the other hand, since ∂Rη0 = −(R/2)η0 and L is a convolution
+operator, which therefore commutes with derivatives, we have
+�
+LP1η0
+�
+(R, Z) = R
+2 (Lη0)(R, Z) + L
+�R
+2 η0
+�
+(R, Z) = R
+2 (Lη0)(R, Z) − ∂R
+�
+Lη0
+�
+(R, Z) .
+Recalling that Lη0 = 2πφ0, and that {φ0, η0} = 0 because both φ0, η0 are radially symmetric,
+we thus obtain
+1
+2π
+�
+LP1η0 , η0
+�
+=
+�R
+2 φ0 − ∂Rφ0 , η0
+�
+= 1
+2 φ0 ∂Zη0 +
+�
+φ0 , ∂Rη0
+�
+= 1
+2 φ0 ∂Zη0 −
+�
+φ0 , R
+2 η0
+�
+= 1
+2 φ0 ∂Zη0 + 1
+2 (∂Zφ0)η0 ,
+which concludes the proof of (3.36).
+We next prove formula (3.38) for the vertical velocity. According to the definition of v in
+(3.38), the right-hand side of (3.37) belongs to Y′
+1 = Y ∩ Ker(Λ)⊥ if and only if
+�
+R2
+� v
+2π ∂Zη0 − 3
+2(∂Zφ0)η0 − 1
+2φ0∂Zη0
+�
+Z dR dZ = 0 .
+(A.10)
+Since ∂Zη0 = −(Z/2)η0 and
+�
+R2 Z2η0 dR dZ = 2, it is straightforward to verify that (A.10) is
+equivalent to
+v = π
+�
+R2 φ0η0
+�
+3 − Z2�
+dR dZ = π
+2
+�
+R2 φ0η0
+�
+6 − |X|2�
+dX ,
+(A.11)
+where X = (R, Z) and |X|2 = R2 + Z2.
+To evaluate the right-hand side of (A.11), we temporarily denote ψ0 = 2πφ0 = Lη0, namely
+ψ0(X) =
+1
+4π
+�
+R2 log
+�
+8
+|X − Y |
+�
+e−|Y |2/4 dY ,
+X ∈ R2 .
+This function satisfies −∆ψ0 = 2πη0 = 1
+2 e−|X|2/4, so that
+ψ0(X) = ψ0(0) −
+� |X|
+0
+1 − e−ρ2/4
+ρ
+dρ ≡ ˜ψ0(|X|) ,
+X ∈ R2 ,
+(A.12)
+where
+ψ0(0) = log(8) − 1
+4π
+�
+R2 log(|Y |) e−|Y |2/4 dY = 2 log(2) + γE
+2 .
+(A.13)
+Using (A.12), (A.13) and integrating by parts, we easily find
+�
+R2 ψ0η0 dX = 1
+2
+� ∞
+0
+˜ψ0(ρ)e−ρ2/4ρ dρ = ψ0(0) +
+� ∞
+0
+˜ψ′
+0(ρ)e−ρ2/4 dρ = 3
+2 log(2) + γE
+2 ,
+and similarly
+�
+R2 ψ0η0|X|2 dX = 4ψ0(0) +
+� ∞
+0
+˜ψ′
+0(ρ)e−ρ2/4(ρ2 + 4) dρ = 6 log(2) + 2γE − 1 .
+Returning to (A.11), we conclude that
+v = 1
+4
+�
+R2 ψ0η0
+�
+6 − |X|2�
+dX = 3
+4 log(2) + 1
+4γE + 1
+4 .
+(A.14)
+50
+
+A.3
+Second order calculations
+Our goal here is to prove Lemma 3.11. To establish (3.47), we consider separately the various
+terms in (3.46). As η1 ∈ Y1 has zero mean, we find as in (A.9) that P1η1 is a constant, which
+can be disregarded. Moreover LP1η1 = R
+2 Lη1 + L
+�R
+2 η1
+�
+, hence using the expression (3.40) of η1
+we find that
+LP1η1 = (R2 − Z2)χ1(ρ) + δRZχ2(ρ) + χ3(ρ) ,
+where χ1, χ2, . . . are functions of the radial variable ρ = (R2 + Z2)1/2. As η0 itself is radially
+symmetric, we deduce that
+�
+(βǫ − 1)P1η1 + LP1η1 , η0
+�
+= RZχ4(ρ) + δ(R2 − Z2)χ5(ρ) .
+(A.15)
+Next, using the expression (3.11) of P2, we see that
+�
+P2η0
+�
+(R, Z) =
+1
+16
+�
+R2
+�
+(R−R′)2 + 3(Z−Z′)2�
+η0(R′, Z′) dR′ dZ′ = R2
+16 + 3Z2
+16 + 1
+2 ,
+and a similar calculation gives Q2η0 = 3R2
+16 − Z2
+16 + 1
+4. Moreover,
+�
+LP2η0
+�
+(R, Z) = 1
+16
+�
+R2 log
+� 8
+D
+��
+2D2 + (Z−Z′)2 − (R−R′)2�
+η0(R′, Z′) dR′ dZ′ ,
+where D2 = (R−R′)2 + (Z−Z′)2. Using the fact that η0 given by (3.31) is radially symmetric,
+we easily obtain
+1
+2π
+�
+LP2η0
+�
+(R, Z) = χ6(ρ) + (R2 − Z2)χ7(ρ) .
+Altogether, we arrive at
+1
+2π
+�
+βǫP2η0 + LP2η0 + Q2η0 , η0
+�
+=
+βǫ
+16π RZη0 + RZχ8(ρ) .
+(A.16)
+The remaining terms in (3.46) are easier to treat. In view of (3.38), (3.40), (3.41), we have
+�
+φ1 , η1
+�
+− r0 ˙¯z0
+Γ
+∂Zη1 =
+�
+φ1 − βǫ − 1
+4π
+R , η1
+�
+− v
+2π ∂Zη1
+=
+�R
+2 φ0 − ∂Rφ0 + R φ10(ρ) + δZ φ11(ρ) , R η10(ρ) + δZ η11(ρ)
+�
+= RZ χ9(ρ) + δ
+�
+χ10(ρ) + (R2 − Z2)χ11(ρ)
+�
++ δ2RZ χ12(ρ) .
+(A.17)
+It is also easy to verify that the terms (∂Zφ1)η0 +(∂Zφ0)η1 −2R(∂Zφ0)η0 +δ∂R(Rη0) are exactly
+of the same form. Finally, using again (3.40), (3.41), we obtain
+R
+��
+φ1 , η0
+�
++
+�
+φ0 , η1
+��
+= R
+�βǫ − 1
+4π
+∂Zη0 + Zχ13(ρ) + δRχ14(ρ)
+�
+.
+(A.18)
+If we now combine (A.15), (A.16), (A.17), (A.18), we arrive at (3.47).
+□
+A.4
+Higher order order calculations
+The calculations carried out in Sections 3.5 and 3.6 do not require new ideas, but a more
+compact notation is often helpful. To prove Lemma 3.12 and similar statements, it is important
+to understand how the decomposition (3.21) of the function space Y behaves under the Poisson
+51
+
+bracket. If we use polar coordinates X = (ρ cos ϑ, ρ sin ϑ), we recall that Yn is the subspace of
+Y spanned by functions of the form a(ρ) cos(nϑ) and b(ρ) sin(nϑ). Since
+�
+f , g
+�
+= ∂Rf∂Zg − ∂Zf∂Rg = 1
+ρ
+�
+∂ρf∂ϑg − ∂ϑf∂ρg
+�
+,
+we easily obtain the following result :
+Lemma A.1. If a, b : R+ → R are smooth functions and n, m ∈ N, then
+�
+a(ρ) cos(nϑ) , b(ρ) cos(mϑ)
+�
+= c11(ρ) sin((n−m)ϑ) + c12(ρ) sin((n+m)ϑ) ,
+�
+a(ρ) sin(nϑ) , b(ρ) sin(mϑ)
+�
+= c21(ρ) sin((n−m)ϑ) + c22(ρ) sin((n+m)ϑ) ,
+�
+a(ρ) sin(nϑ) , b(ρ) cos(mϑ)
+�
+= c31(ρ) cos((n−m)ϑ) + c32(ρ) cos((n+m)ϑ) ,
+where cij : R+ → R are smooth functions. In particular {Yn, Ym} ⊂ Yn−m + Yn+m if m ≤ n.
+It is also necessary to compute the homogeneous polynomials Pj, Qj in (3.10) for higher
+values of j than in Lemma 3.3. This is a direct calculation that can be done for instance using
+computer algebra. For j = 3 we find
+P3 = − 1
+32(R + R′)
+�
+(R − R′)2 + 3(Z − Z′)2�
+,
+Q3 = − 1
+48(R + R′)
+�
+(R + R′)2 − 6(Z − Z′)2�
+,
+(A.19)
+and the calculation for j = 4 yields the more complicated expressions
+P4 = −
+15
+1024 (Z−Z′)4 + 21
+512 (R−R′)2(Z−Z′)2 + 3
+16 RR′ (Z−Z′)2
++
+17
+1024 (R2−R′2)2 −
+1
+256 RR′ (R−R′)2 ,
+Q4 = 31
+2048 (Z−Z′)4 −
+89
+1024 (R+R′)2(Z−Z′)2 +
+1
+256 RR′ (Z−Z′)2
+−
+19
+6144 (R2−R′2)2 +
+35
+1536 RR′(R+R′)2 −
+1
+128 R2R′2 .
+(A.20)
+The proof of Lemma 3.12 is similar to that of Lemma 3.11, and the details can be omitted.
+We use the expressions (3.40), (3.49) of the vorticities η1, η2, the formulas (3.41), (3.50) for the
+stream functions φ1, φ2, and the definition (3.15) of the Biot-Savart operators, which involve
+the polynomials (3.11) and (A.19). Using Lemma A.1, it is straightforward to verify that the
+quantity defined in (3.54) satisfies R3 ∈ Y1 + Y3 and takes the form
+R3 = χ1(ρ) sin(ϑ) + χ2(ρ) sin(3ϑ) + δ
+�
+χ3(ρ) cos(ϑ) + χ4(ρ) cos(3ϑ)
+�
++ O(δ2) ,
+where χ1, χ2, χ3, χ4 are radially symmetric functions which may depend on βǫ. To arrive at
+(3.55), it remains to verify that R3 does not contain any term involving β2
+ǫ . Indeed, according
+to (3.11), (3.49), we have
+βǫ
+2π P1η2 = βǫ
+4π
+�
+R2(R + R′) η2(R′, Z′) dR′ dZ′ = βǫR
+4π
+�
+R2 η24(R′, Z′) dR′ dZ′ ,
+so that the first term in (3.54) does not contain β2
+ǫ . The only other terms that we have to check
+are
+�
+φ1 , η2
+�
+− r0
+Γ
+˙¯z0∂Zη2 =
+�
+φ1 − βǫ − 1 + 2v
+4π
+R , η2
+�
+,
+where we the equality results from (3.38), but using the expressions (3.41), (3.49) we immediately
+see that the right-hand side does not contain any factor β2
+ǫ . Altogether we arrive at (3.55).
+□
+52
+
+B
+Appendix to Section 4
+B.1
+Properties of the energy functional
+Proof of Lemma 4.4. We use the first expression of Ekin
+ǫ
+[η] in (4.23) and the representation
+formula (2.20) for the stream function φ. Since supp(η) ⊂ Bǫ by assumption, we have
+Ekin
+ǫ
+[η] =
+1
+4π
+�
+Bǫ
+�
+Bǫ
+Gǫ(R, R′, Z, Z′) η(R, Z) η(R′, Z′) dX dX′ ,
+(B.1)
+where
+Gǫ(R, R′, Z, Z′) =
+�
+(1+ǫR)(1+ǫR′) F
+�
+ǫ2 (R−R′)2 + (Z−Z′)2
+(1+ǫR)(1+ǫR′)
+�
+.
+(B.2)
+As R2 + Z2 ≤ ǫ−2σ1 and R′2 + Z′2 ≤ ǫ−2σ1, the argument of F in (B.2) is of order O(ǫ2−2σ1).
+Using the asymptotic expansion of F(s) as s → 0 and proceeding as in Section 3.1, we easily
+obtain the decomposition
+Gǫ(R, R′, Z, Z′) = βǫ − 2 + log 8
+D + ˜Gǫ(R, R′, Z, Z′) ,
+(B.3)
+where βǫ = log(1/ǫ) and D2 = (R−R′)2 + (Z−Z′)2. The remainder ˜Gǫ satisfies the estimate
+| ˜Gǫ(R, R′, Z, Z′)| ≤ Cǫ
+�
+|R| + |R′|
+��
+βǫ + 1 + log 8
+D
+�
++ O
+�
+βǫǫ2−2σ1�
+.
+(B.4)
+If we insert the decomposition (B.3) into (B.1), the contributions of βǫ − 2 and log(8/D) give
+exactly the first two terms in the right-hand side of (4.25), in view of (4.24). Moreover, taking
+into account estimate (B.4) where ǫ2−2σ1 ≤ ǫ, we see that the contributions of ˜Gǫ to the kinetic
+energy (B.1) are of order O
+�
+ǫβǫ∥η∥2
+Xǫ
+�
+, as stated in (4.25).
+Proof of Proposition 4.6. Given η ∈ Xǫ, we decompose η = η1 + η2 where η1 = η1Bǫ and
+1Bǫ is the indicator function of the ball Bǫ = {(R, Z) ∈ Ωǫ ; R2 + Z2 ≤ ǫ−2σ1}. We thus have
+Eǫ[η] = 1
+2
+�
+Ωǫ
+Wǫ η2
+1 dX + 1
+2
+�
+Ωǫ
+Wǫ η2
+2 dX − 1
+2
+�
+Ωǫ
+�
+φ1 + φ2
+��
+η1 + η2
+�
+dX ,
+(B.5)
+where φj = BSǫ[ηj] for j = 1, 2. We claim that
+1
+2
+�
+Ωǫ
+�
+φ1 + φ2
+��
+η1 + η2
+�
+dX = Ekin
+ǫ
+[η1] + O
+�
+ǫ∞∥η∥2
+Xǫ
+�
+,
+(B.6)
+so that
+Eǫ[η] = Eǫ[η1] + 1
+2∥η2∥2
+Xǫ + O
+�
+ǫ∞∥η∥2
+Xǫ
+�
+.
+(B.7)
+To prove (B.6), we recall that φj(R, Z) =
+1
+2π
+�
+Ωǫ Gǫ(R, Z, R′, Z′)ηj(R′, Z′) dX′, where the
+kernel Gǫ is given by (B.2). Using the crude estimate |F(s)| ≤ C
+�
+| log s| + 1
+�
+, we easily obtain
+��Gǫ(R, R′, Z, Z′)
+�� ≤ C
+�
+1+ǫ|R|
+�a�
+1+ǫ|R′|
+�a�
+βǫ +
+��log D
+�� + 1
+�
+,
+(B.8)
+for some a > 1/2. It follows in particular that
+|φ(R, Z)| ≤ C
+�
+βǫ + 1
+�
+(1 + ρ)b∥η∥Xǫ ,
+ρ =
+�
+R2 + Z2 ,
+53
+
+for some b > 1/2, and using H¨older’s inequality we deduce
+�
+Ωǫ
+|φ(R, Z)| |η2(R, Z)| dX ≤ C
+�
+βǫ + 1
+�
+∥η∥2
+Xǫ
+��
+Bcǫ
+(1 + ρ)2b Wǫ(R, Z)−1 dX
+�1/2
+,
+where the last integral is O(ǫ∞) in view of (4.18). In a similar way we have
+|φ2(R, Z)| ≤ C
+�
+βǫ + 1
+�
+(1 + ρ)b
+�
+Bcǫ
+(1 + ρ′)b|η(R′, Z′)| dX′ = O
+�
+ǫ∞∥η∥Xǫ
+�
+(1 + ρ)b ,
+so that
+�
+Ωǫφ2η1 dx = O
+�
+ǫ∞∥η∥2
+Xǫ
+�
+. Altogether we arrive at (B.6).
+Now, since η1 is supported in the ball Bǫ, it follows from (4.19) and Lemma 4.4 that
+∥η1∥2
+Xǫ = ∥η1∥2
+X0 + O
+�
+ǫγ1∥η∥2
+Xǫ
+�
+,
+Ekin
+ǫ
+[η1] = βǫ−2
+4π
+˜µ2
+0 + Ekin
+0 [η1] + O
+�
+ǫβǫ∥η∥2
+Xǫ
+�
+.
+(B.9)
+Moreover we know from Proposition 4.5 that
+∥η1∥2
+X0 ≤ C4E0[η1] + C5
+�
+˜µ2
+0 + ˜µ2
+1 + ˜µ2
+2
+�
+,
+(B.10)
+where ˜µ0, ˜µ1, ˜µ2 are the moments of η1, which satisfy ˜µj = µj + O
+�
+ǫ∞∥η∥Xǫ
+�
+. Combining both
+estimates in (B.9) we obtain
+E0[η1] = 1
+2∥η1∥2
+X0 − Ekin
+0 [η1] ≤ 1
+2∥η1∥2
+Xǫ − Ekin
+ǫ
+[η1] + βǫ−2
+4π
+˜µ2
+0 + O
+�
+ǫγ1∥η∥2
+Xǫ
+�
+,
+and using in addition (B.10) we deduce
+∥η1∥2
+Xǫ ≤ ∥η1∥2
+X0 + O
+�
+ǫγ1∥η∥2
+Xǫ
+�
+≤ C4Eǫ[η1] + C
+�
+βǫ˜µ2
+0 + ˜µ2
+1 + ˜µ2
+2
+�
++ O
+�
+ǫγ1∥η∥2
+Xǫ
+�
+.
+Finally, invoking (B.7) and recalling that C4 > 2, we find
+∥η∥2
+Xǫ ≤ ∥η1∥2
+Xǫ + C4
+2 ∥η2∥2
+Xǫ ≤ C4Eǫ[η] + C
+�
+βǫ˜µ2
+0 + ˜µ2
+1 + ˜µ2
+2
+�
++ O
+�
+ǫγ1∥η∥2
+Xǫ
+�
+,
+and estimate (4.28) follows, since ˜µj = µj + O
+�
+ǫ∞∥η∥Xǫ
+�
+for j = 0, 1, 2.
+B.2
+Diffusive terms in the energy functional
+We justify here the expression (4.33) of the quantity I4. Integrating by parts as in [31], we find
+�
+Ωǫ
+Wǫ˜η L˜η dX = −
+�
+Ωǫ
+Wǫ|∇˜η|2 dX −
+�
+Ωǫ
+(∇Wǫ · ∇˜η)˜η dX −
+�
+Ωǫ
+˜Vǫ˜η2 dX ,
+where ˜Vǫ = 1
+4(R∂R + Z∂Z)Wǫ − 1
+2Wǫ. Similarly,
+ǫ
+�
+Ωǫ
+Wǫ˜η ∂R ˜ζ dX = ǫ
+�
+Ωǫ
+Wǫ(1 + ǫR)˜ζ ∂R˜ζ dX = − ǫ
+2
+�
+Ωǫ
+∂R
+�
+Wǫ(1 + ǫR)
+�˜ζ2 dX .
+On the other hand, integrating by parts and using the relation (2.19) between ˜φ and ˜η, we obtain
+�
+Ωǫ
+˜φ
+�
+L˜η + ǫ∂R˜ζ
+�
+dX =
+�
+Ωǫ
+˜η
+�
+∆˜φ − ǫ∂R ˜φ
+1 + ǫR
+�
+dX − 1
+2
+�
+Ωǫ
+˜η
+�
+R∂R + Z∂Z
+�˜φ dX
+= −
+�
+Ωǫ
+˜η2(1 + ǫR) dX − 1
+2
+�
+Ωǫ
+˜η
+�
+R∂R + Z∂Z
+�˜φ dX .
+It remains to treat the last term in the right-hand side. Here again, we use the relation (2.19)
+and integrate by parts to obtain
+1
+2
+�
+Ωǫ
+˜η
+�
+R∂R + Z∂Z
+�˜φ dX = ǫ
+4
+�
+Ωǫ
+R|∇˜φ|2
+(1 + ǫR)2 dX .
+Altogether we arrive at (4.33), with Vǫ = ˜Vǫ − (1 + ǫR).
+54
+
+B.3
+Coercivity of the diffusive quadratic form
+This section is devoted to the proof of Proposition 4.15. Given ǫ > 0 sufficiently small, we take
+a smooth partition of unity of the form 1 = χ2
+3 + χ2
+4, where χ3, χ4 are radially symmetric and
+χ3 = 1 when ρ ≤ 1
+2ǫ−σ1, χ3 = 0 when ρ ≥ ǫ−σ1. We can also assume that |∇χ3| + |∇χ4| ≤ Cǫσ1.
+Given η as in the statement of Proposition 4.15, we define η3 = χ3η, η4 = χ4η. We thus have
+the decompositions η2 = η2
+3 + η2
+4, η∇η = η3∇η3 + η4∇η4, and
+|∇η|2 = |∇η3|2 + |∇η4|2 −
+�
+|∇χ3|2 + |∇χ4|2�
+η2 .
+(B.11)
+As a consequence, the quadratic form Qǫ[η] can be decomposed as
+Qǫ[η] = Qǫ[η3] + Qǫ[η4] −
+�
+Ωǫ
+Wǫ
+�
+|∇χ3|2 + |∇χ4|2�
+η2 dX .
+(B.12)
+The last term in (B.12) is bounded by Cǫ2σ1∥η∥2
+Xǫ and is thus negligible when ǫ ≪ 1. So our
+main task is to estimate from below the terms Qǫ[η3] and Qǫ[η4].
+We first consider the function η3 which is supported in the region where ρ ≤ ǫ−σ1. We recall
+that the weight Wǫ in (4.17) satisfies the estimates (4.19), which read
+|∇Wǫ(R, Z) − ∇A(ρ)| + |Wǫ(R, Z) − A(ρ)| ≤ Cǫγ1A(ρ) ,
+when ρ ≤ ǫ−σ1 ,
+(B.13)
+where γ1 > 0. We easily deduce that
+Qǫ[η3] ≥ Q0[η3] − Cǫγ4�
+∥∇η3∥2
+X0 + ∥η3∥2
+X0
+�
+,
+(B.14)
+for some γ4 > 0, where Q0 is the limiting quadratic form (4.63). On the other hand, we know
+from Proposition 4.14 that
+C8Q0[η3] ≥ ∥∇η3∥2
+X0 + ∥ρη3∥2
+X0 + ∥η3∥2
+X0 − C9
+�
+˜µ2
+0 + ˜µ2
+1 + ˜µ2
+2
+�
+,
+(B.15)
+where ˜µ0, ˜µ1, ˜µ2 are the moments of η3, which satisfy ˜µj = µj +O
+�
+ǫ∞∥η∥Xǫ
+�
+. Combining (B.14),
+(B.15) and using (B.13) once again, we arrive at
+∥∇η3∥2
+Xǫ + ∥ρη3∥2
+Xǫ + ∥η3∥2
+Xǫ ≤ C8Qǫ[η3] + C
+�
+˜µ2
+0 + ˜µ2
+1 + ˜µ2
+2
+�
+.
+(B.16)
+We next consider the function η4, which is nonzero only if ρ ≥ 1
+2ǫ−σ1. Our starting point is
+the lower bound
+Qǫ[η4] ≥ 1
+4
+�
+Ωǫ
+Wǫ|∇η4|2 dX +
+�
+Ωǫ
+�
+Vǫ − |∇Wǫ|2
+3Wǫ
+�
+η2
+4 dX ,
+which is obtained from (4.62) by applying Young’s inequality to the middle term in the right-
+hand side. Using the expression (4.17) of the weight function, as well as the estimates (B.13) in
+the inner region Ω′
+ǫ, it is not difficult to verify that
+Vǫ
+Wǫ
+− |∇Wǫ|2
+3W 2ǫ
+≥
+
+
+
+
+
+Cρ2 − ˜C
+in Ω′
+ǫ ,
+− ˜C
+in Ω′′
+ǫ ,
+Cρ2γ
+in Ω′′′
+ǫ ,
+for some positive constants C, ˜C. It follows that
+Qǫ[η4] ≥ 1
+4∥∇η4∥2
+Xǫ + C
+�
+Ω′ǫ∪Ω′′′
+ǫ
+Wǫ ρ2
+γη2
+4 dX − ˜C
+�
+Ω′′ǫ
+Wǫ η2
+4 dX .
+(B.17)
+55
+
+If we now combine (B.16) and (B.17), we obtain
+∥∇η3∥2
+Xǫ + ∥∇η4∥2
+Xǫ + ∥η∥2
+Xǫ +
+�
+Ω′ǫ∪Ω′′′
+ǫ
+Wǫ ρ2
+γη2 dX
+≤ C10
+�
+Qǫ[η3] + Qǫ[η4]
+�
++ C11
+�
+˜µ2 +
+�
+Ω′′ǫ
+Wǫη2�
+,
+(B.18)
+for some positive constants C10, C11, where ˜µ2 = ˜µ2
+0 + ˜µ2
+1 + ˜µ2
+2. Finally, using again (B.11) as
+well as (B.12), and recalling that ˜µj = µj + O
+�
+ǫ∞∥η∥Xǫ
+�
+, we deduce (4.66) from (B.18).
+□
+Acknowledgments. ThG is partially supported by the grant SingFlows ANR-18-CE40-0027
+of the French National Research Agency (ANR). The research of VS is supported in part by
+grant DMS 1956092 from the National Science Foundation.
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+Thierry Gallay
+Institut Fourier, Universit´e Grenoble Alpes, 100 rue des Maths, 38610 Gi`eres, France
+Email : Thierry.Gallay@univ-grenoble-alpes.fr
+Vladim´ır ˇSver´ak
+School of Mathematics, University of Minnesota
+127 Vincent Hall, 206 Church St. SE, Minneapolis, MN 55455, USA
+Email : sverak@math.umn.edu
+58
+
diff --git a/WNAzT4oBgHgl3EQfKPsc/content/tmp_files/load_file.txt b/WNAzT4oBgHgl3EQfKPsc/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d9ad9b167a2595822266684f1a2cffffb239c287
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+++ b/WNAzT4oBgHgl3EQfKPsc/content/tmp_files/load_file.txt
@@ -0,0 +1,2486 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf,len=2485
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='01092v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='AP]  3 Jan 2023  Vanishing viscosity limit for axisymmetric vortex rings  Thierry Gallay and Vladim´ır ˇSver´ak  January 4, 2023  Abstract  For the incompressible Navier-Stokes equations in R3 with low viscosity ν > 0, we con-  sider initial data ω0 that represent an infinitely thin vortex filament of arbitrary strength Γ  supported on a closed circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The vorticity field ω(x, t) of the fluid is smooth at any positive  time and corresponds to a vortex ring with thickness of order  √  νt that is translated along its  symmetry axis due to self-induction, an effect anticipated by Helmholtz in 1858 and quanti-  fied by Kelvin in 1867.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' For small viscosities, we show that ω(x, t) is well approximated on a  large time interval by ωlin(x−a(t), t), where ωlin(·, t) = exp(νt∆)ω0 is the solution of the heat  equation with initial data ω0, and ˙a(t) is the instantaneous velocity of a vortex ring of thick-  ness  √  νt given by Kelvin’s formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The proof relies on the construction of a fairly precise  approximate solution, using a perturbative expansion in self-similar variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' To verify the  stability of this approximation, one needs to rule out potential instabilities coming from very  large convection terms in the linearized operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This is done by adapting V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Arnold’s  geometric stability methods developed in the inviscid case ν = 0 to the slightly viscous situ-  ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It turns out that although the geometric structures behind Arnold’s approach are no  longer preserved by the equation for ν > 0, the relevant quadratic forms behave nicely on  larger subspaces than those originally used in Arnold’s theory and interact favorably with  the viscous terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  1  Introduction and main result  We consider the Cauchy problem for the 3d incompressible Navier-Stokes equations in the vor-  ticity form  ∂tω + u · ∇ω − ω · ∇u = ν∆ω  in R3 × (0, ∞) ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1)  ω|t=0 = ω0  in R3 ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2)  where we use the familiar notation ω(x, t) for the vorticity of the fluid, and the velocity u(x, t)  is given by the Biot-Savart law u(x, t) = (4π)−1 �  R3 ω(y, t) ∧ (x − y) |x − y|−3 dy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We focus on  the special case where the initial vorticity ω0 is an idealized vortex filament given by a current1  of strength Γ concentrated on an oriented circle C ⊂ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' More precisely, ω0 is the vector-valued  measure on R3 defined by the identity  ⟨ω0 , ϕ⟩ = Γ  3  �  i=1  �  C  ϕi dxi ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3)  which is assumed to hold for any continuous test function ϕ = (ϕ1, ϕ2, ϕ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We will use the  notation ω0 = ΓδC for this initial vorticity current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In the well-known analogy between fluid  1Here the term current can be understood in its heuristic meaning but also in the technical meaning of the  geometric measure theory, such as in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  1  mechanics and electromagnetism, ω0 can be thought of as an electric current of intensity Γ flowing  through an infinitely thin wire represented by the circle C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The direction of the current is given  by the orientation of the circle and the sign of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Vortex filaments were already considered in  the classical 1858 paper of Helmholtz [36] that deals with the inviscid case ν = 0 corresponding  to the Euler equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Helmholtz argued that a circular vortex filament of zero thickness would  move with infinite speed along its symmetry axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In 1867 Kelvin [40] established the following  formula for the speed of vortex rings of small but finite thickness d > 0 and radius r0 ≫ d :  V ≈  Γ  4πr0  �  log 8r0  d − C  �  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4)  where C ∈ R is a dimensionless constant that depends on the distribution of vorticity inside  a cross section of the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If the vorticity is proportional to the distance from the symmetry  axis, which is the assumption made by Kelvin (although the details of the computations are not  given in [40]), the relevant value is C = 1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  V  r0  thickness ≈  √  νt  t = 0  t > 0  ω0 = ΓδC  =⇒  Figure 1: An illustration our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Starting from a vortex filament supported by an oriented circle  C, the solution of the Navier-Stokes equation evolves into a viscous vortex ring of thickness proportional  to  √  νt which moves along the symmetry axis at a speed V given by Kelvin’s formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In the right  picture, the vortex lines are circles that fill the solid torus depicted in blue, whereas the black arrows  denote the trajectories of the fluid particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  In the viscous case ν > 0, the solution originating from the singular filament ω0 = ΓδC  becomes smooth for any positive time t > 0 and is expected to represent a viscous vortex ring  of thickness  √  νt, as long as that quantity is small compared to the radius r0 of the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Based  on Kelvin’s formula one anticipates that the vortex ring will move with speed  V (t) ≈  Γ  4πr0  �  log 8r0  √  νt − C  �  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5)  where the constant C now corresponds to a Gaussian distribution of vorticity inside the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  A direct calculation shows that C = 3  2 log(2)+ 1  2(1−γE), where γE ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5772.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' is Euler’s constant,  see for example [52,23,51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' When the initial circle C is parametrized by (r0 cos θ, r0 sin θ, 0) for  θ ∈ [0, 2π], with the orientation in the direction of increasing θ, the translational motion will be  in the positive direction along the x3-axis if Γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  It is proved in [30] that the Cauchy problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2) with ω0 = ΓδC has a unique solution  in natural classes of axisymmetric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The main result of the present paper, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1  below, describes the precise behavior of that solution in the low viscosity regime where the  circulation Reynolds number Re := Γ/ν is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Our description is valid on a time interval whose  2  length is intermediate between the advection time and the diffusion time, defined respectively  as  Tadv = r2  0  Γ ,  Tdif = r2  0  ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6)  Note that Tadv ≪ Tdif when Re ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The leading term in our approximation is exactly the  one suggested by Kelvin’s formula together with the simplest diffusion heuristics: The ring  diffuses according to the linear heat equation, and translates with speed (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5) along its symmetry  axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Denoting by ωlin(x, t) the solution of the heat equation ωt = ν∆ω with initial condition  ω|t=0 = ω0 = ΓδC, and defining ||η|| = ||η/r||L1(R3), where r = r(x) is the distance from x to the  symmetry axis, we can state our main result as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' There exist dimensionless constants K > 0, R0 > 0, and σ > 0 such that, for  all Γ > 0, all r0 > 0, and all ν > 0 satisfying Re := Γ/ν ≥ R0, the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If ω0 = Γ δC  where C is an oriented circle of radius r0, the unique axisymmetric solution ω of the Cauchy  problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2) established in [30] can be expressed for t ∈ (0, Tadv Reσ) as  ω(x, t) = ωlin(x − a(t), t) + ωcorr(x, t) ,  with  ||ωcorr(· , t)|| ≤ K Γ  �√  νt  r0  �1−3σ  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7)  where a(t) describes the translation of the ring along its symmetry axis according to Kelvin’s  formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Specifically, if C = {(r0 cos θ, r0 sin θ, 0) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' θ ∈ [0, 2π]} one has a(t) = (0, 0, a3(t))  with a3(0) = 0 and  ˙a3(t) =  Γ  4πr0  �  log 8r0  √  νt − 3  2 log(2) + 1  2(γE − 1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8)  An extended version of our result, including a more precise approximate solution and a much  stronger control of the correction term, is formulated as Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6 below, after the necessary  notation has been introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  In Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1, the constants K and R0 are large, whereas the exponent σ > 0 is taken  small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We conjecture that an approximation result of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7) remains valid on longer  time scales, corresponding to arbitrary values of σ ∈ (0, 1), but currently our proof requires that  σ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In view of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4), the advection time Tadv can be interpreted as the time needed for  a vortex ring of circulation Γ and small but not infinitesimal aspect ratio d/r0 to travel over  a distance comparable to its radius r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In contrast, the diffusion time Tdif = Tadv · Re is the  time at which the diffusion length  √  νt becomes equal to the radius r0, so that the vortex ring  structure is essentially lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The assumption that Re ≫ 1 means that the vortex ring can travel  over a very long distance along its symmetry axis before being destroyed by diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Already  on the intermediate time scale T = Tadv Reσ where Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1 provides a rigorous control we  find, using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8),  |a(T)| =  � T  0  ˙a3(t) dt = r0  4π Reσ�  log  �  Re  1−σ  2 �  + C′�  ,  for some constant C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Obviously the right-hand side grows boundlessly as Re → +∞, even in  the limiting case where σ = 0 and T = Tadv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  It is instructive to compare the situation for vortex rings with the case of a rectilinear  filament, where the vorticity is initially concentrated on a straight line ℓ in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Let us denote  this initial vorticity field by ω0 = Γδℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In that case the solution of the full vorticity equation is  given by ω( · , t) = Γeνt∆δℓ, because the nonlinear terms vanish identically due to symmetries  when evaluated on the solution of the heat equation ωt = ν∆ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Although the evolution of the  3  velocity and the vorticity fields does not look very dramatic, the fluid particles in the vicinity of  ℓ do move at very large speeds when νt is small, and the inertial forces in the fluid are therefore  significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' However, these forces are exactly balanced by the incompressibility constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  When the rectilinear filament is bent into a vortex ring (as already considered in Helmholtz’s  1858 paper), the inertial forces are no longer in balance and the ring has to move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' To achieve a  relatively smooth motion, the bent vortex has to be “well-prepared” so that the inertial forces  generated by the fast-moving fluid particles are still mostly canceled and do not generate fast  oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The initial condition ω0 = ΓδC has the advantage of letting the equation to adjust  the field into a well-prepared state without trying to achieve this “by hand”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Quite remarkably,  the equation adjusts the vorticity field in exactly such a way that the oscillations are avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2  The largest inertial forces still cancel and the situation remains somewhat close to the rectilinear  case with only two significant differences: (a) some motion of the ring along its axis of rotational  symmetry is needed to balance the forces, but the speed of this motion is much lower than the  speed of the fast particles in the fluid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' (b) once the thickness of the ring becomes comparable to  its radius, new effects (not discussed in this work) appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1  Main ideas of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1  Our analysis starts with a construction of a precise approximation of the solution ω(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This is  achieved by writing the solution in suitable self-similar coordinates that capture well the singular  behavior of the solution at t = 0 through explicit rescalings of a smooth “profile” η that can be  thought of as a perturbation of a suitable Gaussian η0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The perturbed profile η is expressed as  an asymptotic series in the time-dependent parameter ǫ =  √  νt/¯r, with ¯r = ¯r(t) denoting the  instantaneous radius of the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' To achieve a precision that is sufficient for our purposes, we need  an expansion up to the fourth order: η = η0+ǫη1+ǫ2η2+ǫ3η3+ǫ4η4+ηcorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The “coefficients” ηj  with j ≥ 1 are obtained by inverting operators containing the small parameter δ = 1/Re = ν/Γ,  and in that sense we really deal with a two-parameter expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' As far as we know, this is  somewhat different from other expansions in the literature, such as [12,51,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' A one-parameter  formal expansion in ǫ would treat δ as ∼ ǫ2, in view of the relation ¯r2ǫ2 = δ Γ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Keeping both  parameters makes it easier to cover the regimes when ǫ2 and δ are not really comparable, as is  the case for very small and very large times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Strictly speaking, the “coefficients” ηj for j ≥ 1  can also depend on log ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' That feature is well-known, and the leading term in the speed of the  ring is related to choosing a moving coordinate system in which the terms with log ǫ in η1 are  eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The main difficulty in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1, however, is not in the computation of an  approximate solution, but in showing that the true solution remains close to this approximation  on a large time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This requires fairly strong stability properties for the linearization of  the vorticity equation at the approximate solution, which is very singular in the low viscosity  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' When the initial condition corresponds to a finite number of parallel rectilinear vortices,  a stability analysis was carried on in [28] by using suitable weighted L2 spaces adapted to the  specific features of the rectilinear vortices with Gaussian profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In the vortex ring case the  nonlinearity of the equations starts affecting the formal expansions earlier and it is unclear  whether the setup in [28] can be used to show that the vortex ring will not disintegrate on  time-scales approaching zero as ν → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' A recent important work [6] extends some of the 2d  methods for proving stability to a relevant 3d situation, but the length of the time interval over  which the solution is under control may approach 0 as ν tends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  In physical flows and numerical experiments one observes a remarkable degree of stability  2In the related situation of interacting vortices in R2, this was already observed in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  4  of vortex rings as well as signs of instabilities with respect to non-axisymmetric perturbations,  see for example [56,48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' At a rigorous mathematical level the stability issues have not been well  understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In fact, when Γ/ν is not small, not only the stability, but even the uniqueness of  the solutions of the Cauchy problem above with ω0 = Γ δC (and also with ω0 = Γ δℓ) is open in  classes of solutions that do not share the symmetry of the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  In the 1960s, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Arnold suggested a variational method for proving stability of steady  solutions to Euler’s equation based on a geometric insight that can be summarized as follows,  using the Hamiltonian setup of [45]:  (a) The incompressible Euler equation can be viewed as a Poisson system with a Hamiltonian  function given by the usual kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (b) The steady states are critical points of the energy on the symplectic leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The latter  coincide with the coadjoint orbits, called just orbits in what follows, of the group of the volume-  preserving diffeomorphisms of the fluid domain acting by push-forward on the vorticity fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (c) When the critical point is a local maximum or a local minimum on an orbit, the corresponding  steady-state should be stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  These ideas fit into a broader family of methods used for proving stability of solutions of  Hamiltonian systems by invoking extremality properties of a conserved quantity under con-  straints given by other conserved quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  For example, a circular planetary orbit in the  three-dimensional Kepler problem is stable because it minimizes energy for a given angular  momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3 In the applications to vortex rings, it is natural to restrict the analysis to axisym-  metric flows with no swirl, which means that the velocity field is invariant under rotations about  a symmetry axis and under reflection across any plane containing that axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Arnold’s method has found many applications to Euler flows in 2d (see, for example, [4]),  and has also been invoked in the work of Benjamin [8] on inviscid vortex rings that is directly  relevant for our purposes here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Although some arguments in [8] may not be fully rigorous, they  provide important suggestions for investigating stability of inviscid vortex rings in the class of  axisymmetric solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In a different direction, the conservation of energy, impulse, and vortex  strength has been used to control the evolution of a general class of concentrated solutions of  the Euler equations describing vortex rings, see for example [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  There is voluminous literature on the stationary vortex ring solutions of the Euler equation,  starting with the explicit solution of Hill [38], see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' [1,2,5,10,13–16,23–26,49,50,54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Many  of these works rely in one way or another on variational aspects of the underlying PDEs that  have connections to the work of Arnold and Benjamin, albeit a somewhat indirect one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Roughly  speaking, if we compare Arnold’s setup to the maximization of a function f(x) under constraints  gj(x) = cj, one can compare some of the variational approaches in the references above to  searching for critical points of f(x) − λ1g1(x) − · · · − λmgm(x) when the Lagrange multipliers  λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' , λm are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Readers interested in related links can find more details in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The works [9,11,7] use very effectively some of the variational principles inherent in Arnold’s  and Benjamin’s approach, essentially at the inviscid level, assuming that the viscosity is suffi-  ciently small for the viscous term to be treated as a lower-order perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In our asymptotic  expansions of the solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2) inviscid vortex ring solutions can also be discerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  For each fixed time t > 0 the third-order expansion in our parameter ǫ =  √  νt/¯r is a good ap-  proximation of an inviscid vortex ring, at least in the limiting case where our second parameter  δ = ν/Γ is taken equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' A part of our stability analysis can be thus understood in terms  of the stability properties of this ring, see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3 and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  3It is well-known that this is no longer the case in dimension four and higher [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  5  If one wishes to apply Arnold’s ideas to the solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2), there appears to be  a non-trivial obstacle: The viscous flows do not preserve the geometric structures that are at  the basis of Arnold’s considerations and the influence of the viscosity is too large to treat the  viscous terms perturbatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' At first this may seem to be serious problem: If the preservation  of the orbits and the Hamiltonian nature of the equations are violated beyond the reach of the  perturbative approach (such as [9,11,7]), can the geometric structure relying on maximization  of the energy on symplectic leaves be helpful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In our previous work [31] we showed, in a much  simpler situation, that the answer to this question can be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It turns out that the quadratic  forms coming up in Arnold’s stability analysis, although originally envisaged as quadratic forms  on the tangent spaces to the orbits, are often well-behaved on much larger subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This point  can still be conceptually explained by the geometry of the Euler equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' What we find more  surprising is that Arnold’s forms also have favorable behavior with respect to the dissipative  term generated by the viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We can show this by direct calculation, but we do not have  a good conceptual explanation of this fortuitous circumstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In the paper [31] we showed  that the above ideas can be used to prove the stability of the rectilinear vortex solution (in  self-similar variables) with respect to perturbations for which the vorticity field stays parallel to  the original vortex line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This result has been established previously by a different method [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The new proof in [31] can be thought of as a proof of concept that the ideas of Arnold can be  applied even in the presence of viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The application to vortex rings presented here is more  complicated, but we are not aware of any simpler approach in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2  Comments on the local induction approximation for general filaments  The problem studied in this paper can be considered as a special case of the viscous version of  the local induction approximation conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In the setup considered here the conjecture could  be formulated as follows: if we replace the circle C be a general smooth closed curve and consider  the Cauchy problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2) with ω0 = ΓδC, the motion of the filament C should still be  determined essentially by two effects: the diffusion, which transforms the filament into a vortex  tube of thickness d(t) ≈  √  νt at time t, and the advection by the self-induced velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The  latter is described by a geometric equation that represents an extension of Kelvin’s formula to  general smooth curves, and was derived by Da Rios [18] in 1906:  V ≈  � Γ  4πr log 8r  d  �  b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9)  Here V is the vector representing the local velocity of the filament, b denotes the local binormal  vector, r is the local radius of the curvature, and d denotes the local thickness of the filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' (All  these quantities may be time- and position-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=') In the limit ν → 0 the approximation  should be valid until the geometric evolution of the curve by the binormal flow leads to a self-  intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' For general initial curves C the time of the first self-intersection may be approaching  zero as ν approaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The first important step towards this general case, a local-in time  well-posedness result for a fixed ν > 0, was obtained in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Some formal computations related  to the conjecture are presented in [12] and we also refer the reader to the important conditional  result in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Our result can be viewed as a proof of the viscous formulation of the conjecture in the special  case where the curve C is a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Important previous works for the axisymmetric case with  very small viscosity include [9,11], where the case of several vortex rings is also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The  setup in these works is somewhat different, in that the initial vortex rings have finite thickness  d and certain relations between d, the vortex strength Γ, the viscosity ν and the maximum of  initial vorticity are assumed so that (among other things) the limiting motion of the rings has  6  uniformly bounded velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In particular, the vortex strength Γ is assumed to approach 0 as  d tends to zero and the viscosity ν has to satisfy smallness conditions related to d, so that the  viscous terms can be treated as a small perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Yet another angle on vortex rings is taken  in the recent work [19] that deals with “leapfrogging” of inviscid vortex rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The general case of the local induction approximation conjecture for the setup considered  in this paper seems to be difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In fact, it is unclear whether the strongest version of the  conjecture is valid even for small perturbations of the circle, as the perturbed filaments may  perhaps become unstable to general 3d perturbations before possible self-intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  For  example, the instabilities studied in [56,48] may be relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  2  Preliminaries and sketch of the proof  In this section we introduce the notation that is necessary to formulate our result in its stronger  form, and we give a pretty detailed sketch of the overall proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The construction of the approx-  imate solution will be performed in Section 3, and the stability analysis in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Technical  calculations are postponed to Appendix A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1  Formulation of the problem in cylindrical coordinates  In a suitable Cartesian coordinate system, the circle of radius r0 > 0 which represents the  support of the initial vorticity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3) is given by C = {(r0 cos θ, r0 sin θ, 0) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' θ ∈ [0, 2π]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Due to  the symmetries of the problem, it is natural to introduce the standard cylindrical coordinates  (r, θ, z) defined by x1 = r cos θ, x2 = r sin θ, x3 = z and to restrict our attention to velocity and  vorticity fields of the form  u(x, t) = ur(r, z, t)er + uz(r, z, t)ez ,  ω(x, t) = ωθ(r, z, t)eθ ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1)  where er, eθ, ez denote unit vectors in the radial, azimuthal, and vertical directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  In the usual terminology, we thus consider axisymmetric flows with no swirl, see [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Due to  the incompressibility condition div u = r−1∂r(rur) + ∂z(uz) = 0, the velocity components ur, uz  can be expressed in terms of the Stokes stream function ψ :  ur = −1  r ∂zψ ,  uz = 1  r ∂rψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2)  With this notation the vorticity formulation of the Navier-Stokes equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1) becomes  ∂tωθ +  �  ψ, ωθ  r  �  = ν  ��  ∂2  r + ∂2  z)ωθ + ∂r  ωθ  r  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3)  where {·, ·} is the Poisson bracket defined by {ψ, φ} = ∂rψ ∂zφ − ∂zψ ∂rφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3) is to be  solved in the domain Ω = {(r, z) ∈ R2 | r > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The smoothness of the fields in the original  variables imposes the “boundary conditions” ωθ(r, z, t) = rζ(r, z, t) and ψ(r, z, t) = r2Ψ(r, z, t)  near r = 0, where ζ and Ψ can be extended to smooth functions on R2 × R+ that are even  functions of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The Biot-Savart law can be expressed in this setup as  ψ(r, z) =  1  2π  �  Ω  √  r¯r F  �(r − ¯r)2 + (z − ¯z)2  r¯r  �  ωθ(¯r, ¯z) d¯r d¯z ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4)  where F : (0, ∞) → (0, ∞) is defined by  F(s) =  � π/2  0  1 − 2 sin2 ψ  �  sin2 ψ + s/4  dψ ,  s > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5)  7  Formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4) also provides a solution to the equation  curl curl  �ψ  r eθ  �  = ωθ eθ ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6)  which is familiar in magnetostatics, see for example [47, §701].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The same expression can also be  found in the classical book [42, §161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It is well-known (and not hard to check) that  F(s) =  �  log  8  √s − 2 + O(s log s)  as s → 0 ,  π  2s3/2 + O(s−5/2)  as s → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7)  Since we wish to solve the Cauchy problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2) with initial data ω0 = ΓδC, we impose  to the scalar function ωθ defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1) the initial condition  ωθ  ���  t=0 = Γδ(r0,0) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8)  where we use δ(r0,z0) to denote the Dirac mass at the location (r0, z0) ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Our starting point  is the following global well-posedness result for the vorticity equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3) with concentrated  initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' [30] For any Γ > 0, any ν > 0, and any (r0, z0) ∈ Ω, the axisymmetric vorticity  equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3) has a unique global mild solution ωθ ∈ C0((0, ∞), L1(Ω) ∩ L∞(Ω)) such that  sup  t>0  ∥ωθ(t)∥L1(Ω) < ∞ ,  and  ωθ(t) dr dz ⇀ Γ δ(r0,z0)  as t → 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9)  Moreover there exists a constant C > 0, depending only on the ratio Γ/ν, such that  �  Ω  ���ωθ(r, z, t) −  Γ  4πνt e− (r−r0)2+(z−z0)2  4νt  ��� dr dz ≤ C Γ  √  νt  r0  log  � r0  √  νt + 1  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10)  whenever t ∈ (0, Tdif), where Tdif = r2  0/ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Here and in what follows, it is understood that L1(Ω) = L1(Ω, dr dz), and similarly for the  other Lebesgue spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1 establishes the existence of a four-dimensional family of  vortex ring solutions to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3) parametrized by the circulation Γ > 0, the viscosity ν > 0, the  initial radius r0 > 0, and the initial vertical position z0 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Due to translation invariance in  the vertical direction, we may assume without loss of generality that z0 = 0, and we can also  take r0 = 1 by rescaling the space variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Then a rescaling of time allows us to change the  values of both ν and Γ, while keeping the ratio Γ/ν fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Hence up to symmetries, the viscous  vortex ring solutions we consider here form a one-parameter family indexed by the circulation  Reynolds number Re := Γ/ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The uniqueness of the vortex ring solution under the minimal assumptions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9) is discussed  in some detail in [30], so we concentrate here on the short-time estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10), which is of  limited use despite appearances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' For a fixed value of the Reynolds number, the right-hand side  of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10) is small whenever t ≪ Tdif, which means that the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3) with initial data  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8) is well approximated by a Gaussian vortex ring of thickness proportional to  √  νt, located  a the initial position (r0, z0) ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' However, since the constant C depends (very badly) on the  Reynolds number, estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10) gives no information on the solution at a fixed time t > 0  in the low viscosity regime ν → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This limitation is not surprising: due to the translational  motion along the vertical axis predicted by Kelvin’s formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5), the vortex ring at time t > 0  is actually located at a new position which is rather far from the initial one if ν is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  8  Our goal in this paper is to replace (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10) by an improved estimate of the form  �  Ω  ���ωθ(r, z, t) −  Γ  4πνt e− (r−¯r(t))2+(z−¯z(t))2  4νt  ��� dr dz ≤ K Γ  √  νt  r0  ,  t ∈ (0, Tadv Reσ) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11)  where the constant K is now independent of the Reynolds number, if Re ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Comparing with  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10), we observe that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11) is valid up to the intermediate time Tadv Reσ, for some σ ∈ (0, 1),  which is shorter than Tdif ≡ Tadv Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  But the main difference is that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11) compares the  solution ωθ(r, z, t) to a vortex ring located at a time-dependent position (¯r(t), ¯z(t)), which has to  be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' As we shall see, we can take ¯r(t), ¯z(t) to be smooth functions of time satisfying  ¯r(0) = r0, ¯z(0) = z0, and  ˙¯r(t) = O  � ν  r0  �  ,  ˙¯z(t) =  Γ  4πr0  �  log  1  ǫ(t) + V  ��  1 + O  �  ǫ(t)2��  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='12)  where ǫ(t) =  √  νt/¯r(t) and V = 3  2 log(2) + 1  2(γE − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The first relation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='12) implies that  ¯r(t) = r0  �  1 + O(ǫ(t)2)  �  , which means that the change in the radius of the vortex ring over the  time interval under consideration is much smaller than the diffusion length  √  νt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The second  equality coincides with the viscous form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5) of Kelvin’s formula, up to higher order corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2  Self-similar variables  From now on, we fix the circulation Γ > 0 and the position (r0, 0) ∈ Ω of the initial filament,  and we consider the vortex ring solution given by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1, in the regime where the viscosity  ν > 0 is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In view of the approximation formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11), which is our objective, it is natural  to make the following self-similar Ansatz for the axisymmetric vorticity and the associated Stokes  stream function :  ωθ(r, z, t) = Γ  νt η  �r − ¯r(t)  √  νt  , z − ¯z(t)  √  νt  , t  �  ,  ψ(r, z, t) = Γ ¯r(t) φ  �r − ¯r(t)  √  νt  , z − ¯z(t)  √  νt  , t  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='13)  where the time-dependent position (¯r(t), ¯z(t)) ∈ Ω has to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We introduce the  important notation  δ = ν  Γ ,  ǫ =  √  νt  ¯r(t) ,  R = r − ¯r(t)  √  νt  ,  Z = z − ¯z(t)  √  νt  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14)  The evolution equation for the rescaled vorticity η(R, Z, t) is found to be  t∂tη + Γ  ν  �  φ ,  η  1 + ǫR  �  −  �  t  ν  �  ˙¯r ∂Rη + ˙¯z ∂Zη  �  = Lη + ∂R  �  ǫη  1 + ǫR  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15)  where  �  φ, χ  �  = ∂Rφ ∂Zχ − ∂Zφ ∂Rχ is the Poisson bracket in the new variables (R, Z), and L is  the Fokker-Planck operator  L = ∂2  R + ∂2  Z + 1  2  �  R∂R + Z∂Z  �  + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='16)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15) is to be solved in the time-dependent domain  Ωǫ =  �  (R, Z) ∈ R2 �� 1 + ǫR > 0  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='17)  9  with the Dirichlet boundary condition η(−1/ǫ, Z, t) = 0 for all (Z, t) ∈ R × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  As in [30], it is useful to introduce the velocity field U = (UR, UZ) defined by  UR = − ∂Zφ  1 + ǫR ,  UZ =  ∂Rφ  1 + ǫR ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='18)  in terms of which the nonlinearity in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15) reads  �  φ ,  η  1+ǫR  �  = ∂R  �  UR η) + ∂Z(UZ η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The  stream function φ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15) satisfies the elliptic equation  η = ∂ZUR − ∂RUZ ≡ −∂R  � ∂Rφ  1 + ǫR  �  −  ∂2  Zφ  1 + ǫR ,  (R, Z) ∈ Ωǫ ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19)  with boundary conditions φ(−1/ǫ, Z, t) = ∂Rφ(−1/ǫ, Z, t) = 0 for all (Z, t) ∈ R × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Using  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4), we easily obtain the representation formula [30]  φ(R, Z) =  1  2π  �  Ωǫ  �  (1+ǫR)(1+ǫR′) F  �  ǫ2 (R−R′)2 + (Z−Z′)2  (1+ǫR)(1+ǫR′)  �  η(R′, Z′) dR′ dZ′ ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='20)  where F is as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In what follows we write φ = BSǫ[η] when (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='20) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The quantities introduced in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14) are all dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The first one is the inverse Reynolds  number δ > 0, a fixed parameter that is assumed to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The second one is the time-  dependent aspect ratio ǫ > 0, which appears in the evolution equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15), in the definition  of the domain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='17), and in the Biot-Savart formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Finally, the variables R, Z are self-  similar coordinates centered at the time-dependent location (¯r(t), ¯z(t)) and normalized according  to the size  √  νt of the vortex core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Note that the rescaled functions η, φ defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='13) are  also dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Recalling that δ = ν/Γ and Tadv = r2  0/Γ, we observe that  ǫ2 = νt  r2  0  r2  0  ¯r(t)2 =  δt  Tadv  r2  0  ¯r(t)2 ≈  δt  Tadv  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='21)  as long as the ratio r0/¯r(t) remains close to unity, which will always be the case thanks to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  It follows in particular that ǫ2 is comparable to δ whenever t is comparable to Tadv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Our goal  is to control the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3) for t ≤ Tadvδ−σ for some σ ∈ (0, 1), and on that interval it  follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='21) that ǫ2 ≲ δ1−σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3  Approximate solution  The first important step in our analysis is the construction of an approximate solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15)  with initial data  η0(R, Z) =  1  4π e−(R2+Z2)/4 ,  (R, Z) ∈ Ω0 = R2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='22)  The associated stream function satisfies −∆0φ0 = η0, where ∆0 = ∂2  R + ∂2  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' As η0, φ0 are both  radially symmetric, it is clear that {φ0, η0} = 0, and the Gaussian profile (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='22) has the property  that Lη0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Since ǫ = 0 when t = 0 in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14), we conclude that equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15) is  indeed satisfied at initial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  For t > 0, we construct our approximate solution as a power series in the time-dependent  parameter ǫ =  √  νt/¯r, the coefficients of which depend on the small parameter δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' To this end,  we multiply both sides of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15) by δ and rewrite the equation in the equivalent form  δ t∂tη +  �  φ ,  η  1 + ǫR  �  − ǫ¯r  Γ  �  ˙¯r ∂Rη + ˙¯z ∂Zη  �  = δ  �  Lη + ∂R  �  ǫη  1 + ǫR  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='23)  10  This equation is defined on the time-dependent domain Ωǫ, but expanding the factors (1+ǫR)−1  in powers of ǫ we get at each order a relation that can be solved in the whole plane Ω0 = R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The corresponding approximation for the stream function φ is obtained in a self-consistent way  by expanding the integrand in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='20) in powers of ǫ, and integrating order by order over the  whole plane R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It turns out one can work with an asymptotic expansion of the form  ηapp(R, Z, t) =  M  �  m=0  ǫm ηm(R, Z, βǫ) ,  φapp(R, Z, t) =  M  �  m=0  ǫm φm(R, Z, βǫ) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='24)  where (R, Z) ∈ R2, 0 < ǫ ≪ 1, with the dependence of the profiles ηm and φm on βǫ := log(1/ǫ)  being polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The profiles also depend on the small parameter δ, but to make the notation  lighter this dependence is not indicated explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The velocity of the vortex center is not known  a priori, but can be expressed in a similar way as a power series in ǫ :  ˙¯r(t) =  M−1  �  m=0  ǫm ˙¯rm(βǫ) ,  ˙¯z(t) =  M−1  �  m=0  ǫm ˙¯zm(βǫ) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='25)  where the quantities ˙¯rm(βǫ), ˙¯zm(βǫ) depend on δ and are polynomials in βǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The outcome of the analysis carried out in Section 3 below is that, if we want our expansions  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='24), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='25) to hold uniformly with respect to the parameter δ in the limit where δ → 0, there  is a unique choice of the profiles ηm, φm and of the velocities ˙¯rm, ˙¯zm such that :  a) Both members of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='23) agree up to order O(ǫM+1), modulo powers of βǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  b) The point (¯r(t), ¯z(t)) ∈ Ω is the center of the vorticity distribution defined by ηapp(R, Z, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The integer M in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='24), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='25) determines the accuracy of our approximate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It turns  out that M = 4 will be sufficient for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The velocities ˙¯r(t), ˙¯z(t) given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='25) are  found to satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If we set δ = ˙¯r = 0, equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='23) reduces to  �  φ ,  η  1 + ǫR  �  − ǫ¯r  Γ  ˙¯z ∂Zη ≡  �  φ − ¯r ˙¯z  2Γ (1 + ǫR)2 ,  η  1 + ǫR  �  = 0 ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='26)  which is exactly the relation satisfied by the vorticity η and the stream function φ of a stationary  solution of the Euler equations in a frame moving with speed ˙¯z ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In that situation the aspect  ratio ǫ > 0 is fixed and, as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14), the dimensionless variables (R, Z) are defined so that  (r, z) = (¯r, ¯z) + ǫ¯r (R, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' An approximate solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='26) can be constructed in the form of  a power series in ǫ, as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='24), where all profiles ηm, φm are even functions of the variable  Z ∈ R, since this is the case for the coefficients of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='26) and for the initial approximation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Returning to the approximate solution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='24), we deduce by uniqueness that ηapp, φapp  are even functions of Z in the limit δ → 0, and that ˙¯r = Γ  r0 O(δ) as δ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='25), the function ǫ(t) is implicitly defined by the relation  √  νt  ǫ(t) = ¯r(t) = r0 +  M−1  �  m=0  � t  0  ǫ(s)m ˙¯rm  �  βǫ(s)  �  ds ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='27)  which should hold when 0 < t ≪ Tdif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' As was mentioned in the previous remark, the radial  velocities ˙¯ri are small when δ ≪ 1, so that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='27) will be easy to solve, see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  11  The asymptotic approximation ηapp(R, Z, t) is defined on the whole plane and does not vanish  on the boundary ∂Ωǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' To obtain a valid approximate solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15), we fix σ0 ∈ (0, 1) and  we truncate ηapp outside a large ball of radius ǫ−σ0 by setting  η∗(R, Z, t) = χ0  �  ǫσ0(R2+Z2)1/2�  ηapp(R, Z, t) ,  φ∗(·, t) = BSǫ[η∗(·, t)] ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28)  where χ0 : R+ → [0, 1] is a smooth function such that χ0(r) = 1 for r ≤ 1 and χ0(r) = 0 for  r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The remainder of that approximation is defined as  Rem(R, Z, t) = Lη∗ + ∂R  � ǫη∗  1+ǫR  �  − t∂tη∗ − 1  δ  �  φ∗ ,  η∗  1+ǫR  �  + ǫ¯r  δΓ  �  ˙¯r ∂Rη∗ + ˙¯z ∂Zη∗  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='29)  By construction, this quantity depends on time only through the parameter ǫ =  √  νt/¯r(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The accuracy of our approximate solution is quantified by the following result, which is  established in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7 below :  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Given any γ0 < 1 and any γ5 < 5, there exist a constant C > 0 such that the  remainder (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='29) satisfies  sup  (R,Z)∈Ωǫ  eγ0(R2+Z2)/4 |Rem(R, Z, t)| ≤ C  �  ǫδ + ǫγ5δ−1�  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='30)  whenever the parameters ǫ, δ are small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4  Stability estimates  In our previous work [30], the evolution equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15) was carefully studied in the particular  case where ¯r(t) = r0 and ¯z(t) = z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This does not make any substantial difference for the initial  value problem at fixed viscosity, and we can thus infer from the results of [30] that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15) has  a unique solution η(R, Z, t) with initial data η0 given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Our purpose is to show that, if  the inverse Reynolds number δ = ν/Γ is sufficiently small, the solution η(R, Z, t) remains close  to the approximation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28) on a long time interval of the form (0, Tadvδ−σ), for some small  σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We use the following decomposition :  η(R, Z, t) = η∗(R, Z, t) + δ ˜η(R, Z, t) ,  φ(R, Z, t) = φ∗(R, Z, t) + δ ˜φ(R, Z, t) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='31)  where ˜φ = BSǫ[˜η] in the sense of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The equation satisfied by the perturbation ˜η reads  t∂t˜η + 1  δ  �  φ∗ ,  ˜η  1 + ǫR  �  + 1  δ  �  ˜φ ,  η∗  1 + ǫR  �  +  �  ˜φ ,  ˜η  1 + ǫR  �  − ǫ¯r  δΓ  �  ˙¯r ∂R˜η + ˙¯z ∂Z ˜η  �  = L˜η + ∂R  �  ǫ˜η  1 + ǫR  �  + 1  δ Rem(R, Z, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='32)  Since η∗(R, Z, 0) = η0(R, Z), the nonlinear evolution equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='32) is to be solved with  zero initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The solution is therefore driven by the source term δ−1Rem(R, Z, t), which is  small in view of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' As long as ˜η stays small, the nonlinear term  {˜φ, (1+ǫR)−1˜η} is of course harmless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The most serious difficulty in controlling ˜η using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='32)  comes from the linear terms with a large prefactor δ−1 = Γ/ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' These terms could conceivably  trigger violent instabilities that might lead to strong amplification of ˜η in a short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Our goal  is to show that this scenario does not occur, due to the special structure of the advection terms  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' A similar strategy was applied in the previous work [28] devoted to the vanishing  viscosity limit of interacting vortices in the plane, but the specific estimates used there do not  seem to be easily adaptable to the present situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  12  To control the time evolution of the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='32), we use the energy functional  Eǫ(t) = 1  2  �  Ωǫ  Wǫ ˜η2 dR dZ − 1  2  �  Ωǫ  ˜φ ˜η dR dZ ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='33)  where Wǫ : Ωǫ → (0, +∞) is a weight function that will be described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The first term in  the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='33) is a weighted L2 integral of the vorticity ˜η, similar to weighted  enstrophies that were used for the same purposes in [33,28,30], for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The second term is  just the kinetic energy associated with the vorticity perturbation ˜η, as can be seen by invoking  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='18), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19) and integrating by parts :  1  2  �  Ωǫ  ˜φ ˜η dR dZ = 1  2  �  Ωǫ  |∂R ˜φ|2 + |∂Z ˜φ|2  1 + ǫR  dR dZ = 1  2  �  Ωǫ  �  | ˜UR|2 + | ˜UZ|2�  (1 + ǫR) dR dZ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  To construct the weight Wǫ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='33), we consider three different regions :  1) The inner region where ρ := (R2+Z2)1/2 ≲ ǫ−σ1, for some small σ1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Here we choose  Wǫ =  1  1 + ǫR Φ′  ǫ  �  η∗  1 + ǫR  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='34)  where η∗ is the approximate solution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28) and Φǫ : (0, +∞) → R is a smooth function with  the property that, in the region under consideration,  φ∗ − ¯r ˙¯z  2Γ (1 + ǫR)2 = Φǫ  �  η∗  1 + ǫR  �  + O(ǫδ + ǫγ3) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='35)  for some γ3 < 3 that can be arbitrarily close to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It is not difficult to understand intuitively why  such a function should exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Indeed, in the dimensionless variables (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14), the left-hand side  of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='35) is nothing but the stream function of the approximate solution φ∗ in a frame moving  with constant speed ˙¯z in the vertical direction, see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If we drop the remainder term  O(ǫδ + ǫγ3) and consider ǫ > 0 as a fixed parameter, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='35) expresses a functional relation  between the potential vorticity ζ∗ := (1+ǫR)−1η∗ and the stream function, which implies that η∗  represents a stationary solution of the Euler equation in the moving frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This is not exactly  true, of course, but the estimate on the remainder Rem(R, Z, t) in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5 ensures that  the approximate solution η∗ (for a fixed value of ǫ > 0) is not far from a stationary solution of  Euler, and in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8 we verify that this implies the existence of a function Φǫ satisfying  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Moreover, an easy calculation shows that  1  1 + ǫR Φ′  ǫ  �  η∗  1 + ǫR  �  = 4  ρ2  �  eρ2/4 − 1  �  + O(ǫ) ,  ρ =  �  R2 + Z2 ≤ ǫ−σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  2) The intermediate region where ǫ−σ1 ≲ ρ ≤ ǫ−σ2, for some σ2 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In this area we assume  that the weight is approximately constant in space, with value Wǫ ≈ exp(ǫ−2σ1/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  3) The far field region where ρ ≥ ǫ−σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Here we take Wǫ ≈ exp(ρ2γ/4), where γ = σ1/σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The actual construction of the weight is more complicated, and ensures that Wǫ is Lipschitz  continuous at the boundaries of the three regions under consideration, see Section 4 below  for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' For the moment, we just mention that our choice of the energy functional in the  inner region is related to Arnold’s variational characterization of the steady states of the Euler  equation, as discussed in our previous work [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In fact, if we suppose that ζ∗ is a stationary  solution of the axisymmetric Euler equation in a moving frame (which not exactly true), then  the functional (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='33) with the weight (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='34) corresponds, up to a constant factor, to the second  13  ≈ ǫ−σ1  ǫ−σ0  ǫ−1  ǫ−σ2  ρ  1  exp(ǫ−2σ1/4)  Wǫ(R, Z)  A(ρ)  exp(ρ2γ/4)  Figure 2: When ǫ > 0 is small, the weight Wǫ(R, Z) entering the energy functional (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='33) is close to a  piecewise smooth radially symmetric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In the inner region where ρ := (R2+Z2)1/2 ≲ ǫ−σ1, we  have Wǫ ≈ A(ρ) := (4/ρ2)  �  eρ2/4 − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' When Wǫ reaches the threshold value exp(ǫ−2σ1/4), it remains  approximately constant until ρ = ǫ−σ2, and outside that region we have Wǫ ≈ exp(ρ2γ/4) with γ = σ1/σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The dashed lines reflect the fact that exp(ρ2γ/4) ≲ Wǫ ≲ A(ρ) where the implicit constants do not depend  on the parameter ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The intermediate scales ǫ−σ0, where the truncation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28) occurs, and ǫ−1, which is  the distance from the origin to the boundary ∂Ωǫ, are indicated for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  variation of the kinetic energy on the isovortical surface, which is the set of (potential) vorticities  ζ := (1 + ǫR)−1η that are measure-preserving rearrangements of ζ∗ [3, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Since the kinetic  energy is conserved under the inviscid dynamics, the advection terms involving δ−1 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='32),  which originate from the linearization of Euler’s equation at the “steady state” ζ∗, do not  contribute to the time evolution of the functional Eǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  In reality ζ∗ is only an approximate  steady state of Euler, and the cancellations alluded to above only occur up to correction terms  of order O(ǫδ + ǫγ3), but this is sufficient to cancel the dangerous factors δ−1 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' On the  other hand, away from the inner region, the last term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='33) is extremely small, so that our  functional Eǫ reduces to a weighted enstrophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We assume that the weight Wǫ is approximately  constant in the intermediate region, so that the advection terms in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='32) do not contribute  to the evolution of Eǫ, and in the far field region the dynamics is dominated by the diffusion  operator L in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='32) so that we can just take any radially symmetric weight with sufficiently  fast growth at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  A technical difficulty inherent to our approach is the fact that the functional Eǫ is not  coercive, unless the perturbed vorticity ˜η satisfies some moment conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The problem comes  from the inner region, where the last term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='33) plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The results  established in [31, Section 2] indicate that Eǫ is positive definite provided ˜η has zero mean and  vanishing first order moments with respect to the space variables R, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In practice this means  that, in addition to the information provided by the energy Eǫ, we must control the integral  and the first order moments of the perturbed vorticity ˜η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It turns out that  �  ˜η dR dZ is always  extremely small, of the order of O(exp(−c/ǫ2)) for some c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The radial moment  �  R ˜η dR dZ  may take larger values, but can be controlled using the conservation of the total impulse of the  vortex ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Finally, to get rid of the vertical moment  �  Z ˜η dR dZ, we exploit the translation  invariance of the system in the vertical direction, and we introduce a “modulation parameter”  in the same spirit as for the stability analysis of solitary waves [55,46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' A precise description of  our approach to control the moments of the vorticity ˜η can be found in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  14  Disregarding these technical questions for the moment, we briefly indicate how the argument  is concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If we differentiate Eǫ with respect to time, and use the evolution equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='32)  together with the estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='30) on the source term, we obtain after lengthy calculations a  differential inequality of the form  tE′  ǫ(t) ≤ −c1Eǫ(t) + c2  �  ǫ2 + ǫ2γ3  δ2  �  ,  t ∈ (0, Tadvδ−σ) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='36)  for some positive constants c1, c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Integrating (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='36) with initial condition Eǫ(0) = 0, we find  Eǫ(t) ≤ c3  �  ǫ2 + ǫ2γ3  δ2  �  ,  t ∈ (0, Tadvδ−σ) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='37)  and using in addition the bounds on the moments of ˜η that are obtained by a different argument  we arrive at an estimate of the form δ∥˜η(t)∥Xǫ ≤ c(ǫδ + ǫγ3), where Xǫ is the weighted L2 space  equipped with the norm  ∥˜η∥Xǫ =  ��  Ωǫ  Wǫ(R, Z) |˜η(R, Z)|2 dR dZ  �1/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='38)  This space depends on time through the parameter ǫ > 0, but we recall that the weight function  satisfies a uniform lower bound of the form Wǫ(R, Z) ≳ exp(ρ2γ/4), see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The main result of Section 4 can now be formulated as follows :  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' For any γ3 ∈ (2, 3), there exist constants K > 0, δ0 > 0, and σ ∈ (0, 1) such  that, for all Γ > 0, all r0 > 0, and all ν > 0 satisfying δ := ν/Γ ≤ δ0, the unique solution η of  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15) with initial data (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='22) satisfies  ∥η(t) − η∗(t)∥Xǫ ≤ K  �  ǫδ + ǫγ3�  ,  t ∈ (0, Tadvδ−σ) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='39)  where ǫ(t) =  √  νt/¯r(t) and η∗ is the approximate solution defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='24), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  In the statement above, it is understood that the location (¯r(t), ¯z(t)) of the vortex ring is  determined by the expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In particular the velocities ˙¯r(t), ˙¯z(t) satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It is  not difficult to verify that Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6 implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9 below for some  details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Here we just show how to derive estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11), which is essentially a reformulation of  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' By construction we have ∥η∗(t) − η0∥Xǫ = O(ǫ), where η0 is the Gaussian function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='22),  and the lower bound Wǫ(R, Z) ≳ exp(ρ2γ/4) implies that Xǫ ֒→ L1(Ωǫ) uniformly in ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It thus  follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='39) that  ∥η(t) − η0∥L1(Ωǫ) ≤ C1  �  ∥η(t) − η∗(t)∥Xǫ + ∥η∗(t) − η0∥Xǫ  �  ≤ C2ǫ ,  for any t ∈ (0, Tadvδ−σ), and returning to the original variables (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='13) we arrive at (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='24) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='39) that the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15) satisfies  η(R, Z, t) = η0(R, Z) + ǫη1(R, Z) + ǫ2η2(R, Z, βǫ) + O  �  δǫ + ǫγ3�  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='40)  where the remainder term is understood in the topology of Xǫ as ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Here η0 is the Gaussian  function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='22), and the vorticity profiles η1, η2 are explicitly constructed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Since  δ ≲ ǫ2 except for very small times, see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2, we see that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='40) determines the deformation  of the vortex core due to self-induction up to third order in ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  15  3  Construction of the approximate solution  In this section we construct perturbatively an approximate solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='23) such that the  corresponding remainder satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Approximations of vortex rings with varying degrees of  accuracy were obtained by many authors, and typically rely on matched asymptotics expansions  where the inner core of the vortex and the outer region are considered separately, see [40,37,20,  23,24] in the inviscid case and [52,12,27] in the viscous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Here we rather follow the direct  approach introduced in [28] for interacting vortices in the plane, which does not rely on matched  asymptotics techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1  Expansion of the Biot-Savart formula  Our first task is to compute an accurate asymptotic expansion of the function F(s) defined by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5) in the limit where s → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This can be done by expressing F in terms of elliptic integrals,  a procedure initiated in the early references [36,47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' For 0 < s < 4 we have the power series representation  F(s) = log  � 8  √s  �  ∞  �  m=0  Amsm +  ∞  �  m=0  Bmsm ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1)  where Am, Bm are real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Moreover  A0 = 1 ,  A1 = 3  16 ,  A2 = − 15  1024 ,  B0 = −2 ,  B1 = − 1  16 ,  B2 =  31  2048 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If s > 0 and k = 2/√s + 4 ∈ (0, 1), it is straightforward to verify that  F(s) =  � π/2  0  1 − 2 sin2 ψ  �  sin2 ψ + s/4  dψ = 2 − k2  k  K(k) − 2  k E(k) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3)  where K(k), E(k) are the complete elliptic integrals with modulus k :  K(k) =  � π/2  0  1  �  1 − k2 sin2 θ  dθ ,  E(k) =  � π/2  0  �  1 − k2 sin2 θ dθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  We are interested in the limit where s → 0, namely k → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Introducing the complementary  modulus κ =  √  1 − k2, we have the power series expansions (see [17])  K(k) =  ∞  �  m=0  a2  m κ2m�  log 1  κ + 2bm  �  ,  E(k) = 1 +  ∞  �  m=0  2m + 1  2m + 2 a2  m κ2m+2�  log 1  κ + bm + bm+1  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4)  where a0 = 1, b0 = log(2), and  am = 1  2 · 3  4 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' · 2m−1  2m  ,  bm = log(2) +  2m  �  ℓ=1  (−1)ℓ  ℓ  ,  m ∈ N∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4), we obtain a representation of the form  F(s) =  1 + κ2  √  1 − κ2 K(k) −  2  √  1 − κ2 E(k) = log  � 4  κ  �  ∞  �  m=0  Cmκ2m +  ∞  �  m=0  Dmκ2m ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5)  16  which converges for 0 < κ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Moreover, a direct calculation shows that  C0 = 1 ,  C1 = 3  4 ,  C2 = 33  64 ,  D0 = −2 ,  D1 = −3  4 ,  D2 = − 81  128 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6)  As κ2 = s/(s + 4), the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5) can be written in the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1), and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6)  we see that the first coefficients satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Various asymptotic expansions of the stream function given by the Biot-Savart law  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4) can be found in the literature [37,20,42,52,27], and are easily recovered using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  We next consider the rescaled Biot-Savart formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='20), which can be written in the equiv-  alent form  φ(R, Z) =  1  2π  �  Ωǫ  Kǫ(R, Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' R′, Z′) η(R′, Z′) dR′ dZ′ ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7)  where  Kǫ =  �  (1+ǫR)(1+ǫR′) F  �  ǫ2D2  (1+ǫR)(1+ǫR′)  �  ,  D2 = (R−R′)2 + (Z−Z′)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8)  To simplify the notations below, we define  βǫ = log 1  ǫ ,  L(R, Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' R′, Z′) = log  � 8  D  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9)  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' For any (R, Z), (R′, Z′) ∈ R2 with (R, Z) ̸= (R′, Z′) and any sufficiently small  ǫ > 0, we have the expansion  Kǫ = (βǫ + L)  ∞  �  m=0  ǫmPm +  ∞  �  m=0  ǫmQm ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10)  where Pm(R, Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' R′, Z′), Qm(R, Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' R′, Z′) are homogeneous polynomials of degree m in the three  variables R, R′, and Z − Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Moreover  P0  = 1  P1  =  1  2(R + R′)  P2  =  1  16(R − R′)2 + 3  16(Z − Z′)2  Q0 = −2  Q1 = − 1  2(R + R′)  Q2 =  1  4(R2 + R′2) − 1  16 D2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If (R, Z), (R′, Z′) are as in the statement, we take ǫ > 0 small enough so that  max  �  |R|, |R′|  �  < 1  ǫ ,  and  s :=  ǫ2D2  (1+ǫR)(1+ǫR′) < 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='12)  As D ̸= 0 by assumption, we have 0 < s < 4, so that we can apply expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1) to the  quantity F(s) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In view of definitions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9) we have  log  � 8  √s  �  = βǫ + L + 1  2 log(1 + ǫR) + 1  2 log(1 + ǫR′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='13)  We observe that the last two terms in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='13), as well as the prefactor  �  (1+ǫR)(1+ǫR′) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8)  and each monomial sm in the series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1), can be expanded into a power series in the three  variables ǫR, ǫR′, and ǫ(Z − Z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Thus, combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8), we obtain a representation  of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10), where the first homogeneous polynomials Pm, Qm are easily computed using  the explicit values (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  17  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In what follows, with a slight abuse of notation, we denote by L the integral  operator on R2 given by the kernel (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' For any continuous and rapidly decreasing function  η : R2 → R, we thus have  �  Lη  �  (R, Z) =  �  R2 log  �  8  �  (R−R′)2 + (Z−Z′)2  �  η(R′, Z′) dR′ dZ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14)  Similarly, we associate integral operators to the homogeneous polynomials Pm, Qm in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10), as  well as to the functions LPm for all m ∈ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Using the notation introduced in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4, we define the linear operators  BS0 =  1  2π L ,  and  BSm =  1  2π  �  βǫPm + LPm + Qm  �  ,  for all m ∈ N∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15)  Note that, for m ≥ 1, the linear operator BSm depends on the parameter ǫ through the  constant factor βǫ = log(1/ǫ), but for simplicity this mild dependence is not indicated explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  For convenience, we do not include the constant term βǫP0 + Q0 ≡ βǫ − 2 in the definition of  BS0, because the stream function is only defined up to an additive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It is important  to observe that, in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14) and in the corresponding definition of the integral operators Pm, Qm,  the integration is performed on the whole plane R2, rather than on the half-plane Ωǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This  is justified because these operators will always be applied to functions that decay rapidly at  infinity, so that the integration on R2 \\ Ωǫ and gives a contribution of order O(ǫ∞) as ǫ → 0,  which can be neglected in our perturbative expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If η : R2 → R is compactly supported,  then according to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3 the following equality holds in any bounded region of R2:  BSǫ[η] = βǫ − 2  2π  �  R2 η(R′, Z′) dR′ dZ′ +  ∞  �  m=0  ǫm BSm[η] ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='16)  provided ǫ > 0 is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' As was already mentioned, the first term in the right-hand  side is a constant that can be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2  Function spaces and linear operators  We next introduce the function spaces in which we shall construct our approximate solution of  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' These spaces consist of functions that are defined on the whole space R2, and not just  on the half-plane Ωǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Indeed, at each step of the approximation, the vorticity profile ηj(R, Z, βǫ)  and the stream function φj(R, Z, βǫ) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='24) are defined for all (R, Z) ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' To simplify the  writing we often denote X = (R, Z), and we use polar coordinates (ρ, ϑ) in R2 defined by the  relations R = ρ cos ϑ, Z = ρ sin ϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Following [32,33] we introduce the weighted L2 space  Y =  �  η ∈ L2(R2)  ���  �  R2 |η(X)|2 e|X|2/4 dX < ∞  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='17)  equipped with the scalar product (η1, η2)Y =  �  R2 η1(X)η2(X) e|X|2/4 dX and the associated  norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  We also introduce the differential operator L : D(L) → Y corresponding to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='16),  namely  Lη = ∆η + 1  2 X · ∇η + η ,  η ∈ D(L) =  �  η ∈ Y  ��� ∆η ∈ Y , X · ∇η ∈ Y  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='18)  18  as well as the integro-differential operator Λ : D(Λ) → Y defined by  Λη =  1  2π  ��  Lη0 , η  �  +  �  Lη , η0  ��  ,  η ∈ D(Λ) =  �  η ∈ Y  ���  �  Lη0 , η  �  ∈ Y  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19)  where η0 is the Gaussian function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='22) and L denotes the integral operator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Here and  in what follows the Poisson bracket is understood with respect to the rescaled variables (R, Z),  so that {φ, η} = ∂Rφ ∂Zη − ∂Zφ ∂Rη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We recall the following well-known properties :  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' [32, 33, 43]  1) The linear operator L is self-adjoint in Y, with purely discrete spectrum  σ(L) =  �  −n  2  ��� n = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The kernel of L is one-dimensional and spanned by the Gaussian function η0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' More generally,  for any n ∈ N, the eigenspace corresponding to the eigenvalue λn = −n/2 is spanned by the n+1  Hermite functions ∂αη0 where α = (α1, α2) ∈ N2 and α1 + α2 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  2) The linear operator Λ is skew-adjoint in Y, so that Λ∗ = −Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Moreover,  Ker(Λ) = Y0 ⊕  �  β1∂Rη0 + β2∂Zη0  �� β1, β2 ∈ R  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='20)  where Y0 ⊂ Y is the subspace of all radially symmetric elements of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  A crucial observation is that both operators L, Λ are invariant under rotations about the  origin in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It is therefore advantageous to decompose the space Y into a direct sum  Y =  ∞⊕  n=0 Yn ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='21)  where Y0 ⊂ Y is as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6 and, for all n ≥ 1, the subspace Yn ⊂ Y consists  of all functions η ∈ Y such that η(ρ cos ϑ, ρ sin ϑ) = a1(ρ) cos(nϑ) + a2(ρ) sin(nϑ) for some  a1, a2 : R+ → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It is clear that Yn ⊥ Yn′ if n ̸= n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In particular, in view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='20), we have  Yn ∈ Ker(Λ)⊥ for all n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' When n = 1, the functions ∂Rη0, ∂Zη0 belong to Y1 ∩ Ker(Λ), and  we define  Y′  1 = Y1 ∩ Ker(Λ)⊥ =  �  η ∈ Y1  ����  �  R2 η(R, Z)R dR dZ =  �  R2 η(R, Z)Z dR dZ = 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='22)  Since Λ is skew-adjoint, we have Ker(Λ)⊥ = Ran(Λ), but the image of Λ is not dense in  Y and, therefore, we cannot solve the equation Λη = f for any f ∈ Ker(Λ)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' As is shown  in [34,28], the problem disappears if one assumes in addition that f belongs to a smaller space  such as  Z =  �  η : R2 → R  ��� e|X|2/4η ∈ S∗(R2)  �  ⊂ Y ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='23)  where S∗(R2) denotes the space of all smooth functions which are slowly growing at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  More precisely, a smooth function w : R2 → R belongs to S∗(R2) if, for any α = (α1, α2) ∈ N2,  there exist C > 0 and N ∈ N such that |∂αw(X)| ≤ C(1 + |X|)N for all X ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  To formulate the main technical result of this section, we introduce the notation  ϕ(ρ) =  1  2πρ2  �  1 − e−ρ2/4�  ,  h(ρ) =  ρ2/4  eρ2/4 − 1 ,  ρ > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='24)  The following statement is a slight extension of [28, Lemma 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' For the reader’s convenience, we  give a short proof of it in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1, emphasizing the case n = 1 which was not treated in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  19  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If n ≥ 2 and f ∈ Yn ∩ Z, or if n = 1 and f ∈ Y′  1 ∩ Z, there exists a  unique η ∈ Yn ∩ Z (respectively, a unique η ∈ Y′  1 ∩ Z) such that Λη = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In particular, if  f = a(ρ) sin(nϑ), then η = ω(ρ) cos(nϑ), where  ω(ρ) = h(ρ)Ω(ρ) + a(ρ)  nϕ(ρ) ,  ρ > 0 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='25)  and where Ω : (0, ∞) → R is the unique solution of the differential equation  − Ω′′(ρ) − 1  ρ Ω′(ρ) +  �n2  ρ2 − h(ρ)  �  Ω(ρ) =  a(ρ)  nϕ(ρ) ,  ρ > 0 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='26)  such that Ω(ρ) = O(ρn) as ρ → 0 and Ω(ρ) = O(ρ−n) as ρ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' As was observed in [28], if f = a(ρ) cos(nϑ), then η = −ω(ρ) cos(nϑ), where ω  is still given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='25), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The general case where f = a1(ρ) sin(nϑ) + a2(ρ) cos(nϑ) follows  by linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  In the construction of an approximate solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='23), we shall encounter linear equations  of the form δ(κ− L)η + Λη = f, where κ > 0 is fixed and the parameter δ > 0 can be arbitrarily  small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6 implies that the linear operator δ(κ − L) + Λ is invertible in Y for any  δ > 0, so that our problem has a unique solution η for any f ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' However, the norm of  the inverse is O(δ−1), and it is intuitively clear that η has a regular limit when δ → 0 only if  f ∈ Ran(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Under similar assumptions as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7, we have the following result:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Assume that n ≥ 2 and f ∈ Yn ∩ Z, or that n = 1 and f ∈ Y′  1 ∩ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' For any  fixed κ > 0 and all δ ≥ 0, the equation  δ  �  κ − L  �  ηδ + Ληδ = f  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='27)  has a unique solution ηδ ∈ Yn (respectively, ηδ ∈ Y′  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Moreover there exists a constant C > 0,  depending only on f, such that  ∥ηδ − η∥Y ≤ Cδ  and  ∥ηδ − η − δˆη∥Y ≤ Cδ2 ,  for all δ ≥ 0 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28)  where  1) η ∈ Yn ∩ Z (respectively, η ∈ Y′  1 ∩ Z) is the unique solution of Λη = f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  2) ˆη ∈ Yn ∩ Z (respectively, ˆη ∈ Y′  1 ∩ Z) is the unique solution of Λˆη = (L − κ)η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9 is similar to the one of Lemma 5 in [28], and can therefore be  omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We just add here the following comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' First, since f ∈ Yn ∩ Z or f ∈ Y′  1 ∩ Z,  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7 shows that there is a unique η in the same class such that Λη = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It is then  easily verified that (κ−L)η still belongs to Yn ∩Z (respectively, to Y′  1 ∩Z), so that the equation  Λˆη = −(κ − L)η also has a unique solution ˆη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This shows that the approximations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28) of  ηδ are well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' On the other hand, proceeding as in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7, it is  possible to verify that the solution ηδ of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='27) belongs to the space Z, and that the expansions  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28) also hold in the (Fr´echet) topology of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' But for our purposes it will be sufficient to use  η + δˆη ∈ Z as an approximation of ηδ, observing that  �  δ  �  κ − L  �  + Λ  �  (η + δˆη) − f = δ2(κ − L)ˆη = OZ(δ2) ,  as δ → 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='29)  20  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3  First order approximation  We now begin the construction of an approximate solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='23) in the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We recall  that, for an exact solution, the stream function is determined by the relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='20), which we  write in the compact form φ = BSǫ[η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' For our approximate solution, we expand the Biot-Savart  operator as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='16), omitting the constant term in the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We thus obtain the  formal relation  �  ∞  �  m=0  ǫm BSm  ��  M  �  m=0  ǫmηm  �  =  M  �  m=0  ǫmφm + O  �  ǫM+1�  ,  which we assume to be satisfied order by order in ǫ, up to order M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This leads to the relations  φ0 = BS0[η0], φ1 = BS0[η1] + BS1[η0], and more generally  φm = BS0[ηm] + BS1[ηm−1] + · · · + BSm−1[η1] + BSm[η0] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='30)  In particular, in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15), the leading order of our approximation is  η0(R, Z) =  1  4π e−(R2+Z2)/4 ,  φ0(R, Z) =  1  2π  �  Lη0  �  (R, Z) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='31)  where L is the integral operator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The stream function φ0 has the expression  φ0(R, Z) = φ0(0) − 1  4πEin  �R2+Z2  4  �  ,  where  Ein(x) =  � x  0  1 − e−t  t  dt ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='32)  so that φ0 is radially symmetric and φ0(R, Z) ∼ −(2π)−1 log ρ as ρ := (X2 + Z2)1/2 → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The value at the origin does not play a big role in our analysis, but can be computed too, see  Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2 :  φ0(0) = log(2)  π  + γE  4π ,  where γE is Euler’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Before proceeding further, we estimate the time derivative of the quantity ǫ =  √  νt/¯r(t)  introduced in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='25), we have  t˙ǫ = ǫ  2 − ǫt ˙¯r  ¯r  = ǫ  2 − ǫt  ¯r  �  ˙¯r0 + ǫ ˙¯r1 + ǫ2 ˙¯r2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='33)  At this stage the radial velocity profiles ˙¯rj are not determined yet, but in view of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3 we  can anticipate the fact that | ˙¯r| = (Γ/r0)·O(δ) as δ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Since δt = (r2  0/Γ)·O(ǫ2) by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2,  it follows that ¯r(t) = r0  �  1 + O(ǫ2)  �  and that t˙ǫ = ǫ/2 + O(ǫ3) as ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  With that observation in mind, we substitute the expansions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='24), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='25) into the evolution  equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='23), keeping only the terms that are exactly of order ǫ or ǫβǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This gives the relation  �  φ1 , η0  �  +  �  φ0 , η1  �  + η0∂Zφ0 − r0  Γ  �  ˙¯r0 ∂Rη0 + ˙¯z0 ∂Zη0  �  = δ  �  ∂Rη0 +  �  L − 1  2  �  η1 − t∂tη1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='34)  To solve (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='34) we first impose the relation  ˙¯r0 = −Γδ  r0  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='35)  which ensures that the terms involving ∂Rη0 cancel exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We also assume that η1 does not  depend on βǫ, so that ∂tη1 = 0 (this property will be verified later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' On the other hand, using  21  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='30) with m = 1, we deduce that {φ1 , η0} = {BS0[η1] , η0}+{BS1[η0] , η0}, where BS0, BS1 are  defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='31) and the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19) of the linear operator Λ, we thus find  �  φ1 , η0  �  +  �  φ0 , η1  �  =  1  2π  ��  Lη1 , η0  �  +  �  Lη0 , η1  ��  +  �  BS1[η0] , η0  �  = Λη1 + βǫ − 1  2π  �  P1η0 , η0  �  + 1  2π  �  LP1η0 , η0  �  ,  where in the second line we used the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15) of BS1 and the fact that Q1 = −P1 in  view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Now, elementary calculations that are reproduced in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2 show that  �  P1η0 , η0  �  = 1  2 ∂Zη0 ,  and  1  2π  �  LP1η0 , η0  �  = 1  2 ∂Z  �  φ0η0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='36)  It follows that we can write (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='34) in the equivalent form  Λη1 + δ  � 1  2 − L  �  η1 =  �r0  Γ  ˙¯z0 − βǫ − 1  4π  �  ∂Zη0 − 3  2 (∂Zφ0)η0 − 1  2 φ0∂Zη0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='37)  Using the explicit expressions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='31), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='32) of the profiles η0, φ0, it is straightforward to  verify that the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='37), which we denote by −R1, belongs to Y1 ∩ Z, where  Y1, Z are the function spaces defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='21), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Therefore, according to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9,  the linear equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='37) has a unique solution in Y1 for all δ > 0, and that solution has a  well-defined limit as δ → 0 if and only if R1 ∈ (ker Λ)⊥, namely if R1 ∈ Y′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='22),  this gives the solvability condition  �  R2 R1Z dR dZ = 0, which determines uniquely the value of  the constant ˙¯z0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The calculations are presented in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2, and yield the following  expression of the vertical velocity to leading order :  ˙¯z0 =  Γ  4πr0  �  βǫ − 1 + 2v  �  ,  where  v = 3  4 log(2) + 1  4 γE + 1  4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='38)  Here again γE = 0, 5772 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' denotes Euler’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='38), including the leading term βǫ = log(1/ǫ) and the correct  value of the constant 2v − 1, was established by Saffman [51], see also Fukumoto & Moffatt [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  We assume henceforth that ˙¯z0 is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='38), so that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='37) reduces to  Λη1 + δ  � 1  2 − L  �  η1 =  v  2π ∂Zη0 − 3  2 (∂Zφ0)η0 − 1  2 φ0∂Zη0 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='39)  where the right-hand side −R1 now belongs to Y′  1 ∩ Z and is independent of ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The solution of  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='39) given by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9 can be expanded as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28), and if is sufficient for our purposes  to consider only the approximate solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='29), which reads  η1(R, Z) = R η10(ρ) + δZ η11(ρ) ,  ρ =  �  R2 + Z2 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='40)  where Λ(R η10) = −R1 and Λ(Z η11) = (L − 1  2)(R η10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Note that η1 ∈ Y′ ∩ Z, which implies  in particular that the functions η10, η11 are smooth and have a Gaussian decay at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The  corresponding stream function φ1 = BS0[η1] + BS1[η0] has a similar expression :  φ1(R, Z, βǫ) = βǫ − 1  4π  R + R  2 φ0 − ∂Rφ0 + R φ10(ρ) + δZ φ11(ρ) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='41)  which can be established using the calculations in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Here the functions φ10, φ11 are  smooth and decay at least like 1/ρ2 as ρ → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Note that φ1 involves the time-dependent term  βǫ = log(1/ǫ), so that ∂tφ1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' With the choices (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='35), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='38), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='40), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='41), the relation  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='34) is not satisfied exactly, but the difference of both members is O(δ2) in the topology of Z,  which is all we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  22  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4  Second order approximation  We next compute the second order terms in the asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' As we shall see,  it is consistent at this stage to take  ˙¯r1 = ˙¯z1 = 0 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='42)  so we make that assumption from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' As before, we deduce from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='33), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='35), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='42) that  ¯r(t) = r0  �  1 + O(ǫ2)  �  and t˙ǫ = ǫ/2 + O(ǫ3) as ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='24), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='25) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='23) and  keeping only the terms involving ǫ2 or ǫ2βǫ, we obtain the relation  �  φ2 , η0  �  +  �  φ1 , η1 − Rη0  �  +  �  φ0 , η2 − Rη1 + R2η0  �  − r0  Γ  �  ˙¯r0 ∂Rη1 + ˙¯z0 ∂Zη1  �  = δ  ��  L − 1  �  η2 + ∂R(η1 − Rη0) − t∂tη2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='43)  In view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='35), the terms involving ∂Rη1 cancel exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Moreover, we know from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='30) that  φ2 =  1  2π  �  Lη2 +  �  βǫP1 + LP1 + Q1  �  η1 +  �  βǫP2 + LP2 + Q2  �  η0  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='44)  where the notations are introduced in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Recalling the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19) of the operator  Λ, we can thus write (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='43) in the equivalent form  Λη2 + δ  �  t∂tη2 +  �  1 − L  �  η2  �  + R2 = 0 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='45)  where  R2 = 1  2π  �  (βǫ − 1)P1η1 + LP1η1 , η0  �  + 1  2π  �  βǫP2η0 + LP2η0 + Q2η0 , η0  �  +  �  φ1 , η1  �  + (∂Zφ1)η0 + (∂Zφ0)η1 − R  ��  φ1 , η0  �  +  �  φ0 , η1  �  + 2(∂Zφ0)η0  �  + δ∂R(Rη0) − r0 ˙¯z0  Γ  ∂Zη1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='46)  We have the following result, whose proof is postponed to Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3 :  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The function R2 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='46) belongs to (δY0 + Y2) ∩ Z and satisfies  R2 = 3βǫ  16π RZη0 + RZχ20 + δ  �  χ21 + (R2 − Z2)χ22  �  + δ2RZχ23 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='47)  for some (time-independent) radially symmetric functions χ20, χ21, χ22, χ23 ∈ Y0 ∩ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  In view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='47), we look for a solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='45) in the form η2 = βǫˆη20 + ˆη21 + ˆη22, where  ˆη20, ˆη21 ∈ Y2 and ˆη22 ∈ Y0 do not depend on βǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Inserting this ansatz into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='45) and using the  fact that t∂tβǫ = −1/2 + O(ǫ2), we obtain the system  Λˆη20 + δ  �  1 − L  �  ˆη20 +  3  16π RZη0 = 0 ,  Λˆη21 + δ  �  1 − L  �  ˆη21 − δ  2 ˆη20 + P2  �  R2 − 3βǫ  16π RZη0  �  = 0 ,  δ  �  1 − L  �  ˆη22 + P0R2 = 0 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='48)  where Pn denotes the orthogonal projection in Y onto the subspace Yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The first two equations  in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='48) have a unique solution by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9, and as in the previous section we are satisfied  23  with the approximate solutions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28) which are valid up to O(δ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Since P0R2 = δχ21 by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='47),  the third equation reduces to (1 − L)ˆη22 + χ21 = 0, which also has a unique solution due to  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We conclude that we can choose η2 in the form  η2(R, Z, βǫ) = βǫ  �  (R2−Z2)η20 + δRZη21  �  + (R2−Z2)η22 + δRZη23 + η24 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='49)  where all functions η2j belong to Y0 ∩ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='44) and the calculations at the beginning of  Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3, we obtain a similar expression for the corresponding stream function  φ2(R, Z, βǫ) = βǫ  �  (R2−Z2)φ20 + δRZφ21  �  + (R2−Z2)φ22 + δRZφ23 + βǫφ24 + φ25 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='50)  where the functions φ2j are radially symmetric and belong to S∗(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' With these choices, the  difference of both members in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='43) is O(δ2 + ǫ2δ) in the topology of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5  Third order approximation  The third order in the asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='24) can be computed in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' According  to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='35), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='42) and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3, we have ¯r(t) = r0  �  1 − ǫ2 + O(ǫ4−)  �  as ǫ → 0, and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='33)  we deduce that t˙ǫ = ǫ/2 + ǫ3 + O(ǫ5−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' So, if we substitute (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='24), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='25) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='23) and keep  only the terms involving ǫ3 or ǫ3βǫ, we find  �  φ3 , η0  �  +  �  φ2 , η1 − Rη0  �  +  �  φ1 , η2 − Rη1 + R2η0  �  +  �  φ0 , η3 − Rη2 + R2η1 − R3η0  �  − r0  Γ  �  ˙¯r0 ∂Rη2 +  � ˙¯r2− ˙¯r0  �  ∂Rη0 + ˙¯z0 ∂Zη2 +  � ˙¯z2− ˙¯z0  �  ∂Zη0  �  = δ  ��  L − 3  2  �  η3 + ∂R(η2 − Rη1 + R2η0) − t∂tη3 − η1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='51)  On the other hand, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='30) with m = 3 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15), we obtain  φ3 =  3  �  m=0  BSm[η3−m] =  1  2π Lη3 + 1  2π  3  �  m=1  �  (βǫ + L)Pm + Qm  �  η3−m ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='52)  where the polynomials Pm, Qm are defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11) for m ≤ 2 and in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19) for m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We can  thus write (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='51) in the form  Λη3 + δ  �  t∂tη3 +  � 3  2 − L  �  η3  �  + R3 = 0 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='53)  where  R3 =  1  2π  �  3  �  m=1  �  (βǫ + L)Pm + Qm  �  η3−m , η0  �  +  �  φ2 , η1 − Rη0  �  +  �  φ1 , η2 − Rη1 + R2η0  �  −  �  φ0 , Rη2 − R2η1 + R3η0  �  − r0  Γ  �� ˙¯r2 − ˙¯r0  �  ∂Rη0 +  � ˙¯z2 − ˙¯z0  �  ∂Zη0 + ˙¯z0∂Zη2  �  + δ∂R  �  Rη1 − R2η0  �  + δη1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='54)  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The function R3 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='54) belongs to (Y1 + Y3) ∩ Z and satisfies  R3 = βǫ  �  R2Zχ30 + Zχ31  �  + R2Zχ32 + Zχ33 + O(δ) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='55)  for some (time-independent) radially symmetric functions χ30, χ31, χ32, χ33 ∈ Y0 ∩ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  24  The proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='12 is a direct calculation that is briefly outlined in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In  particular we verify there that the quantity R3 does not contain any factor β2  ǫ , which is rather  surprising since φ1, φ2, and η2 all contain at least one term multiplied by βǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We do not need  the expression of the O(δ) terms in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='55), but they can be computed too and are found to be  of the form δβǫ  �  R3 ˜χ30 + R˜χ31  �  + δ  �  R3 ˜χ32 + R˜χ33  �  , where ˜χ3j are radially symmetric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Finally we mention that R3 also includes terms of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='55) that are multiplied by δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  As can be seen from the last line of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='54), there is a unique way to choose the quantities  ˙¯r2 and ˙¯z2 so that R3 ∈ Y′  1 + Y3, where Y′  1 is the subspace defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='55),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='35), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='38), the velocities obtained in this way satisfy  r0  Γ  ˙¯r2 =  �  c1βǫ + c2  �  δ ,  r0  Γ  ˙¯z2 = c3βǫ + c4 + O(δ2) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='56)  for some constants c1, c2, c3, c4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Now, decomposing R3 = βǫR31 + R32 where R31, R32 are  independent of βǫ, we look for a solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='53) in the form η3 = βǫˆη31 + ˆη32 where  Λˆη31 + δ  � 3  2 − L  �  ˆη31 + R31 = 0 ,  Λˆη32 + δ  � 3  2 − L  �  ˆη32 − δ  2 ˆη31 + R32 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='57)  Since R31, R32 ∈ Y′  1+Y3, both equations in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='57) can be solved using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' However,  at this stage, we are satisfied with the crude approximation which consists in neglecting the terms  proportional to δ, so we really solve Λˆη3j + R3j = 0 for j = 1, 2 using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We thus  obtain an approximate solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='53) of the form  η3(R, Z, βǫ) = βǫ  �  R3η30 + Rη31  �  + R3η32 + Rη33 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='58)  where all functions η3j belong to Y0 ∩ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='52) we deduce the corresponding expression  of the stream function  φ3(R, Z, βǫ) = βǫ  �  R3φ30 + Rφ31  �  + R3φ32 + Rφ33 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='59)  where the functions φ3j are radially symmetric and belong to S∗(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='59) does not  contain any factor β2  ǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' With the choices (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='58), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='59), the difference of both members in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='53)  is O(δ) in the topology of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6  Fourth order approximation  Finally we compute the fourth order approximation, which is the final step in our construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  No modification of the vortex speed is needed at this stage, so we can take  ˙¯r3 = ˙¯z3 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='60)  The full expansion of the vortex speed is therefore  ˙¯r(t) = ˙¯r0(βǫ) + ǫ2 ˙¯r2(βǫ) ,  ˙¯z(t) = ˙¯z0(βǫ) + ǫ2 ˙¯z2(βǫ) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='61)  where ˙¯r0, ˙¯z0 are defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='35), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='38) and ˙¯r2, ˙¯z2 satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' As is easily verified, this implies  that ¯r(t) = r0  �  1 − ǫ2 + O(ǫ4βǫ)  �  and t˙ǫ = ǫ/2 + ǫ3 + O(ǫ5βǫ) as ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  We look for an approximate solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='23) of the form  ηapp(R, Z, t) =  4  �  m=0  ǫmηm(R, Z, βǫ) ,  φapp(R, Z, t) =  4  �  m=0  ǫmφm(R, Z, βǫ) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='62)  25  where the profiles ηm, φm for m ≤ 3 have been constructed in the previous steps, and η0, η1, φ0  are actually independent of βǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In analogy with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='52), we have  φ4 =  1  2π Lη4 + 1  2π  4  �  m=1  ��  βǫ + L  �  Pm + Qm  �  η4−m ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='63)  where the operators Pm, Qm are defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11) for m ≤ 2, in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19) for m = 3, and in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='20)  for m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Replacing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='61), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='62), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='63) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='23) and proceeding as in the previous sections,  we obtain the following equation for the profile η4 :  Λη4 + δ  �  t∂tη4 +  �  2 − L  �  η4  �  + R4 = 0 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='64)  where  R4 =  1  2π  �  4  �  m=1  �  (βǫ + L)Pm + Qm  �  η4−m , η0  �  +  �  φ3 , η1 − Rη0  �  +  �  φ2 , η2 − Rη1 + R2η0  �  +  �  φ1 , η3 − Rη2 + R2η1 − R3η0  �  −  �  φ0 , Rη3 − R2η2 + R3η1 − R4η0  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='65)  − r0  Γ  �� ˙¯r2 − ˙¯r0  �  ∂Rη1 +  � ˙¯z2 − ˙¯z0  �  ∂Zη1 + ˙¯z0∂Zη3  �  + δ∂R  �  Rη2 − R2η1 + R3η0  �  + 2δη2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The function R4 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='65) belongs to (δY0 + Y2 + Y4) ∩ Z and satisfies  R4 =  2  �  k=0  βk  ǫ  �  R3Zχ4k + RZχ5k  �  + O(δ) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='66)  for some (time-independent) radially symmetric functions χ4k, χ5k ∈ Y0 ∩ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='13 is the same as that of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='12, and can therefore be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The only important observation is that the projection of R4 onto the subspace Y0 is of order  O(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This can be seen as a consequence of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3 : when δ = ˙¯r = 0, all profiles ηm, φm  are even functions of Z, so that the quantities Rm are odd in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  We now project Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='64) on the subspace Ym for m = 0, 2, 4, and compute an (approximate)  solution Pmη4 invoking either Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7 (for m = 2, 4) or Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6 (for m = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In  the latter case, we use the observation that P0R4 = O(δ) to show that P0η4 is regular in the  limit δ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Altogether, we obtain an approximate solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='64) in the form  η4(R, Z, βǫ) =  2  �  k=0  βk  ǫ  �  R2Z2η4k +  �  R2 − Z2�  η5k + η6k  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='67)  where the functions ηjk ∈ Y0 ∩ Z are radially symmetric and time-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='63)  we deduce a similar expression for the stream function  φ4(R, Z, βǫ) =  2  �  k=0  βk  ǫ  �  R2Z2φ4k +  �  R2 − Z2�  φ5k + φ6k  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='68)  and with these choices the difference of both members in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='64) is O(δ) in the topology of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Since we have now completed the construction of our approximate solution, we explain pre-  cisely how to define the vortex radius ¯r(t) and the time-dependent aspect ratio ǫ(t) =  √  νt/¯r(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  In view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='35), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='56), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='61), the function ¯r(t) satisfies the differential equation  ˙¯r(t) = −Γδ  r0  �  1 − ǫ(t)2�  c1βǫ(t) + c2  ��  = −Γδ  r0  �  1 −  νt  ¯r(t)2  �  c1 log ¯r(t)  √  νt + c2  ��  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='69)  26  with initial condition ¯r(0) = r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='69) a smooth function of ¯r > 0,  uniformly in t ∈ (0, Tdif), and also a C1,α function of time for any α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Applying the Cauchy-  Lipschitz theorem, we obtain a unique local solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='69), which can be extended as long  as ¯r(t) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Now, if we define ǫ(t) =  √  νt/¯r(t), it follows that ¯r(t) = r0  �  1 − ǫ(t)2 + O(ǫ4βǫ)  �  ,  which implies that the solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='69) is well defined at least as long as ǫ(t) ≪ 1, namely  when t ≪ Tdif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It is useful to notice that the approximate solution ηapp given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='62) satisfies,  for all t > 0,  �  R2 ηapp(R, Z, t) dR dZ = 1 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='70)  �  R2 R ηapp(R, Z, t) dR dZ =  �  R2 Z ηapp(R, Z, t) dR dZ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='71)  Indeed, at each step m ≥ 1, the vorticity profile ηm is constructed by solving equations of the form  Ληm +  �m  2 − L  �  ηm + Rm = 0, where the source term Rm has vanishing integral (by definition)  and zero first order moments (due to the choice of the speeds ˙¯rm−1, ˙¯zm−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' These properties are  inherited by the profile ηm, due to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6, and in view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='31) this leads to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='70),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7  Estimate of the remainder  This section is devoted to the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Our task is to estimate the remainder  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='29), where η∗, φ∗ are defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28), and for this we need bounds on the derivatives of  the stream function in terms of the vorticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If φ = BSǫ[η], where the Biot-Savart operator is  defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='20), we have the formulas  ∂Zφ(R, Z) = − 1  2π  �  Ωǫ  �  (1+ǫR)(1+ǫR′) ˜F(s)  (Z−Z′) η(R′, Z′)  (R−R′)2 + (Z−Z′)2 dR′ dZ′ ,  ∂Rφ(R, Z) = − 1  2π  �  Ωǫ  �  (1+ǫR)(1+ǫR′) ˜F(s)  (R−R′) η(R′, Z′)  (R−R′)2 + (Z−Z′)2 dR′ dZ′  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='72)  + ǫ  4π  �  Ωǫ  √  1+ǫR′  √  1 + ǫR  �  F(s) + ˜F(s)  �  η(R′, Z′) dR′ dZ′ ,  where ˜F(s) = −2sF ′(s), see [30, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Here, as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='12), we use the shorthand notation  s =  ǫ2D2  (1+ǫR)(1+ǫR′) ≡ ǫ2 (R−R′)2 + (Z−Z′)2  (1+ǫR)(1+ǫR′)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='73)  In view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7), we have ˜F(s) → 1 as s → 0 and ˜F(s) = O(s−3/2) as s → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Throughout the proof, we fix t > 0 and we assume that the parameters ǫ =  √  νt/¯r(t) and  δ = ν/Γ are small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' By construction the vorticity η∗(R, Z, t) defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28) vanishes  identically when ρ := (R2+Z2)1/2 ≥ 2ǫ−σ0, so we can assume henceforth that ρ ≤ 2ǫ−σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In  that region, we have for any γ ∈ (0, 1) the a priori bounds  �  |α|≤2  |∂αη∗(R, Z, t)| ≤ C e−γρ2/4 ,  �  |α|=1  |∂αφ∗(R, Z, t)| ≤ C ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='74)  where α = (α1, α2) ∈ N2 and ∂α = ∂α1  R ∂α2  Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Indeed, the first estimate in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='74) holds because η∗  is obtained by truncating the asymptotic approximation ηapp(R, Z, t) which belongs to the space  27  Z defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The second estimate can then be obtained using the expressions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='72) with  φ = φ∗ and η = η∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' To see this, we first observe that 1 + ǫR ≈ 1 and 1 + ǫR′ ≈ 1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='72),  because both quantities ρ and ρ′ := (R′2+Z′2)1/2 are smaller than 2ǫ−σ0 ≪ ǫ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If we use the  estimates | ˜F(s)| ≤ C in the first two lines of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='72) and |F(s) + ˜F(s)| ≤ Cs−1/2 in the third  line, we thus obtain  |∂Rφ∗(R, Z, t)| + |∂Zφ∗(R, Z, t)| ≤ C  �  R2  |η∗(R′, Z′, t)|  �  (R−R′)2 + (Z−Z′)2 dR′ dZ′ ≤ C ,  which concludes the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Finally, since  t∂tη∗(R, Z, t) = χ0  �  ǫσ0ρ  �  t∂tηapp(R, Z, t) + σ0 ǫσ0ρχ′  0  �  ǫσ0ρ  �  ηapp(R, Z, t) t∂t log(ǫ) ,  it follows from the expressions given in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6 that t∂tη∗ satisfies the same bound as  η∗ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Summarizing, in view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='74), the remainder Rem(R, Z, t) satisfies  eγ0ρ2/4 |Rem(R, Z, t)| ≤ C δ−1(1 + ρ) e−(γ−γ0)ρ2/4 ,  when ρ ≤ 2ǫ−σ0 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='75)  for any γ0 ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If we assume that γ ∈ (γ0, 1), we conclude that the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='75)  is O(δ−1ǫ∞) if ρ ≥ ǫ−σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' So from now on we may concentrate on the inner region ρ ≤ ǫ−σ0,  where η∗ = ηapp is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  In that region we decompose the stream function as φ∗ = BSǫ[χ0 ηapp] = φ0  ∗ − φ1  ∗ + φ2  ∗, where  φ0  ∗ =  4  �  m=0  ǫm BSm[ηapp] ,  φ1  ∗ =  4  �  m=0  ǫm BSm[(1−χ0) ηapp] ,  φ2  ∗ =  ∞  �  m=5  ǫm BSm[χ0 ηapp] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Here χ0 is a shorthand notation for χ0(ǫσ0ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The convergence of the series defining φ2  ∗ is easily  justified using Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3, if we observe that both inequalities in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='12) are satisfied  since ρ, ρ′ ≪ ǫ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In fact, proceeding as in the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='74), one can verify that  |∂Rφ2  ∗(R, Z, t)| + |∂Zφ2  ∗(R, Z, t)| ≤ Cǫ5βǫ (1 + ρ)5 ,  ρ ≤ ǫ−σ0 ,  where βǫ = log(1/ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Moreover we have |∂Rφ1  ∗|+|∂Zφ1  ∗| = O(ǫ∞) because (1−χ0)ηapp = O(ǫ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Finally, in view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='30), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='62), it is clear that  |∂R  �  φ0  ∗ − φapp  �  | + |∂Z  �  φ0  ∗ − φapp  �  | ≤ Cǫ5β3  ǫ (1 + ρ)5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Summarizing, we have shown that the remainder (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='29) satisfies  ��Rem(R, Z, t) − �  Rem(R, Z, t)  �� ≤ Cδ−1ǫ5β3  ǫ (1 + ρ)5 e−γρ2/4 ,  ρ ≤ ǫ−σ0 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='76)  where �  Rem(R, Z, t) is the quantity defined for all (R, Z) ∈ R2 by the formula  Lηapp + ǫ∂R  �  S4ηapp  �  − t∂tηapp − 1  δ  �  φapp , S4ηapp  �  + ǫ¯r  δΓ  �  ˙¯r ∂Rηapp + ˙¯z ∂Zηapp  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='77)  with S4 = 1 − ǫR + (ǫR)2 − (ǫR)3 + (ǫR)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Now the crucial observation is that the asymptotic approximation ηapp was constructed  precisely so as to make the quantity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='77) small in the topology of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' More precisely, the  results of Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6 can be rephrased as follows:  δ �  Rem(R, Z, t) = OZ  �  ǫδ2 + ǫ2βǫδ2 + ǫ3βǫδ + ǫ4β2  ǫ δ + ǫ5β3  ǫ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='78)  28  Inside the parenthesis in the right-hand side, the first four terms represent what remains from  the quantities ǫm�  Ληm + δ  �  t∂t + m  2 − L  �  ηm + Rm) for m = 1, 2, 3, 4 after the profiles ηm have  been determined, and the last one corresponds to those terms in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='77) which are of order O(ǫ5)  or higher and were not considered in the construction of ηapp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='76), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='78) and  using Young’s inequality, we obtain  sup  ρ≤ǫ−σ0  eγ0ρ2/4 |Rem(R, Z, t)| ≤ C  δ  �  ǫδ2 + ǫ3βǫδ + ǫ5β3  ǫ  �  ≤ C  �  ǫδ + ǫγ5δ−1�  ,  for any γ5 < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This concludes the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8  The Eulerian approximation  As was already observed in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3, if we set δ = ˙¯r = 0 in the expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='24), we obtain  an approximate solution ηE  app, φE  app of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='26), which is nothing but the stationary Euler  equation in a frame moving with (constant) velocity ˙¯z ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' As is well known [3], steady states of  the Euler system are often characterized by a global functional relation between the vorticity and  the stream function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In our case, in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='26), we expect finding a function Φǫ : R+ → R  such that  φE  app(R, Z) − r0 ˙¯z  2Γ (1 + ǫR)2 = Φǫ  �ηE  app(R, Z)  1 + ǫR  �  + O  �  ǫM+1−�  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='79)  for all (R, Z) ∈ R2 such that ρ :=  √  R2 + Z2 ≪ ǫ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  In this section, we first verify that a relation of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='79) holds to second order, namely  with M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Using the expressions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='40), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='41), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='49), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='50) with δ = 0 and simplifying  somehow the notation, we can write our approximate solution in the form  ηE  app(R, Z) = η0 + ǫRη1 + ǫ2(R2−Z2)η2 + ǫ2η3 ,  φE  app(R, Z) = φ0 + ǫRφ1 + ǫ2(R2−Z2)φ2 + ǫ2φ3 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='80)  where η0, φ0 are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='31), and the profiles η1, η2, η3 ∈ Z and φ1, φ2, φ3 ∈ S∗(R2) are all  radially symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Note that ηm, φm may include factors of βǫ = log(1/ǫ) when m ≥ 1, but  this dependence is not explicitly indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We also expand the unknown function Φǫ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='79)  in powers of ǫ :  Φǫ(s) = Φ0(s) + ǫΦ1(s) + ǫ2Φ2(s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='81)  Finally, to simplify the writing, we denote  ω =  1  4π  �  βǫ − 1 + 2v  �  = r0 ˙¯z  Γ + O(ǫ2−) ,  where the last equality follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='38), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  If we consider equality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='79) to leading order in ǫ, thus neglecting all terms that are O(ǫ)  or O(ǫβǫ), we obtain the relation φ0 − ω/2 = Φ0(η0), which determines the principal term Φ0 in  the expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='31), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='32) we thus have  Φ0(s) = φ0(0) − ω  2 − 1  4π Ein  �  log  1  4πs  �  ,  s > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='82)  The constant in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='82) has no relevance, but it is important to note that Φ0(s) ∼ − 1  4π log log 1  s  as s → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' For later use we define  A(ρ) = Φ′  0  �  η0(ρ)  �  = ∂Rφ0  ∂Rη0  = ∂Zφ0  ∂Zη0  = 4  ρ2  �  eρ2/4 − 1  �  ,  ρ > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='83)  29  Incidentally we observe that A(ρ) = 1/h(ρ) where h is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  To the next order in ǫ, we deduce from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='79) the relation  (φ1 − ω)R = Φ′  0(η0)(η1 − η0)R + Φ1(η0) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='84)  which can be satisfied only if Φ1 = 0, because Φ1(η0) is the only radially symmetric term in  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='84).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Dividing by R, we obtain the equality φ1 −ω = A(η1 −η0), which happens to be satisfied  in view of our definitions of the profiles η1, φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This fact can be verified by following carefully  the calculations in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Finally we exploit (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='79) to order ǫ2, keeping in mind that Φ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We obtain the more  complex relation  (R2−Z2)φ2 + φ3 − ω  2 R2 = Φ′  0(η0)  �  (R2−Z2)η2 + η3 + (η0 − η1)R2�  + 1  2Φ′′  0(η0)(η0 − η1)2R2 + Φ2(η0) ,  where it is useful to substitute R2 = 1  2(R2+Z2) + 1  2(R2−Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The terms containing R2−Z2  cancel exactly due to the identity  φ2 − 1  2Ψ − Aη2 = 0 ,  where  Ψ = ω  2 + Φ′  0(η0)(η0 − η1) + 1  2Φ′′  0(η0)(η0 − η1)2 ,  which is satisfied by definition of the profiles φ2, η2, as can be verified by following the calculations  in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We are thus left with a relation involving only radially symmetric terms  φ3 − 1  2(R2+Z2)Ψ − Aη3 = Φ2(η0) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='85)  which provides the definition of the second order correction Φ2 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Summarizing, if Φǫ is  defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='81) with Φ1 = 0, Φ0 given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='82) and Φ2 by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='85), we have shown that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='79)  holds with M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  We now come back to the approximate solution η∗, φ∗ of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='23) constructed in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3–  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6, and we show that it also satisfies a relation of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='79), in a sufficiently small region  near the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' To formulate that result, we denote  Θ(R, Z, t) = φ∗(R, Z, t) − ¯r ˙¯z  2Γ (1 + ǫR)2 − Φǫ  �η∗(R, Z, t)  1 + ǫR  �  ,  (R, Z) ∈ Ωǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='86)  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' There exist σ1 ∈ (0, σ0) and N ∈ N such that, for any γ3 < 3, the quantity  Θ defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='86) satisfies, for some C > 0,  |∂RΘ(R, Z, t)| + |∂ZΘ(R, Z, t)| ≤ C(ǫδ + ǫγ3)(1 + ρ)N ,  ρ ≤ ǫ−σ1 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='87)  whenever ǫ and δ are small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The idea is to compare Θ with the second order Eulerian approximation  ΘE  app(R, Z, t) = φE  app(R, Z, t) − r0 ˙¯zE  2Γ (1 + ǫR)2 − Φǫ  �ηE  app(R, Z, t)  1 + ǫR  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='88)  which is of size O(ǫ3−) in view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='79).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Here we consider both quantities ηE  app, φE  app as time-  dependent, because we deal with the viscous case where ǫ =  √  νt/¯r(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We already estimated  30  the difference φ∗ − φapp (assuming M = 4) in the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5, and by construction  we know that φapp = φE  app + O(ǫδ + ǫ3β2  ǫ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' These arguments lead to the bound  |∂R  �  φ∗ − φE  app  �  | + |∂Z  �  φ∗ − φE  app  �  | ≤ C  �  ǫδ + ǫ3β2  ǫ  �  (1 + ρ)3 ,  ρ ≤ ǫ−σ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='89)  On the other hand, we have already observed that ¯r(t) = r0(1 + O(ǫ2)), and in view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='42),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='56) the difference between the vertical speed ˙¯z and its second order approximation ˙¯zE is of  size (Γ/r0) · O(ǫ2βǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We thus find  ��� ¯r ˙¯z  2Γ − r0 ˙¯zE  2Γ  ���  ��∂R(1 + ǫR)2�� ≤ Cǫ3βǫ ,  ρ ≤ ǫ−σ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='90)  Finally η∗ is just a truncation of ηapp and by definition ηapp−ηE  app = O(ǫδ+ǫ3βǫ) in the topology  of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This gives the following bound  �  |α|≤1  ��∂α�  η∗ − ηE  app  �  (R, Z, t)  �� ≤ C  �  ǫδ + ǫ3βǫ  �  (1 + ρ)3e−ρ2/4 ,  ρ ≤ ǫ−σ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='91)  At this point we observe that, since η∗ and ηE  app are O(ǫ) perturbations of η0 in the topology  of Z, one can choose σ1 > 0 small enough so that  1  2 η0(ρ) ≤ η∗(R, Z, t)  1 + ǫR  ≤ 2 η0(ρ) ,  ρ ≤ ǫ−σ1 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='92)  and similarly for ηE  app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' To conclude the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15, we need estimates on the  derivatives of the function Φǫ defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We begin with the leading order term Φ0 which  is given by the explicit formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We have  Φ′  0  � s  4π  �  =  1 − s  s log(1/s) ,  1  4π Φ′′  0  � s  4π  �  = −s − 1 + log(1/s)  s2�  log(1/s)  �2  ,  s > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Thanks to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='92) we only need to evaluate these expressions when the argument s/(4π) takes  its value in the interval  � 1  2η0(ρ), 2η0(ρ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In view of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='16 below, there exists C > 1 such  that, for all λ ∈ [1/2, 2] and all ρ > 0,  A(ρ)  C  ≤ Φ′  0  �  λη0(ρ)  �  ≤ CA(ρ) ,  ��Φ′′  0  �  λη0(ρ)  ��� ≤ CB(ρ) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='93)  where A(ρ) is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='83) and  B(ρ) = −Φ′′  0(η0(ρ)) = 16π  ρ4  �  (ρ2 − 4)eρ2/2 + 4eρ2/4�  ,  ρ > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='94)  The second order contribution Φ2 is not known explicitly, but from the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='85), where  the left-hand side belongs to S∗(R2), we deduce that there exist C > 0 and N ∈ N such that  ��Φ′  2  �  λη0(ρ)  ��� ≤ CA(ρ)(1 + ρ)N ,  ��Φ′′  2  �  λη0(ρ)  ��� ≤ CB(ρ)(1 + ρ)N ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='95)  for all ρ > 0 and all λ ∈ [1/2, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Now, if ∂α = ∂R or ∂Z, we decompose  ∂αΦǫ  �  η∗  1+ǫR  �  − ∂αΦǫ  � ηE  app  1+ǫR  �  = Φ′  ǫ  �  η∗  1+ǫR  ��  ∂α�  η∗  1+ǫR  �  − ∂α� ηE  app  1+ǫR  ��  +  �  Φ′  ǫ  �  η∗  1+ǫR  �  − Φ′  ǫ  � ηE  app  1+ǫR  ��  ∂α� ηE  app  1+ǫR  �  ,  31  and we estimate the right-hand side using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='91), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='93), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='95).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Taking into account the  preliminary bounds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='89), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='90), we arrive at an estimate of the form  �  |α|=1  ��∂α�  Θ(R, Z, t) − ΘE  app(R, Z, t)  ��� ≤ C(ǫδ + ǫγ3)(1 + ρ)N ,  ρ ≤ ǫ−σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  As was already mentioned, the approximation ΘE  app(R, Z, t) is O(ǫ3−) in the topology of S∗(R2),  so altogether we arrive at (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='87).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  In the argument above we used the following elementary result, whose proof can be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Let f, g : (0, +∞) → (0, +∞) be defined by  f(s) =  1 − s  s log(1/s) ,  g(s) = s − 1 + log(1/s)  s2�  log(1/s)  �2  = −f ′(s) ,  s > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Then given any Λ > 1 there exists C > 1 such that, for any λ ∈ [Λ−1, Λ] and any s > 0,  1  C ≤ f(λs)  f(s)  ≤ C ,  1  C ≤ g(λs)  g(s)  ≤ C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  4  Energy estimates and stability proof  In the previous section we constructed an approximate solution η∗(R, Z, t) of the rescaled vortic-  ity equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15) which corresponds, in the original variables, to a sharply concentrated vortex  ring of radius ¯r(t) located at the vertical position ¯z(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Our goal is now to control the difference  between this approximation and the actual solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15) with initial data η0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This will  conclude the proof of our main results, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  For technical reasons that were mentioned in the introduction, it is convenient for the stability  analysis to center the vertical coordinate Z not at the point ¯z(t), which is associated with  the approximate solution η∗, but at a point ¯z(t) + ˜z(t) where ˜z(t) is a small correction to be  determined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This can be viewed as the analogue of the modulation technique that is used,  for instance, in the stability analysis of solitary waves [55, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Thus, instead of the variables  (R, Z) defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14), we use henceforth the slightly modified coordinates  R = r − ¯r(t)  √  νt  ,  Z = z − ¯z(t) − ˜z(t)  √  νt  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1)  where the velocities ˙¯r(t), ˙¯z(t) are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' As is easily verified, the evolution equation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15) still holds if ˙¯z is replaced by ˙¯z + ˙˜z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' As a consequence, if the solution η is decomposed as  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='31), the perturbation ˜η(R, Z, t) satisfies the equation  t∂t˜η + 1  δ  �  φ∗ , ˜ζ  �  + 1  δ  �˜φ , ζ∗  �  +  �˜φ , ˜ζ  �  − ǫ¯r  δΓ  �  ˙¯r ∂R˜η + ˙¯z ∂Z ˜η  �  = L˜η + ǫ∂R˜ζ + 1  δ Rem(R, Z, t) + ǫ¯r ˙˜z  δ2Γ  �  ∂Zη∗ + δ∂Z ˜η  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2)  where to simplify the writing we use the letter ζ to denote the potential vorticity, namely  ˜ζ(R, Z, t) = ˜η(R, Z, t)  1 + ǫR  ,  ζ∗(R, Z, t) = η∗(R, Z, t)  1 + ǫR  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3)  32  Clearly, if ˙˜z = 0, the last term in the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2) disappears, and we recover the  perturbation equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  From our previous work [30] we know that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2) has a unique solution ˜η, in an appropriate  weighted L2 space, with zero initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Our goal is to control the evolution of that solution  on a large time interval, uniformly with respect to the viscosity in the limit ν → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This is not  an easy task, because several terms in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2) are multiplied by the Reynolds number δ−1 = Γ/ν,  which becomes arbitrarily large in the regime we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' As was explained in the introduction,  we shall use energy estimates to control the solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2), but a few preliminary steps are  necessary before starting the actual calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1  Control of the lowest order moments  To implement our strategy based on energy estimates, we need a precise information on the  lowest order moments of the solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We first define, for all t > 0,  µ0(t) =  �  Ωǫ  ˜η(R, Z, t) dX ,  µ1(t) =  �  Ωǫ  �  R + ǫR2/2  �  ˜η(R, Z, t) dX ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4)  where dX = dR dZ denotes the Lebesgue measure in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The moments defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4) satisfy µ0(t) = O(ǫ∞δ−1) and µ1(t) = O(ǫ+ǫγ5δ−2)  for any γ5 < 1, whenever ǫ and δ are small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The conclusion can be obtained by direct calculations, but we find it more illuminating  to use the conserved quantities of the original equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The first one is the total circulation  M(t) =  �  Ω  ωθ(r, z, t) dr dz = Γ  �  Ωǫ  �  η∗ + δ˜η  �  (R, Z, t) dX = Γ  �  Ωǫ  η∗ dX + Γδµ0(t) ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5)  which satisfies M(0) = Γ and is almost constant in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In fact it is proved in [30, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4]  that 0 ≤ 1 − M(t)/Γ ≤ C exp(−c/ǫ2) for some positive constants C and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Moreover, since the  approximate solution ηapp lies in the space Z defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='23), it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='70)  that  �  Ωǫ η∗ dX = 1 + O(exp(−c/ǫ2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Therefore µ0(t) = O(exp(−c/ǫ2) δ−1) by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  We next consider the total impulse in the vertical direction  I =  �  Ω  r2ωθ(r, z, t) dr dz = Γ¯r(t)2  �  Ωǫ  (1 + ǫR)2�  η∗ + δ˜η  �  (R, Z, t) dX ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6)  which is known to be exactly conserved [44, 29], so that I = Γr2  0 for all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6)  can be rephrased as I/Γ = I∗(t) + δ¯r(t)2µ(t), where  I∗(t) = ¯r(t)2  �  Ωǫ  (1 + ǫR)2η∗(R, Z, t) dX ,  µ(t) = µ0(t) + 2ǫµ1(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7)  It is not difficult to show that  tI′  ∗(t) = −¯r(t)2  �  Ωǫ  (1 + ǫR)2 Rem(R, Z, t) dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8)  The easiest way to establish (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8) is to observe that the impulse I∗(t) would be conserved if  η∗ was an exact solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15), so that the remainder Rem(R, Z, t) defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='29) is  the only term that contributes to the evolution of I∗(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' However equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8) can also be  33  verified by a direct calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In any case, since Rem(R, Z, t) satisfies estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='30) and  �  Ωǫ Rem(R, Z, t) dx = O(ǫ∞), we deduce from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8) that |tI′  ∗(t)| ≤ Cr2  0  �  ǫ2δ + ǫγ5+1δ−1�  , hence  |I∗(t) − r2  0| ≤  � t  0  |I′  ∗(s)| ds ≤ Cr2  0  � t  0  ǫ(s)2δ + ǫ(s)γ5+1δ−1  s  ds ≤ Cr2  0  �  ǫ2δ + ǫγ5+1δ−1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  As r2  0 − I∗(t) = δ¯r(t)2µ(t), we conclude that µ(t) = O  �  ǫ2 + ǫγ5+1δ−2�  , which gives the desired  estimate for µ1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  It is not clear if the strategy above can be applied to control the first order moment of the  perturbation ˜η with respect to the vertical variable Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In particular, we are not aware of any  (approximately) conserved quantity that we could invoke as in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Instead  we use the modulation parameter ˜z(t) introduced in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1) to impose that the vertical moment  vanishes identically :  µ2(t) :=  �  Ωǫ  Z ˜η(R, Z, t) dX = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9)  Differentiating (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9) with respect to time and using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2), we obtain the relation  ˙˜z(t)  �  Ωǫ  Z  �  ∂Zη∗ + δ∂Z ˜η  �  dX = δ2Γ  ǫ¯r  �  Ωǫ  Z R(R, Z, t) dX ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10)  where  R = 1  δ  �  φ∗ , ˜ζ  �  + 1  δ  �˜φ , ζ∗  �  +  �˜φ , ˜ζ  �  − ǫ¯r  δΓ  �  ˙¯r ∂R˜η + ˙¯z ∂Z ˜η  �  − L˜η − ǫ∂R˜ζ − 1  δ Rem(R, Z, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11)  In view of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1 the integral in the left-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10) is equal to −1 + O(ǫ∞), and is  therefore bounded away from zero if ǫ is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The integral in the right-hand side is a  priori of size O(δ−1), but we observe that R = δ−1Λ˜η + O(ǫδ−1), where Λ is the linear operator  defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Using the properties established in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6, we see that the leading  term gives no contribution :  1  4π  �  R2 Z Λ˜η dX =  �  Zη0 , Λ˜η  �  Y = −  �  Λ(Zη0) , ˜η  �  Y = 0 ,  since Zη0 = −2∂Zη0 is in the kernel of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' These considerations, which will be made rigorous  in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8 below, show that the modulation speed ˙˜z is uniquely determined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10) and  satisfies ˙˜z(t) = O(δ) as long as ˜η remains O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In particular ˜z(t) is indeed a small correction  to the vertical position of the vortex ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2  Definition and properties of the weight function  We now provide the precise definition of the weight function Wǫ : Ωǫ → (0, +∞) which appears  in the energy functional (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We give ourselves three positive numbers σ1, σ2, γ such that  0 < σ1 < σ0 < 1 < σ2 ,  γ = σ1/σ2 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='12)  where σ0 ∈ (0, 1) is the cut-off exponent already introduced in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' As we shall see σ2 > 1  can be chosen arbitrarily, but σ1 > 0 has to be taken sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In particular σ1 should  be small enough so that Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  34  As in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3), if ǫ > 0 and δ > 0 are sufficiently small, we denote ζ∗ = η∗/(1 + ǫR), where  η∗ is the approximate solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15) given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We recall that ζ∗ and φ∗ := BSǫ[η∗]  satisfy the relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='35), where Φǫ : R+ → R is the function constructed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We  decompose the domain Ωǫ =  �  (R, Z) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' 1 + ǫR > 0  �  into a disjoint union Ω′  ǫ ∪ Ω′′  ǫ ∪ Ω′′′  ǫ , where  Ω′  ǫ =  �  (R, Z) ∈ Ωǫ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Φ′  ǫ(ζ∗(R, Z)) < exp  �  ǫ−2σ1/4  ��  ,  Ω′′  ǫ =  �  (R, Z) ∈ Ωǫ \\ Ω′  ǫ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' ρ ≤ ǫ−σ2  �  ,  Ω′′′  ǫ =  �  (R, Z) ∈ Ωǫ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' ρ > ǫ−σ2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='13)  Here and in what follows, if (R, Z) ∈ R2, we denote ρ = (R2+Z2)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The domains Ω′  ǫ, Ω′′  ǫ also  depend (mildly) on δ, but for simplicity this dependence is not indicated explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If ǫ > 0 is small enough, the inner region Ω′  ǫ defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='13) is diffeomorphic to  a open disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Moreover, if κ is any fixed positive number, one has  �  (R, Z) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' ρ ≤ ǫ−σ1�  ⊂ Ω′  ǫ ⊂  �  (R, Z) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' ρ ≤ ǫ−σ1 + κ log 1  ǫ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The main properties of the function Φǫ are established in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  In particular, using estimates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='92), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='93), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='95), it is easy to verify that  1  2 A(ρ) ≤ Φ′  ǫ  �  ζ∗(R, Z)  �  ≤ 2A(ρ) ,  when ρ ≤ 2ǫ−σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15)  Here A(ρ) = (4/ρ2)  �  eρ2/4 − 1  �  , see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='83).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Since 2A(ǫ−σ1) < exp(ǫ−2σ1/4) as soon as ǫ−σ1 ≥ 3,  we deduce that (R, Z) ∈ Ω′  ǫ if ρ ≤ ǫ−σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Similarly, using the lower bound in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15), it is easy  to verify that, for any κ > 0, the inner region Ω′  ǫ is contained in the disk ρ ≤ ǫ−σ1 + κ log 1  ǫ if  ǫ > 0 is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Finally Ω′  ǫ is diffeomorphic to a disk because Φ′  ǫ(ζ∗) is close to a strictly  increasing radially symmetric function when ǫ > 0 is small, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  We next choose a smooth cut-off function χ1 : R → [1  2, 3] such that  χ1(x) =  1  1 + x  for |x| ≤ 1  2 ,  χ′  1(x) = 0  for |x| ≥ 3  4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='16)  The weight Wǫ : Ωǫ → (0, +∞) is defined by  Wǫ(R, Z) = χ1(ǫR) ×  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  Φ′  ǫ  �  ζ∗(R, Z)  �  in Ω′  ǫ ,  exp  �  ǫ−2σ1/4  �  in Ω′′  ǫ ,  exp  �  ρ2γ/4  �  in Ω′′′  ǫ ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='17)  where γ = σ1/σ2 < 1 and Ω′  ǫ, Ω′′  ǫ , Ω′′′  ǫ are the regions defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In other words, we assume  that Wǫ = Φ′  ǫ(ζ∗)/(1+ǫR) as long as the numerator remains smaller than the threshold value  exp(ǫ−2σ1/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Outside this inner region, the weight is radially symmetric except for the geometric  factor χ1(ǫR), and the radial profile remains constant as long as ρ ≤ ǫ−σ2 before increasing again  like exp(ρ2γ/4) when ρ > ǫ−σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' By construction the function Wǫ is locally Lipschitz continuous  in Ωǫ, and smooth in the interior of all three regions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The (mild) dependence of Wǫ upon  the parameter δ > 0 is not indicated explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' A schematic representation of the graph of Wǫ  is given in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Further properties of the weight Wǫ are collected in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  35  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' There exist positive constants C1, C2 such that, if ǫ, δ, and σ1 are small enough,  the weight Wǫ satisfies the uniform bounds  C1 exp  �  ρ2γ/4  �  ≤ Wǫ(R, Z) ≤ C2A(ρ) ,  (R, Z) ∈ Ωǫ ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='18)  where ρ = (R2+Z2)1/2 and A(ρ) is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='83).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Moreover, given γ1 < 1 there exists C3 > 0  such that  ��Wǫ(R, Z) − A(ρ)  �� +  ��∇Wǫ(R, Z) − ∇A(ρ)  �� ≤ C3 ǫγ1A(ρ) ,  when ρ ≤ ǫ−σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Since 1  2 ≤ χ1(ǫR) ≤ 3 and exp(ρ2γ/4) ≤ CA(ρ), we deduce from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15) that the bounds  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='18) hold in the inner region Ω′  ǫ, and the same inequalities obviously hold in the far field region  Ω′′′  ǫ too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In the intermediate region Ω′′  ǫ , we know that ρ ≤ ǫ−σ2 so that exp(ρ2γ/4) ≤ exp(ǫ−2σ1/4)  because γ = σ1/σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If ρ ≥ 2ǫ−σ1, it is clear that exp(ǫ−2σ1/4) ≤ A(ρ), which gives the desired  upper bound in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Finally if (R, Z) ∈ Ω′′  ǫ and ρ ≤ 2ǫ−σ1, we deduce from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15) that  exp(ǫ−2σ1/4) ≤ Φ′  ǫ  �  ζ∗(R, Z)  �  ≤ 2A(ρ), which concludes the proof of the upper bound in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  To prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19), we start from the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='17) of the weight Wǫ in the inner region Ω′  ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  We know from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='83) that A(ρ) = Φ′  0(η0), where η0 is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We thus find  |Wǫ(R, Z) − A(ρ)  �� ≤  ��χ1(ǫR) − 1  ��Φ′  ǫ(ζ∗) +  ��Φ′  ǫ(ζ∗) − Φ′  ǫ(η0)  �� +  ��Φ′  ǫ(η0) − Φ′  0(η0)  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='20)  Since χ1(ǫR) = (1 + ǫR)−1 when ρ ≤ ǫ−σ1, the first term in the right-hand of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='20) is smaller  than Cǫ|R| Φ′  ǫ(ζ∗) ≤ Cǫ1−σ1A(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' For the second term, we use the bounds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='92), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='93), and  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='95) to obtain  ��Φ′  ǫ(ζ∗) − Φ′  ǫ(η0)  �� ≤  sup  1  2≤λ≤2  ��Φ′′  ǫ(λη0)  �� |ζ∗ − η0| ≤ CB(ρ)(1 + ρ)Nǫη0 ≤ Cǫγ1A(ρ) ,  where in the last inequality we assumed that σ1 > 0 is small enough so that Nσ1 ≤ 1 − γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The  last term in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='20) is bounded by ǫ2|Φ′  2(η0)| ≤ Cǫγ1A(ρ) in view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='95).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Altogether we arrive  at the estimate |Wǫ(R, Z) − A(ρ)  �� ≤ Cǫγ1A(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The corresponding inequality for the first order  derivatives can be obtained in a similar way, and we omit the details  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3  Coercivity of the energy functional  For ǫ ≥ 0 small enough, we introduce the weighted L2 space Xǫ =  �  η ∈ L2(Ωǫ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' ∥η∥Xǫ < ∞  �  defined by the norm  ∥η∥2  Xǫ =  �  Ωǫ  Wǫ(R, Z) |η(R, Z)|2 dR dZ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='21)  In the limiting case ǫ = 0, it is understood that Ω0 = R2 and W0(R, Z) = A(ρ), in agreement  with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Assuming that ǫ > 0, we consider the energy functional (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='33), namely  Eǫ[η] = 1  2 ∥η∥2  Xǫ − Ekin  ǫ  [η] ,  η ∈ Xǫ ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='22)  where Ekin  ǫ  is the kinetic energy defined by  Ekin  ǫ  [η] = 1  2  �  Ωǫ  φ η dR dZ = 1  2  �  Ωǫ  |∇φ|2  1 + ǫR dR dZ ,  φ = BSǫ[η] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='23)  Since we are interested in the regime where ǫ is small, it is important to observe that Ekin  ǫ  [η]  becomes singular in the limit ǫ → 0, if the vorticity η has nonzero mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This divergence is  36  related to the well-known fact that any (nontrivial) nonnegative vorticity distribution in R2  has infinite kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The regular part of Ekin  ǫ  [η] is given, to leading order, by the two-  dimensional energy  Ekin  0 [η] =  1  4π  �  R2  �  Lη)η dX =  1  4π  �  R2  �  R2 log  � 8  D  �  η(R, Z)η(R′, Z′) dX dX′ ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='24)  where L is the integral operator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14) and D2 = (R−R′)2 + (Z−Z′)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' More precisely, we have  the following statement, whose proof is postponed to Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If ǫ > 0 is small and η ∈ Xǫ satisfies supp(η) ⊂ Bǫ := {(R, Z) ∈ Ωǫ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' ρ ≤ ǫ−σ1},  we have the expansion  Ekin  ǫ  [η] = βǫ − 2  4π  µ2  0 + Ekin  0 [η] + O  �  ǫβǫ∥η∥2  Xǫ  �  ,  as ǫ → 0 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='25)  where βǫ = log(1/ǫ) and µ0 =  �  Ωǫ η dR dZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  We now consider the (formal) limit of the functional Eǫ[η] as ǫ → 0, assuming that η has  zero mean to avoid the logarithmic divergence in the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19)  and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4, we obtain the limiting functional  E0[η] = 1  2  �  R2 A(ρ) η(R, Z)2 dR dZ − Ekin  0 [η] = 1  2 ∥η∥2  X0 − Ekin  0 [η] ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='26)  which is studied in detail in our previous work [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In particular, we have the following property :  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' There exists constants C4 > 2 and C5 > 0 such that, for all η ∈ X0,  ∥η∥2  X0 ≤ C4E0[η] + C5  �  µ2  0 + µ2  1 + µ2  2  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='27)  where µ0 =  �  R2 η dX, µ1 =  �  R2 Rη dX, µ2 =  �  R2 Zη dX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The results of [31, Section 2] show that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='27) holds when µ0 = µ1 = µ2 = 0, and the  general case is easily deduced by the following argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Given η ∈ X0 we define  ˆη = η − µ0η0 + µ1∂Rη0 + µ2∂Zη0 ,  ˆφ = φ − µ0φ0 + µ1∂Rφ0 + µ2∂Zφ0 ,  where φ = (2π)−1Lη and η0, φ0 are as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' By construction the integral and the first order  moments of the new function ˆη ∈ X0 vanish, so that we can apply the results of [31] which give  the bound ∥ˆη∥2  X0 ≤ C4E0[ˆη].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' On the other hand, expanding the quadratic expressions ∥ˆη∥2  X0  and E0[ˆη] and using H¨older’s inequality, it is straightforward to verify that  ∥ˆη∥2  X0 ≥ 1  2 ∥η∥2  X0 − C  �  µ2  0 + µ2  1 + µ2  2) ,  E0[ˆη] ≤ E0[η] +  1  4C4  ∥η∥2  X0 + C  �  µ2  0 + µ2  1 + µ2  2) ,  for some C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If we combine these estimates, we arrive at the bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='27) with a deteriorated  constant C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Using Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5, we now establish a similar coercivity property for the functional Eǫ  when ǫ > 0 is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The proof of the following proposition is again postponed to Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If the weight Wǫ satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='18) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19), there exist constants C6 > 0 and  C7 > 0 such that, for all sufficiently small ǫ > 0 and all η ∈ Xǫ, we have the estimate  ∥η∥2  Xǫ ≤ C6Eǫ[η] + C7  �  βǫµ2  0 + µ2  1 + µ2  2  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28)  where βǫ = log(1/ǫ) and µ0 =  �  Ωǫ η dX, µ1 =  �  Ωǫ Rη dX, µ2 =  �  Ωǫ Zη dX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  In what follows we use the bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28) to estimate the vorticity perturbation ˜η introduced  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The corresponding moments µ0, µ1 are under control thanks to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1, and µ2 = 0  according to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' So it remains to bound the energy functional Eǫ[˜η], which is the purpose of  the remaining sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  37  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4  Time evolution of the energy  Let ˜η be the solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2) with zero initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Assuming that δ > 0 and σ > 0 are  sufficiently small, we consider for t ∈ (0, Tadvδ−σ) the energy function  Eǫ(t) = 1  2  �  Ωǫ  Wǫ(R, Z) ˜η(R, Z, t)2 dX − 1  2  �  Ωǫ  ˜φ(R, Z, t) ˜η(R, Z, t) dX ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='29)  where ǫ =  √  νt/¯r(t) and Wǫ is the weight function defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The first term in the  right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='29) is equal to 1  2∥˜η∥2  Xǫ, and the second one is the kinetic energy Ekin  ǫ  [˜η],  which satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='23) and involves the stream function ˜φ = BSǫ[˜η] defined by the Biot-Savart  formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Differentiating (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='29) with respect to time and using in particular the relations  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='33), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='23), we obtain by direct calculation  t∂tEǫ =  �  Ωǫ  �  Wǫ˜η t∂t˜η + 1  2t(∂tWǫ)˜η2�  dX −  �  Ωǫ  �  ˜φ t∂t˜η + t˙ǫ  2  R|∇˜φ|2  (1 + ǫR)2  �  dX  = I1 + I2 + I3 + I4 + I5 + I6 ,  where the quantities I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' , I6 gather the following terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Local advection terms :  I1 = −1  δ  �  Ωǫ  Wǫ˜η  �  φ∗ , ˜ζ  �  dX + ǫ¯r ˙¯z  δΓ  �  Ωǫ  Wǫ˜η ∂Z ˜η dX  = −1  δ  �  Ωǫ  Wǫ˜η  �  φ∗ − ¯r ˙¯z  2Γ (1 + ǫR)2 , ˜ζ  �  dX  = − 1  2δ  �  Ωǫ  �  Wǫ(1 + ǫR) , φ∗ − ¯r ˙¯z  2Γ (1 + ǫR)2�  ˜ζ2 dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='30)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Nonlocal advection terms :  I2 = 1  δ  �  Ωǫ  ˜φ  �  φ∗ , ˜ζ  �  dX − ǫ¯r ˙¯z  δΓ  �  Ωǫ  ˜φ ∂Z ˜η dX − 1  δ  �  Ωǫ  �  Wǫ˜η − ˜φ  ��˜φ , ζ∗  �  dX  = 1  δ  �  Ωǫ  ˜φ  �  φ∗ − ¯r ˙¯z  2Γ (1 + ǫR)2 , ˜ζ  �  dX − 1  δ  �  Ωǫ  Wǫ˜η  �˜φ , ζ∗  �  dX  = 1  δ  �  Ωǫ  �  ˜φ , φ∗ − ¯r ˙¯z  2Γ (1 + ǫR)2�  ˜ζ dX − 1  δ  �  Ωǫ  Wǫ(1 + ǫR)  �˜φ , ζ∗  �˜ζ dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='31)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Nonlinear terms :  I3 = −  �  Ωǫ  �  Wǫ˜η − ˜φ  ��˜φ , ˜ζ  �  dX = −  �  Ωǫ  �  Wǫ˜η , ˜φ  �˜ζ dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='32)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Diffusive terms :  I4 =  �  Ωǫ  �  Wǫ˜η − ˜φ  ��  L˜η + ǫ∂R˜ζ  �  dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Integrating by parts as indicated in Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2, we obtain the equivalent expression  I4 = −  �  Ωǫ  Wǫ|∇˜η|2 dX −  �  Ωǫ  (∇Wǫ · ∇˜η)˜η dX −  �  Ωǫ  Vǫ˜η2 dX  − ǫ  2  �  Ωǫ  ∂R  �  Wǫ(1 + ǫR)  �˜ζ2 dX + ǫ  4  �  Ωǫ  R|∇˜φ|2  (1 + ǫR)2 dX ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='33)  38  where  Vǫ = 1  4(R∂R + Z∂Z)Wǫ − 1  2Wǫ − (1 + ǫR) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='34)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Remainder term :  I5 = 1  δ  �  Ωǫ  �  Wǫ˜η − ˜φ  �  Rem(R, Z, t) dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='35)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Additional terms :  I6 = 1  2  �  Ωǫ  t(∂tWǫ)˜η2 dX + ǫ¯r ˙¯r  δΓ  �  Ωǫ  �  Wǫ˜η − ˜φ  �  ∂R˜η dX  − t˙ǫ  2  �  Ωǫ  R|∇˜φ|2  (1 + ǫR)2 dX + ǫ¯r ˙˜z  δ2Γ  �  Ωǫ  �  Wǫ˜η − ˜φ  � �  ∂Zη∗ + δ∂Z ˜η  �  dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='36)  For the purposes of our analysis, it is useful to reorganize some terms appearing in the  quantities I4 and I6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' First, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19) and integrating by parts, it is easy to verify that  −  �  Ωǫ  ˜φ ∂R˜η dX =  �  Ωǫ  ˜η ∂R ˜φ dX = ǫ  2  �  Ωǫ  |∇˜φ|2  (1 + ǫR)2 dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='37)  So, if we collect all terms involving |∇˜φ|2 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='33), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='36), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='37), we obtain the quantity  � ǫ  4 − t˙ǫ  2  � �  Ωǫ  R|∇˜φ|2  (1 + ǫR)2 dX + ǫ2¯r ˙¯r  2δΓ  �  Ωǫ  |∇˜φ|2  (1 + ǫR)2 dX = t ˙¯r  2¯r  �  Ωǫ  |∇˜φ|2  1 + ǫR dX ,  where we used the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='33) of t˙ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Next, we prefer including the term involving t∂tWǫ in  I4 rather than I6, because it will be combined with the diffusive terms in I4 to obtain negative  quantities that will allow us to control the evolution of the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Summarizing, if we define  ˆI4 = −  �  Ωǫ  Wǫ|∇˜η|2 dX −  �  Ωǫ  (∇Wǫ · ∇˜η)˜η dX −  �  Ωǫ  Vǫ˜η2 dX  − ǫ  2  �  Ωǫ  ∂R  �  Wǫ(1 + ǫR)  �˜ζ2 dX + 1  2  �  Ωǫ  t(∂tWǫ)˜η2 dX ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='38)  and  ˆI6 = ǫ¯r ˙¯r  δΓ  �  Ωǫ  Wǫ˜η∂R˜η dX + t ˙¯r  ¯r Ekin  ǫ  [˜η] + ǫ¯r ˙˜z  δ2Γ  �  Ωǫ  �  Wǫ˜η − ˜φ  � �  ∂Zη∗ + δ∂Z ˜η  �  dX ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='39)  we obtain the identity t∂tEǫ = I1 + I2 + I3 + ˆI4 + I5 + ˆI6, which we exploit in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5  Bounds on the stream function  In this section we collect a few estimates on the stream function φ = BSǫ[η], where BSǫ is the  ǫ-dependent Biot-Savart operator (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We are especially interested in bounds on the velocity  field U = (UR, UZ) defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' There exists a constant C > 0 such that, for all ǫ ∈ (0, 1),  ��� ∂Rφ  1 + ǫR  ��� +  ��� ∂Zφ  1 + ǫR  ��� ≤  �  Ωǫ  C  �  (R−R′)2 + (Z−Z′)2 |η(R′, Z′)| dX′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='40)  In particular, for any q > 2, we have ∥U∥Lq ≤ Cq∥η∥Xǫ where U is the velocity field (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  39  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='40) is established in [30, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1], which in turns relies on [29, Proposi-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Using the Hardy-Littlewood-Sobolev inequality, we deduce from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='40) that ∥U∥Lq ≤  Cq∥η∥Lp if q > 2 and p ∈ (1, 2) satisfy the relation 1/p = 1/q + 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Finally, the lower bound  on Wǫ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='18) implies that ∥η∥Lp ≤ C∥η∥Xǫ for any p ∈ [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The particular case where η = η∗ is the approximate solution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28) plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The following estimates hold for the stream function φ∗ = BSǫ[η∗] :  ��� ∂Rφ∗  1 + ǫR  ��� +  ��� ∂Zφ∗  1 + ǫR  ��� ≤  C  1 + ρ + ǫ2ρ3 ,  ���  ∂Zφ∗  (1 + ǫR)2  ��� ≤  C  1 + ρ + ǫ3ρ4 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='41)  where ρ = (R2+Z2)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In the region where ρ ≤ 1/(2ǫ), we can use estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='40) with η = η∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Since η∗ satisfies  the Gaussian bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='74), we easily deduce that |U| ≤ C(1 + ρ)−1, which gives estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='41)  in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  We now concentrate on the region ρ ≥ 1/(2ǫ), where a more careful analysis  is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We start from the formulas (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='72) with η = η∗, and we first estimate the vertical  derivative ∂Zφ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Since | ˜F(s)| ≤ Cs−3/2 for all s > 0, we see that  ���  ∂Zφ∗  (1 + ǫR)2  ��� ≤ C  ǫ3  �  Ωǫ  (1+ǫR′)2 |η∗(R′, Z′)|  �  (R−R′)2 + (Z−Z′)2�2 dR′ dZ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='42)  Note that the integral is, in fact, taken over the support of η∗, which is included in the ball  ρ′ := (R′2+Z′2)1/2 ≤ 2ǫ−σ0 where σ0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In particular we can disregard the factor (1+ǫR′)2 in  the numerator, and the denominator is always larger that ρ4/2 if ǫ is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' So the  right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='42) is bounded by Cǫ−3ρ−4 when ρ ≥ 1/(2ǫ), which concludes the proof of  the second inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Since 1 + ǫR ≤ 1 + ǫρ, the estimate on ∂Zφ∗/(1+ǫR) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='41)  follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  To conclude the proof of the first inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='41), we must estimate the quantity ∂Rφ∗  which contains an additional term given by the last line in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In the region where ρ ≥ 1/(2ǫ),  using the fact that |F(s)| + | ˜F(s)| ≤ Cs−3/2, we see that the contribution of that term to the  vertical speed UZ = ∂Rφ∗/(1+ǫR) is bounded by  C  ǫ2  �  Ωǫ  (1+ǫR′)2 |η∗(R′, Z′)|  �  (R−R′)2 + (Z−Z′)2�3/2 dR′ dZ′ ≤  C  ǫ2ρ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='41) is thus complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6  Control of the advection terms  In what follows we always assume that δ > 0 is sufficiently small and that ǫ2 ≲ δ1−σ for some  small σ > 0, see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We first estimate the advection terms I1, I2 defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='30),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' These terms are potentially dangerous because they include a factor 1/δ which is very  large in the vanishing viscosity limit, but the energy functional (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='33) was designed precisely so  that these contributions can be controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' There exist γ1 > 0 and C > 0 such that  |I1| ≤ Cǫγ1 ∥˜η∥2  Xǫ + Cǫ2  δ  �  Ω′′ǫ  Wǫ|˜η|2 dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='43)  40  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' To exploit the properties of the weight Wǫ, we decompose the integral (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='30) defining  I1 in three pieces, which correspond to the subdomains (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If (R, Z) ∈ Ω′  ǫ, we know from  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='17), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='86) that  Wǫ = Φ′  ǫ(ζ∗)  1 + ǫR ,  φ∗ − ¯r ˙¯z  2Γ (1 + ǫR)2 = Φǫ(ζ∗) + Θ ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='44)  where Θ is a remainder term that is studied in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It follows that  �  Wǫ(1 + ǫR) , φ∗ − ¯r ˙¯z  2Γ (1 + ǫR)2�  =  �  Φ′  ǫ(ζ∗) , Φǫ(ζ∗) + Θ  �  =  �  Φ′  ǫ(ζ∗) , Θ  �  ,  where the right-hand side can be controlled using the bounds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='87) on Θ and the estimates  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='93), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='95) on Φǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This gives, for some N ∈ N and any γ3 ∈ (2, 3),  ���  Φ′  ǫ(ζ∗) , Θ  ��� ≤ C  �  ǫδ + ǫγ3�  (1 + ρ)N Wǫ ≤ C  �  ǫδ + ǫγ3�  ǫ−Nσ1 Wǫ ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='45)  where we used the fact that 1 + ρ ≤ 2ǫ−σ1 when (R, Z) ∈ Ω′  ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Since ˜ζ ≈ ˜η in Ω′  ǫ and since  δ−1 ≲ ǫ−2/(1−σ) in the regime we consider, it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='45) that  1  δ  �  Ω′ǫ  ���  Φ′  ǫ(ζ∗) , Θ  ��� ˜ζ2 dX ≤ C  �  ǫ + ǫγ3  δ  �  ǫ−Nσ1  �  Ω′ǫ  Wǫ˜η2 dX ≤ Cǫγ1 ∥˜η∥2  Xǫ ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='46)  where γ1 is taken so that 0 < γ1 < γ3 − 2/(1−σ) − Nσ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' As γ3 < 3 is arbitrary, such a choice is  always possible if we assume that σ > 0 and σ1 > 0 are small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  We next consider the intermediate region Ω′′  ǫ in which Wǫ(1 + ǫR) = χ2(ǫR) exp  �  ǫ−2σ1/4  �  ,  where χ2(x) = (1 + x)χ1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In that region, we thus have  Jǫ :=  �  Wǫ(1 + ǫR) , φ∗ − ¯r ˙¯z  2Γ (1 + ǫR)2�  = ǫχ′  2(ǫR) exp  �  ǫ−2σ1/4  �  ∂Zφ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Since χ2(x) = 1 when |x| ≤ 1  2, the quantity Jǫ vanishes when ρ := (R2+Z2)1/2 ≤ 1/(2ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In the  region where 1/(2ǫ) ≤ ρ ≤ ǫ−σ2, we know from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='41) that |∂Zφ∗/(1+ǫR)2| ≤ Cǫ−3ρ−4 ≤ Cǫ,  and that Wǫ ≈ exp  �  ǫ−2σ1/4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Since χ′  2 is a bounded function, we deduce  1  δ  �  Ω′′ǫ  |Jǫ| ˜ζ2 dX = 1  δ  �  Ω′′ǫ  |Jǫ| ˜η2  (1+ǫR)2 dX ≤ Cǫ2  δ  �  Ω′′ǫ  Wǫ|˜η|2 dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='47)  Finally, in Ω′′′  ǫ we have Wǫ(1 + ǫR) = χ2(ǫR) ˆWǫ where ˆWǫ = exp(ρ2γ/4), so that  Jǫ = ǫχ′  2(ǫR) ˆWǫ ∂Zφ∗ + ǫ¯r ˙¯z  Γ χ1(ǫR)(1 + ǫR)2∂Z ˆWǫ + χ2(ǫR)  � ˆWǫ , φ∗  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The first term in the right-hand side is estimated as above, with the difference that we now have  the improved bound |∂Zφ∗/(1+ǫR)2| ≤ Cǫ−3ρ−4 ≤ Cǫ4σ2−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' For the second one we observe that  ��∂R ˆWǫ  �� +  ��∂Z ˆWǫ  �� ≤ γ ρ2γ−1 ˆWǫ ≤ γ ǫσ2−2σ1 ˆWǫ ,  since ρ ≥ ǫ−σ2 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='48)  and the last term is estimated using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='48) and the first bound in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Altogether we find  1  δ  �  Ω′′′  ǫ  |Jǫ| ˜ζ2 dX ≤ C  δ  �  Ω′′′  ǫ  � 1  ǫ2ρ4 + ¯r| ˙¯z|  Γ  ǫ  ρ1−2γ +  1  ǫ2ρ4−2γ  �  Wǫ|˜η|2 dX ≤ Cǫγ1 ∥˜η∥2  Xǫ ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='49)  provided 0 < γ1 < σ2 + 1 − 2σ1 − 2/(1−σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Since σ2 > 1, such a choice is again possible if σ > 0  and σ1 > 0 are small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='46), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='47), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='49), we arrive at (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  41  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' There exist γ1 > 0 and C > 0 such that  |I2| ≤ Cǫγ1 ∥˜η∥2  Xǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='50)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In Ω′  ǫ we have Wǫ(1 + ǫR) = Φ′  ǫ(ζ∗) by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='44), hence Wǫ(1 + ǫR)  �˜φ , ζ∗  �  =  �˜φ , Φǫ(ζ∗)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Using the second relation in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='44), we deduce that  �  ˜φ , φ∗ − ¯r ˙¯z  2Γ (1 + ǫR)2�  − Wǫ(1 + ǫR)  �˜φ , ζ∗  �  =  �˜φ , Θ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='51)  The first-order derivatives of Θ are estimated in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Proceeding as in the previous  lemma, we thus obtain  1  δ  �  Ω′ǫ  ���˜φ , Θ  ��� |˜ζ| dX ≤ C  �  ǫ + ǫγ3  δ  �  ǫ−Nσ1  �  Ω′ǫ  |∇˜φ|  1+ǫR |˜η| dX ≤ Cǫγ1 ∥˜η∥2  Xǫ ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='52)  where 0 < γ1 < γ3 − 2/(1−σ) − Nσ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In the last inequality, we used estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='40) for ∇˜φ and  we applied H¨older’s inequality with exponents 3 and 3/2 (for instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  In Dǫ := Ωǫ \\ Ω′  ǫ, we consider both terms in the left-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='51) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The  contribution of the first one to I2 is estimated by  1  δ  �  Dǫ  |∇˜φ| |∇φ∗|  1 + ǫR  | ˜η| dX + ǫ¯r| ˙¯z|  δΓ  �  Dǫ  |∂Z ˜φ| |˜η| dX = O  �  ǫ∞∥˜η∥2  Xǫ  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='53)  because |∇φ∗| ≤ C by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='41), ∥∇˜φ/(1+ǫR)∥L3 ≤ C∥˜η∥Xǫ by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7, and  ∥˜η∥L3/2(Dǫ) ≤  ��  Dǫ  Wǫ|˜η|2 dX  �1/2��  Dǫ  W −3  ǫ  dX  �1/6  = O  �  ǫ∞∥˜η∥Xǫ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The second term in the left-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='51) is nonzero only if ρ ≤ 2ǫ−σ0, in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28),  and we know from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='74) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='18) that Wǫ|∇ζ∗| is a bounded function in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The  contribution of that term to I2 can therefore be estimated in the same way as above:  1  δ  �  Dǫ  Wǫ |  �˜φ, ζ∗  �  | |˜ζ| dX ≤ C  δ  �  Dǫ  |∇˜φ| |˜η|  1 + ǫR dX = O  �  ǫ∞∥˜η∥2  Xǫ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='54)  Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='52), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='53), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='54), we obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7  Control of the diffusive terms  Our next task is to estimate the diffusive terms collected in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' To formulate the result, we  introduce the continuous function ργ : R2 × R+ → R+ defined by  ργ(R, Z, ǫ) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  ρ  if  ρ ≤ ǫ−σ1 ,  ǫ−σ1  if  ǫ−σ1 < ρ < ǫ−σ2 ,  ργ  if  ρ ≥ ǫ−σ2 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='55)  where as usual ρ = (R2 + Z2)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The goal of this section is to obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' There exist κ > 0 and C > 0 such that  ˆI4 ≤ −κ  �  Ωǫ  Wǫ  �  |∇˜η|2 + ρ2  γ ˜η2 + ˜η2�  dX + C  �  µ2  0 + µ2  1 + µ2  2  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='56)  where µ0, µ1, µ2 are defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  42  The proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11 requires several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We first control the term in ˆI4 that  involves the time derivative of the weight function Wǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' There exist C > 0 and γ1 > 0 such that  �  Ωǫ  t(∂tWǫ)˜η2 dX ≤ −σ1  5  �  Ω′′ǫ  Wǫρ2  γ ˜η2 dX + C  �  Ω′′′  ǫ  Wǫ˜η2 dX + Cǫγ1∥˜η∥2  Xǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='57)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Following (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='17) we decompose Wǫ(R, Z) = χ1(ǫR) ˆWǫ(R, Z), so that  t∂tWǫ = χ1(ǫR) t∂t ˆWǫ(R, Z) + t˙ǫRχ′  1(ǫR) ˆWǫ(R, Z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='58)  We first estimate the right-hand side in the region Ω′  ǫ defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='13), where ˆWǫ = Φ′  ǫ(ζ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' As  Φǫ = Φ0 + ǫ2Φ2 according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='81), we have t∂t ˆWǫ = Φ′′  ǫ(ζ∗) t∂tζ∗ + 2tǫ˙ǫ Φ′  2(ζ∗) in that region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  We recall that 2t˙ǫ = ǫ(1 + O(ǫ2)) by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='33), and that the functions Φ0, Φ2 satisfy the estimates  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='93), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='95).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It follows immediately that |tǫ˙ǫ Φ′  2(ζ∗)| ≤ Cǫ2−Nσ1 ˆWǫ ≤ CǫWǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Moreover, since  ζ∗ = η∗/(1+ǫR) with η∗ given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28), we also have |Φ′′  ǫ(ζ∗)t∂tζ∗| ≤ Cǫ(1 + ρ)N ˆWǫ ≤ Cǫγ1Wǫ,  for some γ1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Finally, the last term in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='58) is bounded by CǫρWǫ ≤ Cǫ1−σ1Wǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Altogether  we have shown that |t∂tWǫ| ≤ Cǫγ1Wǫ in Ω′  ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  In the intermediate region Ω′′  ǫ we have ˆWǫ = exp  �  ǫ−2σ1/4  �  and ργ = ǫ−σ1, so that  t∂t ˆWǫ = −σ1  2 exp  �  ǫ−2σ1/4  �  t˙ǫ  ǫ2σ1+1 = −σ1  2  ˆWǫ ρ2  γ  t˙ǫ  ǫ ≈ −σ1  4  ˆWǫ ρ2  γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Since |t˙ǫRχ′  1(ǫR)| ≤ |ǫRχ′  1(ǫR)| ≤ C, it follows that t∂tWǫ ≤ −(σ1/5)Wǫρ2  γ in Ω′′  ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Finally, in  the exterior region Ω′′′  ǫ , the weight function ˆWǫ = exp(ρ2γ/4) does not depend on time, and we  deduce from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='58) that |t∂tWǫ| ≤ CWǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Collecting all these estimates, we arrive at (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  We next consider the term involving ˜ζ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' There exist C > 0 and γ1 > 0 such that  − ǫ  2  �  Ωǫ  ∂R  �  Wǫ(1 + ǫR)  �˜ζ2 dX ≤ −ǫ2  4  �  Ωǫ  Wǫ˜ζ2 dX + Cǫγ1∥˜η∥2  Xǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='59)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If Dǫ denotes one of the three regions defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='13), we have  − ǫ  2  �  Dǫ  ∂R  �  Wǫ(1 + ǫR)  �˜ζ2 dX = −ǫ2  2  �  Dǫ  Wǫ˜ζ2 dX − ǫ  2  �  Dǫ  �  ∂RWǫ  �˜ζ˜η dX  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='60)  ≤ −ǫ2  4  �  Dǫ  Wǫ˜ζ2 dX + 1  4  �  Dǫ  (∂RWǫ)2  Wǫ  ˜η2 dX ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='61)  where in the second line we used Young’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In the interior region Ω′  ǫ we observe that  ˜ζ ≈ ˜η, because |ǫR| ≤ 2ǫ1−σ1 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Moreover we have ǫ|∂RWǫ| ≤ Cǫγ1Wǫ for some γ1 > 0, so  taking Dǫ = Ω′  ǫ and using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='60) we obtain the analogue of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='59) in that region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Outside Ω′  ǫ,  we cannot directly compare ˜ζ and ˜η, so we prefer using inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In the intermediate  region Ω′′  ǫ, we have |∂RWǫ| ≤ CǫWǫ by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='17), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='59) easily follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Finally, in the exterior  region Ω′′′  ǫ , we observe that  ∂RWǫ =  �ǫχ′  1(ǫR)  χ1(ǫR) + γR  2 ρ2γ−2  �  Wǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  As γ < 1/2 and ρ ≥ ǫ−σ2 in that region, we deduce that |∂RWǫ| ≤ Cǫγ1Wǫ for some γ1 > 0, and  this leads to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The proof is thus complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  43  To conclude the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11, we consider the quadratic form given by the first  line of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='38), namely  Qǫ[η] =  �  Ωǫ  Wǫ|∇η|2 dX +  �  Ωǫ  (∇Wǫ · ∇η)η dX +  �  Ωǫ  Vǫη2 dX ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='62)  where Vǫ is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Taking formally the limit ǫ → 0 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='62), we obtain  Q0[η] =  �  R2 A|∇η|2 dX +  �  R2(∇A · ∇η)η dX +  �  R2 V η2 dX ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='63)  where A is defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='83) and V = 1  4(R∂R + Z∂Z)A − 1  2A − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The limiting quadratic form  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='63) is carefully studied in our previous work [31], and we have the following result :  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' There exists constants C8 > 2 and C9 > 0 such that, for all η ∈ X0 with  ρη ∈ X0 and ∇η ∈ X 2  0 , we have  ∥∇η∥2  X0 + ∥ρη∥2  X0 + ∥η∥2  X0 ≤ C8Q0[η] + C9  �  µ2  0 + µ2  1 + µ2  2  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='64)  where µ0 =  �  R2 η dX, µ1 =  �  R2 Rη dX, µ2 =  �  R2 Zη dX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In [31, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2] we prove that there exists δ0 > 0 such that Q0[η] ≥ δ0∥η∥2  X0 for  any η ∈ X0 such that µ0 = µ1 = µ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' On the other hand, if we apply Young’s inequality to  the middle term in the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='63), we obtain the lower bound  Q0[η] ≥ 1  4  �  R2 A|∇η|2 dX +  �  R2  �  V − |∇A|2  3A  �  η2 dX ≥ 1  4∥∇η∥2  X0 + 1  24∥ρη∥2  X0 − C∥η∥2  X0 ,  because a direct calculation reveals that V/A − |∇A|2/(3A2) ≥ ρ2/(24) − C for some constant  C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Taking a convex combination of both estimates, we see that there exists C8 > 0 such  that  ∥∇η∥2  X0 + ∥ρη∥2  X0 + ∥η∥2  X0 ≤ C8Q0[η] ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='65)  whenever η ∈ X0 satisfies µ0 = µ1 = µ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It remains to deduce (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='64) from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='65), which is  easily done using exactly the same arguments as in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The analogue of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14 for the full quadratic form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='62) is the following statement,  whose proof is postponed to Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' There exists constants C10 > 2 and C11 > 0 such that, for all sufficiently  small ǫ > 0 and all η ∈ Xǫ with ργη ∈ Xǫ and ∇η ∈ X 2  ǫ , we have  ∥∇η∥2  Xǫ + ∥η∥2  Xǫ +  �  Ω′ǫ∪Ω′′′  ǫ  Wǫρ2  γη2 dX ≤ C10Qǫ[η] + C11  �  µ2 +  �  Ω′′ǫ  Wǫη2�  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='66)  where µ2 = µ2  0 + µ2  1 + µ2  2 and µ0 =  �  Ωǫ η dX, µ1 =  �  Ωǫ Rη dX, µ2 =  �  Ωǫ Zη dX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  End of the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='38) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='62) we have  ˆI4 = −Qǫ[˜η] − ǫ  2  �  Ωǫ  ∂R  �  Wǫ(1 + ǫR)  �˜ζ2 dX + 1  2  �  Ωǫ  t(∂tWǫ)˜η2 dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  The three terms in the right-hand side are estimated using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='66), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='59), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='57), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Taking ǫ > 0 sufficiently small and recalling that ργ ≥ ǫ−σ1 ≫ 1 outside the interior region  Ω′  ǫ, we arrive at (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The slight discrepancy between the definitions of µ1 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4) and in  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15 is completely harmless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  44  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8  Control of the remaining terms  In this section, we estimate the remaining terms I3, I5, and ˆI6 defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='32), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='35), and  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='39), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Control of I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We deduce from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='32) that  |I3| ≤  �  Ωǫ  |∇˜φ|  1+ǫR |˜η|  ��∇(Wǫ˜η)  �� dX ≤  �  Ωǫ  |∇˜φ|  1+ǫR |˜η|  �  |˜η||∇Wǫ| + Wǫ|∇˜η|  �  dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='67)  To estimate the right-hand side, we use (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='40) to obtain the uniform bound  ��� |∇˜φ|  1+ǫR  ���  L∞ ≤ C∥˜η∥1/2  L4/3∥˜η∥1/2  L4 ≤ C∥˜η∥1/2  Xǫ  �  ∥˜η∥1/2  Xǫ + ∥∇˜η∥1/2  Xǫ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  On the other hand it is easy to verify that |∇Wǫ| ≤ C(1 + ργ)Wǫ where ργ is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  It follows that  |I3| ≤ C∥˜η∥3/2  Xǫ  �  ∥˜η∥1/2  Xǫ + ∥∇˜η∥1/2  Xǫ  ��  ∥˜η∥Xǫ + ∥ργ ˜η∥Xǫ + ∥∇˜η∥Xǫ  �  ≤ C∥˜η∥XǫDǫ[˜η] ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='68)  where for convenience we denote  Dǫ[˜η] = ∥∇˜η∥2  Xǫ + ∥ργ ˜η∥2  Xǫ + ∥˜η∥2  Xǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='69)  Control of I5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5 asserts that the remainder Rem(R, Z, t) satisfies the pointwise  estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='30), which implies in particular that Rem ∈ Xǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='35), we thus find  |I5| ≤ 1  δ ∥Rem∥Xǫ  �  ∥˜η∥Xǫ + ∥W −1  ǫ  ˜φ∥Xǫ  �  ≤ C  �  ǫ + ǫγ5  δ2  ��  ∥˜η∥Xǫ + ∥W −1  ǫ  ˜φ∥Xǫ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  It remains to estimate the norm of W −1  ǫ  ˜φ in the space Xǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This can be done by decomposing the  Biot-Savart kernel as in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4, see in particular Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Neglecting  contributions of order O(ǫ∞), we can restrict the integrals to the region where R2 + Z2 ≤ ǫ−2σ1  and R′2 + Z′2 ≤ ǫ−2σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Invoking (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4) and recalling that µ0(t) = O(ǫ∞) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1, we find  that ∥W −1  ǫ  ˜φ∥Xǫ = ∥W −1/2  ǫ  ˜φ∥L2(Ωǫ) ≤ C∥˜η∥Xǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We conclude that  |I5| ≤ C  �  ǫ + ǫγ5  δ2  �  ∥˜η∥Xǫ ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='70)  Control of ˆI6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The first two terms in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='39) are easily estimated, because ˙¯r = O(δ) by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='69).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Proceeding as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4 to control the kinetic energy, and recalling that µ0(t) = O(ǫ∞), we  find  I0 :=  ����  ǫ¯r ˙¯r  δΓ  �  Ωǫ  Wǫ˜η∂R˜η dX + t ˙¯r  ¯r Ekin  ǫ  [˜η]  ���� ≤ Cǫ∥˜η∥Xǫ∥∇˜η∥Xǫ + Cǫ2∥˜η∥2  Xǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  So it remains to estimate the last term in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='39), which involves the correction ˙˜z(t) to the vertical  speed introduced in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19) and integrating by parts we first observe that  I1 : =  �  Ωǫ  �  Wǫ˜η − ˜φ  �  ∂Zη∗ dX =  �  Ωǫ  �  Wǫ∂Zη∗ − ∂Zφ∗  �  ˜η dX  = −  �  Ω′ǫ  (∂ZΘ)˜η dX +  �  Ω′′ǫ ∪Ω′′′  ǫ  �  Wǫ∂Zη∗ − ∂Zφ∗  �  ˜η dX ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='71)  45  where Θ is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='86).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In the second line, we used the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='17) of Wǫ in the  inner region Ω′  ǫ to obtain the identity Wǫ∂Zη∗ − ∂Zφ∗ = Φ′  ǫ(ζ∗)∂Zζ∗ − ∂Zφ∗ = −∂ZΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The  last integral in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='71) is of order O(ǫ∞∥˜η∥Xǫ), and the integral over Ω′  ǫ can be controlled using  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We thus obtain |I1| ≤ C(ǫδ + ǫγ3)∥˜η∥Xǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Moreover, we obviously have  I2 :=  ����  �  Ωǫ  �  Wǫ˜η − ˜φ  �  ∂Z ˜η dX  ���� ≤ C∥˜η∥Xǫ∥∇˜η∥Xǫ + C∥˜η∥2  Xǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Finally, to control the velocity ˙˜z(t), we need the following lemma:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Let J(t) =  �  Ωǫ ZR(R, Z, t) dX where R is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Then there exists a  constant C > 0 such that  |J| ≤ Cǫβǫ  δ  �  ∥˜η∥Xǫ + δ∥˜η∥2  Xǫ  �  + C  �  ǫ + ǫγ5  δ2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='72)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We consider separately the various terms in the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Integrating  by parts, we find  J1 := 1  δ  �  Ωǫ  Z  ��  φ∗ , ˜ζ  �  +  �˜φ , ζ∗  ��  dX = −1  δ  �  Ωǫ  � ˜η∂Rφ∗  1+ǫR + η∗∂R ˜φ  1+ǫR  �  dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  In the right-hand side, we can restrict the integration to the region where ρ ≤ ǫ−σ0, because the  integral on the complement is of order O  �  ǫ∞∥˜η∥Xǫ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Thus, expanding the Biot-Savart formula  as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1, we obtain  − δJ1 =  1  2π  �  R2  �  ˜η∂R(Lη∗) + η∗∂R(L˜η)  �  dX + O  �  ǫβǫ∥˜η∥Xǫ  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='73)  where L is the convolution operator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Since L is symmetric in L2(R2) and commutes with  ∂R, the integral in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='73) vanishes and we conclude that |J1| ≤ δ−1ǫβǫ∥˜η∥Xǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Similarly, we have  J2 :=  �  Ωǫ  Z  �˜φ , ˜ζ  �  dX =  �  Ωǫ  �  Z , ˜φ  �˜ζ dX = −  �  Ωǫ  ˜η∂R ˜φ  1+ǫR dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Here again, up to a negligible error, we can assume that ˜η is supported in the ball ρ ≤ ǫ−σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Proceeding as before, we thus find  J2 = − 1  2π  �  R2 ˜η∂R(L˜η) dX + O  �  ǫβǫ∥˜η∥2  Xǫ  �  = O  �  ǫβǫ∥˜η∥2  Xǫ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='74)  The remaining terms in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11) are easier to treat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9) we have  �  Ωǫ  Z  �  L˜η + ǫ∂R˜ζ  �  dx = 0 ,  and  �  Ωǫ  Z  �  ˙¯r ∂R˜η + ˙¯z ∂Z ˜η  �  dX = − ˙¯zµ0 ,  where µ0(t) = O(ǫ∞) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Finally, using estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='30), we obtain  1  δ  �  Ωǫ  |Z| |Rem(R, Z, t)| dX ≤ C  �  ǫ + ǫγ5  δ2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='75)  Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='73), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='74), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='75), we arrive at (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  46  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' There exists a constant C > 0 such that the velocity ˙˜z defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10) satisfies  ¯r| ˙˜z|  Γ  ≤ Cδβǫ  �  ∥˜η∥Xǫ + δ∥˜η∥2  Xǫ  �  + C  �  δ2 + ǫγ5−1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='76)  We now conclude the estimate of the term ˆI6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  To simplify the writing, we assume that  ∥˜η∥Xǫ ≤ 1 and we use the shorthand notation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='69).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Also, since ǫ2 ≲ δ1−σ we observe that  ǫ + ǫγ5  δ2 ≲ Rǫ(t) ,  where  Rǫ(t) := ǫ + ǫγ3  δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='77)  Here γ3 = γ5 − 2/(1−σ) < 3, so that γ3 can be chosen arbitrary close to γ5 − 2 if σ > 0 is small  enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='39) we have |ˆI6| ≤ I0 + |J|  �  |I1| + δI2  �  , so that  |ˆI6| ≤ Cǫ∥˜η∥XǫD1/2  ǫ  + C  �ǫβǫ  δ ∥˜η∥Xǫ + Rǫ  ��  δRǫ∥˜η∥Xǫ + δ∥˜η∥XǫD1/2  ǫ  �  ≤ C∥˜η∥Xǫ  �  D1/2  ǫ  + Rǫ  ��  ǫβǫ + δRǫ  �  ≤ Cǫβǫ∥˜η∥Xǫ  �  D1/2  ǫ  + Rǫ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='78)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9  Conclusion of the proof  We are now in position to conclude the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6, hence also of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Let  ˜η be the unique solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='32) with zero initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The associated energy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='33) satisfies  the evolution equation  t∂tEǫ(t) = I1 + I2 + I3 + ˆI4 + I5 + ˆI6 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='79)  where the various terms in the right-hand side are defined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4 and estimated in  Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='43), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='50), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='68), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='56), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='70), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='78), we find that, as long as  t ≤ Tadvδ−σ and ∥˜η∥Xǫ ≤ 1, there exist positive constants C, C∗ such that  t∂tEǫ(t) ≤ −κDǫ + C∗∥˜η∥XǫDǫ + C∥˜η∥Xǫ  �  Rǫ + ǫβǫD1/2  ǫ  �  + Cǫ2  δ  �  Ω′′ǫ  Wǫ|˜η|2 dX + Cµ2 ,  where Dǫ is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='69), Rǫ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='77), and µ2 := µ2  0 + µ2  1 + µ2  2 ≤ C R2  ǫ by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Since  ργ ≥ ǫ−σ1 in the region Ω′′  ǫ, the integral term can be estimated as follows  ǫ2  δ  �  Ω′′ǫ  Wǫ|˜η|2 dX ≤ ǫ2+2σ1  δ  �  Ω′′ǫ  Wǫρ2  γ|˜η|2 dX ≲ ǫγ∗Dǫ ,  where γ∗ = 2+2σ1−2/(1−σ) > 0 if σ > 0 is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' So, if we assume that C∗∥˜η∥Xǫ ≤ κ/4  and that ǫ is sufficiently small, we obtain by Young’s inequality  t∂tEǫ(t) ≤ −κ  2 Dǫ + C Rǫ∥˜η∥Xǫ + Cµ2 ≤ −κ  4 Dǫ + C R2  ǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Integrating that differential inequality over the time interval (0, t) and recalling that Eǫ(0) = 0,  we arrive at  Eǫ(t) ≤ C  � t  0  Rǫ(s)2  s  ds ≤ C Rǫ(t)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Finally we invoke (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28) as well as Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9) to conclude that  ∥˜η(t)∥2  Xǫ ≤ C6Eǫ(t) + C7  �  βǫµ0(t)2 + µ1(t)2�  ≤ C Rǫ(t)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='80)  Inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='80) holds as long as ∥˜η(t)∥Xǫ ≤ 1 and t < Tadvδ−σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' But on that time interval we  know that Rǫ ≲ ǫγ3−2/(1−σ) ≪ 1, so (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='80) is actually valid for all t ∈ (0, Tadvδ−σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Returning  to the the solution (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15) with initial data (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='22), we obtain in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='31), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='80)  ∥η(t) − η∗(t)∥Xǫ = δ∥˜η(t)∥Xǫ ≤ Cδ Rǫ(t) = C  �  ǫδ + ǫγ3�  ,  t ∈ (0, Tadvδ−σ) ,  which gives (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This concludes the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  □  47  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Strictly speaking, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6 deals with the rescaled vorticity η defined by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='13), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14), whereas the proof above uses the modified self-similar variables defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  However, the shift in the vertical position induced by the nonlinear correction ˜z(t) is very small,  and produces negligible corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Indeed, it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='76) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='80) that  ¯r| ˙˜z(t)|  Γ  ≲  �  δβǫ Rǫ + δ2 + ǫγ5−1�  ,  hence  |˜z(t)| ≲ ǫ2¯r(t)  �  δ + βǫ Rǫ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  When rephrased in self-similar coordinates, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1 compares the solution η of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15)  to the solution ˆη of the linear heat equation  t∂tˆη −  �  t  ν  �  ˙¯r ∂Rˆη + ˙¯z ∂Z ˆη  �  = Lˆη + ∂R  �  ǫˆη  1 + ǫR  �  ,  with the same initial data η0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It is straightforward to verify that ∥ˆη − η0∥Xǫ = O(ǫ) as ǫ → 0, so  that ∥ˆη − η∥Xǫ = O(ǫ) too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The discrepancy between the velocities ˙¯r, ˙¯z constructed in Section 3  and the approximate values corresponding to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8) is responsible for the deterioration of the  error term in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7), which is of order ǫ + ǫβǫδ−σ ≲ ǫ1−3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  A  Appendix to Section 3  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1  Inverting the operator Λ  Following [28], we give here a short proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Assume that n ≥ 2 and f ∈ Yn∩Z,  or that n = 1 and f ∈ Y′  1 ∩ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In both cases, we have f ∈ Ker(Λ)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We want to show that  there exists a unique η ∈ Yn ∩ Z (or η ∈ Y′  1 ∩ Z if n = 1) such that Λη = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  To make things concrete, we assume without loss of generality that f = a(ρ) sin(nϑ), for  some function a : R+ → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Our hypotheses imply that a is smooth, that a(ρ) = O(ρn) as ρ → 0,  and that eρ2/4a(ρ) grows at most polynomially as ρ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We look for a solution of the form  η = ω(ρ) cos(nϑ), where ω : R+ → R has to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19), we have  Λη =  �  φ0 , η  �  + {Ψ , η0  �  ,  where  φ0 =  1  2π Lη0 ,  Ψ =  1  2π Lη .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1)  The function φ0 is radially symmetric and satisfies ∂ρφ0 = −ρϕ, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='24) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='12) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It  follows that  �  φ0 , η  �  = ∂ρφ0  1  ρ ∂ϑη = nϕ(ρ)ω(ρ) sin(nϑ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2)  On the other hand, as −∆Ψ = η, we have Ψ = Ω(ρ) cos(nϑ), where Ω is the unique regular  solution of the differential equation  − Ω′′(ρ) − 1  ρ Ω′(ρ) + n2  ρ2 Ω(ρ) = ω(ρ) ,  ρ > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3)  Since η0 is radially symmetric and ∂ρη0 = −(ρ/2)η0 = −ρϕh, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='24), we deduce  {Ψ , η0  �  = −∂ρη0  1  ρ ∂ϑΨ = −nϕ(ρ)h(ρ)Ω(ρ) sin(nϑ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4)  In view of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4), the equation Λη = f is equivalent to the relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='25), and  using in addition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3) we obtain the differential equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='26) for the unknown function Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  48  The main step in the proof is to show that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='26) has a unique solution that is regular at  the origin and decays to zero at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Here we distinguish two cases according to the value  of the angular Fourier mode n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If n ≥ 2, the homogeneous equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='26) with a ≡ 0 has two linearly independent solutions  ψ+, ψ− which satisfy  ψ−(ρ) ∼  �  ρn  as ρ → 0 ,  κρn  as ρ → ∞ ,  ψ+(ρ) ∼  �  κρ−n  as ρ → 0 ,  ρ−n  as ρ → ∞ ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5)  for some κ > 0, see [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Here we use the crucial observation that (n2/ρ2)−h(ρ) > 0 when n ≥ 2,  so that the differential operator in the left-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='26) satisfies the Maximum Principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  We deduce the following representation formula for the solution of the inhomogeneous equation :  Ω(ρ) = ψ+(ρ)  � ρ  0  r  w0  ψ−(r) a(r)  nϕ(r) dr + ψ−(ρ)  � ∞  ρ  r  w0  ψ+(r) a(r)  nϕ(r) dr ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6)  where w0 = 2nκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It is then straightforward to verify that Ω(ρ) = O(ρn) as ρ → 0 and Ω(ρ) =  O(ρ−n) as ρ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Moreover, if ω is defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='25), the function η = ω(ρ) cos(nϑ) lies in  Yn ∩ Z and satisfies Λη = f by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The details can be found in [28, Lemma 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The situation is quite different when n = 1, because the lower order term 1/ρ2 −h(ρ) in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3)  is no longer positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In that case, it happens that the homogeneous equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='26) with a ≡ 0  has a solution ψ(ρ) = ρϕ(ρ) which satisfies ψ(ρ) ∼ ρ/(8π) as ρ → 0 and ψ(ρ) ∼ 1/(2πρ) as  ρ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In other words, the linear operator in the left-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='26) has a one-dimensional  kernel, and for that reason we have to impose the solvability condition  f ∈ Y′  1 ⊂ Ker(Λ)⊥ ,  or equivalently  � ∞  0  a(ρ)ρ2 dρ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7)  To solve (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='26) for n = 1, we look for a solution of the form Ω(ρ) = b(ρ)ψ(ρ), which leads to a  first-order differential equation for b(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In view of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7), we thus find  b′(ρ) = −  1  ρψ(ρ)2  � ρ  0  a(r)r2 dr =  1  ρψ(ρ)2  � ∞  ρ  a(r)r2 dr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8)  Integrating (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8) gives the representation formula  b(ρ) = b0 −  � ρ  0  a(r)r2�  F(ρ) − F(r)  �  dr ,  for some b0 ∈ R ,  where  F(ρ) = 8π2  �  log  �  eρ2/4 − 1  �  −  1  eρ2/4 − 1  �  ,  F′(ρ) =  1  ρψ(ρ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  We now substitute Ω(ρ) = b(ρ)ψ(ρ) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='25) with n = 1, and we choose the constant b0 so  that  � ∞  0 ω(ρ)ρ2 dρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This is always possible in a unique way, since  � ∞  0  ρ2h(ρ)ψ(ρ) dρ =  � ∞  0  ρ3h(ρ)ϕ(ρ) dρ =  1  8π  � ∞  0  ρ3 e−ρ2/4 dρ = 1  π ̸= 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  To conclude the proof, it remains to verify that the function η = ω(ρ) sin(ϑ) constructed above  belongs to Y′  1 ∩ Z and satisfies Λη = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' These are straightforward calculations, which can be  omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  □  49  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2  First order calculations  We first establish the relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' As η0 ∈ Y0 has unit mass we find, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11),  �  P1η0  �  (R, Z) =  �  R2  R+R′  2  η0(R′, Z′) dR′ dZ′ = R  2 ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9)  hence {P1η0 , η0} = 1  2 ∂Zη0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' On the other hand, since ∂Rη0 = −(R/2)η0 and L is a convolution  operator, which therefore commutes with derivatives, we have  �  LP1η0  �  (R, Z) = R  2 (Lη0)(R, Z) + L  �R  2 η0  �  (R, Z) = R  2 (Lη0)(R, Z) − ∂R  �  Lη0  �  (R, Z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Recalling that Lη0 = 2πφ0, and that {φ0, η0} = 0 because both φ0, η0 are radially symmetric,  we thus obtain  1  2π  �  LP1η0 , η0  �  =  �R  2 φ0 − ∂Rφ0 , η0  �  = 1  2 φ0 ∂Zη0 +  �  φ0 , ∂Rη0  �  = 1  2 φ0 ∂Zη0 −  �  φ0 , R  2 η0  �  = 1  2 φ0 ∂Zη0 + 1  2 (∂Zφ0)η0 ,  which concludes the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  We next prove formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='38) for the vertical velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' According to the definition of v in  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='38), the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='37) belongs to Y′  1 = Y ∩ Ker(Λ)⊥ if and only if  �  R2  � v  2π ∂Zη0 − 3  2(∂Zφ0)η0 − 1  2φ0∂Zη0  �  Z dR dZ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10)  Since ∂Zη0 = −(Z/2)η0 and  �  R2 Z2η0 dR dZ = 2, it is straightforward to verify that (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10) is  equivalent to  v = π  �  R2 φ0η0  �  3 − Z2�  dR dZ = π  2  �  R2 φ0η0  �  6 − |X|2�  dX ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11)  where X = (R, Z) and |X|2 = R2 + Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  To evaluate the right-hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11), we temporarily denote ψ0 = 2πφ0 = Lη0, namely  ψ0(X) =  1  4π  �  R2 log  �  8  |X − Y |  �  e−|Y |2/4 dY ,  X ∈ R2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  This function satisfies −∆ψ0 = 2πη0 = 1  2 e−|X|2/4, so that  ψ0(X) = ψ0(0) −  � |X|  0  1 − e−ρ2/4  ρ  dρ ≡ ˜ψ0(|X|) ,  X ∈ R2 ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='12)  where  ψ0(0) = log(8) − 1  4π  �  R2 log(|Y |) e−|Y |2/4 dY = 2 log(2) + γE  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='13)  Using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='12), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='13) and integrating by parts, we easily find  �  R2 ψ0η0 dX = 1  2  � ∞  0  ˜ψ0(ρ)e−ρ2/4ρ dρ = ψ0(0) +  � ∞  0  ˜ψ′  0(ρ)e−ρ2/4 dρ = 3  2 log(2) + γE  2 ,  and similarly  �  R2 ψ0η0|X|2 dX = 4ψ0(0) +  � ∞  0  ˜ψ′  0(ρ)e−ρ2/4(ρ2 + 4) dρ = 6 log(2) + 2γE − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Returning to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11), we conclude that  v = 1  4  �  R2 ψ0η0  �  6 − |X|2�  dX = 3  4 log(2) + 1  4γE + 1  4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14)  50  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3  Second order calculations  Our goal here is to prove Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' To establish (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='47), we consider separately the various  terms in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' As η1 ∈ Y1 has zero mean, we find as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9) that P1η1 is a constant, which  can be disregarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Moreover LP1η1 = R  2 Lη1 + L  �R  2 η1  �  , hence using the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='40) of η1  we find that  LP1η1 = (R2 − Z2)χ1(ρ) + δRZχ2(ρ) + χ3(ρ) ,  where χ1, χ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' are functions of the radial variable ρ = (R2 + Z2)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' As η0 itself is radially  symmetric, we deduce that  �  (βǫ − 1)P1η1 + LP1η1 , η0  �  = RZχ4(ρ) + δ(R2 − Z2)χ5(ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15)  Next, using the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11) of P2, we see that  �  P2η0  �  (R, Z) =  1  16  �  R2  �  (R−R′)2 + 3(Z−Z′)2�  η0(R′, Z′) dR′ dZ′ = R2  16 + 3Z2  16 + 1  2 ,  and a similar calculation gives Q2η0 = 3R2  16 − Z2  16 + 1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Moreover,  �  LP2η0  �  (R, Z) = 1  16  �  R2 log  � 8  D  ��  2D2 + (Z−Z′)2 − (R−R′)2�  η0(R′, Z′) dR′ dZ′ ,  where D2 = (R−R′)2 + (Z−Z′)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Using the fact that η0 given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='31) is radially symmetric,  we easily obtain  1  2π  �  LP2η0  �  (R, Z) = χ6(ρ) + (R2 − Z2)χ7(ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Altogether, we arrive at  1  2π  �  βǫP2η0 + LP2η0 + Q2η0 , η0  �  =  βǫ  16π RZη0 + RZχ8(ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='16)  The remaining terms in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='46) are easier to treat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='38), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='40), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='41), we have  �  φ1 , η1  �  − r0 ˙¯z0  Γ  ∂Zη1 =  �  φ1 − βǫ − 1  4π  R , η1  �  − v  2π ∂Zη1  =  �R  2 φ0 − ∂Rφ0 + R φ10(ρ) + δZ φ11(ρ) , R η10(ρ) + δZ η11(ρ)  �  = RZ χ9(ρ) + δ  �  χ10(ρ) + (R2 − Z2)χ11(ρ)  �  + δ2RZ χ12(ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='17)  It is also easy to verify that the terms (∂Zφ1)η0 +(∂Zφ0)η1 −2R(∂Zφ0)η0 +δ∂R(Rη0) are exactly  of the same form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Finally, using again (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='40), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='41), we obtain  R  ��  φ1 , η0  �  +  �  φ0 , η1  ��  = R  �βǫ − 1  4π  ∂Zη0 + Zχ13(ρ) + δRχ14(ρ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='18)  If we now combine (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='16), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='17), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='18), we arrive at (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  □  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4  Higher order order calculations  The calculations carried out in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6 do not require new ideas, but a more  compact notation is often helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' To prove Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='12 and similar statements, it is important  to understand how the decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='21) of the function space Y behaves under the Poisson  51  bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If we use polar coordinates X = (ρ cos ϑ, ρ sin ϑ), we recall that Yn is the subspace of  Y spanned by functions of the form a(ρ) cos(nϑ) and b(ρ) sin(nϑ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Since  �  f , g  �  = ∂Rf∂Zg − ∂Zf∂Rg = 1  ρ  �  ∂ρf∂ϑg − ∂ϑf∂ρg  �  ,  we easily obtain the following result :  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' If a, b : R+ → R are smooth functions and n, m ∈ N, then  �  a(ρ) cos(nϑ) , b(ρ) cos(mϑ)  �  = c11(ρ) sin((n−m)ϑ) + c12(ρ) sin((n+m)ϑ) ,  �  a(ρ) sin(nϑ) , b(ρ) sin(mϑ)  �  = c21(ρ) sin((n−m)ϑ) + c22(ρ) sin((n+m)ϑ) ,  �  a(ρ) sin(nϑ) , b(ρ) cos(mϑ)  �  = c31(ρ) cos((n−m)ϑ) + c32(ρ) cos((n+m)ϑ) ,  where cij : R+ → R are smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In particular {Yn, Ym} ⊂ Yn−m + Yn+m if m ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  It is also necessary to compute the homogeneous polynomials Pj, Qj in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10) for higher  values of j than in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' This is a direct calculation that can be done for instance using  computer algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' For j = 3 we find  P3 = − 1  32(R + R′)  �  (R − R′)2 + 3(Z − Z′)2�  ,  Q3 = − 1  48(R + R′)  �  (R + R′)2 − 6(Z − Z′)2�  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19)  and the calculation for j = 4 yields the more complicated expressions  P4 = −  15  1024 (Z−Z′)4 + 21  512 (R−R′)2(Z−Z′)2 + 3  16 RR′ (Z−Z′)2  +  17  1024 (R2−R′2)2 −  1  256 RR′ (R−R′)2 ,  Q4 = 31  2048 (Z−Z′)4 −  89  1024 (R+R′)2(Z−Z′)2 +  1  256 RR′ (Z−Z′)2  −  19  6144 (R2−R′2)2 +  35  1536 RR′(R+R′)2 −  1  128 R2R′2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='20)  The proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='12 is similar to that of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11, and the details can be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  We use the expressions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='40), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='49) of the vorticities η1, η2, the formulas (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='41), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='50) for the  stream functions φ1, φ2, and the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15) of the Biot-Savart operators, which involve  the polynomials (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1, it is straightforward to verify that the  quantity defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='54) satisfies R3 ∈ Y1 + Y3 and takes the form  R3 = χ1(ρ) sin(ϑ) + χ2(ρ) sin(3ϑ) + δ  �  χ3(ρ) cos(ϑ) + χ4(ρ) cos(3ϑ)  �  + O(δ2) ,  where χ1, χ2, χ3, χ4 are radially symmetric functions which may depend on βǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' To arrive at  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='55), it remains to verify that R3 does not contain any term involving β2  ǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Indeed, according  to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='49), we have  βǫ  2π P1η2 = βǫ  4π  �  R2(R + R′) η2(R′, Z′) dR′ dZ′ = βǫR  4π  �  R2 η24(R′, Z′) dR′ dZ′ ,  so that the first term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='54) does not contain β2  ǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The only other terms that we have to check  are  �  φ1 , η2  �  − r0  Γ  ˙¯z0∂Zη2 =  �  φ1 − βǫ − 1 + 2v  4π  R , η2  �  ,  where we the equality results from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='38), but using the expressions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='41), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='49) we immediately  see that the right-hand side does not contain any factor β2  ǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Altogether we arrive at (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  □  52  B  Appendix to Section 4  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1  Properties of the energy functional  Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We use the first expression of Ekin  ǫ  [η] in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='23) and the representation  formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='20) for the stream function φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Since supp(η) ⊂ Bǫ by assumption, we have  Ekin  ǫ  [η] =  1  4π  �  Bǫ  �  Bǫ  Gǫ(R, R′, Z, Z′) η(R, Z) η(R′, Z′) dX dX′ ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1)  where  Gǫ(R, R′, Z, Z′) =  �  (1+ǫR)(1+ǫR′) F  �  ǫ2 (R−R′)2 + (Z−Z′)2  (1+ǫR)(1+ǫR′)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2)  As R2 + Z2 ≤ ǫ−2σ1 and R′2 + Z′2 ≤ ǫ−2σ1, the argument of F in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2) is of order O(ǫ2−2σ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Using the asymptotic expansion of F(s) as s → 0 and proceeding as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1, we easily  obtain the decomposition  Gǫ(R, R′, Z, Z′) = βǫ − 2 + log 8  D + ˜Gǫ(R, R′, Z, Z′) ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3)  where βǫ = log(1/ǫ) and D2 = (R−R′)2 + (Z−Z′)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The remainder ˜Gǫ satisfies the estimate  | ˜Gǫ(R, R′, Z, Z′)| ≤ Cǫ  �  |R| + |R′|  ��  βǫ + 1 + log 8  D  �  + O  �  βǫǫ2−2σ1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4)  If we insert the decomposition (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3) into (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1), the contributions of βǫ − 2 and log(8/D) give  exactly the first two terms in the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='25), in view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Moreover, taking  into account estimate (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4) where ǫ2−2σ1 ≤ ǫ, we see that the contributions of ˜Gǫ to the kinetic  energy (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='1) are of order O  �  ǫβǫ∥η∥2  Xǫ  �  , as stated in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Given η ∈ Xǫ, we decompose η = η1 + η2 where η1 = η1Bǫ and  1Bǫ is the indicator function of the ball Bǫ = {(R, Z) ∈ Ωǫ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' R2 + Z2 ≤ ǫ−2σ1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We thus have  Eǫ[η] = 1  2  �  Ωǫ  Wǫ η2  1 dX + 1  2  �  Ωǫ  Wǫ η2  2 dX − 1  2  �  Ωǫ  �  φ1 + φ2  ��  η1 + η2  �  dX ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5)  where φj = BSǫ[ηj] for j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We claim that  1  2  �  Ωǫ  �  φ1 + φ2  ��  η1 + η2  �  dX = Ekin  ǫ  [η1] + O  �  ǫ∞∥η∥2  Xǫ  �  ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6)  so that  Eǫ[η] = Eǫ[η1] + 1  2∥η2∥2  Xǫ + O  �  ǫ∞∥η∥2  Xǫ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7)  To prove (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6), we recall that φj(R, Z) =  1  2π  �  Ωǫ Gǫ(R, Z, R′, Z′)ηj(R′, Z′) dX′, where the  kernel Gǫ is given by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Using the crude estimate |F(s)| ≤ C  �  | log s| + 1  �  , we easily obtain  ��Gǫ(R, R′, Z, Z′)  �� ≤ C  �  1+ǫ|R|  �a�  1+ǫ|R′|  �a�  βǫ +  ��log D  �� + 1  �  ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='8)  for some a > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It follows in particular that  |φ(R, Z)| ≤ C  �  βǫ + 1  �  (1 + ρ)b∥η∥Xǫ ,  ρ =  �  R2 + Z2 ,  53  for some b > 1/2, and using H¨older’s inequality we deduce  �  Ωǫ  |φ(R, Z)| |η2(R, Z)| dX ≤ C  �  βǫ + 1  �  ∥η∥2  Xǫ  ��  Bcǫ  (1 + ρ)2b Wǫ(R, Z)−1 dX  �1/2  ,  where the last integral is O(ǫ∞) in view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' In a similar way we have  |φ2(R, Z)| ≤ C  �  βǫ + 1  �  (1 + ρ)b  �  Bcǫ  (1 + ρ′)b|η(R′, Z′)| dX′ = O  �  ǫ∞∥η∥Xǫ  �  (1 + ρ)b ,  so that  �  Ωǫφ2η1 dx = O  �  ǫ∞∥η∥2  Xǫ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Altogether we arrive at (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Now, since η1 is supported in the ball Bǫ, it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='4 that  ∥η1∥2  Xǫ = ∥η1∥2  X0 + O  �  ǫγ1∥η∥2  Xǫ  �  ,  Ekin  ǫ  [η1] = βǫ−2  4π  ˜µ2  0 + Ekin  0 [η1] + O  �  ǫβǫ∥η∥2  Xǫ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9)  Moreover we know from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='5 that  ∥η1∥2  X0 ≤ C4E0[η1] + C5  �  ˜µ2  0 + ˜µ2  1 + ˜µ2  2  �  ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10)  where ˜µ0, ˜µ1, ˜µ2 are the moments of η1, which satisfy ˜µj = µj + O  �  ǫ∞∥η∥Xǫ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Combining both  estimates in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='9) we obtain  E0[η1] = 1  2∥η1∥2  X0 − Ekin  0 [η1] ≤ 1  2∥η1∥2  Xǫ − Ekin  ǫ  [η1] + βǫ−2  4π  ˜µ2  0 + O  �  ǫγ1∥η∥2  Xǫ  �  ,  and using in addition (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='10) we deduce  ∥η1∥2  Xǫ ≤ ∥η1∥2  X0 + O  �  ǫγ1∥η∥2  Xǫ  �  ≤ C4Eǫ[η1] + C  �  βǫ˜µ2  0 + ˜µ2  1 + ˜µ2  2  �  + O  �  ǫγ1∥η∥2  Xǫ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Finally, invoking (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='7) and recalling that C4 > 2, we find  ∥η∥2  Xǫ ≤ ∥η1∥2  Xǫ + C4  2 ∥η2∥2  Xǫ ≤ C4Eǫ[η] + C  �  βǫ˜µ2  0 + ˜µ2  1 + ˜µ2  2  �  + O  �  ǫγ1∥η∥2  Xǫ  �  ,  and estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='28) follows, since ˜µj = µj + O  �  ǫ∞∥η∥Xǫ  �  for j = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='2  Diffusive terms in the energy functional  We justify here the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='33) of the quantity I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Integrating by parts as in [31], we find  �  Ωǫ  Wǫ˜η L˜η dX = −  �  Ωǫ  Wǫ|∇˜η|2 dX −  �  Ωǫ  (∇Wǫ · ∇˜η)˜η dX −  �  Ωǫ  ˜Vǫ˜η2 dX ,  where ˜Vǫ = 1  4(R∂R + Z∂Z)Wǫ − 1  2Wǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Similarly,  ǫ  �  Ωǫ  Wǫ˜η ∂R ˜ζ dX = ǫ  �  Ωǫ  Wǫ(1 + ǫR)˜ζ ∂R˜ζ dX = − ǫ  2  �  Ωǫ  ∂R  �  Wǫ(1 + ǫR)  �˜ζ2 dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  On the other hand, integrating by parts and using the relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19) between ˜φ and ˜η, we obtain  �  Ωǫ  ˜φ  �  L˜η + ǫ∂R˜ζ  �  dX =  �  Ωǫ  ˜η  �  ∆˜φ − ǫ∂R ˜φ  1 + ǫR  �  dX − 1  2  �  Ωǫ  ˜η  �  R∂R + Z∂Z  �˜φ dX  = −  �  Ωǫ  ˜η2(1 + ǫR) dX − 1  2  �  Ωǫ  ˜η  �  R∂R + Z∂Z  �˜φ dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  It remains to treat the last term in the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Here again, we use the relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19)  and integrate by parts to obtain  1  2  �  Ωǫ  ˜η  �  R∂R + Z∂Z  �˜φ dX = ǫ  4  �  Ωǫ  R|∇˜φ|2  (1 + ǫR)2 dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Altogether we arrive at (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='33), with Vǫ = ˜Vǫ − (1 + ǫR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  54  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='3  Coercivity of the diffusive quadratic form  This section is devoted to the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Given ǫ > 0 sufficiently small, we take  a smooth partition of unity of the form 1 = χ2  3 + χ2  4, where χ3, χ4 are radially symmetric and  χ3 = 1 when ρ ≤ 1  2ǫ−σ1, χ3 = 0 when ρ ≥ ǫ−σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We can also assume that |∇χ3| + |∇χ4| ≤ Cǫσ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Given η as in the statement of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15, we define η3 = χ3η, η4 = χ4η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We thus have  the decompositions η2 = η2  3 + η2  4, η∇η = η3∇η3 + η4∇η4, and  |∇η|2 = |∇η3|2 + |∇η4|2 −  �  |∇χ3|2 + |∇χ4|2�  η2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11)  As a consequence, the quadratic form Qǫ[η] can be decomposed as  Qǫ[η] = Qǫ[η3] + Qǫ[η4] −  �  Ωǫ  Wǫ  �  |∇χ3|2 + |∇χ4|2�  η2 dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='12)  The last term in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='12) is bounded by Cǫ2σ1∥η∥2  Xǫ and is thus negligible when ǫ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' So our  main task is to estimate from below the terms Qǫ[η3] and Qǫ[η4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  We first consider the function η3 which is supported in the region where ρ ≤ ǫ−σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We recall  that the weight Wǫ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='17) satisfies the estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='19), which read  |∇Wǫ(R, Z) − ∇A(ρ)| + |Wǫ(R, Z) − A(ρ)| ≤ Cǫγ1A(ρ) ,  when ρ ≤ ǫ−σ1 ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='13)  where γ1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' We easily deduce that  Qǫ[η3] ≥ Q0[η3] − Cǫγ4�  ∥∇η3∥2  X0 + ∥η3∥2  X0  �  ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14)  for some γ4 > 0, where Q0 is the limiting quadratic form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' On the other hand, we know  from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14 that  C8Q0[η3] ≥ ∥∇η3∥2  X0 + ∥ρη3∥2  X0 + ∥η3∥2  X0 − C9  �  ˜µ2  0 + ˜µ2  1 + ˜µ2  2  �  ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15)  where ˜µ0, ˜µ1, ˜µ2 are the moments of η3, which satisfy ˜µj = µj +O  �  ǫ∞∥η∥Xǫ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Combining (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='14),  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='15) and using (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='13) once again, we arrive at  ∥∇η3∥2  Xǫ + ∥ρη3∥2  Xǫ + ∥η3∥2  Xǫ ≤ C8Qǫ[η3] + C  �  ˜µ2  0 + ˜µ2  1 + ˜µ2  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='16)  We next consider the function η4, which is nonzero only if ρ ≥ 1  2ǫ−σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Our starting point is  the lower bound  Qǫ[η4] ≥ 1  4  �  Ωǫ  Wǫ|∇η4|2 dX +  �  Ωǫ  �  Vǫ − |∇Wǫ|2  3Wǫ  �  η2  4 dX ,  which is obtained from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='62) by applying Young’s inequality to the middle term in the right-  hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Using the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='17) of the weight function, as well as the estimates (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='13) in  the inner region Ω′  ǫ, it is not difficult to verify that  Vǫ  Wǫ  − |∇Wǫ|2  3W 2ǫ  ≥  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  Cρ2 − ˜C  in Ω′  ǫ ,  − ˜C  in Ω′′  ǫ ,  Cρ2γ  in Ω′′′  ǫ ,  for some positive constants C, ˜C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' It follows that  Qǫ[η4] ≥ 1  4∥∇η4∥2  Xǫ + C  �  Ω′ǫ∪Ω′′′  ǫ  Wǫ ρ2  γη2  4 dX − ˜C  �  Ω′′ǫ  Wǫ η2  4 dX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='17)  55  If we now combine (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='16) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='17), we obtain  ∥∇η3∥2  Xǫ + ∥∇η4∥2  Xǫ + ∥η∥2  Xǫ +  �  Ω′ǫ∪Ω′′′  ǫ  Wǫ ρ2  γη2 dX  ≤ C10  �  Qǫ[η3] + Qǫ[η4]  �  + C11  �  ˜µ2 +  �  Ω′′ǫ  Wǫη2�  ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='18)  for some positive constants C10, C11, where ˜µ2 = ˜µ2  0 + ˜µ2  1 + ˜µ2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Finally, using again (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='11) as  well as (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='12), and recalling that ˜µj = µj + O  �  ǫ∞∥η∥Xǫ  �  , we deduce (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='66) from (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  □  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' ThG is partially supported by the grant SingFlows ANR-18-CE40-0027  of the French National Research Agency (ANR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' The research of VS is supported in part by  grant DMS 1956092 from the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
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+page_content=' Sullivan, On the stability of vortex rings, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' A  332 1973, 335–353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='  Thierry Gallay  Institut Fourier, Universit´e Grenoble Alpes, 100 rue des Maths, 38610 Gi`eres, France  Email : Thierry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='Gallay@univ-grenoble-alpes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='fr  Vladim´ır ˇSver´ak  School of Mathematics, University of Minnesota  127 Vincent Hall, 206 Church St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content=' SE, Minneapolis, MN 55455, USA  Email : sverak@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
+page_content='edu  58' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfKPsc/content/2301.01092v1.pdf'}
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+Topological order in interacting semimetals
+Jinmin Yi,∗ Xuzhe Ying,∗ Lei Gioia, and A.A. Burkov
+Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada and
+Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
+(Dated: January 11, 2023)
+It has recently been demonstrated that it is possible to open a gap in a magnetic Weyl semimetal,
+while preserving the chiral anomaly along with the charge conservation and translational symmetries,
+which all protect the gapless nodes in a weakly interacting semimetal. The resulting state was shown
+to be a nontrivial generalization of a nonabelian fractional quantum Hall liquid to three dimensions.
+Here we point out that a second fractional quantum Hall state exists in this case.
+This state
+has exactly the same electrical and thermal Hall responses as the first, but a distinct (fracton)
+topological order. Moreover, the existence of this second fractional quantum Hall state necessarily
+implies a gapless phase, which has identical topological response to a noninteracting Weyl semimetal,
+but is distinct from it. This may be viewed as a generalization (in a weaker form) of the known
+duality between a noninteracting two-dimensional Dirac fermion and QED3 to 3 + 1 dimensions. In
+addition we discuss a (3 + 1)-dimensional topologically ordered state, obtained by gapping a nodal
+line semimetal without breaking symmetries.
+I.
+INTRODUCTION
+Topological order, a concept that originated in the
+study of the fractional quantum Hall effect (FQHE) in
+two dimensional (2D) electron gas systems,1 continues
+to be a subject of intense interest.
+From the funda-
+mental physics prospective, topologically ordered states
+provide perfect examples of emergent macroscopic quan-
+tum phenomena, with fractionally-quantized electromag-
+netic and thermal responses, that are impossible to ex-
+plain based on textbook models of weakly-interacting
+electrons.
+Instead, such fractionally-quantized observ-
+able responses necessarily imply excitations with frac-
+tional charges, fractional and nonabelian statistics, which
+can not be constructed out of any finite number of ele-
+mentary constituents.2 In addition, such exotic excita-
+tions may have future potential practical uses in quan-
+tum computing and quantum simulation, as their non-
+local topological nature makes them highly resistant to
+decoherence and noise.3
+Topologically-ordered states in 2D are by now well-
+understood.
+Various theoretical models,4–7 as well as
+complete formal classifications of 2D topological orders
+exist.8 Although significant progress has been made in re-
+cent years,9–19 less is known about topologically-ordered
+states in three dimensions (3D). 3D topologically ordered
+states are significantly different from the 2D ones. On
+the one hand, fractional statistics is impossible in 3D and
+quasiparticle excitations may only be bosons or fermions.
+This could make one doubt that, for example, fractional
+quantum Hall (FQH) states may even in principle be gen-
+eralized to 3D, as the existence of anyons, i.e. quasipar-
+ticles with fractional statistics, is an essential feature of
+the 2D FQHE. On the other hand, in addition to point
+quasiparticles, one-dimensional loop excitations exist in
+3D, which both adds complexity and opens up new in-
+teresting possibilities.
+We recently demonstrated that a promising way to
+achieve 3D topologically ordered states is through gap-
+ping topological semimetals without breaking the pro-
+tecting symmetries20–22 (see Refs. 23–29 for related
+work).
+Topological semimetals30–35 are intermediate
+phases between insulators of different electronic structure
+topology.
+They may be characterized by unquantized
+anomalies,36,37 i.e. topological terms with noninteger and
+continuously-tunable coefficients, similar to the electron
+filling parameter, characterizing ordinary Fermi liquids.
+Much like fractional filling in a Fermi liquid mandates the
+existence of a Fermi surface of gapless particle-hole ex-
+citations,38 these unquantized anomalies necessarily im-
+ply gapless modes and corresponding long-range entan-
+glement. The only way gaplessness may be circumvented
+in the absence of broken symmetries is through the for-
+mation of a topologically-ordered state, which preserves
+the anomaly and the long-range entanglement of the gap-
+less semimetal.
+Specifically, in Ref. 20 we presented an explicit con-
+struction of a 3D topologically-ordered state in a gapped
+magnetic Weyl semimetal, which exhibits a nontrivial
+generalization of the FQHE to 3D. This state is ob-
+tained starting from a magnetic Weyl semimetal with
+a single pair of nodes, separated by half a reciprocal lat-
+tice vector.
+These nodes may be gapped by breaking
+the U(1) charge conservation symmetry while forming a
+superconducting state with intra-nodal pairing. In gen-
+eral, such states break translational symmetry since the
+Weyl nodes exist at nontrivial momenta in the first Bril-
+louin zone (BZ). However, when the nodes are separated
+by exactly half a reciprocal lattice vector, such a pair-
+ing leads to density modulation at the reciprocal lattice
+vector, which does not break the crystal translational
+symmetry.
+Restoring the charge conservation symme-
+try by proliferating flux 2hc/e = 4π (we will be using
+ℏ = c = e = 1 units throughout this paper) vortices in
+the superconducting order parameter leads to a feature-
+less fractionalized insulator with Z4 topological order,
+that has the same electrical and thermal Hall conductiv-
+ities as the original noninteracting Weyl semimetal, i.e.
+arXiv:2301.03628v1  [cond-mat.str-el]  9 Jan 2023
+
+2
+exhibits FQHE in 3D. Unlike in 2D FQH liquids, quasi-
+particle excitations in this state are bosons and fermions.
+What plays the role of the anyons in the 2D FQHE are
+intersections of the vortex-loop excitations with atomic
+planes.
+These behave as fractionally-charged particles
+with semionic statistics, which may be sharply defined
+by considering three-loop braiding processes,15 involving
+a line defect of translational symmetry, i.e. an edge dis-
+location.
+In this paper we show that, in addition to the 3D FQH
+state of Ref. 20, another state exists, which has identi-
+cal topological response, but distinct topological order,
+which turns out to be of a fracton type. The existence
+of these two distinct states turns out to be closely re-
+lated to a very similar property of gapped symmetric
+2D Dirac surface states of 3D time-reversal (TR) in-
+variant topological insulators (TI).39–43 In this case, two
+distinct topologically-ordered states exist.
+One, called
+Pfaffian-antisemion,40,42 is closely related to the 3D FQH
+states of Ref. 20 (more precisely, the relation is with the
+TR-broken version of this state).
+The second one, T-
+Pfaffian,39,41 is related to the other 3D state we will con-
+struct in the present paper (again, more precisely, the
+relation is with the TR-breaking version of this state,
+which is usually called PH, which stands for particle-
+hole-symmetric, Pfaffian44,45).
+Another interesting consequence emerges from these
+analogies to the 2D TR-invariant TI surface state topo-
+logical orders. It is well-known that the PH-Pfaffian is
+closely related to the recently discovered duality relation
+between a massless noninteracting 2D Dirac fermion and
+QED3.44,46–51 Namely, the PH-Pfaffian state is obtained
+when the dual Dirac fermion of QED3 is gapped by pair-
+ing, which does not break the charge conservation sym-
+metry since the dual fermion is neutral. The existence of
+the analog of the PH-Pfaffian state in our 3D system then
+also implies the existence of a gapless state, which is re-
+lated to the noninteracting Weyl semimetal via a duality
+relation, somewhat similar to the 2D Dirac duality. We
+demonstrate that this is indeed the case. However, we
+find that the duality only applies to topological response
+in this case and not to the dynamics and is weaker than
+the 2D duality in this sense.
+The path to topologically ordered insulators through
+gapping topological semimetals is quite general and is
+not limited to the magnetic Weyl semimetal case.
+To
+emphasize this point, here we also discuss a topologically-
+ordered state, which is obtained by gapping a nodal
+line semimetal without breaking symmetries. This state
+has a topological order, distinct from a gapped Weyl
+semimetal, and is characterized by a fractional elec-
+tric polarization, impossible in an ordinary weakly-
+interacting insulator.
+The rest of the paper is organized as follows.
+In
+section II, after a preliminary discussion of topological
+field theory description of the electromagnetic response
+of Weyl semimetals, we recap the construction of the 3D
+analog of the Pfaffian-antisemion state of Refs. 20 and
+21. In section III, we demonstrate the existence of a du-
+ality relation (which applies to topological response only)
+between a noninteracting Weyl semimetal and a QED4,
+which describes a time-reversed Weyl semimetal, coupled
+to a dynamical gauge field. In section IV we discuss a
+topologically-ordered state, obtained by gapping a nodal
+line semimetal without breaking symmetries. This state
+is characterized by a fractional electric polarization, im-
+possible in an ordinary insulator. We conclude in sec-
+tion V with a brief discussion of our results.
+II.
+GAPPED SYMMETRY-PRESERVING
+STATES IN WEYL SEMIMETALS
+A.
+Preliminaries
+To keep the paper self-contained, we will start by re-
+capping the construction of the 3D FQH state of Ref. 20
+and 21, which, as will be explained below, may be viewed
+as a TR-breaking 3D analog of the Pfaffian-antisemion
+state on a strongly-interacting 3D TI surface. We will
+also put the theory of Ref. 21 on a more rigorous foot-
+ing by introducing the language of translation gauge
+fields,36,52–55 which allows one to use proper coordinate-
+free notation for the corresponding topological field the-
+ories.
+We start from the simplest cubic lattice model of a
+magnetic Weyl semimetal with a pair of nodes34
+H =
+�
+k
+ψ†
+k [σx sin(kxd) + σy sin(kyd) + σzm(k)] ψk.
+(1)
+Here σi are Pauli matrices, describing the pair of touching
+bands and
+m(k) = cos(kzd) − cos(Qd) − ˜m[2 − cos(kxd) − cos(kyd)],
+(2)
+where d is the lattice constant, ˜m > 1 and m(k) van-
+ishes at two points on the z-axis with kz = ±Q, which
+correspond to the locations of the Weyl nodes.
+Such a Weyl semimetal is characterized by the anoma-
+lous Hall conductivity
+σxy = e2
+h
+2Q
+2π = 1
+2π
+2Q
+2π .
+(3)
+This may be expressed as a topological term in the effec-
+tive action for probe electromagnetic gauge fields when
+the fermions are integrated out56
+L = i2Q
+8π ϵzναβAν∂αAβ.
+(4)
+In its primitive form above, Eq. (4) does not actually
+look like a topological term, since it explicitly contains a
+preferred direction in space (z) and depends on a nonuni-
+versal microscopic lattice constant d through the Weyl
+node separation 2Q.
+
+3
+To fix these issues, it proves useful to introduce the
+concept of a translation gauge field.36,52–55 Recall that
+Bravais lattice points R of a perfect crystal may be
+described as intersections of families of crystal planes,
+perpendicular to primitive reciprocal lattice vectors bi,
+where i = 1, 2, 3 (or x, y, z for a cubic crystal). Mathe-
+matically, this is expressed by the equation
+θi(r, t) = bi · r = 2πni,
+(5)
+where ni are sets of integers, labeling the crystal planes
+in a family i and the Bravais lattice vectors r = R are the
+solutions of this equations. Eq. (5) implies that the recip-
+rocal lattice vectors in a perfect crystal may be expressed
+as gradients of the phases bi
+j = ∂jθi. This may be gen-
+eralized to a distorted crystal, including time-dependent
+distortions, by introducing translation “gauge fields”
+ei
+µ = 1
+2π ∂µθi.
+(6)
+The fields ei
+µ may in fact be viewed as true (strictly
+speaking, integer valued) gauge fields, if one explicitly
+takes account of the fact that the phases θi on crys-
+tal planes may be relabelled in arbitrary 2π × Z incre-
+ments.54,55 This will not make a significant difference in
+our case and either viewpoint is acceptable.
+In a convenient differential form language, we may view
+ei as a one-form
+ei = ei
+µdxµ.
+(7)
+In a crystal without dislocations,
+dei = 1
+2(∂µei
+ν − ∂νei
+µ)dxµ ∧ dxν = 0,
+(8)
+as clearly follows from the definition Eq. (6).
+On the
+other hand, if a dislocation with a Burgers vector along
+bi is present, the integral of ei around a loop, enclosing
+the dislocation line is
+�
+ei = 1.
+The benefit of introducing translation gauge fields be-
+comes apparent if we now replace a reciprocal lattice vec-
+tor along the z-direction in Eq. (4) with the correspond-
+ing translation gauge field
+2π
+d δz
+µ → 2πez
+µ.
+(9)
+Then Eq. (4) becomes
+L = i λ
+4π ϵµναβez
+µAν∂αAβ = i λ
+4π ez ∧ A ∧ dA,
+(10)
+where λ = 2Q/(2π/d) is a dimensionless separation be-
+tween the Weyl nodes in units of the reciprocal lattice
+vector. Now Eq. (10) looks like a proper topological term,
+which only contains gauge fields and a universal coeffi-
+cient. The nonuniversal and variable lattice constant d
+has been absorbed into the definition of the translation
+gauge field and we will henceforth set d = 1 for simplicity.
+Varying Eq. (10) with respect to ez
+z produces response
+per atomic xy-plane, which is determined by a universal
+numerical coefficient λ. A noninteger value of the coef-
+ficient λ requires gapless modes in the form of a pair of
+Weyl nodes to be present,36,37 since a fractional value (in
+units of e2/h) of the Hall conductance per atomic plane
+is impossible in a noninteracting gapped insulator.
+B.
+3D analog of the Pfaffian-antisemion state
+To derive the field theory of the gapped 3D FQH state
+of Ref. 20 we first move to a dual description of the nonin-
+teracting Weyl semimetal of Eq. (1), in which the electric
+charge is separated from the fermions and is represented
+in terms of a two-form gauge potential, which couples
+to the vortex loop excitations.21,57,58 This approach is
+essentially equivalent to what is known as “functional
+bosonization”,59–62 apart from unimportant technical de-
+tails. We start by representing the fermion operators in
+Eq. (1) (after Fourier transforming them to real space)
+as
+ψr = eiθrfr,
+(11)
+where r label the sites of a cubic lattice, eiθr represents
+a spinless charged boson (chargon) and fr is a neutral
+fermion (spinon).
+After straightforward and standard
+manipulations,21,63 one obtains the following exact rep-
+resentation of the Weyl semimetal Lagrangian L
+L = Lf + Lb
+(12)
+where Lf is the Lagrangian of the spinons fr, which has
+a form, identical to the lattice Lagrangian of the original
+electrons ψr, except that fr are coupled to a compact
+statistical gauge field aµ rather than the probe electro-
+magnetic field Aµ. The statistical field expresses U(1)
+gauge invariance of the parton decomposition Eq. (11)
+and serves the purpose of gluing together the spinons
+and the chargons. The chargon Lagrangian has the form
+Lb = i
+4π (Aµ − aµ)ϵµναβ∆νbαβ +
+1
+8π2χ(ϵµναβ∆νbαβ)2.
+(13)
+Here bµν = −bνµ is a two-form 2π×Z valued lattice gauge
+field, which represents integer chargon currents Jµ as
+Jµ = 1
+4π ϵµναβ∆νbαβ,
+(14)
+∆µ is a lattice derivative and χ is a positive constant.
+Lattice site indices r have been suppressed everywhere
+for brevity.
+To avoid dealing with discrete variables, we may imple-
+ment the 2πZ constraint on bµν by adding a term i
+2 ˜Jµνbµν
+to Lb and summing over integer-valued variables ˜Jµν,
+which have the meaning of vortex loop currents. Gauge
+invariance of Eq. (14) with respect to a transformation
+bµν → bµν + ∆µgν − ∆νgµ implies a conservation law
+∆µ ˜Jµν = 0,
+(15)
+
+4
+which may be solved as
+˜Jµν = 1
+2π ϵµναβ∆αcβ,
+(16)
+where cµ are 2πZ valued one-form gauge fields. The con-
+straint on cµ may, in turn, be softened by adding a term
+−t cos(∆µφ + cµ), where the presence of a new compact
+angular variable φ takes account of the gauge invariance
+of Eq. (16) with respect to cµ → cµ + ∆µφ. In essence,
+the particle created by eiφ, is the original chargon.
+Then, after taking the continuum limit, the chargon
+Lagrangian takes the dual form
+Lb = i
+4π (Aµ − aµ + cµ)ϵµναβ∂νbαβ + . . . ,
+(17)
+where . . . contain both the higher-derivative terms for
+bµν and the additional terms for cµ whose form depends
+on the value of the parameter t. In particular, when t
+is large, eiφ boson is condensed, leading to a mass term
+for cµ (i.e. gap for vortices), which may then be ignored.
+Integration over bµν the simply sets Aµ = aµ, i.e. the
+electric charge is re-attached to the spinons and we re-
+cover the original Weyl semimetal. In contrast, when t
+is small, eiφ particle is gapped, which leads to a Maxwell
+term, (ϵ∂c)2, for the gauge field cµ. In this case, integra-
+tion over cµ produces a mass term b2 for the two-form
+gauge field, which corresponds to a charge gap.
+This
+state is a Mott insulator, which has gapless spinons that
+retain the Weyl semimetal band structure.
+To obtain a fully gapped state, which preserves topo-
+logical response of the Weyl semimetal Eq. (10) and does
+not break any symmetries, we place the spinons into
+a paired state. For weak pairing, only the intra-nodal
+pairing state opens a gap.64–67 Such a pairing generally
+breaks translational symmetry, except when 2Q = π or
+λ = 1/2,20 to which we now specialize. With such an
+intra-nodal pairing term, the spinon Hamiltonian may
+be brought to the form
+H = 1
+2
+�
+k
+f †
+k {σx sin(kx) + σy sin(ky)
++
+��
+∆2 + cos2(kz) − ˜m(2 − cos(kx) − cos(ky))
+�
+σz
+�
+fk,
+(18)
+where ∆ is the pairing amplitude. This Hamiltonian de-
+scribes a 3D topological p-wave superconductor, which
+has a chiral Majorana mode, spanning the full extent of
+the BZ. This may also be viewed as a stack of 2D topo-
+logical superconductors, since the pairing gap does not
+close at any value of kz.
+The spinon pairing produces a term ∝ − cos(2aµ)
+for the statistical gauge field, which leaves only aµ =
+0, π mod 2π possible values at low energies and makes it
+a Z2 gauge field. While nontrivial π-flux configurations of
+aµ (visons68) are still possible, these may be easily shown
+to bind gapless 1D Majorana mode in their cores, which
+is a direct consequence of the fact that the spinon super-
+conductor is topologically nontrivial. This means that in
+any fully gapped symmetry-preserving state such vison
+loop excitations must be gapped, which means that we
+may set aµ = 0 mod 2π at low energies. This detaches
+the boson and fermion sectors of the theory. The fermion
+sector thus contributes the same thermal Hall response
+as the noninteracting Weyl semimetal at λ = 1/2, which
+arises from the chiral Majorana mode, spanning the full
+BZ. The electrical response must entirely come from the
+boson sector.
+In order to reproduce the electrical response of the non-
+interacting Weyl semimetal, it is necessary to condense
+double (i.e. flux 4π) vortices of the boson field eiθ. This
+is accomplished by the following modification of the field
+theory Eq. (17)
+Lb = i
+4π (Aµ + 2cµ)ϵµναβ∂νbαβ + 2i
+4π ϵµναβez
+µcν∂αcβ
++ 1
+2g (ϵµναβ∂αcβ)2 + i
+2bµν˜jµν + icµjµ.
+(19)
+The extra factor of 2 in front of cµ, compared to Eq. (17),
+expresses the fact that double (flux 4π) vortices are be-
+ing condensed. This also means that the quasiparticle,
+which is minimally coupled to the gauge field cµ, carries
+a charge 1/2.
+The second term is a topological term,
+which will give rise to the correct electrical Hall con-
+ductivity, as will be shown below.
+This term may be
+viewed as describing a layered integer quantum Hall state
+of the charge-1/2 bosonic quasiparticles. The third term
+is the Maxwell term. It is subdominant to the topological
+term at long wavelengths, but has been included explic-
+itly since the topological term only contains components
+of cµ, transverse to the translation gauge field ez. In par-
+ticular, if ez
+µ = δz
+µ, cz does not enter into the topological
+term and its dynamics is governed by the Maxwell term.
+Let us now demonstrate that Eq. (19) indeed describes
+the correct physics. Let us set ˜jµν = 0 and integrate out
+bµν. This gives cµ = −Aµ/2. Substituting this back into
+Eq. (19), we obtain
+Lb = i
+8π ϵµναβez
+µAν∂αAβ − i
+2Aµjµ.
+(20)
+The first term in Eq. (20) correctly reproduces the electri-
+cal Hall conductivity of a noninteracting Weyl semimetal
+with λ = 1/2, which is half conductance quantum σxy =
+1/4π per atomic plane. The second term tells us that
+quasiparticle excitations in the gapped state, described
+by Eq. (19), are bosons with electric charge 1/2. To es-
+tablish gapped nature of this state it is important to note
+the following. If we reinsert the statistical gauge field aµ
+into Eq. (19), it is clear that fluctuations of bµν effec-
+tively constrain cµ = (aµ − Aµ)/2. This implies that,
+since aµ is made a Z2 gauge field by spinon pairing, cµ
+becomes a discrete Z4 gauge field.
+This is important,
+since, unlike in 2 + 1 dimensions, a (3 + 1)-dimensional
+Maxwell-Chern-Simons theory with U(1) gauge fields is
+gapless.69,70
+The most straightforward way to see that this the-
+ory also correctly captures the thermal Hall conductivity
+
+5
+κxy = 0 is to consider the boundary theory, that cor-
+responds to Eq. (19).
+To derive the boundary theory
+we follow the standard method.2 We choose a gauge, in
+which on the boundary, taken to be in the xz-plane, we
+set the temporal components of all the gauge fields to
+zero, i.e. c0 = 0, b0µ = 0. Then, integrating out c0, we
+obtain
+ϵ0νλρ∂νbλρ = ϵ0νλρ∂ν(ez
+λcρ − ez
+ρcλ),
+(21)
+while integrating b0ν gives
+ϵ0νλρ∂λcρ = 0.
+(22)
+Eqs. (21) and (22) along with dez = 0 imply that
+ϵ0νλρ∂νbλρ = 0.
+(23)
+Eq. (23) may then be solved as
+bij = ∂igj − ∂jgi,
+(24)
+where i, j = x, z refer to spatial coordinates on the
+boundary, while Eq. (22) is solved as
+ci = ∂iϕ.
+(25)
+Plugging this back into what remains of Eq. (19) after
+integrating out c0 and b0µ, we obtain
+Lb = i
+2π ϵ0νλρez
+ν∂λϕ∂τ∂ρϕ − i
+π ϵ0νλρ∂νϕ∂τ∂λgρ,
+(26)
+where ∂τ ≡ ∂0.
+Integrating this in the presence of a
+boundary, perpendicular to the y-direction, gives
+Lsurf = i
+2π ϵijez
+i ∂τϕ∂jϕ − i
+π ϵij∂τϕ∂igj,
+(27)
+where i, j
+=
+x, z.
+Adding symmetry-allowed non-
+topological terms and the electromagnetic field, we finally
+obtain the following surface state Lagrangian
+Lsurf = i
+2π ϵijez
+i ∂τϕ∂jϕ − i
+π ϵij∂τϕ∂igj + vϕ
+2π (∂iϕ)2
++ vg
+2π (∂igj − ∂jgi)2 + i
+2π ϵµνλAµ∂νgλ.
+(28)
+Setting ez
+µ = δz
+µ and Fourier transforming, we obtain the
+following expression for the excitation spectrum of the
+surface modes
+ϵ(k) = −vgkx
+2
++
+��v2g
+4 + vgvϕ
+�
+k2x + vgvϕk2z.
+(29)
+This looks like an ordinary anisotropic 2D superfluid dis-
+persion, except for a “tilt” in the x-direction due to the
+first term. However, the dispersion is still nonchiral, since
+there is always a pair of left- and right-handed modes for
+every value of kz. Consequently, a straightforward calcu-
+lation gives a vanishing thermal Hall conductivity in this
+state
+κxy ∼
+�
+dkxdkzvx(k)ϵ(k)∂nB[ϵ(k)]
+∂T
+= 0,
+(30)
+where vx(k) = ∂ϵ(k)
+∂kx
+and nB(ϵ) is the Bose-Einstein dis-
+tribution. The integral over kx in Eq. (30) vanishes since
+the left-handed (kx < 0) and right-handed (kx > 0)
+modes give a contribution that is equal in magnitude but
+opposite in sign.
+By construction, this state is a fully gapped symmet-
+ric state, which has an identical topological response to
+a noninteracting Weyl semimetal at λ = 1/2. Note again
+that, while there does exist a close connection between
+this state and the 2D Pfaffian-antisemion state, it may
+not be viewed as a simple stack of such 2D states. In par-
+ticular, there are no semion quasiparticles, but isolated
+intersections of 2π vortex loop excitations with atomic
+xy-planes do behave as semions.
+C.
+3D analog of the PH-Pfaffian state
+Now we note that a second distinct gapped symmet-
+ric state, reproducing topological response of a nonin-
+teracting Weyl semimetal, actually exists. This state is,
+in a way, simpler than the 3D analog of the Pfaffian-
+antisemion above and, as we will demonstrate, may be
+viewed as a 3D analog of the PH-Pfaffian.39,41,44,45
+To construct this state, we take a time-reversed copy
+of our Weyl semimetal with λ = 1/2. Writing its La-
+grangian in terms of spinon and chargon variables, we
+have
+L = ¯fγµ(∂µ+iaµ)f − i
+8π ez∧a∧da+ i
+4π (A−a)∧db, (31)
+where the first term is the contribution of the gapless
+Weyl fermions while the second term is the topological
+contribution from all the filled negative-energy states.
+We will switch to the index-free notation henceforth.
+We now place the chargons into a stack of independent
+ν = 1/2 quantum Hall states in each xy-atomic plane.
+Technically, this means that we take the two-form gauge
+field b to be “foliated”71–74
+b = ez ∧ ˜b,
+(32)
+where ˜b = ˜b0dτ +˜bxdx+˜bydy is a one-form field that lacks
+the z-component, and add a term − 2i
+4πez ∧ ˜b ∧ d˜b to the
+Lagrangian Eq. (31). Furthermore, we place the spinons
+into the intra-nodal pairing state of Eq. (18), which leads
+to a 3D p + ip topological superconductor with a chiral
+Majorana mode, spanning the surface BZ, whose chirality
+is, however, opposite to the chirality of the Fermi-arc
+state of the original noninteracting Weyl semimetal. This
+gaps out the gauge field aµ and decouples the boson and
+fermion sectors.
+The boson sector Lagrangian now reads
+Lb = − 2i
+4π ez ∧ ˜b ∧ d˜b + i
+2π ez ∧ A ∧ d˜b.
+(33)
+Integrating over ˜b leaves the effective action
+Lb = i
+8π ez ∧ A ∧ dA,
+(34)
+
+6
+which describes topological electrical response, which is
+identical to the original noninteracting Weyl semimetal.
+The thermal Hall effect, coming from Lb, is twice that
+of the noninteracting Weyl semimetal, however a mi-
+nus a half is contributed by the opposite-chirality Ma-
+jorana surface state of the paired time-reversed spinons.
+Thus we fully reproduce both electrical and thermal
+topological responses of the noninteracting gapless Weyl
+semimetal.
+This state may be viewed as a 3D generalization of
+the 2D PH-Pfaffian state. Note that, unlike the 3D ana-
+log of the Pfaffian-antisemion state, described above, this
+state is not a 3D incompressible liquid, but exhibits a
+fracton-type order.71–74 If we ignore fermions, Eq. (33)
+describes a stack of independent 2D PH-Pfaffian states.
+The charge-1/2 anyon excitations in these 2D states are
+only able to move within a given plane and can not tun-
+nel between the planes. Neutral fermions propagate in
+3D and connect the individual layers together, but the
+anyons remain confined within 2D layers.
+III.
+“DUAL” WEYL SEMIMETAL
+The existence of a 3D analog of the PH-Pfaffian has an
+important implication, which we will now discuss. Let us
+first return back to the 3D Pfaffian-antisemion state. Let
+us note that, in this case, the topological response of a
+noninteracting Weyl semimetal is only reproduced when
+the fermionic spinons are gapped by pairing and vison
+vortex loops excitations are gapped. If the pairing gap is
+taken to zero, the statistical field a is no longer massive
+and its coupling to the gapless Weyl spinons produces a
+topological term
+i
+8πez ∧ a ∧ da, so that the Lagrangian
+may be written as
+L = ¯fγµ(∂µ + iaµ)f + i
+8π ez ∧ a ∧ da
++ i
+4π (A − a + 2c) ∧ db + i
+2π ez ∧ c ∧ dc.
+(35)
+Integrating out b and c gives
+L = ¯fγµ(∂µ + iaµ)f + i
+4π ez ∧ a ∧ da
+− i
+4π ez ∧ A ∧ da + i
+8π ez ∧ A ∧ dA.
+(36)
+To obtain the electromagnetic response, we now integrate
+out a. This may be done perturbatively, treating the re-
+sponse of the gapless low-energy modes, i.e.
+the first
+term in Eq. (36) as a perturbation, compared to the sec-
+ond term. This is possible, because the response of the
+gapless modes, treated in the random phase approxima-
+tion (RPA), is given by
+Sf = 1
+2
+�
+q
+aµ(q)Πµν(q)aν(−q),
+(37)
+where
+Πµν(q) = (q2δµν − qµqν)f(q2),
+(38)
+is the polarization operator of the massless 3D Dirac
+fermion and
+f(q2) =
+1
+12π2 ln
+�4Λ2
+q2
+�
++ O(1).
+(39)
+Here Λ ≫ q is the cutoff momentum, and a convention
+q0 = −Ω is used (Ω is the Matsubara frequency). Note
+that Πµν(q) is almost the same as the polarization op-
+erator of the massive 3D Dirac fermion, in which case
+f(q2) would have been a constant at small q. Even with
+the log nonanalyticity, Πµν(q) is still much smaller, in
+the long wavelength limit, than the topological contribu-
+tions, which are of first order in q.
+At leading order we may then ignore the gapless
+fermions and vary the Lagrangian with respect to a,
+which gives at the saddle point a = A/2 and leaves the
+Lagrangian
+L = ¯fγµ(∂µ + iAµ/2)f +
+i
+16π ez ∧ A ∧ dA,
+(40)
+which clearly corresponds to half of the Hall conductivity
+of a noninteracting Weyl semimetal, i.e. the theory with
+gapless spinons does not reproduce topological response
+of the noninteracting Weyl semimetal.
+In contrast, let us return to Eq. (31), which describes a
+time-reversed Weyl semimetal and add to it the foliated
+topological term of Eq. (33), without opening the spinon
+pairing gap
+L = ¯fγµ(∂µ + iaµ)f − i
+8π ez ∧ a ∧ da
++ i
+2π ez ∧ (A − a) ∧ d˜b − 2i
+4π ez ∧ ˜b ∧ d˜b.
+(41)
+Integrating out ˜b now, we obtain
+L = ¯fγµ(∂µ+iaµ)f − i
+4π ez∧A∧da+ i
+8π ez∧A∧dA. (42)
+This has identical electrical and thermal Hall responses to
+the original noninteracting Weyl semimetal. This means
+that Eq. (41) describes a distinct gapless state, which
+reproduces the topological response of a noninteracting
+Weyl semimetal. This statement is very closely analo-
+gous to the statement of duality between noninteract-
+ing 2D Dirac fermion and QED3.44,46–51 However, note
+that, in contrast to the 2D Dirac duality case, dynami-
+cally this system is quite different from a noninteracting
+Weyl semimetal. Indeed, integrating out f and then a
+in Eq. (42) using RPA produces a Meissner term for the
+components of A, transverse to z. The coefficient of the
+Meissner term, however, vanishes in the long-wavelength
+limit (it is equal to the inverse of the function f(q2), in-
+troduced in Eq. (39)).
+The system thus behaves as a
+superconductor at finite length scales and in directions,
+transverse to z, but with a phase stiffness that vanishes
+in the thermodynamic limit. In contrast, it behaves as
+an insulator along z.
+
+7
+IV.
+TOPOLOGICAL ORDER IN A GAPPED
+NODAL LINE SEMIMETAL
+We will now extend the ideas, developed above, to
+the case of nodal line semimetals, which realize a dis-
+tinct kind of (3 + 1)-dimensional topological order, when
+gapped without breaking the protecting symmetries. In
+the nodal line semimetals, only nodal lines which arise
+from touchings of pairs of nondegenerate bands, are topo-
+logically nontrivial. In this case, TR symmetry may be
+taken to be broken, while the nodal line is then protected
+by the mirror reflection symmetry in the plane, contain-
+ing the line.75 This may be described by the following
+two-band cubic-lattice Hamiltonian76,77
+H(k) = [6 − t1 − 2(cos kx + cos ky + cos kz)] σx
++ 2t2 sin kzσy.
+(43)
+The nodal line in this model appears in the xy-plane
+of the momentum space and is protected by the mirror
+reflection symmetry within this plane, where the mirror
+reflection operator is σx. The band-touching line in the
+xy-plane is given by the solution of the equation
+4 − t1 − 2(cos kx + cos ky) = 0.
+(44)
+In order construct a gapped symmetric state, it is use-
+ful to reinterpret Eq. (43) as a stacking of alternating
+electron and hole-like Fermi liquids with the band dis-
+persions (see Fig. 1)
+ϵ±(k) = ±[4 − t1 − 2(cos kx + cos ky)],
+(45)
+where ± are the two eigenvalues of the mirror reflection
+operator σx.37 The Luttinger volumes of the two Fermi
+liquids ±VF are equal in magnitude to the area in mo-
+mentum space, enclosed by the nodal line. For the two
+Fermi liquids, the topological response describes the fill-
+ing of the charged particles
+L = ±iVF
+4π2 ex ∧ ey ∧ A,
+(46)
+Consequently, the topological response of the nodal line,
+takes the form of fractional electric polarization37,78,79
+L = ±iVF
+8π2 ex ∧ ey ∧ dA,
+(47)
+impossible in an ordinary insulator without topological
+order.
+The simplest way to obtain a gapped mirror-symmetric
+insulator with the same topological response Eq. (47)
+is to stack gapped 2D Fermi liquid states in a mirror-
+symmetric fashion.
+To gap the 2D Fermi liquids, we
+follow the same procedure as above.
+We represent an
+electron as a product of a neutral spinon f and a bosonic
+chargon eiθ and place the spinons into a fully-gapped
+paired state.
+The simplest fully gapped paired spinon
++
+-
++
++
++
++
+-
+-
+-
+-
+d
+d
+FIG. 1.
+(Color online) Construction of the nodal line
+semimetal as a stack of alternating coupled electron and hole-
+like Fermi liquids. The Luttinger volume of each 2D Fermi
+liquid is equal in magnitude to the area in momentum space,
+enclosed by the nodal line. The lattice constant d is set equal
+to unity in all the equations.
+state is p-wave (since the Fermi liquids are spinless), de-
+scribed by the following Hamiltonian
+Hf =
+�
+k
+�
+ϵ±(k)f †
+kfk + ∆
+2 (sin kx + i sin ky)f †
+kf †
+−k + h.c.
+�
+.
+(48)
+Introducing Nambu spinor notation ψk = (fk, f †
+−k), this
+may be represented as a massive 2D Dirac Hamiltonian
+H = 1
+2
+�
+k
+ψ†
+k [ϵ±(k)τz + ∆(τx sin kx − τy sin ky)] ψk,
+(49)
+where τa are Pauli matrices in the particle-hole space.
+This is the Hamiltonian of a Read-Green topological su-
+perconductor,80 which hosts chiral Majorana modes at
+the edges, with opposite chirality for electron and hole-
+like Fermi liquid states. Consequently, an elementary flux
+hc/2e = π vortex hosts a zero-energy localized Majorana
+bound state and can not be condensed.
+To condense higher-flux vortices, we need to consider
+the chargon sector of the theory. Suppose we attempt
+to condense flux-2π vortices. The chargon sector will be
+described by the following theory81,82
+Lb = i
+2π (A + c) ∧ db ± iVF
+(2π)2 ex ∧ ey ∧ c.
+(50)
+Here b is a one-form gauge field, which determines the
+charge current
+Jµ = 1
+2π ϵµνλ∂νbλ,
+(51)
+while c is a one-form gauge field, which determines the
+vortex current
+˜Jµ = 1
+2π ϵµνλ∂νcλ.
+(52)
+The last term of the Lagrangian produces the correct
+electromagnetic response of a system with charge ν =
+
+8
+±VF /(2π)2 per unit cell when b is integrated out, setting
+c = −A. However, when the filling ν is not an integer,
+Eq. (50) can not be the correct theory of a featureless
+insulator since the last term is not gauge invariant. With
+ν = ±p/q, a featureless insulator may be obtained only
+by condensing flux 2πq vortices, which is described by
+the theory
+Lb = i
+2π (A + qc) ∧ db ± ipex ∧ ey ∧ c,
+(53)
+where all terms now have properly quantized integer co-
+efficients and are gauge invariant. This is because the
+quasiparticle, minimally coupled to cµ, carries charge
+1/q, as seen from the first term. Therefore, the filling
+of such quasiparticles is qν = p (i.e. an integer), which is
+what the second term expresses.
+Stacking such insulators with alternating sign of ν in
+the z-direction in a mirror-symmetric fashion, we obtain
+Lb = i
+4π (A + qc) ∧ db ± ip
+2 ex ∧ ey ∧ dc,
+(54)
+where the factor of 1/2 in front of the last term arises
+due to the fact that the unit cell of the stack contains a
+pair of electron and hole-like 2D Fermi liquids and the
+mirror symmetry requires that all neighboring 2D Fermi
+liquids in the stack are separated by an equal distance.
+The gauge field b in Eq. (54) has now been promoted to a
+two-form field, such that the (3 + 1)-dimensional charge
+current is given by
+Jµ = 1
+4π ϵµναβ∂νbαβ,
+(55)
+while the two-form vortex current is
+˜Jµν = 1
+2π ϵµναβ∂αcβ.
+(56)
+Integrating out b in Eq. (54) gives c = −A/q and the
+electromagnetic response described by
+L = ± ip
+2q ex ∧ ey ∧ dA = ±iVF
+8π2 ex ∧ ey ∧ dA,
+(57)
+which coincides with Eq. (47). Thus we obtain a feature-
+less insulator with topological order, which has an iden-
+tical topological response to a weakly-interacting nodal
+line semimetal. Note that the nodal line semimetal has
+no topological thermal response, which is also the case in
+the fractionalized insulator that we have constructed.
+V.
+DISCUSSION AND CONCLUSIONS
+In this paper we have presented a theory of (3 + 1)-
+dimensional topologically ordered states, obtained by
+gapping 3D topological semimetals without breaking pro-
+tecting symmetries.
+We started by pointing out that
+a second gapped symmetric topologically-ordered state,
+preserving the chiral anomaly of magnetic Weyl semimet-
+als, exists, in addition to the state, originally proposed
+in Ref. 20.
+We have shown that, while the state of
+Ref. 20 may be viewed as a 3D TR-breaking analog of
+the Pfaffian-antisemion state in gapped 3D TI surface
+states, the new state is the 3D analog of the PH-Pfaffian.
+In contrast to the 3D Pfaffian-antisemion state, the 3D
+PH-Pfaffian does not exhibit a true 3D topological order,
+but a fracton-like order instead, with independent layers
+of 2D PH-Pfaffian liquid immersed in a 3D p + ip topo-
+logical superconductor of neutral composite fermions.
+We then demonstrated that an interesting consequence
+of the existence of the 3D PH-Pfaffian, is a duality
+relation between a noninteracting Weyl semimetal and
+QED4, in which a time-reversed electrically-neutral Weyl
+semimetal is coupled to a dynamical gauge field, whose
+topological defects (intersections of flux lines with atomic
+planes) carry the electric charge. This duality relation
+may be viewed as a 3D generalization of the known Dirac
+fermion to QED3 duality relation, but is weaker than in
+the 2D case, since the duality only applies to the topo-
+logical response and not to the dynamics.
+Finally, we have extended the theory to include topo-
+logical orders in a gapped nodal line semimetal. Other
+extensions, in particular to TR-invariant Weyl and Dirac
+semimetals are also possible, but do not lead to any fun-
+damentally new structure. One lesson we may highlight
+is that gapped symmetric topological semimetals provide
+a very simple and natural setting for (3 + 1)-dimensional
+topologically-ordered states to appear.
+The simplicity
+stems, in part, from the fact that, due to the existence
+of a preferred direction, selected by either the separa-
+tion between the Weyl points in momentum space, or
+the plane of the nodal line, there exists a natural con-
+nection to well-studied (2 + 1)-dimensional topological
+orders. The connection manifests either directly, in the
+form of layered fracton-like order, or less directly, when
+intersections of (3 + 1)-dimensional vortex-loop excita-
+tions with atomic planes behave as fractionally-charged
+and sometimes anyonic (2 + 1)-dimensional quasiparticle
+excitations.
+ACKNOWLEDGMENTS
+We acknowledge useful discussions with Chong Wang.
+Financial support was provided by the Natural Sciences
+and Engineering Research Council (NSERC) of Canada.
+AAB was also supported by Center for Advancement
+of Topological Semimetals, an Energy Frontier Research
+Center funded by the U.S. Department of Energy Office
+of Science, Office of Basic Energy Sciences, through the
+Ames Laboratory under contract DE-AC02-07CH11358.
+Research at Perimeter Institute is supported in part by
+the Government of Canada through the Department of
+Innovation, Science and Economic Development and by
+the Province of Ontario through the Ministry of Eco-
+nomic Development, Job Creation and Trade.
+
+9
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diff --git a/XNE2T4oBgHgl3EQfDwZ0/content/tmp_files/load_file.txt b/XNE2T4oBgHgl3EQfDwZ0/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..4cde9cf9068cc7b893fe7aa22b23664e1bae8007
--- /dev/null
+++ b/XNE2T4oBgHgl3EQfDwZ0/content/tmp_files/load_file.txt
@@ -0,0 +1,859 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf,len=858
+page_content='Topological order in interacting semimetals  Jinmin Yi,∗ Xuzhe Ying,∗ Lei Gioia, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Burkov  Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada and  Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada  (Dated: January 11, 2023)  It has recently been demonstrated that it is possible to open a gap in a magnetic Weyl semimetal,  while preserving the chiral anomaly along with the charge conservation and translational symmetries,  which all protect the gapless nodes in a weakly interacting semimetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' The resulting state was shown  to be a nontrivial generalization of a nonabelian fractional quantum Hall liquid to three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Here we point out that a second fractional quantum Hall state exists in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  This state  has exactly the same electrical and thermal Hall responses as the first, but a distinct (fracton)  topological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Moreover, the existence of this second fractional quantum Hall state necessarily  implies a gapless phase, which has identical topological response to a noninteracting Weyl semimetal,  but is distinct from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' This may be viewed as a generalization (in a weaker form) of the known  duality between a noninteracting two-dimensional Dirac fermion and QED3 to 3 + 1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' In  addition we discuss a (3 + 1)-dimensional topologically ordered state, obtained by gapping a nodal  line semimetal without breaking symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  INTRODUCTION  Topological order, a concept that originated in the  study of the fractional quantum Hall effect (FQHE) in  two dimensional (2D) electron gas systems,1 continues  to be a subject of intense interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  From the funda-  mental physics prospective, topologically ordered states  provide perfect examples of emergent macroscopic quan-  tum phenomena, with fractionally-quantized electromag-  netic and thermal responses, that are impossible to ex-  plain based on textbook models of weakly-interacting  electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Instead, such fractionally-quantized observ-  able responses necessarily imply excitations with frac-  tional charges, fractional and nonabelian statistics, which  can not be constructed out of any finite number of ele-  mentary constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='2 In addition, such exotic excita-  tions may have future potential practical uses in quan-  tum computing and quantum simulation, as their non-  local topological nature makes them highly resistant to  decoherence and noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='3  Topologically-ordered states in 2D are by now well-  understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Various theoretical models,4–7 as well as  complete formal classifications of 2D topological orders  exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='8 Although significant progress has been made in re-  cent years,9–19 less is known about topologically-ordered  states in three dimensions (3D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' 3D topologically ordered  states are significantly different from the 2D ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' On  the one hand, fractional statistics is impossible in 3D and  quasiparticle excitations may only be bosons or fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  This could make one doubt that, for example, fractional  quantum Hall (FQH) states may even in principle be gen-  eralized to 3D, as the existence of anyons, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' quasipar-  ticles with fractional statistics, is an essential feature of  the 2D FQHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' On the other hand, in addition to point  quasiparticles, one-dimensional loop excitations exist in  3D, which both adds complexity and opens up new in-  teresting possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  We recently demonstrated that a promising way to  achieve 3D topologically ordered states is through gap-  ping topological semimetals without breaking the pro-  tecting symmetries20–22 (see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' 23–29 for related  work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Topological semimetals30–35 are intermediate  phases between insulators of different electronic structure  topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  They may be characterized by unquantized  anomalies,36,37 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' topological terms with noninteger and  continuously-tunable coefficients, similar to the electron  filling parameter, characterizing ordinary Fermi liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Much like fractional filling in a Fermi liquid mandates the  existence of a Fermi surface of gapless particle-hole ex-  citations,38 these unquantized anomalies necessarily im-  ply gapless modes and corresponding long-range entan-  glement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' The only way gaplessness may be circumvented  in the absence of broken symmetries is through the for-  mation of a topologically-ordered state, which preserves  the anomaly and the long-range entanglement of the gap-  less semimetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Specifically, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' 20 we presented an explicit con-  struction of a 3D topologically-ordered state in a gapped  magnetic Weyl semimetal, which exhibits a nontrivial  generalization of the FQHE to 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' This state is ob-  tained starting from a magnetic Weyl semimetal with  a single pair of nodes, separated by half a reciprocal lat-  tice vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  These nodes may be gapped by breaking  the U(1) charge conservation symmetry while forming a  superconducting state with intra-nodal pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' In gen-  eral, such states break translational symmetry since the  Weyl nodes exist at nontrivial momenta in the first Bril-  louin zone (BZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' However, when the nodes are separated  by exactly half a reciprocal lattice vector, such a pair-  ing leads to density modulation at the reciprocal lattice  vector, which does not break the crystal translational  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Restoring the charge conservation symme-  try by proliferating flux 2hc/e = 4π (we will be using  ℏ = c = e = 1 units throughout this paper) vortices in  the superconducting order parameter leads to a feature-  less fractionalized insulator with Z4 topological order,  that has the same electrical and thermal Hall conductiv-  ities as the original noninteracting Weyl semimetal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='03628v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='str-el]  9 Jan 2023  2  exhibits FQHE in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Unlike in 2D FQH liquids, quasi-  particle excitations in this state are bosons and fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  What plays the role of the anyons in the 2D FQHE are  intersections of the vortex-loop excitations with atomic  planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  These behave as fractionally-charged particles  with semionic statistics, which may be sharply defined  by considering three-loop braiding processes,15 involving  a line defect of translational symmetry, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' an edge dis-  location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  In this paper we show that, in addition to the 3D FQH  state of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' 20, another state exists, which has identi-  cal topological response, but distinct topological order,  which turns out to be of a fracton type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' The existence  of these two distinct states turns out to be closely re-  lated to a very similar property of gapped symmetric  2D Dirac surface states of 3D time-reversal (TR) in-  variant topological insulators (TI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='39–43 In this case, two  distinct topologically-ordered states exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  One, called  Pfaffian-antisemion,40,42 is closely related to the 3D FQH  states of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' 20 (more precisely, the relation is with the  TR-broken version of this state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  The second one, T-  Pfaffian,39,41 is related to the other 3D state we will con-  struct in the present paper (again, more precisely, the  relation is with the TR-breaking version of this state,  which is usually called PH, which stands for particle-  hole-symmetric, Pfaffian44,45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Another interesting consequence emerges from these  analogies to the 2D TR-invariant TI surface state topo-  logical orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' It is well-known that the PH-Pfaffian is  closely related to the recently discovered duality relation  between a massless noninteracting 2D Dirac fermion and  QED3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='44,46–51 Namely, the PH-Pfaffian state is obtained  when the dual Dirac fermion of QED3 is gapped by pair-  ing, which does not break the charge conservation sym-  metry since the dual fermion is neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' The existence of  the analog of the PH-Pfaffian state in our 3D system then  also implies the existence of a gapless state, which is re-  lated to the noninteracting Weyl semimetal via a duality  relation, somewhat similar to the 2D Dirac duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' We  demonstrate that this is indeed the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' However, we  find that the duality only applies to topological response  in this case and not to the dynamics and is weaker than  the 2D duality in this sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  The path to topologically ordered insulators through  gapping topological semimetals is quite general and is  not limited to the magnetic Weyl semimetal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  To  emphasize this point, here we also discuss a topologically-  ordered state, which is obtained by gapping a nodal  line semimetal without breaking symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' This state  has a topological order, distinct from a gapped Weyl  semimetal, and is characterized by a fractional elec-  tric polarization, impossible in an ordinary weakly-  interacting insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  In  section II, after a preliminary discussion of topological  field theory description of the electromagnetic response  of Weyl semimetals, we recap the construction of the 3D  analog of the Pfaffian-antisemion state of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' 20 and  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' In section III, we demonstrate the existence of a du-  ality relation (which applies to topological response only)  between a noninteracting Weyl semimetal and a QED4,  which describes a time-reversed Weyl semimetal, coupled  to a dynamical gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' In section IV we discuss a  topologically-ordered state, obtained by gapping a nodal  line semimetal without breaking symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' This state  is characterized by a fractional electric polarization, im-  possible in an ordinary insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' We conclude in sec-  tion V with a brief discussion of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  GAPPED SYMMETRY-PRESERVING  STATES IN WEYL SEMIMETALS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Preliminaries  To keep the paper self-contained, we will start by re-  capping the construction of the 3D FQH state of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' 20  and 21, which, as will be explained below, may be viewed  as a TR-breaking 3D analog of the Pfaffian-antisemion  state on a strongly-interacting 3D TI surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' We will  also put the theory of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' 21 on a more rigorous foot-  ing by introducing the language of translation gauge  fields,36,52–55 which allows one to use proper coordinate-  free notation for the corresponding topological field the-  ories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  We start from the simplest cubic lattice model of a  magnetic Weyl semimetal with a pair of nodes34  H =  �  k  ψ†  k [σx sin(kxd) + σy sin(kyd) + σzm(k)] ψk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (1)  Here σi are Pauli matrices, describing the pair of touching  bands and  m(k) = cos(kzd) − cos(Qd) − ˜m[2 − cos(kxd) − cos(kyd)],  (2)  where d is the lattice constant, ˜m > 1 and m(k) van-  ishes at two points on the z-axis with kz = ±Q, which  correspond to the locations of the Weyl nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Such a Weyl semimetal is characterized by the anoma-  lous Hall conductivity  σxy = e2  h  2Q  2π = 1  2π  2Q  2π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (3)  This may be expressed as a topological term in the effec-  tive action for probe electromagnetic gauge fields when  the fermions are integrated out56  L = i2Q  8π ϵzναβAν∂αAβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (4)  In its primitive form above, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (4) does not actually  look like a topological term, since it explicitly contains a  preferred direction in space (z) and depends on a nonuni-  versal microscopic lattice constant d through the Weyl  node separation 2Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  3  To fix these issues, it proves useful to introduce the  concept of a translation gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='36,52–55 Recall that  Bravais lattice points R of a perfect crystal may be  described as intersections of families of crystal planes,  perpendicular to primitive reciprocal lattice vectors bi,  where i = 1, 2, 3 (or x, y, z for a cubic crystal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Mathe-  matically, this is expressed by the equation  θi(r, t) = bi · r = 2πni,  (5)  where ni are sets of integers, labeling the crystal planes  in a family i and the Bravais lattice vectors r = R are the  solutions of this equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (5) implies that the recip-  rocal lattice vectors in a perfect crystal may be expressed  as gradients of the phases bi  j = ∂jθi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' This may be gen-  eralized to a distorted crystal, including time-dependent  distortions, by introducing translation “gauge fields”  ei  µ = 1  2π ∂µθi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (6)  The fields ei  µ may in fact be viewed as true (strictly  speaking, integer valued) gauge fields, if one explicitly  takes account of the fact that the phases θi on crys-  tal planes may be relabelled in arbitrary 2π × Z incre-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='54,55 This will not make a significant difference in  our case and either viewpoint is acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  In a convenient differential form language, we may view  ei as a one-form  ei = ei  µdxµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (7)  In a crystal without dislocations,  dei = 1  2(∂µei  ν − ∂νei  µ)dxµ ∧ dxν = 0,  (8)  as clearly follows from the definition Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  On the  other hand, if a dislocation with a Burgers vector along  bi is present, the integral of ei around a loop, enclosing  the dislocation line is  �  ei = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  The benefit of introducing translation gauge fields be-  comes apparent if we now replace a reciprocal lattice vec-  tor along the z-direction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (4) with the correspond-  ing translation gauge field  2π  d δz  µ → 2πez  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (9)  Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (4) becomes  L = i λ  4π ϵµναβez  µAν∂αAβ = i λ  4π ez ∧ A ∧ dA,  (10)  where λ = 2Q/(2π/d) is a dimensionless separation be-  tween the Weyl nodes in units of the reciprocal lattice  vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Now Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (10) looks like a proper topological term,  which only contains gauge fields and a universal coeffi-  cient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' The nonuniversal and variable lattice constant d  has been absorbed into the definition of the translation  gauge field and we will henceforth set d = 1 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Varying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (10) with respect to ez  z produces response  per atomic xy-plane, which is determined by a universal  numerical coefficient λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' A noninteger value of the coef-  ficient λ requires gapless modes in the form of a pair of  Weyl nodes to be present,36,37 since a fractional value (in  units of e2/h) of the Hall conductance per atomic plane  is impossible in a noninteracting gapped insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  3D analog of the Pfaffian-antisemion state  To derive the field theory of the gapped 3D FQH state  of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' 20 we first move to a dual description of the nonin-  teracting Weyl semimetal of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (1), in which the electric  charge is separated from the fermions and is represented  in terms of a two-form gauge potential, which couples  to the vortex loop excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='21,57,58 This approach is  essentially equivalent to what is known as “functional  bosonization”,59–62 apart from unimportant technical de-  tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' We start by representing the fermion operators in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (1) (after Fourier transforming them to real space)  as  ψr = eiθrfr,  (11)  where r label the sites of a cubic lattice, eiθr represents  a spinless charged boson (chargon) and fr is a neutral  fermion (spinon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  After straightforward and standard  manipulations,21,63 one obtains the following exact rep-  resentation of the Weyl semimetal Lagrangian L  L = Lf + Lb  (12)  where Lf is the Lagrangian of the spinons fr, which has  a form, identical to the lattice Lagrangian of the original  electrons ψr, except that fr are coupled to a compact  statistical gauge field aµ rather than the probe electro-  magnetic field Aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' The statistical field expresses U(1)  gauge invariance of the parton decomposition Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (11)  and serves the purpose of gluing together the spinons  and the chargons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' The chargon Lagrangian has the form  Lb = i  4π (Aµ − aµ)ϵµναβ∆νbαβ +  1  8π2χ(ϵµναβ∆νbαβ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (13)  Here bµν = −bνµ is a two-form 2π×Z valued lattice gauge  field, which represents integer chargon currents Jµ as  Jµ = 1  4π ϵµναβ∆νbαβ,  (14)  ∆µ is a lattice derivative and χ is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Lattice site indices r have been suppressed everywhere  for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  To avoid dealing with discrete variables, we may imple-  ment the 2πZ constraint on bµν by adding a term i  2 ˜Jµνbµν  to Lb and summing over integer-valued variables ˜Jµν,  which have the meaning of vortex loop currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Gauge  invariance of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (14) with respect to a transformation  bµν → bµν + ∆µgν − ∆νgµ implies a conservation law  ∆µ ˜Jµν = 0,  (15)  4  which may be solved as  ˜Jµν = 1  2π ϵµναβ∆αcβ,  (16)  where cµ are 2πZ valued one-form gauge fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' The con-  straint on cµ may, in turn, be softened by adding a term  −t cos(∆µφ + cµ), where the presence of a new compact  angular variable φ takes account of the gauge invariance  of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (16) with respect to cµ → cµ + ∆µφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' In essence,  the particle created by eiφ, is the original chargon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Then, after taking the continuum limit, the chargon  Lagrangian takes the dual form  Lb = i  4π (Aµ − aµ + cµ)ϵµναβ∂νbαβ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' ,  (17)  where .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' contain both the higher-derivative terms for  bµν and the additional terms for cµ whose form depends  on the value of the parameter t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' In particular, when t  is large, eiφ boson is condensed, leading to a mass term  for cµ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' gap for vortices), which may then be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Integration over bµν the simply sets Aµ = aµ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' the  electric charge is re-attached to the spinons and we re-  cover the original Weyl semimetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' In contrast, when t  is small, eiφ particle is gapped, which leads to a Maxwell  term, (ϵ∂c)2, for the gauge field cµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' In this case, integra-  tion over cµ produces a mass term b2 for the two-form  gauge field, which corresponds to a charge gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  This  state is a Mott insulator, which has gapless spinons that  retain the Weyl semimetal band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  To obtain a fully gapped state, which preserves topo-  logical response of the Weyl semimetal Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (10) and does  not break any symmetries, we place the spinons into  a paired state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' For weak pairing, only the intra-nodal  pairing state opens a gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='64–67 Such a pairing generally  breaks translational symmetry, except when 2Q = π or  λ = 1/2,20 to which we now specialize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' With such an  intra-nodal pairing term, the spinon Hamiltonian may  be brought to the form  H = 1  2  �  k  f †  k {σx sin(kx) + σy sin(ky)  +  ��  ∆2 + cos2(kz) − ˜m(2 − cos(kx) − cos(ky))  �  σz  �  fk,  (18)  where ∆ is the pairing amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' This Hamiltonian de-  scribes a 3D topological p-wave superconductor, which  has a chiral Majorana mode, spanning the full extent of  the BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' This may also be viewed as a stack of 2D topo-  logical superconductors, since the pairing gap does not  close at any value of kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  The spinon pairing produces a term ∝ − cos(2aµ)  for the statistical gauge field, which leaves only aµ =  0, π mod 2π possible values at low energies and makes it  a Z2 gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' While nontrivial π-flux configurations of  aµ (visons68) are still possible, these may be easily shown  to bind gapless 1D Majorana mode in their cores, which  is a direct consequence of the fact that the spinon super-  conductor is topologically nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' This means that in  any fully gapped symmetry-preserving state such vison  loop excitations must be gapped, which means that we  may set aµ = 0 mod 2π at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' This detaches  the boson and fermion sectors of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' The fermion  sector thus contributes the same thermal Hall response  as the noninteracting Weyl semimetal at λ = 1/2, which  arises from the chiral Majorana mode, spanning the full  BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' The electrical response must entirely come from the  boson sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  In order to reproduce the electrical response of the non-  interacting Weyl semimetal, it is necessary to condense  double (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' flux 4π) vortices of the boson field eiθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' This  is accomplished by the following modification of the field  theory Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (17)  Lb = i  4π (Aµ + 2cµ)ϵµναβ∂νbαβ + 2i  4π ϵµναβez  µcν∂αcβ  + 1  2g (ϵµναβ∂αcβ)2 + i  2bµν˜jµν + icµjµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (19)  The extra factor of 2 in front of cµ, compared to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (17),  expresses the fact that double (flux 4π) vortices are be-  ing condensed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' This also means that the quasiparticle,  which is minimally coupled to the gauge field cµ, carries  a charge 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  The second term is a topological term,  which will give rise to the correct electrical Hall con-  ductivity, as will be shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  This term may be  viewed as describing a layered integer quantum Hall state  of the charge-1/2 bosonic quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' The third term  is the Maxwell term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' It is subdominant to the topological  term at long wavelengths, but has been included explic-  itly since the topological term only contains components  of cµ, transverse to the translation gauge field ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' In par-  ticular, if ez  µ = δz  µ, cz does not enter into the topological  term and its dynamics is governed by the Maxwell term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Let us now demonstrate that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (19) indeed describes  the correct physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Let us set ˜jµν = 0 and integrate out  bµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' This gives cµ = −Aµ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Substituting this back into  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (19), we obtain  Lb = i  8π ϵµναβez  µAν∂αAβ − i  2Aµjµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (20)  The first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (20) correctly reproduces the electri-  cal Hall conductivity of a noninteracting Weyl semimetal  with λ = 1/2, which is half conductance quantum σxy =  1/4π per atomic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' The second term tells us that  quasiparticle excitations in the gapped state, described  by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (19), are bosons with electric charge 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' To es-  tablish gapped nature of this state it is important to note  the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' If we reinsert the statistical gauge field aµ  into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (19), it is clear that fluctuations of bµν effec-  tively constrain cµ = (aµ − Aµ)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' This implies that,  since aµ is made a Z2 gauge field by spinon pairing, cµ  becomes a discrete Z4 gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  This is important,  since, unlike in 2 + 1 dimensions, a (3 + 1)-dimensional  Maxwell-Chern-Simons theory with U(1) gauge fields is  gapless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='69,70  The most straightforward way to see that this the-  ory also correctly captures the thermal Hall conductivity  5  κxy = 0 is to consider the boundary theory, that cor-  responds to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  To derive the boundary theory  we follow the standard method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='2 We choose a gauge, in  which on the boundary, taken to be in the xz-plane, we  set the temporal components of all the gauge fields to  zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' c0 = 0, b0µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Then, integrating out c0, we  obtain  ϵ0νλρ∂νbλρ = ϵ0νλρ∂ν(ez  λcρ − ez  ρcλ),  (21)  while integrating b0ν gives  ϵ0νλρ∂λcρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (22)  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (21) and (22) along with dez = 0 imply that  ϵ0νλρ∂νbλρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (23)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (23) may then be solved as  bij = ∂igj − ∂jgi,  (24)  where i, j = x, z refer to spatial coordinates on the  boundary, while Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (22) is solved as  ci = ∂iϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (25)  Plugging this back into what remains of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (19) after  integrating out c0 and b0µ, we obtain  Lb = i  2π ϵ0νλρez  ν∂λϕ∂τ∂ρϕ − i  π ϵ0νλρ∂νϕ∂τ∂λgρ,  (26)  where ∂τ ≡ ∂0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Integrating this in the presence of a  boundary, perpendicular to the y-direction, gives  Lsurf = i  2π ϵijez  i ∂τϕ∂jϕ − i  π ϵij∂τϕ∂igj,  (27)  where i, j  =  x, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Adding symmetry-allowed non-  topological terms and the electromagnetic field, we finally  obtain the following surface state Lagrangian  Lsurf = i  2π ϵijez  i ∂τϕ∂jϕ − i  π ϵij∂τϕ∂igj + vϕ  2π (∂iϕ)2  + vg  2π (∂igj − ∂jgi)2 + i  2π ϵµνλAµ∂νgλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (28)  Setting ez  µ = δz  µ and Fourier transforming, we obtain the  following expression for the excitation spectrum of the  surface modes  ϵ(k) = −vgkx  2  +  ��v2g  4 + vgvϕ  �  k2x + vgvϕk2z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (29)  This looks like an ordinary anisotropic 2D superfluid dis-  persion, except for a “tilt” in the x-direction due to the  first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' However, the dispersion is still nonchiral, since  there is always a pair of left- and right-handed modes for  every value of kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Consequently, a straightforward calcu-  lation gives a vanishing thermal Hall conductivity in this  state  κxy ∼  �  dkxdkzvx(k)ϵ(k)∂nB[ϵ(k)]  ∂T  = 0,  (30)  where vx(k) = ∂ϵ(k)  ∂kx  and nB(ϵ) is the Bose-Einstein dis-  tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' The integral over kx in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (30) vanishes since  the left-handed (kx < 0) and right-handed (kx > 0)  modes give a contribution that is equal in magnitude but  opposite in sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  By construction, this state is a fully gapped symmet-  ric state, which has an identical topological response to  a noninteracting Weyl semimetal at λ = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Note again  that, while there does exist a close connection between  this state and the 2D Pfaffian-antisemion state, it may  not be viewed as a simple stack of such 2D states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' In par-  ticular, there are no semion quasiparticles, but isolated  intersections of 2π vortex loop excitations with atomic  xy-planes do behave as semions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  3D analog of the PH-Pfaffian state  Now we note that a second distinct gapped symmet-  ric state, reproducing topological response of a nonin-  teracting Weyl semimetal, actually exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' This state is,  in a way, simpler than the 3D analog of the Pfaffian-  antisemion above and, as we will demonstrate, may be  viewed as a 3D analog of the PH-Pfaffian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='39,41,44,45  To construct this state, we take a time-reversed copy  of our Weyl semimetal with λ = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Writing its La-  grangian in terms of spinon and chargon variables, we  have  L = ¯fγµ(∂µ+iaµ)f − i  8π ez∧a∧da+ i  4π (A−a)∧db, (31)  where the first term is the contribution of the gapless  Weyl fermions while the second term is the topological  contribution from all the filled negative-energy states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  We will switch to the index-free notation henceforth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  We now place the chargons into a stack of independent  ν = 1/2 quantum Hall states in each xy-atomic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Technically, this means that we take the two-form gauge  field b to be “foliated”71–74  b = ez ∧ ˜b,  (32)  where ˜b = ˜b0dτ +˜bxdx+˜bydy is a one-form field that lacks  the z-component, and add a term − 2i  4πez ∧ ˜b ∧ d˜b to the  Lagrangian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Furthermore, we place the spinons  into the intra-nodal pairing state of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (18), which leads  to a 3D p + ip topological superconductor with a chiral  Majorana mode, spanning the surface BZ, whose chirality  is, however, opposite to the chirality of the Fermi-arc  state of the original noninteracting Weyl semimetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' This  gaps out the gauge field aµ and decouples the boson and  fermion sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  The boson sector Lagrangian now reads  Lb = − 2i  4π ez ∧ ˜b ∧ d˜b + i  2π ez ∧ A ∧ d˜b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (33)  Integrating over ˜b leaves the effective action  Lb = i  8π ez ∧ A ∧ dA,  (34)  6  which describes topological electrical response, which is  identical to the original noninteracting Weyl semimetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  The thermal Hall effect, coming from Lb, is twice that  of the noninteracting Weyl semimetal, however a mi-  nus a half is contributed by the opposite-chirality Ma-  jorana surface state of the paired time-reversed spinons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Thus we fully reproduce both electrical and thermal  topological responses of the noninteracting gapless Weyl  semimetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  This state may be viewed as a 3D generalization of  the 2D PH-Pfaffian state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Note that, unlike the 3D ana-  log of the Pfaffian-antisemion state, described above, this  state is not a 3D incompressible liquid, but exhibits a  fracton-type order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='71–74 If we ignore fermions, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (33)  describes a stack of independent 2D PH-Pfaffian states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  The charge-1/2 anyon excitations in these 2D states are  only able to move within a given plane and can not tun-  nel between the planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Neutral fermions propagate in  3D and connect the individual layers together, but the  anyons remain confined within 2D layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  “DUAL” WEYL SEMIMETAL  The existence of a 3D analog of the PH-Pfaffian has an  important implication, which we will now discuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Let us  first return back to the 3D Pfaffian-antisemion state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Let  us note that, in this case, the topological response of a  noninteracting Weyl semimetal is only reproduced when  the fermionic spinons are gapped by pairing and vison  vortex loops excitations are gapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' If the pairing gap is  taken to zero, the statistical field a is no longer massive  and its coupling to the gapless Weyl spinons produces a  topological term  i  8πez ∧ a ∧ da, so that the Lagrangian  may be written as  L = ¯fγµ(∂µ + iaµ)f + i  8π ez ∧ a ∧ da  + i  4π (A − a + 2c) ∧ db + i  2π ez ∧ c ∧ dc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (35)  Integrating out b and c gives  L = ¯fγµ(∂µ + iaµ)f + i  4π ez ∧ a ∧ da  − i  4π ez ∧ A ∧ da + i  8π ez ∧ A ∧ dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (36)  To obtain the electromagnetic response, we now integrate  out a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' This may be done perturbatively, treating the re-  sponse of the gapless low-energy modes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  the first  term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (36) as a perturbation, compared to the sec-  ond term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' This is possible, because the response of the  gapless modes, treated in the random phase approxima-  tion (RPA), is given by  Sf = 1  2  �  q  aµ(q)Πµν(q)aν(−q),  (37)  where  Πµν(q) = (q2δµν − qµqν)f(q2),  (38)  is the polarization operator of the massless 3D Dirac  fermion and  f(q2) =  1  12π2 ln  �4Λ2  q2  �  + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (39)  Here Λ ≫ q is the cutoff momentum, and a convention  q0 = −Ω is used (Ω is the Matsubara frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Note  that Πµν(q) is almost the same as the polarization op-  erator of the massive 3D Dirac fermion, in which case  f(q2) would have been a constant at small q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Even with  the log nonanalyticity, Πµν(q) is still much smaller, in  the long wavelength limit, than the topological contribu-  tions, which are of first order in q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  At leading order we may then ignore the gapless  fermions and vary the Lagrangian with respect to a,  which gives at the saddle point a = A/2 and leaves the  Lagrangian  L = ¯fγµ(∂µ + iAµ/2)f +  i  16π ez ∧ A ∧ dA,  (40)  which clearly corresponds to half of the Hall conductivity  of a noninteracting Weyl semimetal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' the theory with  gapless spinons does not reproduce topological response  of the noninteracting Weyl semimetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  In contrast, let us return to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (31), which describes a  time-reversed Weyl semimetal and add to it the foliated  topological term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (33), without opening the spinon  pairing gap  L = ¯fγµ(∂µ + iaµ)f − i  8π ez ∧ a ∧ da  + i  2π ez ∧ (A − a) ∧ d˜b − 2i  4π ez ∧ ˜b ∧ d˜b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (41)  Integrating out ˜b now, we obtain  L = ¯fγµ(∂µ+iaµ)f − i  4π ez∧A∧da+ i  8π ez∧A∧dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (42)  This has identical electrical and thermal Hall responses to  the original noninteracting Weyl semimetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' This means  that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (41) describes a distinct gapless state, which  reproduces the topological response of a noninteracting  Weyl semimetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' This statement is very closely analo-  gous to the statement of duality between noninteract-  ing 2D Dirac fermion and QED3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='44,46–51 However, note  that, in contrast to the 2D Dirac duality case, dynami-  cally this system is quite different from a noninteracting  Weyl semimetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Indeed, integrating out f and then a  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (42) using RPA produces a Meissner term for the  components of A, transverse to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' The coefficient of the  Meissner term, however, vanishes in the long-wavelength  limit (it is equal to the inverse of the function f(q2), in-  troduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (39)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  The system thus behaves as a  superconductor at finite length scales and in directions,  transverse to z, but with a phase stiffness that vanishes  in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' In contrast, it behaves as  an insulator along z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  7  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  TOPOLOGICAL ORDER IN A GAPPED  NODAL LINE SEMIMETAL  We will now extend the ideas, developed above, to  the case of nodal line semimetals, which realize a dis-  tinct kind of (3 + 1)-dimensional topological order, when  gapped without breaking the protecting symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' In  the nodal line semimetals, only nodal lines which arise  from touchings of pairs of nondegenerate bands, are topo-  logically nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' In this case, TR symmetry may be  taken to be broken, while the nodal line is then protected  by the mirror reflection symmetry in the plane, contain-  ing the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='75 This may be described by the following  two-band cubic-lattice Hamiltonian76,77  H(k) = [6 − t1 − 2(cos kx + cos ky + cos kz)] σx  + 2t2 sin kzσy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (43)  The nodal line in this model appears in the xy-plane  of the momentum space and is protected by the mirror  reflection symmetry within this plane, where the mirror  reflection operator is σx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' The band-touching line in the  xy-plane is given by the solution of the equation  4 − t1 − 2(cos kx + cos ky) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (44)  In order construct a gapped symmetric state, it is use-  ful to reinterpret Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (43) as a stacking of alternating  electron and hole-like Fermi liquids with the band dis-  persions (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' 1)  ϵ±(k) = ±[4 − t1 − 2(cos kx + cos ky)],  (45)  where ± are the two eigenvalues of the mirror reflection  operator σx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='37 The Luttinger volumes of the two Fermi  liquids ±VF are equal in magnitude to the area in mo-  mentum space, enclosed by the nodal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' For the two  Fermi liquids, the topological response describes the fill-  ing of the charged particles  L = ±iVF  4π2 ex ∧ ey ∧ A,  (46)  Consequently, the topological response of the nodal line,  takes the form of fractional electric polarization37,78,79  L = ±iVF  8π2 ex ∧ ey ∧ dA,  (47)  impossible in an ordinary insulator without topological  order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  The simplest way to obtain a gapped mirror-symmetric  insulator with the same topological response Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (47)  is to stack gapped 2D Fermi liquid states in a mirror-  symmetric fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  To gap the 2D Fermi liquids, we  follow the same procedure as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  We represent an  electron as a product of a neutral spinon f and a bosonic  chargon eiθ and place the spinons into a fully-gapped  paired state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  The simplest fully gapped paired spinon  + +  +  +  + d  d  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (Color online) Construction of the nodal line  semimetal as a stack of alternating coupled electron and hole-  like Fermi liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' The Luttinger volume of each 2D Fermi  liquid is equal in magnitude to the area in momentum space,  enclosed by the nodal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' The lattice constant d is set equal  to unity in all the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  state is p-wave (since the Fermi liquids are spinless), de-  scribed by the following Hamiltonian  Hf =  �  k  �  ϵ±(k)f †  kfk + ∆  2 (sin kx + i sin ky)f †  kf †  −k + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (48)  Introducing Nambu spinor notation ψk = (fk, f †  −k), this  may be represented as a massive 2D Dirac Hamiltonian  H = 1  2  �  k  ψ†  k [ϵ±(k)τz + ∆(τx sin kx − τy sin ky)] ψk,  (49)  where τa are Pauli matrices in the particle-hole space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  This is the Hamiltonian of a Read-Green topological su-  perconductor,80 which hosts chiral Majorana modes at  the edges, with opposite chirality for electron and hole-  like Fermi liquid states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Consequently, an elementary flux  hc/2e = π vortex hosts a zero-energy localized Majorana  bound state and can not be condensed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  To condense higher-flux vortices, we need to consider  the chargon sector of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Suppose we attempt  to condense flux-2π vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' The chargon sector will be  described by the following theory81,82  Lb = i  2π (A + c) ∧ db ± iVF  (2π)2 ex ∧ ey ∧ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (50)  Here b is a one-form gauge field, which determines the  charge current  Jµ = 1  2π ϵµνλ∂νbλ,  (51)  while c is a one-form gauge field, which determines the  vortex current  ˜Jµ = 1  2π ϵµνλ∂νcλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (52)  The last term of the Lagrangian produces the correct  electromagnetic response of a system with charge ν =  8  ±VF /(2π)2 per unit cell when b is integrated out, setting  c = −A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' However, when the filling ν is not an integer,  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (50) can not be the correct theory of a featureless  insulator since the last term is not gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' With  ν = ±p/q, a featureless insulator may be obtained only  by condensing flux 2πq vortices, which is described by  the theory  Lb = i  2π (A + qc) ∧ db ± ipex ∧ ey ∧ c,  (53)  where all terms now have properly quantized integer co-  efficients and are gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' This is because the  quasiparticle, minimally coupled to cµ, carries charge  1/q, as seen from the first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Therefore, the filling  of such quasiparticles is qν = p (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' an integer), which is  what the second term expresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Stacking such insulators with alternating sign of ν in  the z-direction in a mirror-symmetric fashion, we obtain  Lb = i  4π (A + qc) ∧ db ± ip  2 ex ∧ ey ∧ dc,  (54)  where the factor of 1/2 in front of the last term arises  due to the fact that the unit cell of the stack contains a  pair of electron and hole-like 2D Fermi liquids and the  mirror symmetry requires that all neighboring 2D Fermi  liquids in the stack are separated by an equal distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  The gauge field b in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (54) has now been promoted to a  two-form field, such that the (3 + 1)-dimensional charge  current is given by  Jµ = 1  4π ϵµναβ∂νbαβ,  (55)  while the two-form vortex current is  ˜Jµν = 1  2π ϵµναβ∂αcβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  (56)  Integrating out b in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (54) gives c = −A/q and the  electromagnetic response described by  L = ± ip  2q ex ∧ ey ∧ dA = ±iVF  8π2 ex ∧ ey ∧ dA,  (57)  which coincides with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Thus we obtain a feature-  less insulator with topological order, which has an iden-  tical topological response to a weakly-interacting nodal  line semimetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Note that the nodal line semimetal has  no topological thermal response, which is also the case in  the fractionalized insulator that we have constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  DISCUSSION AND CONCLUSIONS  In this paper we have presented a theory of (3 + 1)-  dimensional topologically ordered states, obtained by  gapping 3D topological semimetals without breaking pro-  tecting symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  We started by pointing out that  a second gapped symmetric topologically-ordered state,  preserving the chiral anomaly of magnetic Weyl semimet-  als, exists, in addition to the state, originally proposed  in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  We have shown that, while the state of  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' 20 may be viewed as a 3D TR-breaking analog of  the Pfaffian-antisemion state in gapped 3D TI surface  states, the new state is the 3D analog of the PH-Pfaffian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  In contrast to the 3D Pfaffian-antisemion state, the 3D  PH-Pfaffian does not exhibit a true 3D topological order,  but a fracton-like order instead, with independent layers  of 2D PH-Pfaffian liquid immersed in a 3D p + ip topo-  logical superconductor of neutral composite fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  We then demonstrated that an interesting consequence  of the existence of the 3D PH-Pfaffian, is a duality  relation between a noninteracting Weyl semimetal and  QED4, in which a time-reversed electrically-neutral Weyl  semimetal is coupled to a dynamical gauge field, whose  topological defects (intersections of flux lines with atomic  planes) carry the electric charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' This duality relation  may be viewed as a 3D generalization of the known Dirac  fermion to QED3 duality relation, but is weaker than in  the 2D case, since the duality only applies to the topo-  logical response and not to the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Finally, we have extended the theory to include topo-  logical orders in a gapped nodal line semimetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Other  extensions, in particular to TR-invariant Weyl and Dirac  semimetals are also possible, but do not lead to any fun-  damentally new structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' One lesson we may highlight  is that gapped symmetric topological semimetals provide  a very simple and natural setting for (3 + 1)-dimensional  topologically-ordered states to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  The simplicity  stems, in part, from the fact that, due to the existence  of a preferred direction, selected by either the separa-  tion between the Weyl points in momentum space, or  the plane of the nodal line, there exists a natural con-  nection to well-studied (2 + 1)-dimensional topological  orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' The connection manifests either directly, in the  form of layered fracton-like order, or less directly, when  intersections of (3 + 1)-dimensional vortex-loop excita-  tions with atomic planes behave as fractionally-charged  and sometimes anyonic (2 + 1)-dimensional quasiparticle  excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  We acknowledge useful discussions with Chong Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Financial support was provided by the Natural Sciences  and Engineering Research Council (NSERC) of Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  AAB was also supported by Center for Advancement  of Topological Semimetals, an Energy Frontier Research  Center funded by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content=' Department of Energy Office  of Science, Office of Basic Energy Sciences, through the  Ames Laboratory under contract DE-AC02-07CH11358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
+page_content='  Research at Perimeter Institute is supported in part by  the Government of Canada through the Department of  Innovation, Science and Economic Development and by  the Province of Ontario through the Ministry of Eco-  nomic Development, Job Creation and Trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content=' X 4, 041043 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content=' X 9, 021005 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content=' Mele,  and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content=' Research  3, 043067 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content=' Wang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content=' X 5,  011011 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content=' Son, Annual Review of Condensed Matter Physics  9, 397 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content=' Seiberg, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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+page_content=' Witten, Annals  of Physics 374, 395 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE2T4oBgHgl3EQfDwZ0/content/2301.03628v1.pdf'}
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diff --git a/XdAzT4oBgHgl3EQf1v7O/content/tmp_files/2301.01805v1.pdf.txt b/XdAzT4oBgHgl3EQf1v7O/content/tmp_files/2301.01805v1.pdf.txt
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@@ -0,0 +1,1736 @@
+Unsupervised Manifold Linearizing and Clustering
+Tianjiao Ding1 Shengbang Tong2 Kwan Ho Ryan Chan1 Xili Dai3 Yi Ma2 Benjamin D. Haeffele1
+Abstract
+Clustering data lying close to a union of low-
+dimensional manifolds, with each manifold as a cluster,
+is a fundamental problem in machine learning. When the
+manifolds are assumed to be linear subspaces, many meth-
+ods succeed using low-rank and sparse priors, which have
+been studied extensively over the past two decades. Un-
+fortunately, most real-world datasets can not be well ap-
+proximated by linear subspaces. On the other hand, several
+works have proposed to identify the manifolds by learning
+a feature map such that the data transformed by the map
+lie in a union of linear subspaces, even though the original
+data are from non-linear manifolds. However, most works
+either assume knowledge of the membership of samples to
+clusters, or are shown to learn trivial representations. In
+this paper, we propose to simultaneously perform cluster-
+ing and learn a union-of-subspace representation via Max-
+imal Coding Rate Reduction. Experiments on synthetic and
+realistic datasets show that the proposed method achieves
+clustering accuracy comparable with state-of-the-art alter-
+natives, while being more scalable and learning geometri-
+cally meaningful representations.
+1. Introduction
+1.1. Motivation and Contributions
+Clustering is a fundamental problem in machine learn-
+ing, allowing one to group data into clusters based on as-
+sumptions about the geometry of clusters. For example,
+when data are concentrated around distinct centroids, classi-
+cal k-means clustering [14,17,26,29] and its variants [2,3,5]
+are able to find the cluster centroids and assign membership
+to each data point. More generally4, subspace clustering
+methods [11, 13, 16, 25, 27, 50] are designed to cluster data
+that lie close to a union of different low-dimensional linear
+(or affine) subspaces, where each subspace defines a cluster.
+1Mathematical Institute for Data Science, Johns Hopkins University,
+USA 2Department of Electrical Engineering and Computer Sciences, Uni-
+versity of California, Berkeley, USA 3The Hong Kong University of Sci-
+ence and Technology (Guangzhou), PRC
+4This includes k-means-based methods, since a centroid is a 0-
+dimensional affine subspace.
+Overall, those methods often enjoy theoretical guarantees of
+correct clustering [13,22,27,37,40,41,44,47–51] and find
+applications in various problems such as image clustering,
+face recognition, motion segmentation, and recently in pop-
+ular Transformer architectures in deep learning [38].
+Despite the wide range of applications and theoretical
+guarantees, subspace clustering methods rely on a crucial
+assumption that each cluster can be well approximated by
+a linear/affine subspace, which is often not valid for many
+real-world datasets. For instance, even in a dataset as sim-
+ple as MNIST hand-written digits, images of a single digit
+do not lie close to a low-dimensional linear subspace, thus
+directly applying subspace clustering will fail. Instead, it
+is more natural to assume the clusters are from non-linear
+low-dimensional manifolds (one manifold per cluster), and
+attempt to learn or design a non-linear embedding of the
+data so that the transformed data lies close to distinct linear
+subspaces, with points from one manifold mapped to the
+same subspace. For example, [24] shows that a subspace
+clustering method can achieve 99% clustering accuracy on
+MNIST images after embedding the data with the scattering
+transform [6].
+Beyond the above example, numerous other subspace
+clustering methods have explored hand-designing an appro-
+priate feature embedding (or kernel) such as polynomial or
+exponential mappings [34]. However, these embeddings as-
+sume specific families of manifolds, thus they need to be
+hand-crafted for various tasks and datasets using domain
+knowledge, which makes their application challenging for
+complicated data such as natural images.
+On the other
+hand, [12] proposes to cluster data based on treating a lo-
+cal neighborhood of the manifold approximately as a linear
+subspace. However, for this to succeed sufficient sampling
+density is required, which implies a prohibitive number of
+samples when the manifolds are of moderate dimension or
+are highly curved. Further, for a new sample unseen at train-
+ing time one needs to run the algorithm with all samples
+to embed it or assign a membership to it, which is expen-
+sive computationally. More recently, numerous works pro-
+pose to learn an appropriate linear embedding of the data
+via deep networks and then perform subspace clustering in
+the feature space [1, 18, 20, 35, 54]. Unfortunately, it has
+been shown that many of these formulations are provably
+arXiv:2301.01805v1  [cs.LG]  4 Jan 2023
+
+(a)
+(b)
+(c)
+(d)
+Figure 1. (a) Input data X of two manifolds each containing 100 points. (b) Features Zθ at random initialization. (c) Zθ after self-
+supervised initialization. (d) Zθ after MLC (4) training. Details are in the Appendix.
+ill-posed and learn trivial representations5, with much of the
+claimed benefit coming from ad-hoc post-processing rather
+than the method itself [15].
+This leads to the following
+question:
+Question 1. For data approximately supported on an un-
+derlying union of manifolds, can we learn a transformation
+of the data, so that the transformed data lie in distinct linear
+subspaces to be easily clustered?
+Meanwhile, learning a representation from multi-modal
+data has been a topic of its own interest in machine learning.
+An ideal property of the learned representation often pur-
+sued is between-cluster discrimination, i.e., features from
+different clusters should be well separated. Further, an im-
+portant yet often ignored property of the learned representa-
+tion is that it maintains within-cluster diversity. This is de-
+sirable as it allows distances of samples within a cluster to
+be preserved under the learned transformation, which could
+facilitate downstream tasks such as denoising, hierarchical
+clustering and semantic interpretation. In the supervised
+setting, training with the cross-entropy (CE) classification
+objective fails to achieve the second property, as it has been
+shown empirically [32] and theoretically [43, 56] that the
+representation learned by CE has the property that features
+from one cluster tend to collapse to a single point. On the
+other hand, recent work has proposed the principle of Max-
+imal Coding Rate Reduction (MCR2) [53] as one of the few
+methods that are able to achieve the two ideal properties
+by learning a representation where features from each clus-
+ter are expected to lie close to a low-dimensional subspace
+(within-cluster diverse), and the subspaces from different
+clusters are orthogonal to each other (between-cluster dis-
+criminative). However, for MCR2 to learn such orthogonal
+subspaces each corresponding to one cluster, one needs the
+annotation of which sample belong to which cluster. Such
+annotation might be expensive or impossible to acquire for
+5In this paper, we use ‘representation’ and ‘feature’ interchangeably to
+mean the image of data under a (learned) transformation.
+large-scale datasets. This motivates another question of in-
+terest:
+Question 2. Can we learn a union-of-orthogonal-subspace
+representation of data coming from an underlying union of
+manifolds without access to the labels?
+This paper gives positive answers to the two interrelated
+questions by making the following contributions.
+1. We propose to simultaneously cluster the data and
+learn a union-of-orthogonal-subspace representation
+via MCR2, when data is assumed to lie close to a
+union of manifolds. This is achieved by formulation
+(4), which optimizes over both the representation and
+a doubly stochastic membership formulation inspired
+by the state-of-the-art subspace clustering result [24].
+2. Since the membership has as many entries as the
+square of the batch size of the input data, we give a
+parameterization of the membership (Figure 2). Fur-
+ther, as problem (4) is highly non-convex, we give a
+meta-algorithm (Algorithm 1) on how to initialize the
+variables and to optimize it.
+3. We conduct experiments on simulation and CIFAR10
+to demonstrate some desirable properties of the pro-
+posed method.
+We further experiment on datasets
+with larger number of clusters and imbalanced clus-
+ters such as CIFAR100-20, CIFAR100-100, and Tiny-
+ImageNet200, and show that the proposed method
+achieves state-of-the-art performance.
+1.2. Additional Related Work
+Beyond the above, we make connections to a few impor-
+tant works that are related to this paper.
+Deep Clustering and Representation Learning.
+Re-
+cently, there is an interesting line of research in representa-
+tion learning and clustering that takes advantage of pseudo-
+labelling and semi/self-supervised learning [7, 30, 33, 45].
+
+1.0
+0.5
+0.0
+-1.0
+-0.5
+-0.5
+0.0
+-1.0
+0.5
+-1.0
+-0.5
+0.0
+1.0
+0.5
+1.01.0
+0.5
+0.0
+-1.0
+-0.5
+-0.5
+0.0
+-1.0
+0.5
+-1.0
+-0.5
+0.0
+1.0
+0.5
+1.01.0
+0.5
+0.0
+-1.0
+-0.5
+-0.5
+0.0
+-1.0
+0.5
+-1.0
+-0.5
+0.0
+1.0
+0.5
+1.01.0
+0.5
+0.0
+-1.0
+-0.5
+-0.5
+0.0
+-1.0
+0.5
+-1.0
+-0.5
+0.0
+1.0
+0.5
+1.0Specifically, one first identifies a subset of samples (often
+termed reliable samples) based on geometric or statistical
+criteria in the learned representation and cluster prediction,
+and then uses the predicted labels for those reliable sam-
+ples as if they are ground-truth labels to refine the rep-
+resentation and cluster prediction of other samples.
+De-
+spite the promising clustering performance, the represen-
+tation learned by these methods are not constrained to be
+both between-cluster discriminative and within-cluster di-
+verse. In contrast, the proposed method learns a represen-
+tation with these two ideal properties (see Figure 4) and
+also achieves state-of-the-art clustering performance (see
+Tables 2 and 4).
+Neural Manifold Clustering and Embedding (NMCE).
+A recent preprint [23] also proposes a solution to the same
+problem we study, i.e., clustering the data and learning an
+union-of-orthogonal-subspace representation.
+In particu-
+lar, [23] proposes to model the point-to-cluster membership
+and optimize MCR2 [53] over both the representation and
+the membership. In this paper, we adopt a similar formula-
+tion, but we propose to model the point-to-point affinity us-
+ing a doubly stochastic matrix, inspired by the state-of-the-
+art subspace clustering methods (§2.2). Aside from having
+different conceptual formulations and algorithms, our for-
+mulation is much more stable with respect to initialization
+and is naturally suitable for hierarchical clustering. We de-
+tail these distinctions in §2.2. Experiments (Table 2) further
+show that the proposed method (MLC) achieves higher ac-
+curacy than [23] (NMCE) on large scale realistic datasets.
+2. Problem Formulation
+We start by defining the problem that we study. Suppose
+X = [x1, . . . , xn] ∈ RD×n is a dataset of n samples drawn
+from a union of k underlying manifolds �k
+j=1 Mj and y ∈
+Rn their memberships to the manifolds, i.e., xi ∈ My(i).
+Problem 1 (Unsupervised Manifold Linearizing and Clus-
+tering). Given the dataset X, can we simultaneously 1)
+cluster the samples, i.e., estimate y, and 2) learn a lin-
+ear representation for manifolds, i.e., find a transformation
+f : RD → Rd, such that the image of each manifold f(Mi)
+is a low-dimensional linear subspace of Rd, and the sub-
+spaces satisfy desired properties (§1), i.e., they are between-
+cluster discriminative and within-cluster diverse?
+Here we base our approach on the principle of Maximal
+Coding Rate Reduction (MCR2) which is designed to learn
+ideal representations in the supervised case, i.e., when the
+membership y is given (§2.1). Then we discuss the chal-
+lenges of simultaneously clustering and learning represen-
+tation (Problem 1), and propose our MCR2 clustering ob-
+jective to solve Problem 1 with those challenges in mind
+(§2.2). We further give an algorithm to optimize the pro-
+posed objective (§2.3).
+2.1. Supervised Manifold Linearizing via MCR2
+In the case when the labels y are given as supervision,
+MCR2 [53] aims to address part 2) of Problem 1.
+Let
+fθ : RD → Sd−1 be a featurizer parameterized by a
+neural network, which in turn gives an embedding Zθ :=
+[z1, . . . , zn] ∈ Rd×n of data with zi := fθ(xi) ∈ Sd−1.
+MCR2 aims to learn an ideal representation by optimizing
+max
+θ
+R(Zθ; ϵ) − Rc(Zθ, Π; ϵ)
+s.t.
+Zθ ∈ S
+(1)
+where R(Zθ; ϵ) := log det
+�
+I +
+d
+nϵ2 ZθZ⊤
+θ
+�
+,
+and Rc(Zθ, Π; ϵ) :=
+k
+�
+j=1
+⟨Πj, 1⟩
+n
+log det
+�
+I +
+d
+⟨Πj, 1⟩ϵ2 Zθ Diag(Πj)Z⊤
+θ
+�
+.
+Here S is the set of matrices whose columns all have unit
+ℓ2 norm6, Π ∈ Rn×k is a given membership matrix such
+that Πij = 1 if j = y(i) and Πij = 0 otherwise, ϵ > 0
+is a prescribed precision parameter, Πj ∈ Rn denotes the
+jth column of Π, 1 is a vector of all ones so that ⟨Πj, 1⟩
+is the number of points in cluster j, and finally for v ∈
+Rn, Diag(v) denotes a diagonal matrix with the entries of
+v along the diagonal.
+Intuitively7, the R(Zθ; ϵ) term of (1) measures the vol-
+ume of Zθ, and maximizing it would diversify features from
+all samples, which we will refer to as the expansion term
+Likewise, the Rc(Zθ, Π; ϵ) term measures the sum of vol-
+umes of each cluster of Zθ and is referred to as the compres-
+sion term, since minimizing it would push features within
+each cluster to stay close. It has been shown that given
+Π, the features obtained by maximizing the rate reduction
+R(Zθ; ϵ)−Rc(Zθ, Π; ϵ) has the property that the features
+of each cluster spread uniformly within a subspace (within-
+cluster diverse), and the subspaces from different clusters
+are orthogonal (between-cluster discriminative), under rel-
+atively mild assumptions [53].
+2.2. Unsupervised Manifold Linearizing and Clus-
+tering via MCR2
+While the MCR2 formulation is designed to learn ideal
+representations (§1) when the membership y (or equiva-
+lently Π) is given, here we are interested in the unsuper-
+vised setting where one does not have access to membership
+annotations. Thus, we propose to simultaneously perform
+both parts 1) and 2) of Problem 1 by also optimizing over
+6This can be easily achieved by having the last layer of the neural net-
+work fθ be a normalization layer.
+7More formally, terms of the form log det
+�
+I +
+d
+nϵ2 W W ⊤�
+esti-
+mate the average number of bits needed to code n i.i.d. samples W ∈
+Rd×n from a zero-mean d-dimensional Gaussian up to a distortion ϵ [28],
+hence the name coding rate.
+
+the membership Π of data. This naturally leads to
+max
+θ,Π∈Ω◦ R(Zθ; ϵ) − Rc(Zθ, Π; ϵ)
+s.t.
+Zθ ∈ S,
+(2)
+where Ω◦ := {Π ∈ Rn×k : ∀i ∈ [n], ∃ˆy(i)
+s.t. Πiˆy(i) =
+1 and Πij = 0 for j ̸= ˆy(i)} is the set of all ‘hard’ assign-
+ments, i.e., each row of Π is a one-hot vector. However,
+this optimization is in general combinatorial: its complex-
+ity grows exponentially in n and k, and it does not allow
+smooth and gradual changes of Π. Further, a second chal-
+lenge is the chicken-and-egg nature of this problem: If one
+already has an ideal representation Z, then existing sub-
+space clustering methods can be applied on Z to estimate
+the membership. Likewise, if one is given the membership
+Π of clusters, then solving (1) would lead to an ideal rep-
+resentation. However, the Zθ and Π at the beginning of
+optimization is typically far from ideal.
+Doubly Stochastic Subspace Clustering. To address the
+combinatorial of estimating the cluster memberships, we
+draw inspiration from the closely related problem of sub-
+space clustering, where the goal is to cluster n samples as-
+sumed to lie close to a union of k low-dimensional sub-
+spaces (§1). In this case, one typically does not directly
+learn an n × k matrix denoting memberships of n points
+into k subspaces. Instead, one first learns an affinity ma-
+trix Π ∈ Rn×n signaling the similarity between pairs of
+points, and then applies spectral clustering on the learned
+Π to obtain a final clustering [11,13,16,25,27,50]. In par-
+ticular, requiring doubly-stochastic constraints on the affin-
+ity Π is shown theoretically to suppress false inter-cluster
+connections for clustering problems [9] along with state-of-
+the-art empirical performance for subspace clustering prob-
+lems [24].
+Inspired by the above, we propose a constraint set Ω for
+matrix Π to be the set of n × n doubly stochastic matrices,
+Ω = {Π ∈ Rn×n : Π ≥ 0,
+Π1 = Π⊤1 = 1}.
+(3)
+However, this constraint alone is insufficient for strong clus-
+tering performance: Consider the optimization of (2) with
+respect to Π ∈ Ω only, and note that the objective is
+strongly convex with respect to Π. Since we maximize a
+convex function with respect to convex constraints Ω, an
+optimal Π would lie at an extreme point of Ω, which for
+doubly stochastic matrices is a permutation matrix. This is
+not ideal for clustering, as it implies that every point is as-
+signed to its own distinct cluster, and there is no incentive
+to merge points into larger clusters. To resolve this issue,
+we follow the approach in [24] and add ℓ2 regularization8
+γ
+2 ∥Π∥2
+F to Π which biases Π toward the uniform matrix
+1
+n11⊤, so by tuning γ we can also tune the sparsity level of
+8Other choices of regularization are also possible: Essentially any func-
+tion which achieves its minimum over Ω at the uniform matrix could po-
+tentially be used, e.g., the negative entropy function �
+ij Πij log(Πij).
+Π. This results in our final proposed formulation, dubbed
+Manifold Linearizing and Clustering (MLC):
+max
+θ
+R(Zθ; ϵ) − Rc(Zθ, Πθ; ϵ) − γ
+2 ∥Πθ∥2
+F
+(4)
+s.t.
+Zθ ∈ S, Πθ ∈ Ω,
+where R(Zθ; ϵ) = log det
+�
+I +
+d
+nϵ2 ZθZ⊤
+θ
+�
+, and
+Rc(Zθ, Πθ; ϵ) = 1
+n
+n
+�
+j=1
+log det
+�
+I + d
+ϵ2 Zθ Diag((Πθ)j)Z⊤
+θ
+�
+.
+Note that here Πθ = Πθ(X) is now also parameterized
+by a neural network. While this is constrained optimization
+which may appear difficult to handle, we explain in §2.3
+how we parameterize Zθ and Πθ via neural networks so
+that the constraints are satisfied by construction. Below, we
+note a few advantages of the proposed formulation.
+Parameterizing Π via a Neural Network versus Free
+Variables. An alternative way to parameterize the mem-
+bership would be to directly take Π as decision variables in
+Ω, as opposed to outputs of a neural network. However, this
+leads to maintaining O(n2) variables which is prohibitive
+for large datasets (e.g., n = 106 for ImageNet). In contrast,
+this is not the case if one parameterizes Π as a neural net-
+work, since one can do stochastic gradient descent such that
+for each batch both the memory and computational com-
+plexity is at most square of the batch size (Figure 2).
+Comparison with NMCE. As mentioned in §1.2, NMCE
+[23] approaches Problem 1 also by optimizing MCR2 over
+both the representation and membership.
+However, in
+NMCE the membership is parameterized by an n × k ma-
+trix Πn×k that models the point-cluster membership, which
+is different from our doubly stochastic point-point member-
+ship matrix Πθ inspired from the state-of-the-art subspace
+clustering. Note further that for NMCE the initialization
+of Πn×k is arbitrary and has nothing to do with the struc-
+tures in the initialized representation Πθ, and a bad initial-
+ization of Πn×k could lead to the features from different
+true clusters being compressed. On the other hand, the pro-
+posed doubly stochastic membership Πθ can be initialized
+deterministically using structures from self-supervised ini-
+tialized features Zθ (§2.3).
+Interestingly, optimizing (4)
+allows an interpretation of linearizing each point with its
+neighbors. Empirically as seen in (Table 2), the proposed
+MLC yields a higher clustering accuracy than NMCE [23].
+2.3. Algorithms
+As mentioned, in the MLC objective (4), we parameter-
+ize both the representation Zθ and doubly stochastic mem-
+bership Πθ via a neural network. Below we elaborate on
+how this is done. We summarize the network architecture
+in Figure 2, and the meta algorithm in Algorithm 1.
+Parameterizing Zθ. We follow [53] and take some existing
+network architecture as the backbone. We append a few
+
+Figure 2. Overall architecture for optimizing the proposed manifold linearizing and clustering (MLC) objective (4). Given n input samples
+X each lying in RD, their d-dimensional representation is given by Zθ(X), where θ denotes network parameters. Further, their doubly
+stochastic membership matrix Πθ(X) is given by taking an inner product kernel of the output of the cluster head Cθ(X) followed by a
+doubly stochastic projection.
+affine layers with non-linearity as the representation head to
+further transform the output in Rd, followed by a projection
+layer to respect the unit sphere Sd−1 constraint.
+Parameterizing Πθ. In subspace clustering, the member-
+ship Π given data X often takes the form of g(X)⊤g(X)
+for some (linear) transformation g, such as in the inner
+product kernel [9, 16] where g = I or the least square re-
+gression [27] where g = (I + λX⊤X)−1/2. This moti-
+vates us to parameterize gθ by a neural network, and take
+C⊤
+θ Cθ ∈ Rn×n as the membership where Cθ is shorthand
+for gθ(X). Nevertheless, such an n×n matrix is in general
+not doubly stochastic, i.e., C⊤
+θ Cθ /∈ Ω. To obtain a doubly
+stochastic membership, we further apply a Sinkhorn projec-
+tion layer PΩ,η(·) [10,39], which gives our final parameter-
+ization of the membership as Πθ = PΩ,η(C⊤
+θ Cθ) ∈ Ω.
+Initializing Zθ: Self-supervised Representation Learn-
+ing via MCR2. Since the proposed MCR2 clustering objec-
+tive (4) is non-convex, it is important to properly initialize
+both Z and Π to converge to good (local) minimum. On
+the other hand, randomly initialized features are typically
+far from being ideal, since they may not satisfy the idealized
+properties (§1), and further may not respect the invariance
+to augmentation, i.e., the augmented samples should have
+their representation close to each other. Thus, we adopt the
+self-supervised training strategy [23]
+max
+θ
+R
+�Zθ + Z′
+θ
+2
+; ϵ
+�
++ λ
+n
+�
+i=1
+|z⊤
+i z′
+i|,
+s.t.
+z′
+i, zi ∈ Sd−1,
+∀i ∈ [n],
+(5)
+where for every i, zi and z′
+i are features of different aug-
+mentations of the i-th sample. This essentially requires that
+features from different augmentations of the same sample
+should be as close as possible, whereas features from dif-
+ferent samples should be as uncorrelated as possible.
+Initializing Πθ. An ideal initialization of Πθ would be
+such that if (Πθ)ij has a high value then points i, j are
+likely to be from the same true cluster and vice versa. On the
+other hand, after the self-supervised feature initialization
+mentioned above, Zθ already have some structures which
+we can utilize.
+Thus, we propose to initialize Πθ with
+PΩ,η(Z⊤
+θ Zθ), which is easily implemented by copying the
+parameters from Zθ to Cθ once after the self-supervised
+initialization of the former, i.e., from the feature head to the
+cluster head in Figure 2.
+Data Augmentation. Beyond initializing Zθ, it is often
+desirable to incorporate augmentation in optimizing (4).
+Specifically, from {X(a) ∈ RD×n}A
+a=1 the dataset X
+under A different augmentations, one computes (Z(a)
+θ
+∈
+Rd×n, Π(a)
+θ
+∈ Rd×n) for each augmentation a, and use in
+(4)
+Zθ = PSd−1
+�
+1
+A
+A
+�
+a=1
+Z(a)
+θ
+�
+,
+Πθ = 1
+A
+A
+�
+a=1
+Π(a)
+θ
+∈ Ω.
+(6)
+Note that one can benefit from parallelization by putting
+X(a), Z(a)
+θ , Π(a)
+θ
+for each augmentation a on one comput-
+ing device, since Π(a)
+θ
+only depends on X(a) but not from
+other augmentations.
+3. Experiments on Real Datasets
+Metrics. To evaluate the clustering quality, we run spec-
+tral clustering on learned membership matrix Π, and re-
+port the normalized mutual information (NMI, [42]) and
+clustering accuracy (ACC, [21]), as are commonly used in
+clustering tasks.
+To evaluate the learned representation,
+we define the following metric: for a collection of points
+W = [w1, . . . , wl] ∈ Rd×l (l > d) with associated sin-
+gular values {σi}d
+i=1, define the numerical rank of W as
+arg minr
+�
+r : �r
+i=1 σ2
+i / �d
+i=1 σ2
+i > 0.95
+�
+. Now, one can
+measure the numerical rank of the learned representaion Z,
+
+BackboneAlgorithm 1 MLC: Unsupervised Manifold Linearizing
+and Clustering
+Input: X ∈ RD×n,
+ϵ, γ, η, λ > 0,
+d, k, nb, T, A ∈ Z≥0
+1: initialize Zθ by self-supervised representation learning
+via MCR2
+▷ (5)
+2: initialize Πθ
+3: for t = 1, . . . , T do
+4:
+¯
+X ∈ RD×nb ← sample a batch from X
+5:
+¯
+X(1), . . . , ¯
+X(A) ← apply A augmentations to ¯
+X
+6:
+¯Zθ, ¯Πθ ← forward pass with { ¯
+X(a)}A
+a=1 and net-
+work parameters θ
+▷
+(6)
+7:
+∇θ(4) ← backward pass with respect objective (4)
+8:
+θ ←update θ using some optimizer on ∇θ(4)
+9: end for
+10: run spectral clustering on Πθ to estimate labels ˆy of
+samples
+Output: Zθ, ˆy
+as well as that of each ground-truth cluster9 of Z. A low
+numerical rank of W implies that points in W lie close to
+a low-dimensional subspace. We further report the cosine
+similarity of learned representation, which is simply |z⊤
+i zj|
+for points i and j, since ∥zi∥ = 1 by construction in (4).
+Finally, to compare the efficiency of methods we report the
+training time in §3.2, where the experiments are run on 2
+Nvidia RTX3090 GPUs.
+3.1. Comparison with Subspace Clustering
+To demonstrate the ability of MLC to cluster the sam-
+ples and linearize the manifolds, we conduct experiments on
+CIFAR10, which consists of RGB images from 10 classes
+such as planes, birds, and deers. As mentioned in §1 sub-
+space clustering methods rely crucially on the assumption
+that data lie close to a union of linear subspaces, which
+many real-world dataset may not satisfy. To show that this
+is the case, we additionally compare the proposed method
+with subspace clustering methods. As we shall see, apply-
+ing subspace clustering directly on self-supervised features
+of CIFAR10 will yield low clustering accuracy. In contrast,
+MLC is able to achieve high clustering accuracy, and more-
+over, produce a union-of-orthogonal-subspace representa-
+tion on which subspace clustering methods can also achieve
+high accuracy.
+Data.
+We use the training split of CIFAR10 containing
+50000 RGB images, each of size 3 × 32 × 32. We use
+the augmentation specified in the Appendix to perform self-
+supervised representation learning (5) and get Zself. For
+a fair comparison, the so-learned Zself are used both as ini-
+tialization for MLC (line 1 of Algorithm 1), and as the input
+9They are defined by the true labels y (§2), so that the numerical rank
+metric is decoupled from the quality of learned membership Π.
+Table 1. Clustering accuracy and normalized mutual information
+for subspace clustering (EnSC, SSC-OMP) on self-supervised fea-
+tures Zself, features ZMLC learned by MLC, and manifold clus-
+tering (MLC) on X, where X is 6 · 104 images from 10 classes
+of CIFAR10.
+Method
+Input Data
+ACC
+NMI
+EnSC
+Zself
+72.2
+67.9
+ZMLC
+81.5
+79.2
+SSC-OMP
+Zself
+67.8
+64.5
+ZMLC
+78.4
+76.3
+MLC
+X
+86.3
+78.3
+for subspace clustering methods10. In MLC, for each image
+in each batch we randomly sample A = 2 augmentations
+to apply on the image. As an additional comparison, we
+also run subspace clustering methods on the features ZMLC
+learned by MLC.
+Methods. We compare with the elastic-net subspace clus-
+tering with active-set solver (EnSC, [50]) and sparse sub-
+space clustering with orthogonal matching pursuit solver
+(SSC-OMP, [51]), using off-the-shelf implementation pro-
+vided by the authors11.
+We search the parameters of
+EnSC over (γ, τ) ∈ {1, 5, 10, 50, 100} × {0.9, 0.95, 1}
+and those of SSC over (kmax, ϵ)
+∈
+{3, 5, 10, 20} ×
+{10−4, 10−5, 10−6, 10−7}, and report the run with the
+highest clustering accuracy for each method. We summa-
+rize detailed parameters for MLC in the Appendix.
+Results. Figure 3 reports the coding rates (as loss terms
+in (4) and numerical ranks of features learned by MLC as
+epoch varies. As a first note, the coding rate R of all fea-
+tures (the blue curve in 3a) decreases only slightly as epoch
+goes, indicating that the overall representation is diverse in
+the feature space. Indeed, the numerical rank of all features
+(the dark curve in Figure 3b) stays 118 which is close to the
+dimension 128 of the feature space. This is in sharp contrast
+to the deep subspace clustering methods where all the fea-
+tures collapse to a one-dimensional subspace [15]. More-
+over, as the coding rate Rc of clustered features (the orange
+curve in Figure 3a) goes down, the numerical ranks of fea-
+tures from each ground-truth cluster decrease. For instance,
+the representation from true cluster 3 has a numerical rank
+of 37 in the first step and 24 in the last step. This implies
+that most representation gets linearized better and clustered
+more accurately, even though the MLC objective (4) is un-
+supervised, i.e., it does not use ground-truth labels y. Last
+but not the least, note that the features within each ground-
+10The self-supervised features Zself empirically exhibit some union-of-
+subspace structure, and are typically used for subspace clustering, as also
+seen in [53, §3.2] and [55, §4.2].
+11https : / / github . com / ChongYou / subspace -
+clustering
+
+(a) Coding rate of all features R, that of clustered
+features Rc, and the rate reduction ∆R = R−Rc.
+(b) Numerical ranks of all features Zθ and features
+from each ground-truth cluster i, {zj : y(j) = i}.
+Figure 3. Coding rates (as loss terms in (4)) and numerical ranks (§3.1) of the features
+learned by MLC on CIFAR10 as epoch varies.
+Figure 4. Cosine similarity |Z⊤
+MLCZMLC|
+of the features ZMLC learned by MLC.
+truth cluster spread well in a low-dimensional subspace,
+e.g., the numerical ranks for the true clusters at the last step
+are within [13, 23]. This achieves the desired within-cluster
+diverse property (§1), as opposed to the neural collapse phe-
+nomenon that appears with the cross-entropy loss.
+To compare MLC with subspace clustering methods, we
+report clustering accuracy and normalized mutual informa-
+tion for EnSC, SSC-OMP on self-supervised features Zself,
+features ZMLC learned by MLC, and MLC on X, where X
+is 6 · 104 images from 10 classes of CIFAR10. In addition
+we plot the cosine similarity of the features learned by MLC
+in Figure 4. Remarkably, the highest clustering accuracy
+is 86.3% achieved by MLC on X, which surpasses EnSC
+(72.2%) and SSC-OMP (67.8%) on Zself by a large margin,
+even though Zself is used both as initialization for MLC and
+input for EnSC and SSC-OMP. Interestingly, using instead
+the features ZMLC learned by MLC, the clustering perfor-
+mance of EnSC and SSC-OMP increases and even becomes
+comparable to MLC, e.g., EnSC achieves 79.2% normal-
+ized mutual information compared to 78.3% of MLC. This
+suggests that ZMLC has a union-of-subspace structure that
+can be utilized by subspace clustering. Indeed, as seen in
+Figure 4, features from different clusters tend to have a
+small similarity, i.e., being orthogonal to each other. This
+demonstrates the between-cluster discrimination (§1) as de-
+sired.
+3.2. Comparison with Deep Clustering Methods
+We further compare the proposed MLC with state-of-
+the-art deep clustering methods. Note that most methods
+reported (all except NMCE which is discussed in §2.2) do
+not aim to learn a union-of-orthogonal-subspace represen-
+tation, in contrast to MLC. As we will see, MLC achieves
+clustering accuracy comparable to state-of-the-art methods
+on large scale datasets with faster computational time, and
+further surpasses them on extreme yet realistic cases like
+datasets of imbalanced clusters.
+Compared Methods. We conduct experiments with MLC,
+SCAN [45], and IMC-SWAV [31].12 Training details can
+be found in the Appendix. In addition we include the num-
+bers reported from DeepCluster [7], IIC [19], RUC [33] and
+NMCE [23]. For a fair comparison, all methods reported
+use ResNet-18 as the backbone, which is also commonly
+adopted by other methods.
+Datasets.
+Beyond CIFAR10 (§3.1), we further use
+CIFAR100-20, CIFAR100-100 and Tiny Imagenet-200 to
+evaluate the performance of our method. Both CIFAR100-
+100 and CIFAR100-20 contain the same 50000 train images
+and 10000 test images with size 32 × 32 × 3, while the for-
+mer are split into 100 clusters and the latter 20 super clus-
+ters. Finally, Tiny ImageNet contains 100000 train images
+and 10000 test images with size 64 × 64 × 3 split into 200
+clusters.
+Results on Large-scale Datasets. We report clustering ac-
+curacy and normalized mutual information on CIFAR10,
+CIFAR100-20, CIFAR100-100, and TinyImageNet in Ta-
+ble 2, and we further report running time in minutes for
+CIFAR100-100 in Table 3. As seen, the highest cluster-
+ing performance on CIFAR10 is achieved by RUC+SCAN
+(90.3% ACC) and IMC-SWAV (81.1% NMI), where MLC
+yields a slightly lower ACC of 86.3% and NMI of 78.3%.
+We note some interesting semantic interpretation for the
+clustering obtained by MLC in the Appendix. On the other
+hand, MLC performs comparably with other methods on
+CIFAR100-20 by achieving an ACC of 52.2% and NMI of
+54.6%. Notably, MLC outperforms SCAN and IMC-SWAV
+on CIFAR100-100 and TinyImageNet-200 by a large mar-
+gin, while using lower running time: E.g., on CIFAR100-
+12The authors are aware of a preprint [30] which approaches image
+clustering via a combination of self/semi-supervised learning and pseudo-
+labeling. However, to the best of our effort we are unable to reproduce the
+numbers reported in this paper using the implementation provided by the
+authors. We discuss the details in the Appendix and thus do not report their
+numbers here.
+
+Loss terms
+140
+120
+100
+Loss terms
+80
+60
+40 
+△R
+R
+Rc
+20
+0
+500
+1000
+1500
+2000
+2500
+StepNumerical rank of Z
+120
+100
+All
+Class 0
+Class 1
+80
+Class 2
+Class 3
+Class 4
+60
+Class 5
+Class 6
+Class 7
+Class 8
+40
+Class 9
+20
+0
+500
+1000
+1500
+2000
+2500
+Step1.0
+0
+0.8
+2000
+0.6
+4000
+6000
+0.4
+8000
+0.2
+0
+2000
+4000
+6000
+8000
+0.0Table 2. Clustering accuracy and normalized mutual information on large scale datasets. For a fair comparison, all methods use ResNet-18
+as backbone.
+Method / Dataset
+CIFAR10-10
+CIFAR100-20
+CIFAR100-100
+Tiny ImageNet-200
+Metrics
+ACC
+NMI
+ACC
+NMI
+ACC
+NMI
+ACC
+NMI
+DeepCluster (ECCV′18)
+37.4
+-
+18.9
+-
+-
+-
+-
+-
+IIC (ICCV′19)
+61.7
+51.1
+25.7
+22.5
+-
+-
+-
+-
+SCAN (ECCV′20)
+87.6
+78.7
+46.8
+45.9
+34.3
+55.7
+-
+-
+RUC+SCAN (CVPR′21)
+90.3
+-
+53.3
+-
+-
+-
+-
+-
+IMC-SWAV (Arxiv′21)
+89.1
+81.1
+49.0
+50.3
+43.9
+58.3
+28.2
+52.6
+NMCE (Arxiv′22)
+83.0
+76.1
+43.7
+48.8
+-
+-
+-
+-
+MLC
+86.3
+78.3
+52.2
+54.6
+49.4
+68.3
+33.5
+67.5
+Table 3.
+Running time in minutes and clustering accuracy on
+CIFAR100-100. For a fair comparison, all methods use ResNet-18
+as backbone.
+Method / Metric
+Running Time
+ACC
+Stage
+I
+II
+III
+Total
+SCAN (ECCV′20)
+308.3
+33.3
+54.7
+396.3
+34.3
+IMC-SWAV (Arxiv′21)
+529.4
+-
+-
+529.4
+43.9
+MLC
+266.7
+17.7
+-
+284.4
+48.3
+Table 4. Clustering accuracy on imbalanced datasets: (a) Imb-
+CIFAR10, (b) Imb-CIFAR100-100.
+For a fair comparison, all
+methods use ResNet-18 as backbone.
+Method / Dataset
+(a)
+(b)
+IMC-SWAV (Arxiv′21)
+65.7
+38.2
+SCAN (ECCV′20)
+62.9
+31.1
+MLC
+80.0
+46.1
+100, MLC yields an accuracy of 49.4% in 291 minutes,
+whereas IMC-SWAV has 43.9% using 529 minutes, and
+SCAN has 34.3% in 396 minutes.
+Imbalanced Clusters.
+Note that for CIFAR10 or CI-
+FAR100 each cluster contains approximately the same num-
+ber of samples. On the other hand, natural images are typ-
+ically imbalanced, i.e., the clusters have unequal number
+of samples.
+To mimic this setting, we take a naive ap-
+proach to construct the following imbalanced datasets. For
+the 10 clusters of CIFAR10, we remove half of the sam-
+ples from odd-numbered clusters (i.e., clusters 1, 3, . . . , 9)
+from both the training and test split. We refer to the re-
+duced dataset Imb-CIFAR10. Likewise we construct Imb-
+CIFAR100-100. We run two state-of-the-art methods IMC-
+SWAV and SCAN as well as the proposed MLC on Imb-
+CIFAR10 and Imb-CIFAR100-100.
+Table 4 shows clustering accuracy on the imbalanced
+datasets Imb-CIFAR10 and Imb-CIFAR100-100. As a first
+observation, the clustering accuracy of all methods is lower
+on the imbalanced datasets than on the balanced counter-
+parts, as expected. Notably, MLC suffers from the least
+performance drop, e.g., when moving from CIFAR10 to
+Imb-CIFAR10 the accuracy of MLC drops from 86% to
+80%, whereas that of SCAN and IMC-SWAV decreases
+from above 87% to below 66%.
+4. Conclusion
+This paper studies the problem of simultaneously clus-
+tering and learning an union-of-orthogonal-subspace repre-
+sentation for data, when data lies close to a union of low-
+dimensional manifolds. To address the problem we pro-
+pose an objective based on maximal coding rate reduction
+and doubly stochastic membership inspired by the state-of-
+the-art subspace clustering results. We provide an efficient
+and effective parameterization of the membership variables
+as well as a meta-algorithm to optimize the representation
+and membership jointly. We further conduct experiments
+on datasets with larger number of clusters and imbalanced
+clusters and show that the proposed method achieves state-
+of-the-art performance. We believe that our work provides
+a general and unified framework for unsupervised learning
+of structured representations for multi-modal data.
+
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+
+(a) Learned cluster 1
+(b) Learned cluster 2
+(c) Learned cluster 3
+(d) Learned cluster 4
+(e) Learned cluster 5
+(f) Learned cluster 6
+(g) Learned cluster 7
+(h) Learned cluster 8
+(i) Learned cluster 9
+Figure 5. Principal images (defined in §A) of clusters learned by MLC on CIFAR10.
+
+A. Semantic Interpretability of the Learned
+Representation and Clusters on CIFAR10
+Recall that MLC is designed to perform clustering
+while learning a union-of-orthogonal-subspace representa-
+tion (§1), where each cluster defines a low-dimensional sub-
+space. Therefore, we further visualize the different direc-
+tions within each learned cluster or subspace. Specifically,
+after a final clustering is obtained (line 10 of Algorithm 1),
+we take the features from each learned cluster and apply
+Principal Component Analysis (PCA) to them to obtain the
+first 8 principal components. These correspond to the 8
+rows for each cluster in Figure 5. Recall that the princi-
+pal components are mutually orthogonal, indicating uncor-
+related directions within one cluster. To visualize those di-
+rections or principal components in images, we take the fea-
+tures that are closest to the principal components and visu-
+alize the corresponding original images.
+Interestingly, the rows of images corresponding to prin-
+cipal components appear to exhibit some semantic ‘con-
+cepts’. For example in Figure 5b, row 1 and 8 are respec-
+tively white and red trucks, while row 4 are the trucks that
+ship sand or mud; row 1 of Figure 5d are deers with trees
+as background. This further suggests that the learned em-
+bedding seems to preserve distance within each cluster (as
+desired in §1), i.e., images that are close/far in semantic
+meaning will be close/far in the feature space. Note how-
+ever, that some learned clusters do not align fully with the
+ground-truth labels. For instance, rows 1 and 3 of Figure 5h
+are cats while all other rows in this cluster are dogs. On the
+other hand, one may argue that Figure 5h are a cluster of
+cats and dogs of light colors, whereas Figure 5c is a clus-
+ter of those of brown colors, which could be a semantically
+meaningful clustering even though it does not align with the
+ground-truth labels. We believe it would be an interesting
+future work to use MLC to discover new semantics that are
+not present in the given labels.
+Table 5. Ablation study on the roles of different parts of Algo-
+rithm 1 and on using augmentation.
+Ablation Study on CIFAR-10
+Clustering
+Accuracy
+Full Algorithm 1
+86.3%
+Replacing self-supervised initialization (line 1)
+with random initialization
+20.0%
+Replacing updating MLC objective (4) (lines 3-9)
+with subspace clustering (EnSC)
+73.4%
+Not using augmentation in lines 3-9
+80.0%
+B. Role of Augmentation
+Recall that data augmentation was used both in the self-
+supervised initialization (line 1 of Algorithm 1, see ‘Initial-
+izing Zθ’ in §2.3) and in updating the MLC objective (lines
+3-9, ‘Data Augmentation’ in §2.3). Below we give addi-
+tional clarification on the role of augmentation therein.
+B.1. Augmentation for Initializing the Features
+Since the proposed MLC objective (4) is highly non-
+convex, the (local) solution that a first-order optimizer con-
+verges to in general depends on the initialization. However,
+before line 1 of Algorithm 1 is executed, the features Z at
+initialization could be very far from union-of-orthogonal-
+subspace (as desired by Problem 1), since the neural net-
+work has an arbitrary architecture and initialization. To at
+least promote some ideal structures in the features, we con-
+duct line 1 of Algorithm 1 so that the features from an origi-
+nal sample and its augmented copy are close, while features
+from different samples spread out in the feature space. This
+is a common idea used in contrastive learning, and more re-
+lated in [53, §3.2] [23, §3.6] that are both based on MCR2
+as in this paper (even though the formulations are different,
+as argued in §1.2). Empirically, initializing the features us-
+ing augmentation (line 1 of Algorithm 1) is important for
+the following-up steps: as seen in Table 5, on CIFAR-10, if
+one uses random initialization to replace this step, then the
+final clustering accuracy is 20%, in sharp contrast to 86.3%.
+B.2. Augmentation for Updating MLC Objective (4)
+In optimizing MLC (lines 3-9 of Algorithm 1), augmen-
+tation empirically improves clustering performance. As one
+can see in Table 5, on CIFAR-10 using the sample self-
+supervised initialization of the features, MLC achieves only
+80% clustering accuracy without augmentation, in contrast
+to 86.3% with augmentation. We attribute this difference to
+the fact that augmentation enriches the diversity of samples
+the algorithm sees.
+C. Role of Different Parts of Algorithm 1
+C.1. Initialization of the Features (Line 1)
+Please kindly refer to §B.1.
+C.2. Updating the MLC Objective (4) (Lines 3-9)
+The main novelty of this paper lies in updating the MLC
+objective that learns both the representation Zθ and a dou-
+bly stochastic membership Πθ. Note that in this step, clus-
+tering is pursued by modeling the membership Πθ, as op-
+posed to the self-supervised feature initialization step where
+no membership is explicitly pursued. This step is indeed im-
+portant for clustering: as seen in Table 5, on CIFAR-10, the
+
+clustering accuracy on the self-supervised initialized fea-
+tures Zθ is only 73.4%, in contrast to 86.3% obtained after
+updating the MLC objective (4).
+C.3. Spectral Clustering (Line 10)
+Since the proposed MLC learns a doubly stochas-
+tic membership that signals pair-wise similarity between
+points, it is standard to run spectral clustering [46] to com-
+pute a final set of clusters from the learned membership.
+This is done once at the very end of Algorithm 1, and is
+rather efficient compared to the other parts of Algorithm 1:
+for instance, using an unaccelerated implementation from
+SciPy, it takes less than 30 seconds to perform spectral clus-
+tering on a 10000 × 10000 matrix.
+D. Details on Experiment Settings
+D.1. Synthetic Union-of-Manifold Data
+We perform simulations to visualize the properties of the
+proposed manifold learning and clustering method. As seen
+in Figure 1a, we generate data X from two manifolds on
+the sphere S2, each consisting of 200 samples. The points
+from the first manifold (green) take the form
+xi =
+�
+�
+cos
+�
+A sin(ωφi)
+�
+cos φi
+cos
+�
+A sin(ωφi)
+�
+sin φi
+sin
+�
+A sin(ωφi)
+�
+�
+� + ϵi,
+(7)
+where A = 0.2 and ω = 5 sets the curvature of the mani-
+fold, ϵi ∼ N(0, 0.05I3) is the additive noise, and we take
+φi = 2πi
+100 for i = 1, . . . , 100 to generate 100 points. On the
+other hand, the points from the second manifold (blue) are
+simply 100 samples from N([0, 0, 1]⊤, 0.05I3). We take
+the feature dimension d = 3 to be equal to he input di-
+mension D = 3. We paramterize both the feature head
+fθ and the cluster head gθ to be a simple fully-connected
+network with 100 hidden neurons, followed by a Rectified
+Linear Unit as non-linearity and a projection operator onto
+the sphere S2. Figures 1b to 1d report the features Zθ with
+random initialization (i.e., before line 1 of Algorithm 1),
+with self-supervised initialization, and at convergence of
+MLC. Notably, despite Zθ being noisy and only approxi-
+mately piece-wise linear, as epoch goes Zθ gradually trans-
+form to two linear subspaces: the green points converge to
+a 2-dimensional subspace (intersected with S2) and the blue
+points converge to a 1-dimension subspace.
+D.2. Training Details on Real Datasets
+MLC. As said, we use ResNet-18 as the backbone for ex-
+periments on CIFAR10, CIFAR100-20, CIFAR100-100 and
+Tiny-ImageNet-200, and the imbalanced counterparts Imb-
+CIFAR10, Imb-CIFAR100-100. We also fix the batch size
+to be 1024 in all experiments. In self-supervised initializa-
+tion of Zθ (line 1 of Algorithm 1), we use the precision
+(§2.1) parameter ϵ2 = 0.2, a LARS optimizer [52] (as is
+also done in [8, 23]) with a learning rate of 0.3 and trained
+MLC for 1000 epochs. On the other hand, in the train-
+ing of MLC objective, we use ϵ2 = 0.1, γ = 0.05, and
+η = 0.175 for the entropy regularization in the Sinkhorn
+projection [10] layer PΩ,η(·). We fix the backbone and for
+each batch, we perform one update for parameters in the
+feature head Zθ and one update for parameters in the clus-
+ter head Cθ. For each head we use one SGD optimizer [36]
+with a learning rate of 10−2, momentum of 0.9, and weight
+decay of 5 · 10−4. Finally, for all experiments, we use the
+augmentation from [4] detailed below in PyTorch code.
+Augmentation 1 Augmentations for real datasets
+import torchvision.transforms as t
+t.Compose([
+t.RandomResizedCrop(32,scale=(0.04,
+1.0)),
+t.RandomHorizontalFlip(p=0.5),
+t.RandomGrayscale(p=0.2),
+t.RandomApply([t.ColorJitter(0.4,
+0.4, 0.4, 0.1)], p=0.8),
+GaussianBlur(p=0.1)
+])
+SCAN and IMC-SWAV. Recall that we conduct ex-
+periments on CIFAR100-100, Imb-CIFAR10, and Imb-
+CIFAR100-100 with SCAN [45], IMC-SWAV [31] and
+MLC, and report clustering and running time in Tables 3
+and 4. We use off-the-shelf implementation13 provided by
+the authors. For a fair comparison, SCAN, IMC-SWAV and
+MLC all use ResNet-18 as the backbone. Finally, the hyper-
+parameters of SCAN and IMC-SWAV are set to be the ones
+optimally chosen for CIFAR10 and CIFAR100 respectively
+provided by the authors.
+SPICE. As mentioned, the preprint [30] proposed a method
+SPICE that appears to achieve state-of-the-art performance
+in image clustering. We tried to reproduce their results on
+CIFAR-100-20 using the official implementation14. How-
+ever, the provided implementation ran into a few errors,
+which are also observed15. Despite our best effort to fix
+those issues, the experiments yield only 14% clustering ac-
+curacy on CIFAR100-20 as opposed to the 53% reported in
+the paper [30]. Therefore, we note this observation and do
+not include SPICE in the main text.
+13https://github.com/wvangansbeke/Unsupervised-
+Classification,
+https : / / github . com / foiv0s / imc -
+swav-pub
+14https : / / github . com / niuchuangnn / SPICE,
+commit
+5eba538.
+15https://github.com/niuchuangnn/SPICE/issues/
+27,https://github.com/niuchuangnn/SPICE/issues/31
+
+Table 6. Clustering accuracy and normalized mutual information of MLC and NMCE on CIFAR10 over 10 random seeds, using the same
+self-initialized features. For the purpose of comparison, both methods use the same optimization strategy and hyper-parameters optimally
+tuned for NMCE. Consequently, the clustering performance of MLC reported here is lower than that in Table 2.
+Method
+Metric
+Seed
+Mean
+Std.
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+MLC
+ACC
+84.5
+84.8
+84.8
+84.6
+84.4
+84.4
+84.0
+84.3
+84.4
+84.6
+84.5
+0.24
+NMI
+76.6
+77.1
+76.8
+76.8
+76.5
+76.4
+76.1
+76.4
+76.4
+76.5
+76.6
+0.28
+NMCE
+ACC
+83.7
+82.1
+81.6
+73.7
+80.4
+77.9
+81.7
+81.4
+72.7
+80.9
+79.6
+3.69
+NMI
+74.4
+71.2
+70.4
+65.2
+70.0
+68.1
+72.7
+70.8
+69.2
+69.8
+70.2
+2.49
+(a) CIFAR100-20
+(b) CIFAR100-100
+(c) Tiny ImageNet-200
+Figure 6.
+Cosine similarity |Z⊤
+MLCZMLC| of the features ZMLC learned by MLC on more complicated datasets: CIFAR100-20,
+CIFAR100-100, Tiny ImageNet-200.
+E. Additional Comparison of MLC and NMCE
+on Stability with Respect to Random Seeds
+As detailed in §2.2, one of the advantages of the pro-
+posed MLC over NMCE [23] is that MLC has a more sta-
+ble performance with respect to random seeds, since MLC
+is able to initialize the membership deterministically using
+structures from the self-supervised initialized features. Be-
+low we conduct extra experiments to provide empirical ev-
+idence. We first fix a self-supervised initialization of fea-
+tures that is in turn used for both NMCE and MLC. Then,
+using this very same initialization of features, we update
+NMCE and MLC objective respectively with 5 different
+seeds: recall that NMCE initializes the membership ran-
+domly whereas MLC initializes the membership determin-
+istically using the initialized features. To make a valid com-
+parison, for both methods we further use the same opti-
+mization strategy and hyper-parameters that are optimally16
+tuned for NMCE (which are not optimal for MLC): pre-
+cision ϵ2 = 0.2, # epochs 100, LARS optimizer for Zθ
+with an initial learning rate 0.3 decayed to 0 in a cosine
+annealing manner. Table 6 reports clustering accuracy and
+normalized mutual information of MLC and NMCE over
+16For NMCE, we use the implementation as well as the parameters pro-
+vided in https://github.com/zengyi-li/NMCE-release.
+10 random seeds. As expected, MLC has a more stable
+clustering performance by having a standard deviation of
+clustering accuracy and normalized mutual information less
+than 0.28, in contrast to more than 2.49 achieved by NMCE.
+Further, MLC achieves higher mean clustering performance
+than NMCE, as also observed in Table 2. Last but not least,
+we note that the numbers in Table 6 are not comparable to
+those in Table 2, since for MLC the hyper-parameters and
+optimizers are different, and for NMCE an additional step
+that fine tunes the backbone is used in Table 2.
+F. Addtional Visualization on Learned Repre-
+sentation and Clusters
+Figure 6 presents the cosine similarity (as defined in the
+preamble of §3) of the representation learned by MLC on
+CIFAR100-20, CIFAR100-100 and TinyImageNet-200 (for
+the counterpart on CIFAR10 see §3.1). As seen, the cosine
+similarity maps form approximately block diagonal struc-
+tures, showing that the features from different clusters are
+roughly orthogonal to each other. This is desired by the
+between-cluster discrimination (§1).
+Finally, we provide additional visualization of principal
+images on CIFAR100-20 (see §A for definition) in Figure 8.
+
+ZtZepoch:{epoch}batch:{step}
+0
+1.0
+200
+0.8
+400
+0.6
+600
+0.4
+800
+0.2
+1000
+0
+200
+400
+600
+800
+1000ZtZepoch:{epoch}batch:{step)
+0
+1.0
+200
+0.8
+400
+600
+0.6
+800
+0.4
+1000
+1200
+0.2
+1400
+0
+200
+400
+600
+800
+1000
+1200
+1400ZtZepoch:fepoch}batch:(step}
+0
+1.0
+200
+0.8
+400
+600
+0.6
+800
+0.4
+1000
+1200
+0.2
+1400
+0
+200
+400
+600
+800
+1000
+1200
+1400(a) Learned cluster 1
+(b) Learned cluster 2
+(c) Learned cluster 3
+(d) Learned cluster 4
+(e) Learned cluster 5
+(f) Learned cluster 6
+(g) Learned cluster 7
+(h) Learned cluster 8
+(i) Learned cluster 9
+(j) Learned cluster 10
+(k) Learned cluster 11
+(l) Learned cluster 12
+
+9d(m) Learned cluster 13
+(n) Learned cluster 14
+(o) Learned cluster 15
+(p) Learned cluster 16
+(q) Learned cluster 17
+(r) Learned cluster 18
+(s) Learned cluster 19
+(t) Learned cluster 20
+Figure 8. Principal images (defined in §A) of clusters learned by MLC on CIFAR100-20.
+
diff --git a/XdAzT4oBgHgl3EQf1v7O/content/tmp_files/load_file.txt b/XdAzT4oBgHgl3EQf1v7O/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..1eb93b305a4c243f912d70e9f855c26f386c1409
--- /dev/null
+++ b/XdAzT4oBgHgl3EQf1v7O/content/tmp_files/load_file.txt
@@ -0,0 +1,973 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf,len=972
+page_content='Unsupervised Manifold Linearizing and Clustering  Tianjiao Ding1 Shengbang Tong2 Kwan Ho Ryan Chan1 Xili Dai3 Yi Ma2 Benjamin D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Haeffele1  Abstract  Clustering data lying close to a union of low-  dimensional manifolds, with each manifold as a cluster,  is a fundamental problem in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' When the  manifolds are assumed to be linear subspaces, many meth-  ods succeed using low-rank and sparse priors, which have  been studied extensively over the past two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Un-  fortunately, most real-world datasets can not be well ap-  proximated by linear subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' On the other hand, several  works have proposed to identify the manifolds by learning  a feature map such that the data transformed by the map  lie in a union of linear subspaces, even though the original  data are from non-linear manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' However, most works  either assume knowledge of the membership of samples to  clusters, or are shown to learn trivial representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' In  this paper, we propose to simultaneously perform cluster-  ing and learn a union-of-subspace representation via Max-  imal Coding Rate Reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Experiments on synthetic and  realistic datasets show that the proposed method achieves  clustering accuracy comparable with state-of-the-art alter-  natives, while being more scalable and learning geometri-  cally meaningful representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Introduction  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Motivation and Contributions  Clustering is a fundamental problem in machine learn-  ing, allowing one to group data into clusters based on as-  sumptions about the geometry of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' For example,  when data are concentrated around distinct centroids, classi-  cal k-means clustering [14,17,26,29] and its variants [2,3,5]  are able to find the cluster centroids and assign membership  to each data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' More generally4, subspace clustering  methods [11, 13, 16, 25, 27, 50] are designed to cluster data  that lie close to a union of different low-dimensional linear  (or affine) subspaces, where each subspace defines a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  1Mathematical Institute for Data Science, Johns Hopkins University,  USA 2Department of Electrical Engineering and Computer Sciences, Uni-  versity of California, Berkeley, USA 3The Hong Kong University of Sci-  ence and Technology (Guangzhou), PRC  4This includes k-means-based methods, since a centroid is a 0-  dimensional affine subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Overall, those methods often enjoy theoretical guarantees of  correct clustering [13,22,27,37,40,41,44,47–51] and find  applications in various problems such as image clustering,  face recognition, motion segmentation, and recently in pop-  ular Transformer architectures in deep learning [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Despite the wide range of applications and theoretical  guarantees, subspace clustering methods rely on a crucial  assumption that each cluster can be well approximated by  a linear/affine subspace, which is often not valid for many  real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' For instance, even in a dataset as sim-  ple as MNIST hand-written digits, images of a single digit  do not lie close to a low-dimensional linear subspace, thus  directly applying subspace clustering will fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Instead, it  is more natural to assume the clusters are from non-linear  low-dimensional manifolds (one manifold per cluster), and  attempt to learn or design a non-linear embedding of the  data so that the transformed data lies close to distinct linear  subspaces, with points from one manifold mapped to the  same subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' For example, [24] shows that a subspace  clustering method can achieve 99% clustering accuracy on  MNIST images after embedding the data with the scattering  transform [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Beyond the above example, numerous other subspace  clustering methods have explored hand-designing an appro-  priate feature embedding (or kernel) such as polynomial or  exponential mappings [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' However, these embeddings as-  sume specific families of manifolds, thus they need to be  hand-crafted for various tasks and datasets using domain  knowledge, which makes their application challenging for  complicated data such as natural images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  On the other  hand, [12] proposes to cluster data based on treating a lo-  cal neighborhood of the manifold approximately as a linear  subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' However, for this to succeed sufficient sampling  density is required, which implies a prohibitive number of  samples when the manifolds are of moderate dimension or  are highly curved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Further, for a new sample unseen at train-  ing time one needs to run the algorithm with all samples  to embed it or assign a membership to it, which is expen-  sive computationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' More recently, numerous works pro-  pose to learn an appropriate linear embedding of the data  via deep networks and then perform subspace clustering in  the feature space [1, 18, 20, 35, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Unfortunately, it has  been shown that many of these formulations are provably  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='01805v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='LG]  4 Jan 2023  (a)  (b)  (c)  (d)  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' (a) Input data X of two manifolds each containing 100 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' (b) Features Zθ at random initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' (c) Zθ after self-  supervised initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' (d) Zθ after MLC (4) training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Details are in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  ill-posed and learn trivial representations5, with much of the  claimed benefit coming from ad-hoc post-processing rather  than the method itself [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  This leads to the following  question:  Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' For data approximately supported on an un-  derlying union of manifolds, can we learn a transformation  of the data, so that the transformed data lie in distinct linear  subspaces to be easily clustered?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Meanwhile, learning a representation from multi-modal  data has been a topic of its own interest in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  An ideal property of the learned representation often pur-  sued is between-cluster discrimination, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', features from  different clusters should be well separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Further, an im-  portant yet often ignored property of the learned representa-  tion is that it maintains within-cluster diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' This is de-  sirable as it allows distances of samples within a cluster to  be preserved under the learned transformation, which could  facilitate downstream tasks such as denoising, hierarchical  clustering and semantic interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' In the supervised  setting, training with the cross-entropy (CE) classification  objective fails to achieve the second property, as it has been  shown empirically [32] and theoretically [43, 56] that the  representation learned by CE has the property that features  from one cluster tend to collapse to a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' On the  other hand, recent work has proposed the principle of Max-  imal Coding Rate Reduction (MCR2) [53] as one of the few  methods that are able to achieve the two ideal properties  by learning a representation where features from each clus-  ter are expected to lie close to a low-dimensional subspace  (within-cluster diverse), and the subspaces from different  clusters are orthogonal to each other (between-cluster dis-  criminative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' However, for MCR2 to learn such orthogonal  subspaces each corresponding to one cluster, one needs the  annotation of which sample belong to which cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Such  annotation might be expensive or impossible to acquire for  5In this paper, we use ‘representation’ and ‘feature’ interchangeably to  mean the image of data under a (learned) transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  large-scale datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' This motivates another question of in-  terest:  Question 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Can we learn a union-of-orthogonal-subspace  representation of data coming from an underlying union of  manifolds without access to the labels?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  This paper gives positive answers to the two interrelated  questions by making the following contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We propose to simultaneously cluster the data and  learn a union-of-orthogonal-subspace representation  via MCR2, when data is assumed to lie close to a  union of manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' This is achieved by formulation  (4), which optimizes over both the representation and  a doubly stochastic membership formulation inspired  by the state-of-the-art subspace clustering result [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Since the membership has as many entries as the  square of the batch size of the input data, we give a  parameterization of the membership (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Fur-  ther, as problem (4) is highly non-convex, we give a  meta-algorithm (Algorithm 1) on how to initialize the  variables and to optimize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We conduct experiments on simulation and CIFAR10  to demonstrate some desirable properties of the pro-  posed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  We further experiment on datasets  with larger number of clusters and imbalanced clus-  ters such as CIFAR100-20, CIFAR100-100, and Tiny-  ImageNet200, and show that the proposed method  achieves state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Additional Related Work  Beyond the above, we make connections to a few impor-  tant works that are related to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Deep Clustering and Representation Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Re-  cently, there is an interesting line of research in representa-  tion learning and clustering that takes advantage of pseudo-  labelling and semi/self-supervised learning [7, 30, 33, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
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+page_content='0Specifically, one first identifies a subset of samples (often  termed reliable samples) based on geometric or statistical  criteria in the learned representation and cluster prediction,  and then uses the predicted labels for those reliable sam-  ples as if they are ground-truth labels to refine the rep-  resentation and cluster prediction of other samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  De-  spite the promising clustering performance, the represen-  tation learned by these methods are not constrained to be  both between-cluster discriminative and within-cluster di-  verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' In contrast, the proposed method learns a represen-  tation with these two ideal properties (see Figure 4) and  also achieves state-of-the-art clustering performance (see  Tables 2 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Neural Manifold Clustering and Embedding (NMCE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  A recent preprint [23] also proposes a solution to the same  problem we study, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', clustering the data and learning an  union-of-orthogonal-subspace representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  In particu-  lar, [23] proposes to model the point-to-cluster membership  and optimize MCR2 [53] over both the representation and  the membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' In this paper, we adopt a similar formula-  tion, but we propose to model the point-to-point affinity us-  ing a doubly stochastic matrix, inspired by the state-of-the-  art subspace clustering methods (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Aside from having  different conceptual formulations and algorithms, our for-  mulation is much more stable with respect to initialization  and is naturally suitable for hierarchical clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We de-  tail these distinctions in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Experiments (Table 2) further  show that the proposed method (MLC) achieves higher ac-  curacy than [23] (NMCE) on large scale realistic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Problem Formulation  We start by defining the problem that we study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Suppose  X = [x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' , xn] ∈ RD×n is a dataset of n samples drawn  from a union of k underlying manifolds �k  j=1 Mj and y ∈  Rn their memberships to the manifolds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', xi ∈ My(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Problem 1 (Unsupervised Manifold Linearizing and Clus-  tering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Given the dataset X, can we simultaneously 1)  cluster the samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', estimate y, and 2) learn a lin-  ear representation for manifolds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', find a transformation  f : RD → Rd, such that the image of each manifold f(Mi)  is a low-dimensional linear subspace of Rd, and the sub-  spaces satisfy desired properties (§1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', they are between-  cluster discriminative and within-cluster diverse?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Here we base our approach on the principle of Maximal  Coding Rate Reduction (MCR2) which is designed to learn  ideal representations in the supervised case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', when the  membership y is given (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Then we discuss the chal-  lenges of simultaneously clustering and learning represen-  tation (Problem 1), and propose our MCR2 clustering ob-  jective to solve Problem 1 with those challenges in mind  (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We further give an algorithm to optimize the pro-  posed objective (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Supervised Manifold Linearizing via MCR2  In the case when the labels y are given as supervision,  MCR2 [53] aims to address part 2) of Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Let  fθ : RD → Sd−1 be a featurizer parameterized by a  neural network, which in turn gives an embedding Zθ :=  [z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' , zn] ∈ Rd×n of data with zi := fθ(xi) ∈ Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  MCR2 aims to learn an ideal representation by optimizing  max  θ  R(Zθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' ϵ) − Rc(Zθ, Π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' ϵ)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Zθ ∈ S  (1)  where R(Zθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' ϵ) := log det  �  I +  d  nϵ2 ZθZ⊤  θ  �  ,  and Rc(Zθ, Π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' ϵ) :=  k  �  j=1  ⟨Πj, 1⟩  n  log det  �  I +  d  ⟨Πj, 1⟩ϵ2 Zθ Diag(Πj)Z⊤  θ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Here S is the set of matrices whose columns all have unit  ℓ2 norm6, Π ∈ Rn×k is a given membership matrix such  that Πij = 1 if j = y(i) and Πij = 0 otherwise, ϵ > 0  is a prescribed precision parameter, Πj ∈ Rn denotes the  jth column of Π, 1 is a vector of all ones so that ⟨Πj, 1⟩  is the number of points in cluster j, and finally for v ∈  Rn, Diag(v) denotes a diagonal matrix with the entries of  v along the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Intuitively7, the R(Zθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' ϵ) term of (1) measures the vol-  ume of Zθ, and maximizing it would diversify features from  all samples, which we will refer to as the expansion term  Likewise, the Rc(Zθ, Π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' ϵ) term measures the sum of vol-  umes of each cluster of Zθ and is referred to as the compres-  sion term, since minimizing it would push features within  each cluster to stay close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' It has been shown that given  Π, the features obtained by maximizing the rate reduction  R(Zθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' ϵ)−Rc(Zθ, Π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' ϵ) has the property that the features  of each cluster spread uniformly within a subspace (within-  cluster diverse), and the subspaces from different clusters  are orthogonal (between-cluster discriminative), under rel-  atively mild assumptions [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Unsupervised Manifold Linearizing and Clus-  tering via MCR2  While the MCR2 formulation is designed to learn ideal  representations (§1) when the membership y (or equiva-  lently Π) is given, here we are interested in the unsuper-  vised setting where one does not have access to membership  annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Thus, we propose to simultaneously perform  both parts 1) and 2) of Problem 1 by also optimizing over  6This can be easily achieved by having the last layer of the neural net-  work fθ be a normalization layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  7More formally, terms of the form log det  �  I +  d  nϵ2 W W ⊤�  esti-  mate the average number of bits needed to code n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' samples W ∈  Rd×n from a zero-mean d-dimensional Gaussian up to a distortion ϵ [28],  hence the name coding rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  the membership Π of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' This naturally leads to  max  θ,Π∈Ω◦ R(Zθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' ϵ) − Rc(Zθ, Π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' ϵ)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Zθ ∈ S,  (2)  where Ω◦ := {Π ∈ Rn×k : ∀i ∈ [n], ∃ˆy(i)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Πiˆy(i) =  1 and Πij = 0 for j ̸= ˆy(i)} is the set of all ‘hard’ assign-  ments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', each row of Π is a one-hot vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' However,  this optimization is in general combinatorial: its complex-  ity grows exponentially in n and k, and it does not allow  smooth and gradual changes of Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Further, a second chal-  lenge is the chicken-and-egg nature of this problem: If one  already has an ideal representation Z, then existing sub-  space clustering methods can be applied on Z to estimate  the membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Likewise, if one is given the membership  Π of clusters, then solving (1) would lead to an ideal rep-  resentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' However, the Zθ and Π at the beginning of  optimization is typically far from ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Doubly Stochastic Subspace Clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' To address the  combinatorial of estimating the cluster memberships, we  draw inspiration from the closely related problem of sub-  space clustering, where the goal is to cluster n samples as-  sumed to lie close to a union of k low-dimensional sub-  spaces (§1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' In this case, one typically does not directly  learn an n × k matrix denoting memberships of n points  into k subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Instead, one first learns an affinity ma-  trix Π ∈ Rn×n signaling the similarity between pairs of  points, and then applies spectral clustering on the learned  Π to obtain a final clustering [11,13,16,25,27,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' In par-  ticular, requiring doubly-stochastic constraints on the affin-  ity Π is shown theoretically to suppress false inter-cluster  connections for clustering problems [9] along with state-of-  the-art empirical performance for subspace clustering prob-  lems [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Inspired by the above, we propose a constraint set Ω for  matrix Π to be the set of n × n doubly stochastic matrices,  Ω = {Π ∈ Rn×n : Π ≥ 0,  Π1 = Π⊤1 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  (3)  However, this constraint alone is insufficient for strong clus-  tering performance: Consider the optimization of (2) with  respect to Π ∈ Ω only, and note that the objective is  strongly convex with respect to Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Since we maximize a  convex function with respect to convex constraints Ω, an  optimal Π would lie at an extreme point of Ω, which for  doubly stochastic matrices is a permutation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' This is  not ideal for clustering, as it implies that every point is as-  signed to its own distinct cluster, and there is no incentive  to merge points into larger clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' To resolve this issue,  we follow the approach in [24] and add ℓ2 regularization8  γ  2 ∥Π∥2  F to Π which biases Π toward the uniform matrix  1  n11⊤, so by tuning γ we can also tune the sparsity level of  8Other choices of regularization are also possible: Essentially any func-  tion which achieves its minimum over Ω at the uniform matrix could po-  tentially be used, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', the negative entropy function �  ij Πij log(Πij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' This results in our final proposed formulation, dubbed  Manifold Linearizing and Clustering (MLC):  max  θ  R(Zθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' ϵ) − Rc(Zθ, Πθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' ϵ) − γ  2 ∥Πθ∥2  F  (4)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Zθ ∈ S, Πθ ∈ Ω,  where R(Zθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' ϵ) = log det  �  I +  d  nϵ2 ZθZ⊤  θ  �  , and  Rc(Zθ, Πθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' ϵ) = 1  n  n  �  j=1  log det  �  I + d  ϵ2 Zθ Diag((Πθ)j)Z⊤  θ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Note that here Πθ = Πθ(X) is now also parameterized  by a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' While this is constrained optimization  which may appear difficult to handle, we explain in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3  how we parameterize Zθ and Πθ via neural networks so  that the constraints are satisfied by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Below, we  note a few advantages of the proposed formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Parameterizing Π via a Neural Network versus Free  Variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' An alternative way to parameterize the mem-  bership would be to directly take Π as decision variables in  Ω, as opposed to outputs of a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' However, this  leads to maintaining O(n2) variables which is prohibitive  for large datasets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', n = 106 for ImageNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' In contrast,  this is not the case if one parameterizes Π as a neural net-  work, since one can do stochastic gradient descent such that  for each batch both the memory and computational com-  plexity is at most square of the batch size (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Comparison with NMCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' As mentioned in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2, NMCE  [23] approaches Problem 1 also by optimizing MCR2 over  both the representation and membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  However, in  NMCE the membership is parameterized by an n × k ma-  trix Πn×k that models the point-cluster membership, which  is different from our doubly stochastic point-point member-  ship matrix Πθ inspired from the state-of-the-art subspace  clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Note further that for NMCE the initialization  of Πn×k is arbitrary and has nothing to do with the struc-  tures in the initialized representation Πθ, and a bad initial-  ization of Πn×k could lead to the features from different  true clusters being compressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' On the other hand, the pro-  posed doubly stochastic membership Πθ can be initialized  deterministically using structures from self-supervised ini-  tialized features Zθ (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Interestingly, optimizing (4)  allows an interpretation of linearizing each point with its  neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Empirically as seen in (Table 2), the proposed  MLC yields a higher clustering accuracy than NMCE [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Algorithms  As mentioned, in the MLC objective (4), we parameter-  ize both the representation Zθ and doubly stochastic mem-  bership Πθ via a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Below we elaborate on  how this is done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We summarize the network architecture  in Figure 2, and the meta algorithm in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Parameterizing Zθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We follow [53] and take some existing  network architecture as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We append a few  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Overall architecture for optimizing the proposed manifold linearizing and clustering (MLC) objective (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Given n input samples  X each lying in RD, their d-dimensional representation is given by Zθ(X), where θ denotes network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Further, their doubly  stochastic membership matrix Πθ(X) is given by taking an inner product kernel of the output of the cluster head Cθ(X) followed by a  doubly stochastic projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  affine layers with non-linearity as the representation head to  further transform the output in Rd, followed by a projection  layer to respect the unit sphere Sd−1 constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Parameterizing Πθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' In subspace clustering, the member-  ship Π given data X often takes the form of g(X)⊤g(X)  for some (linear) transformation g, such as in the inner  product kernel [9, 16] where g = I or the least square re-  gression [27] where g = (I + λX⊤X)−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' This moti-  vates us to parameterize gθ by a neural network, and take  C⊤  θ Cθ ∈ Rn×n as the membership where Cθ is shorthand  for gθ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Nevertheless, such an n×n matrix is in general  not doubly stochastic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', C⊤  θ Cθ /∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' To obtain a doubly  stochastic membership, we further apply a Sinkhorn projec-  tion layer PΩ,η(·) [10,39], which gives our final parameter-  ization of the membership as Πθ = PΩ,η(C⊤  θ Cθ) ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Initializing Zθ: Self-supervised Representation Learn-  ing via MCR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Since the proposed MCR2 clustering objec-  tive (4) is non-convex, it is important to properly initialize  both Z and Π to converge to good (local) minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' On  the other hand, randomly initialized features are typically  far from being ideal, since they may not satisfy the idealized  properties (§1), and further may not respect the invariance  to augmentation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', the augmented samples should have  their representation close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Thus, we adopt the  self-supervised training strategy [23]  max  θ  R  �Zθ + Z′  θ  2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' ϵ  �  + λ  n  �  i=1  |z⊤  i z′  i|,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  z′  i, zi ∈ Sd−1,  ∀i ∈ [n],  (5)  where for every i, zi and z′  i are features of different aug-  mentations of the i-th sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' This essentially requires that  features from different augmentations of the same sample  should be as close as possible, whereas features from dif-  ferent samples should be as uncorrelated as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Initializing Πθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' An ideal initialization of Πθ would be  such that if (Πθ)ij has a high value then points i, j are  likely to be from the same true cluster and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' On the  other hand, after the self-supervised feature initialization  mentioned above, Zθ already have some structures which  we can utilize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Thus, we propose to initialize Πθ with  PΩ,η(Z⊤  θ Zθ), which is easily implemented by copying the  parameters from Zθ to Cθ once after the self-supervised  initialization of the former, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', from the feature head to the  cluster head in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Data Augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Beyond initializing Zθ, it is often  desirable to incorporate augmentation in optimizing (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Specifically, from {X(a) ∈ RD×n}A  a=1 the dataset X  under A different augmentations, one computes (Z(a)  θ  ∈  Rd×n, Π(a)  θ  ∈ Rd×n) for each augmentation a, and use in  (4)  Zθ = PSd−1  �  1  A  A  �  a=1  Z(a)  θ  �  ,  Πθ = 1  A  A  �  a=1  Π(a)  θ  ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  (6)  Note that one can benefit from parallelization by putting  X(a), Z(a)  θ , Π(a)  θ  for each augmentation a on one comput-  ing device, since Π(a)  θ  only depends on X(a) but not from  other augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Experiments on Real Datasets  Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' To evaluate the clustering quality, we run spec-  tral clustering on learned membership matrix Π, and re-  port the normalized mutual information (NMI, [42]) and  clustering accuracy (ACC, [21]), as are commonly used in  clustering tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  To evaluate the learned representation,  we define the following metric: for a collection of points  W = [w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' , wl] ∈ Rd×l (l > d) with associated sin-  gular values {σi}d  i=1, define the numerical rank of W as  arg minr  �  r : �r  i=1 σ2  i / �d  i=1 σ2  i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='95  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Now, one can  measure the numerical rank of the learned representaion Z,  BackboneAlgorithm 1 MLC: Unsupervised Manifold Linearizing  and Clustering  Input: X ∈ RD×n,  ϵ, γ, η, λ > 0,  d, k, nb, T, A ∈ Z≥0  1: initialize Zθ by self-supervised representation learning  via MCR2  ▷ (5)  2: initialize Πθ  3: for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' , T do  4:  ¯  X ∈ RD×nb ← sample a batch from X  5:  ¯  X(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' , ¯  X(A) ← apply A augmentations to ¯  X  6:  ¯Zθ, ¯Πθ ← forward pass with { ¯  X(a)}A  a=1 and net-  work parameters θ  ▷  (6)  7:  ∇θ(4) ← backward pass with respect objective (4)  8:  θ ←update θ using some optimizer on ∇θ(4)  9: end for  10: run spectral clustering on Πθ to estimate labels ˆy of  samples  Output: Zθ, ˆy  as well as that of each ground-truth cluster9 of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' A low  numerical rank of W implies that points in W lie close to  a low-dimensional subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We further report the cosine  similarity of learned representation, which is simply |z⊤  i zj|  for points i and j, since ∥zi∥ = 1 by construction in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Finally, to compare the efficiency of methods we report the  training time in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2, where the experiments are run on 2  Nvidia RTX3090 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Comparison with Subspace Clustering  To demonstrate the ability of MLC to cluster the sam-  ples and linearize the manifolds, we conduct experiments on  CIFAR10, which consists of RGB images from 10 classes  such as planes, birds, and deers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' As mentioned in §1 sub-  space clustering methods rely crucially on the assumption  that data lie close to a union of linear subspaces, which  many real-world dataset may not satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' To show that this  is the case, we additionally compare the proposed method  with subspace clustering methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' As we shall see, apply-  ing subspace clustering directly on self-supervised features  of CIFAR10 will yield low clustering accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' In contrast,  MLC is able to achieve high clustering accuracy, and more-  over, produce a union-of-orthogonal-subspace representa-  tion on which subspace clustering methods can also achieve  high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  We use the training split of CIFAR10 containing  50000 RGB images, each of size 3 × 32 × 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We use  the augmentation specified in the Appendix to perform self-  supervised representation learning (5) and get Zself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' For  a fair comparison, the so-learned Zself are used both as ini-  tialization for MLC (line 1 of Algorithm 1), and as the input  9They are defined by the true labels y (§2), so that the numerical rank  metric is decoupled from the quality of learned membership Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Clustering accuracy and normalized mutual information  for subspace clustering (EnSC, SSC-OMP) on self-supervised fea-  tures Zself, features ZMLC learned by MLC, and manifold clus-  tering (MLC) on X, where X is 6 · 104 images from 10 classes  of CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Method  Input Data  ACC  NMI  EnSC  Zself  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='9  ZMLC  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='5  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2  SSC-OMP  Zself  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='8  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='5  ZMLC  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3  MLC  X  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3  for subspace clustering methods10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' In MLC, for each image  in each batch we randomly sample A = 2 augmentations  to apply on the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' As an additional comparison, we  also run subspace clustering methods on the features ZMLC  learned by MLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We compare with the elastic-net subspace clus-  tering with active-set solver (EnSC, [50]) and sparse sub-  space clustering with orthogonal matching pursuit solver  (SSC-OMP, [51]), using off-the-shelf implementation pro-  vided by the authors11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  We search the parameters of  EnSC over (γ, τ) ∈ {1, 5, 10, 50, 100} × {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='95, 1}  and those of SSC over (kmax, ϵ)  ∈  {3, 5, 10, 20} ×  {10−4, 10−5, 10−6, 10−7}, and report the run with the  highest clustering accuracy for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We summa-  rize detailed parameters for MLC in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Figure 3 reports the coding rates (as loss terms  in (4) and numerical ranks of features learned by MLC as  epoch varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' As a first note, the coding rate R of all fea-  tures (the blue curve in 3a) decreases only slightly as epoch  goes, indicating that the overall representation is diverse in  the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Indeed, the numerical rank of all features  (the dark curve in Figure 3b) stays 118 which is close to the  dimension 128 of the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' This is in sharp contrast  to the deep subspace clustering methods where all the fea-  tures collapse to a one-dimensional subspace [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' More-  over, as the coding rate Rc of clustered features (the orange  curve in Figure 3a) goes down, the numerical ranks of fea-  tures from each ground-truth cluster decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' For instance,  the representation from true cluster 3 has a numerical rank  of 37 in the first step and 24 in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' This implies  that most representation gets linearized better and clustered  more accurately, even though the MLC objective (4) is un-  supervised, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', it does not use ground-truth labels y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Last  but not the least, note that the features within each ground-  10The self-supervised features Zself empirically exhibit some union-of-  subspace structure, and are typically used for subspace clustering, as also  seen in [53, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2] and [55, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  11https : / / github .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' com / ChongYou / subspace -  clustering  (a) Coding rate of all features R, that of clustered  features Rc, and the rate reduction ∆R = R−Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  (b) Numerical ranks of all features Zθ and features  from each ground-truth cluster i, {zj : y(j) = i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Coding rates (as loss terms in (4)) and numerical ranks (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1) of the features  learned by MLC on CIFAR10 as epoch varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Cosine similarity |Z⊤  MLCZMLC|  of the features ZMLC learned by MLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  truth cluster spread well in a low-dimensional subspace,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', the numerical ranks for the true clusters at the last step  are within [13, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' This achieves the desired within-cluster  diverse property (§1), as opposed to the neural collapse phe-  nomenon that appears with the cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  To compare MLC with subspace clustering methods, we  report clustering accuracy and normalized mutual informa-  tion for EnSC, SSC-OMP on self-supervised features Zself,  features ZMLC learned by MLC, and MLC on X, where X  is 6 · 104 images from 10 classes of CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' In addition  we plot the cosine similarity of the features learned by MLC  in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Remarkably, the highest clustering accuracy  is 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3% achieved by MLC on X, which surpasses EnSC  (72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2%) and SSC-OMP (67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='8%) on Zself by a large margin,  even though Zself is used both as initialization for MLC and  input for EnSC and SSC-OMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Interestingly, using instead  the features ZMLC learned by MLC, the clustering perfor-  mance of EnSC and SSC-OMP increases and even becomes  comparable to MLC, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', EnSC achieves 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2% normal-  ized mutual information compared to 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3% of MLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' This  suggests that ZMLC has a union-of-subspace structure that  can be utilized by subspace clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Indeed, as seen in  Figure 4, features from different clusters tend to have a  small similarity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', being orthogonal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' This  demonstrates the between-cluster discrimination (§1) as de-  sired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Comparison with Deep Clustering Methods  We further compare the proposed MLC with state-of-  the-art deep clustering methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Note that most methods  reported (all except NMCE which is discussed in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2) do  not aim to learn a union-of-orthogonal-subspace represen-  tation, in contrast to MLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' As we will see, MLC achieves  clustering accuracy comparable to state-of-the-art methods  on large scale datasets with faster computational time, and  further surpasses them on extreme yet realistic cases like  datasets of imbalanced clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Compared Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We conduct experiments with MLC,  SCAN [45], and IMC-SWAV [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='12 Training details can  be found in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' In addition we include the num-  bers reported from DeepCluster [7], IIC [19], RUC [33] and  NMCE [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' For a fair comparison, all methods reported  use ResNet-18 as the backbone, which is also commonly  adopted by other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Beyond CIFAR10 (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1), we further use  CIFAR100-20, CIFAR100-100 and Tiny Imagenet-200 to  evaluate the performance of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Both CIFAR100-  100 and CIFAR100-20 contain the same 50000 train images  and 10000 test images with size 32 × 32 × 3, while the for-  mer are split into 100 clusters and the latter 20 super clus-  ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Finally, Tiny ImageNet contains 100000 train images  and 10000 test images with size 64 × 64 × 3 split into 200  clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Results on Large-scale Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We report clustering ac-  curacy and normalized mutual information on CIFAR10,  CIFAR100-20, CIFAR100-100, and TinyImageNet in Ta-  ble 2, and we further report running time in minutes for  CIFAR100-100 in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' As seen, the highest cluster-  ing performance on CIFAR10 is achieved by RUC+SCAN  (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3% ACC) and IMC-SWAV (81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1% NMI), where MLC  yields a slightly lower ACC of 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3% and NMI of 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  We note some interesting semantic interpretation for the  clustering obtained by MLC in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' On the other  hand, MLC performs comparably with other methods on  CIFAR100-20 by achieving an ACC of 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2% and NMI of  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Notably, MLC outperforms SCAN and IMC-SWAV  on CIFAR100-100 and TinyImageNet-200 by a large mar-  gin, while using lower running time: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', on CIFAR100-  12The authors are aware of a preprint [30] which approaches image  clustering via a combination of self/semi-supervised learning and pseudo-  labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' However, to the best of our effort we are unable to reproduce the  numbers reported in this paper using the implementation provided by the  authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We discuss the details in the Appendix and thus do not report their  numbers here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Loss terms  140  120  100  Loss terms  80  60  40  △R  R  Rc  20  0  500  1000  1500  2000  2500  StepNumerical rank of Z  120  100  All  Class 0  Class 1  80  Class 2  Class 3  Class 4  60  Class 5  Class 6  Class 7  Class 8  40  Class 9  20  0  500  1000  1500  2000  2500  Step1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='8  2000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='6  4000  6000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4  8000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2  0  2000  4000  6000  8000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='0Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Clustering accuracy and normalized mutual information on large scale datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' For a fair comparison, all methods use ResNet-18  as backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Method / Dataset  CIFAR10-10  CIFAR100-20  CIFAR100-100  Tiny ImageNet-200  Metrics  ACC  NMI  ACC  NMI  ACC  NMI  ACC  NMI  DeepCluster (ECCV′18)  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='9 IIC (ICCV′19)  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='7  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='7  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='5 SCAN (ECCV′20)  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='6  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='7  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='8  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='9  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='7 RUC+SCAN (CVPR′21)  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3 IMC-SWAV (Arxiv′21)  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='0  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='9  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='6  NMCE (Arxiv′22)  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='0  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='7  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='8 MLC  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='6  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='5  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='5  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Running time in minutes and clustering accuracy on  CIFAR100-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' For a fair comparison, all methods use ResNet-18  as backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Method / Metric  Running Time  ACC  Stage  I  II  III  Total  SCAN (ECCV′20)  308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='7  396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3  IMC-SWAV (Arxiv′21)  529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4 529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='9  MLC  266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='7  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='7 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Clustering accuracy on imbalanced datasets: (a) Imb-  CIFAR10, (b) Imb-CIFAR100-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  For a fair comparison, all  methods use ResNet-18 as backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Method / Dataset  (a)  (b)  IMC-SWAV (Arxiv′21)  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='7  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2  SCAN (ECCV′20)  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='9  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1  MLC  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='0  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1  100, MLC yields an accuracy of 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4% in 291 minutes,  whereas IMC-SWAV has 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='9% using 529 minutes, and  SCAN has 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3% in 396 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Imbalanced Clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Note that for CIFAR10 or CI-  FAR100 each cluster contains approximately the same num-  ber of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' On the other hand, natural images are typ-  ically imbalanced, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', the clusters have unequal number  of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  To mimic this setting, we take a naive ap-  proach to construct the following imbalanced datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' For  the 10 clusters of CIFAR10, we remove half of the sam-  ples from odd-numbered clusters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', clusters 1, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' , 9)  from both the training and test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We refer to the re-  duced dataset Imb-CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Likewise we construct Imb-  CIFAR100-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We run two state-of-the-art methods IMC-  SWAV and SCAN as well as the proposed MLC on Imb-  CIFAR10 and Imb-CIFAR100-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Table 4 shows clustering accuracy on the imbalanced  datasets Imb-CIFAR10 and Imb-CIFAR100-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' As a first  observation, the clustering accuracy of all methods is lower  on the imbalanced datasets than on the balanced counter-  parts, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Notably, MLC suffers from the least  performance drop, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', when moving from CIFAR10 to  Imb-CIFAR10 the accuracy of MLC drops from 86% to  80%, whereas that of SCAN and IMC-SWAV decreases  from above 87% to below 66%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Conclusion  This paper studies the problem of simultaneously clus-  tering and learning an union-of-orthogonal-subspace repre-  sentation for data, when data lies close to a union of low-  dimensional manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' To address the problem we pro-  pose an objective based on maximal coding rate reduction  and doubly stochastic membership inspired by the state-of-  the-art subspace clustering results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We provide an efficient  and effective parameterization of the membership variables  as well as a meta-algorithm to optimize the representation  and membership jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We further conduct experiments  on datasets with larger number of clusters and imbalanced  clusters and show that the proposed method achieves state-  of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We believe that our work provides  a general and unified framework for unsupervised learning  of structured representations for multi-modal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  References  [1] Mahdi Abavisani and Vishal M Patel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Deep multimodal sub-  space clustering networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' IEEE Journal of Selected Topics  in Signal Processing, 12(6):1601–1614, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
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+page_content=' k-means++: The ad-  vantages of careful seeding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' In the Eighteenth Annual ACM-  SIAM Symposium on Discrete Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
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+page_content=' Scalable K-Means++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Pro-  ceedings VLDB Endowment, 5(7), Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
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+page_content='  VI-  CReg:  Variance-Invariance-Covariance regularization for  Self-Supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  In International Conference on  Learning Representations, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
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+page_content=' Deep clustering for unsupervised learning  of visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
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+page_content=' In IEEE Conference on Computer Vision and Pattern  Recognition, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' 6  [56] Jinxin Zhou, Xiao Li, Tianyu Ding, Chong You, Qing Qu,  and Zhihui Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' On the optimization landscape of neural col-  lapse under MSE loss: Global optimality with unconstrained  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' In International Conference in Machine Learning,  Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' 2  (a) Learned cluster 1  (b) Learned cluster 2  (c) Learned cluster 3  (d) Learned cluster 4  (e) Learned cluster 5  (f) Learned cluster 6  (g) Learned cluster 7  (h) Learned cluster 8  (i) Learned cluster 9  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Principal images (defined in §A) of clusters learned by MLC on CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Semantic Interpretability of the Learned  Representation and Clusters on CIFAR10  Recall that MLC is designed to perform clustering  while learning a union-of-orthogonal-subspace representa-  tion (§1), where each cluster defines a low-dimensional sub-  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Therefore, we further visualize the different direc-  tions within each learned cluster or subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Specifically,  after a final clustering is obtained (line 10 of Algorithm 1),  we take the features from each learned cluster and apply  Principal Component Analysis (PCA) to them to obtain the  first 8 principal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' These correspond to the 8  rows for each cluster in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Recall that the princi-  pal components are mutually orthogonal, indicating uncor-  related directions within one cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' To visualize those di-  rections or principal components in images, we take the fea-  tures that are closest to the principal components and visu-  alize the corresponding original images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Interestingly, the rows of images corresponding to prin-  cipal components appear to exhibit some semantic ‘con-  cepts’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' For example in Figure 5b, row 1 and 8 are respec-  tively white and red trucks, while row 4 are the trucks that  ship sand or mud;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' row 1 of Figure 5d are deers with trees  as background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' This further suggests that the learned em-  bedding seems to preserve distance within each cluster (as  desired in §1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', images that are close/far in semantic  meaning will be close/far in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Note how-  ever, that some learned clusters do not align fully with the  ground-truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' For instance, rows 1 and 3 of Figure 5h  are cats while all other rows in this cluster are dogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' On the  other hand, one may argue that Figure 5h are a cluster of  cats and dogs of light colors, whereas Figure 5c is a clus-  ter of those of brown colors, which could be a semantically  meaningful clustering even though it does not align with the  ground-truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We believe it would be an interesting  future work to use MLC to discover new semantics that are  not present in the given labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Ablation study on the roles of different parts of Algo-  rithm 1 and on using augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Ablation Study on CIFAR-10  Clustering  Accuracy  Full Algorithm 1  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3%  Replacing self-supervised initialization (line 1)  with random initialization  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='0%  Replacing updating MLC objective (4) (lines 3-9)  with subspace clustering (EnSC)  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4%  Not using augmentation in lines 3-9  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='0%  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Role of Augmentation  Recall that data augmentation was used both in the self-  supervised initialization (line 1 of Algorithm 1, see ‘Initial-  izing Zθ’ in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3) and in updating the MLC objective (lines  3-9, ‘Data Augmentation’ in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Below we give addi-  tional clarification on the role of augmentation therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Augmentation for Initializing the Features  Since the proposed MLC objective (4) is highly non-  convex, the (local) solution that a first-order optimizer con-  verges to in general depends on the initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' However,  before line 1 of Algorithm 1 is executed, the features Z at  initialization could be very far from union-of-orthogonal-  subspace (as desired by Problem 1), since the neural net-  work has an arbitrary architecture and initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' To at  least promote some ideal structures in the features, we con-  duct line 1 of Algorithm 1 so that the features from an origi-  nal sample and its augmented copy are close, while features  from different samples spread out in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' This  is a common idea used in contrastive learning, and more re-  lated in [53, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2] [23, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='6] that are both based on MCR2  as in this paper (even though the formulations are different,  as argued in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Empirically, initializing the features us-  ing augmentation (line 1 of Algorithm 1) is important for  the following-up steps: as seen in Table 5, on CIFAR-10, if  one uses random initialization to replace this step, then the  final clustering accuracy is 20%, in sharp contrast to 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Augmentation for Updating MLC Objective (4)  In optimizing MLC (lines 3-9 of Algorithm 1), augmen-  tation empirically improves clustering performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' As one  can see in Table 5, on CIFAR-10 using the sample self-  supervised initialization of the features, MLC achieves only  80% clustering accuracy without augmentation, in contrast  to 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3% with augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We attribute this difference to  the fact that augmentation enriches the diversity of samples  the algorithm sees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Role of Different Parts of Algorithm 1  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Initialization of the Features (Line 1)  Please kindly refer to §B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Updating the MLC Objective (4) (Lines 3-9)  The main novelty of this paper lies in updating the MLC  objective that learns both the representation Zθ and a dou-  bly stochastic membership Πθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Note that in this step, clus-  tering is pursued by modeling the membership Πθ, as op-  posed to the self-supervised feature initialization step where  no membership is explicitly pursued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' This step is indeed im-  portant for clustering: as seen in Table 5, on CIFAR-10, the  clustering accuracy on the self-supervised initialized fea-  tures Zθ is only 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4%, in contrast to 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3% obtained after  updating the MLC objective (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Spectral Clustering (Line 10)  Since the proposed MLC learns a doubly stochas-  tic membership that signals pair-wise similarity between  points, it is standard to run spectral clustering [46] to com-  pute a final set of clusters from the learned membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  This is done once at the very end of Algorithm 1, and is  rather efficient compared to the other parts of Algorithm 1:  for instance, using an unaccelerated implementation from  SciPy, it takes less than 30 seconds to perform spectral clus-  tering on a 10000 × 10000 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Details on Experiment Settings  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Synthetic Union-of-Manifold Data  We perform simulations to visualize the properties of the  proposed manifold learning and clustering method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' As seen  in Figure 1a, we generate data X from two manifolds on  the sphere S2, each consisting of 200 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' The points  from the first manifold (green) take the form  xi =  �  �  cos  �  A sin(ωφi)  �  cos φi  cos  �  A sin(ωφi)  �  sin φi  sin  �  A sin(ωφi)  �  �  � + ϵi,  (7)  where A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2 and ω = 5 sets the curvature of the mani-  fold, ϵi ∼ N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='05I3) is the additive noise, and we take  φi = 2πi  100 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' , 100 to generate 100 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' On the  other hand, the points from the second manifold (blue) are  simply 100 samples from N([0, 0, 1]⊤, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='05I3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We take  the feature dimension d = 3 to be equal to he input di-  mension D = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We paramterize both the feature head  fθ and the cluster head gθ to be a simple fully-connected  network with 100 hidden neurons, followed by a Rectified  Linear Unit as non-linearity and a projection operator onto  the sphere S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Figures 1b to 1d report the features Zθ with  random initialization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=', before line 1 of Algorithm 1),  with self-supervised initialization, and at convergence of  MLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Notably, despite Zθ being noisy and only approxi-  mately piece-wise linear, as epoch goes Zθ gradually trans-  form to two linear subspaces: the green points converge to  a 2-dimensional subspace (intersected with S2) and the blue  points converge to a 1-dimension subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Training Details on Real Datasets  MLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' As said, we use ResNet-18 as the backbone for ex-  periments on CIFAR10, CIFAR100-20, CIFAR100-100 and  Tiny-ImageNet-200, and the imbalanced counterparts Imb-  CIFAR10, Imb-CIFAR100-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We also fix the batch size  to be 1024 in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' In self-supervised initializa-  tion of Zθ (line 1 of Algorithm 1), we use the precision  (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1) parameter ϵ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2, a LARS optimizer [52] (as is  also done in [8, 23]) with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3 and trained  MLC for 1000 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' On the other hand, in the train-  ing of MLC objective, we use ϵ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='05, and  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='175 for the entropy regularization in the Sinkhorn  projection [10] layer PΩ,η(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We fix the backbone and for  each batch, we perform one update for parameters in the  feature head Zθ and one update for parameters in the clus-  ter head Cθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' For each head we use one SGD optimizer [36]  with a learning rate of 10−2, momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='9, and weight  decay of 5 · 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Finally, for all experiments, we use the  augmentation from [4] detailed below in PyTorch code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Augmentation 1 Augmentations for real datasets  import torchvision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='transforms as t  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='Compose([  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='RandomResizedCrop(32,scale=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='04,  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='0)),  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='RandomHorizontalFlip(p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='5),  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='RandomGrayscale(p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2),  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='RandomApply([t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='ColorJitter(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1)], p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='8),  GaussianBlur(p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1)  ])  SCAN and IMC-SWAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Recall that we conduct ex-  periments on CIFAR100-100, Imb-CIFAR10, and Imb-  CIFAR100-100 with SCAN [45], IMC-SWAV [31] and  MLC, and report clustering and running time in Tables 3  and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We use off-the-shelf implementation13 provided by  the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' For a fair comparison, SCAN, IMC-SWAV and  MLC all use ResNet-18 as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Finally, the hyper-  parameters of SCAN and IMC-SWAV are set to be the ones  optimally chosen for CIFAR10 and CIFAR100 respectively  provided by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  SPICE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' As mentioned, the preprint [30] proposed a method  SPICE that appears to achieve state-of-the-art performance  in image clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We tried to reproduce their results on  CIFAR-100-20 using the official implementation14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' How-  ever, the provided implementation ran into a few errors,  which are also observed15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Despite our best effort to fix  those issues, the experiments yield only 14% clustering ac-  curacy on CIFAR100-20 as opposed to the 53% reported in  the paper [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Therefore, we note this observation and do  not include SPICE in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  13https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='com/wvangansbeke/Unsupervised-  Classification,  https : / / github .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' com / foiv0s / imc -  swav-pub  14https : / / github .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' com / niuchuangnn / SPICE,  commit  5eba538.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  15https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='com/niuchuangnn/SPICE/issues/  27,https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='com/niuchuangnn/SPICE/issues/31  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Clustering accuracy and normalized mutual information of MLC and NMCE on CIFAR10 over 10 random seeds, using the same  self-initialized features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' For the purpose of comparison, both methods use the same optimization strategy and hyper-parameters optimally  tuned for NMCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Consequently, the clustering performance of MLC reported here is lower than that in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Method  Metric  Seed  Mean  Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  0  1  2  3  4  5  6  7  8  9  MLC  ACC  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='5  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='8  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='8  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='6  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='0  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='6  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='24  NMI  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='6  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='8  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='8  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='5  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='5  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='28  NMCE  ACC  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='7  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='6  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='7  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='9  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='7  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='7  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='9  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='69  NMI  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='0  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='7  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='8  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='8  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='49  (a) CIFAR100-20  (b) CIFAR100-100  (c) Tiny ImageNet-200  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Cosine similarity |Z⊤  MLCZMLC| of the features ZMLC learned by MLC on more complicated datasets: CIFAR100-20,  CIFAR100-100, Tiny ImageNet-200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Additional Comparison of MLC and NMCE  on Stability with Respect to Random Seeds  As detailed in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2, one of the advantages of the pro-  posed MLC over NMCE [23] is that MLC has a more sta-  ble performance with respect to random seeds, since MLC  is able to initialize the membership deterministically using  structures from the self-supervised initialized features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Be-  low we conduct extra experiments to provide empirical ev-  idence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' We first fix a self-supervised initialization of fea-  tures that is in turn used for both NMCE and MLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Then,  using this very same initialization of features, we update  NMCE and MLC objective respectively with 5 different  seeds: recall that NMCE initializes the membership ran-  domly whereas MLC initializes the membership determin-  istically using the initialized features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' To make a valid com-  parison, for both methods we further use the same opti-  mization strategy and hyper-parameters that are optimally16  tuned for NMCE (which are not optimal for MLC): pre-  cision ϵ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2, # epochs 100, LARS optimizer for Zθ  with an initial learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='3 decayed to 0 in a cosine  annealing manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Table 6 reports clustering accuracy and  normalized mutual information of MLC and NMCE over  16For NMCE, we use the implementation as well as the parameters pro-  vided in https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='com/zengyi-li/NMCE-release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  10 random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' As expected, MLC has a more stable  clustering performance by having a standard deviation of  clustering accuracy and normalized mutual information less  than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='28, in contrast to more than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='49 achieved by NMCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Further, MLC achieves higher mean clustering performance  than NMCE, as also observed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Last but not least,  we note that the numbers in Table 6 are not comparable to  those in Table 2, since for MLC the hyper-parameters and  optimizers are different, and for NMCE an additional step  that fine tunes the backbone is used in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Addtional Visualization on Learned Repre-  sentation and Clusters  Figure 6 presents the cosine similarity (as defined in the  preamble of §3) of the representation learned by MLC on  CIFAR100-20, CIFAR100-100 and TinyImageNet-200 (for  the counterpart on CIFAR10 see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' As seen, the cosine  similarity maps form approximately block diagonal struc-  tures, showing that the features from different clusters are  roughly orthogonal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' This is desired by the  between-cluster discrimination (§1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  Finally, we provide additional visualization of principal  images on CIFAR100-20 (see §A for definition) in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='  ZtZepoch:{epoch}batch:{step}  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='0  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='8  400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='6  600  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4  800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2  1000  0  200  400  600  800  1000ZtZepoch:{epoch}batch:{step)  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='0  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='8  400  600  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='6  800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4  1000  1200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2  1400  0  200  400  600  800  1000  1200  1400ZtZepoch:fepoch}batch:(step}  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='0  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='8  400  600  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='6  800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='4  1000  1200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='600  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='800  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='1400(a) Learned cluster 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='(b) Learned cluster 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='(c) Learned cluster 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='(d) Learned cluster 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='(e) Learned cluster 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='(f) Learned cluster 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='(g) Learned cluster 7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='(h) Learned cluster 8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='(i) Learned cluster 9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='(j) Learned cluster 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='(k) Learned cluster 11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='(l) Learned cluster 12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='9d(m) Learned cluster 13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='(n) Learned cluster 14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='(o) Learned cluster 15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='(p) Learned cluster 16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='(q) Learned cluster 17  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='(r) Learned cluster 18  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='(s) Learned cluster 19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='(t) Learned cluster 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content='Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
+page_content=' Principal images (defined in §A) of clusters learned by MLC on CIFAR100-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdAzT4oBgHgl3EQf1v7O/content/2301.01805v1.pdf'}
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+Continuous optical-to-mechanical quantum state transfer in the unresolved sideband
+regime
+Amy Navarathna1,2, James S. Bennett1,2,3, and Warwick P. Bowen1,2∗
+1 ARC Centre of Excellence for Engineered Quantum Systems, St Lucia, Queensland 4072, Australia
+2 School of Mathematics and Physics, University of Queensland, St Lucia, Queensland 4072, Australia
+3 Centre for Quantum Dynamics, Griffith University, Nathan, QLD 4222, Australia
+(Dated: January 11, 2023)
+Optical-to-mechanical quantum state transfer is an important capability for future quantum net-
+works, quantum communication, and distributed quantum sensing. However, existing continuous
+state transfer protocols operate in the resolved sideband regime, necessitating a high-quality optical
+cavity and a high mechanical resonance frequency. Here, we propose a continuous protocol that
+operates in the unresolved sideband regime. The protocol is based on feedback cooling, can be im-
+plemented with current technology, and is able to transfer non-Gaussian quantum states with high
+fidelity. Our protocol significantly expands the kinds of optomechanical devices for which continu-
+ous optical-to-mechanical state transfer is possible, paving the way towards quantum technological
+applications and the preparation of macroscopic superpositions to test the fundamentals of quantum
+science.
+The ability to transfer quantum states between op-
+tical communication channels and quantum computing
+nodes is a necessary ingredient of the emerging quan-
+tum internet [1]. Quantum state transfer also has im-
+portant applications in quantum-enhanced sensing [2, 3],
+quantum-secure communications [4], and fundamental
+tests of macroscopic quantum mechanics [5–10]. A lead-
+ing approach is to mediate the transfer using an optome-
+chanical resonator [11–16]. This is attractive because me-
+chanical resonators interact via radiation pressure with
+electromagnetic fields of all frequencies [1] and can also
+be functionalized to interact with most quantum com-
+puting nodes, such as spins [18–20], superconducting de-
+vices [21–23] and atomic ensembles [24].
+The first step in the transfer process is an optical-
+to-mechanical state transfer, with a subsequent transfer
+to the final computing node [25–27].
+An optical cav-
+ity is employed to enhance the radiation pressure during
+the optical-to-mechanical state transfer.
+Leading pro-
+posals work only in the resolved sideband regime, where
+the decay rate of this cavity is lower than the mechani-
+cal resonance frequency [12, 28]. By contrast, most op-
+tomechanical systems operate in the unresolved sideband
+regime [29].
+In many cases this is due to the benefits
+that low mechanical frequencies convey for applications,
+for instance in precision sensing [30–32].
+In others, it
+is because of the difficulty of simultaneously achieving a
+low decay rate, a high resonance frequency, and sufficient
+radiation pressure coupling [33].
+To date, the only proposals for optical-to-mechanical
+state transfer in the unresolved sideband regime have
+used pulsed, rather than continuous, optomechanical in-
+teractions [34–36].
+This narrows the range of applica-
+tions, introduces significant technical challenges due to
+the additional timing and phase accuracy required [36–
+∗ w.bowen@uq.edu.au
+38], and involves large radiation pressure impulse forces
+that can be problematic [35, 39, 40].
+It is well known that a mechanical resonator can be
+feedback cooled close to its motional ground state in
+the unresolved sideband regime [41]. Here we propose
+a continuous optical-to-mechanical state transfer proto-
+col based on the same concept. By modelling the open
+quantum system dynamics, we show that feedback cool-
+ing can be understood as the transfer of a vacuum state
+of light onto the mechanical resonator. We find that ap-
+propriate choice of the feedback parameters allows the
+transfer of arbitrary quantum states. The requirements
+for successful transfer closely match those for ground-
+state cooling – once the optomechanical cooperativity ex-
+ceeds the thermal occupancy of the mechanical resonator,
+a coherent state can be transferred with near unity fi-
+delity and the Wigner-negativity of non-Gaussian states
+can be preserved.
+Moreover, the feedback parameters
+can be used to phase-sensitively amplify (or squeeze) the
+transferred state, to engineer its temporal profile, and –
+in direct analogy to state-transfer via resolved sideband
+cooling [42] – to achieve the transfer of a single optical
+sideband.
+Our work extends continuous optomechanical state
+transfer beyond the resolved sideband limit, to low-
+quality optical cavities and low frequency mechanical res-
+onators. Feedback cooling of a mechanical resonator to
+near its motional ground state has recently been demon-
+strated, both in cryogenic [43] and room temperature en-
+vironments [44]. As such, our proposal can be directly
+implemented with existing technology, providing a new
+tool for quantum networks and opening a new pathway
+to create and study macroscopic quantum systems. Our
+work also provides new insights into feedback cooling,
+showing that the process is in fact a quantum state trans-
+fer from light to mechanical motion.
+We consider an optomechanical system in the unre-
+solved sideband, high mechanical quality regime (κ ≫
+Ω ≫ Γ) with resonant optical driving, where κ (Γ) is
+arXiv:2301.03855v1  [quant-ph]  10 Jan 2023
+
+2
+FIG. 1.
+Schematic optomechanical system with feedback.
+Light is coupled into an optomechanical cavity. The reflected
+light is measured through homodyne detection. The detected
+photocurrent (Yout(t)) is convolved with a filter f(t) and di-
+rectly fed back to the momentum of the mechanical resonator.
+the optical (mechanical) energy decay rate, and Ω the
+mechanical resonance frequency.
+In this scenario, the
+amplitude quadrature of the input optical field Xin is di-
+rectly imprinted on the mechanical motion via radiation
+pressure. The phase quadrature Yin is not, but is encoded
+on the phase quadrature of the output optical field as [1]:
+Yout = −√ηYin + 2
+�
+ηΓCQ +
+�
+1 − ηYv,
+(1)
+where η is the detection efficiency, C
+=
+4g2
+om/Γκ
+is the optomechanical cooperativity with gom being
+the
+coherent-amplitude-boosted
+optomechanical
+cou-
+pling rate, Yv is the vacuum noise introduced due to
+detection loss, Q (P) is the dimensionless mechanical
+position (momentum) operator with [Q, P] = i, and all
+optical quadrature operators are normalised such that
+[X(t), Y (t′)] = iδ(t−t′). We propose to detect the output
+phase quadrature and use continuous feedback to trans-
+fer it to the mechanical resonator, as shown in Fig. 1. We
+note that feed-forward, similar to our feedback, has been
+applied to improve microwave-to-optical state transfer in
+the resolved sideband regime [45]. In contrast, the feed-
+forward functioned in that experiment to suppress cor-
+related noise terms, while both optical quadratures were
+transferred by radiation pressure.
+Our scheme is analogous to feedback cooling [41, 43,
+44, 46–50], with the detected signal applied as a force
+onto the mechanical resonator. Using quantum Langevin
+equations, we find that it is described by the following
+equations of motion:
+˙Q = ΩP − Γ
+2 Q +
+√
+ΓQin,
+(2)
+and
+˙P = − ΩQ − Γ
+2 P +
+√
+ΓPin − 2
+√
+ΓCXin
+(3)
+− ΓG
+2 f(t) ⊛
+�
+−
+�
+Yin −
+�1 − η
+η
+Yv
+�
+1
+2
+√
+ΓC
++ Q
+�
+,
+where Pin and Qin are white thermal noise operators that
+satisfy [Qin(t), Pin(t′)] = iδ(t−t′), and we have made the
+rotating wave approximation (RWA) with respect to the
+mechanical bath [1, 51]. The last term of Eq. (3) repre-
+sents the feedback force, where the measured photocur-
+rent is convolved with an arbitrary causal filter function
+f(t) ∈ R and amplified by the gain factor G. The fil-
+ter function is normalised so that |f(Ω)| = 1, where
+f(ω) =
+� ∞
+−∞ f (t) eiωtdt is the Fourier transform of f(t).
+The steady-state solutions to Eqs (2) and (3) are found
+by moving into frequency space and adiabatically elimi-
+nating the dynamics of the optical cavity field (Supple-
+mentary Material, Section I [56]).
+This results in the
+quadratures
+Q (ω) =
+√
+Γχ(ω)
+�
+Qin + φ(ω)Pin − 2
+√
+Cφ(ω)Xin + Gf(ω)φ(ω)
+4
+√
+C
+�
+Yin −
+�1 − η
+η
+Yv
+� �
+,
+(4)
+P (ω) =
+√
+Γχ(ω)
+�
+Pin −
+�Gf(ω)Γ
+2Ω
++ 1
+�
+φ(ω)Qin − 2
+√
+CXin + Gf(ω)
+4
+√
+C
+�
+Yin −
+�1 − η
+η
+Yv
+� �
+,
+(5)
+where
+φ(ω) =
+Ω
+Γ/2 − iω ,
+(6)
+the feedback-broadened mechanical susceptibility is
+χ(ω) =
+1
+Ωφ(ω)−1 + (Ω + GΓ f(ω)
+2 )φ(ω)
+,
+(7)
+and the adiabatic elimination is valid in the unresolved
+sideband regime ({Ω, CΓ} ≪ κ) taken throughout this
+paper. From Eq. (7), we see that the mechanical suscep-
+tibility decreases as G increases. This suppresses most
+of the mechanical terms in Eqs (4) and (5). The only
+term that remains is Qin in P(ω), but this is suppressed
+by the large mechanical quality factor (Ω/Γ ≫ 1). It is
+this combined suppression of all mechanical terms that
+enables optical state transfer with high fidelity.
+The optical input field consists of a continuum of op-
+tical modes. To build insight into which of these modes
+is best transferred to the single mechanical mode, as well
+as the gain and noise of the transfer process, we re-write
+
+3
+Eqs (4) and (5) as:
+Q = gXXtrans + Qnoise,optical + Qnoise,mechanical
+(8)
+P = gY Ytrans + Pnoise,optical + Pnoise,mechanical.
+(9)
+Here, Xtrans and Ytrans are the optical quadratures trans-
+ferred to position and momentum, respectively, and gX
+and gY are the transfer gains.
+Terms labelled with a
+subscript ‘noise’ encompass the residual thermal variance
+remaining after feedback, and any optical terms not aris-
+ing from the temporal mode of interest (i.e., inefficient
+detection, mode mismatch).
+The input optical quadratures transferred to Q and P
+in Eqs. (4) and (5) are not perfectly conjugate observ-
+ables. The difference is embodied in φ, and is a result
+of the retarded response of the mechanical position to an
+applied force. The imperfection introduces an ambigu-
+ity in the optical mode that is optimally transferred –
+a different mode is best transferred to P and Q. Here,
+we choose to assess the transfer of the mode that is op-
+timally transferred to P. This mode is described by the
+annihilation operator
+atrans(ω) = u(ω)ain(ω)
+(10)
+and spectral modeshape
+u(ω) = 2
+√
+ΓC
+gY
+χ(ω)
+�Gf(ω)
+8C
+− i
+�
+,
+(11)
+where ain(ω) = (Xin(ω) + iYin(ω))/
+√
+2.
+Using the
+relations Xtrans = (a†
+trans + atrans)/
+√
+2 and Ytrans =
+i(a†
+trans − atrans)/
+√
+2, its amplitude and phase quadra-
+tures are found to be
+Xtrans = 2
+√
+ΓC
+gY
+χ(ω)
+�Gf(ω)
+8C
+Xin + Yin
+�
+(12)
+Ytrans = 2
+√
+ΓC
+gY
+χ(ω)
+�
+−Xin + Gf(ω)
+8C
+Yin
+�
+.
+(13)
+Comparison of Eq. (13) with Eq. (5) confirms that Ytrans
+is reproduced exactly in P(ω), scaled by the momentum
+gain gY .
+The phase quadrature transfer gain, gY , can be de-
+termined by enforcing the boson commutation relation
+[atrans(t), a†
+trans(t)] = 1 on Eq. (10); while that for
+the amplitude quadrature, gX, can be found by requir-
+ing that the optical noise on position commutes with
+both Xtrans and Ytrans, i.e., [Qnoise,optical(t), Xtrans(t)] =
+[Qnoise,optical(t), Ytrans(t)] = 0, where Qnoise,optical is ob-
+tained by rearranging Eq. (8). Together, these give
+gY =
+�4ΓC
+2π
+� ∞
+−∞
+|χ(ω)|2 �
+|f(ω)|2 + 1
+�
+dω
+�1/2
+(14)
+gX = − 1
+gY
+8ΓC
+2π
+� ∞
+−∞
+|χ(ω)|2ℑ(φ(ω))ℑ(f(ω))dω.
+(15)
+The spectral modeshape and quadratures of the trans-
+ferred mode depend on both the feedback-broadened me-
+chanical susceptibility χ(ω) and the feedback filter func-
+tion f(ω), so that the transferred state can be controlled
+through appropriate choice of the filter properties. Thus
+far our results are valid for an arbitrary real-valued causal
+filter function. In the remainder of the paper we choose
+the generalized-Lorentzian filter
+f(ω) =
+Γ′Ω
+ω2 − Ω2 + iΓ′ω ,
+(16)
+where Γ′ is the filter bandwidth. This filter is commonly
+used for feedback cooling [41, 48, 50, 52] and is close
+to the known optimal filter for both momentum estima-
+tion [53] and feedback cooling [54]. Γ′ is chosen to be
+much larger than Ω, so that the filter acts as an integra-
+tor near the mechanical resonance frequency. The gain
+factor G can then be understood as the fractional increase
+in the mechanical decay rate due to the feedback.
+With the filter in Eq. (16) and in the limit of large filter
+bandwidth and mechanical quality factor (Ω/Γ ≫ 1), the
+amplitude and phase quadrature transfer gains can be
+approximated as
+gY = 2
+�
+�
+�
+�C
+�
+1 +
+G2
+64C2
+2 + G
+�
+and gX =
+1
+gY (1 + 2/G).
+(17)
+We define the overall gain of the transfer process as
+√gXgY , so that it is independent of unitary squeezing
+operations on the transferred state [55], and define the
+level of squeezing applied during the transfer as gX/gY .
+The overall gain and squeezing level are plotted as a
+function of the feedback gain factor G in Fig. 2 using
+both numerical calculations and the analytic approxima-
+tions of Eqs (17). For these plots and throughout the
+paper we use the system parameters Ω/2π = 1 MHz,
+Γ/2π = 1 Hz, Γ′/2π = 1.59 MHz, κ/2π = 100 MHz, and
+gom/2π = 395 kHz, T = 30 mK which have been achieved
+in a range of optomechanical experiments [33, 43, 44].
+The overall transfer gain approaches unity for G ≫ 1,
+and the transfer generally involves amplitude quadrature
+squeezing (gX/gY < 1).
+Only at G = 8C do we find
+that the input state is transferred without any squeezing
+(gX/gY = 1). Comparison of Eq. (12) with Eq. (4) shows
+that, in the high quality limit for which f(ω) can be sub-
+stituted with f(±Ω) = ∓i, this choice of gain also results
+in near-agreement between Xtrans and the optical input
+terms in Q. The remaining discrepancy arises from the
+retardation factor φ(ω), and this discrepancy approaches
+zero in the high-quality-factor limit. We therefore select
+G = 8C for the remainder of the paper.
+It is illustrative to consider how our choice of filter
+function and gain factor influences the spectral mode-
+shape u(ω). The frequency dependence of the prefactor
+in Eq. (11) depends only on χ(ω), and is sharply peaked
+at both ±Ω. However, since f(±Ω) = ∓i, for G = 8C the
+term in parentheses is precisely zero at −Ω and equals
+
+4
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+101
+G/C
+0
+0.50
+1.0
+√gXgY
+1
+0.5
+0
+gX/gY
+FIG.
+2.
+Transfer
+gain
+(√gXgY , red)
+and
+squeezing
+(gX/gY , blue) as a function of the feedback strength by co-
+operativity (G/C). The dashed line indicates G = 1 and the
+full grey line indicates the optimal gain (G = 8C), where
+gX/gY = 1. The dots are numerically obtained, and the lines
+are using the analytic expressions derived in the high quality
+factor limit.
+−2i at Ω.
+Our particular choice, therefore, enables a
+single-sideband state transfer, transferring only the lower
+optical sideband and doing this with a modeshape given
+approximately by χ(ω) (see also Supplementary Mate-
+rial, Section II [56]).
+To quantitatively assess the quality of transfer we
+first consider an input vacuum state. We calculate the
+contributions to the position and momentum variances
+from this input and from the noise sources specified in
+Eqs (8) and (9) (see Supplementary Material, Sections
+II & III [56]). We separate the optical noise into con-
+tributions arising from inefficiences and mode mismatch,
+so that the non-ideality of the transfer that arises due
+to φ(ω) can be assessed.
+The results are plotted in
+Fig. 3 (a) as a function of C/nth (with G = 8C). The
+variance of the transferred optical mode increases with C,
+asymptoting to the vacuum variance of 1/2 once C ≫ 1.
+Conversely, the mechanical noise contribution decreases,
+dropping below the vacuum level for C ≫ nth. The vari-
+ance of the optical inefficiency noise has a cooperativity
+dependence that is similar to the optical signal, increas-
+ing with C and asymptoting to a constant value once
+C ≫ 1. As expected, this noise increases as the detec-
+tion efficiency degrades. However, even for η as low as
+0.5 the transferred signal variance still dominates ineffi-
+ciency noise for the whole range of C/nth. The mode-
+mismatch noise on Q is very low for small C, increases
+approximately linearly with C, and eventually exceeds
+the signal variance. Thus, the mode-mismatch ultimately
+constrains the performance of the state transfer.
+Using the analytic expressions for the gains in Eqs (17),
+we derive analytic expressions for the different variance
+contributions that are valid in the same high-quality,
+high-bandwidth limit (see Supplementary Material [56],
+Section III). With the exception of the mismatch noise,
+which is zero in the limit of high quality, these expres-
+sions agree well with the numerical results in Fig. 3 (a).
+From them, we find that when C ≫ 1 the noise variance
+introduced by optical inefficiency is Vη = (1−η)/4η, and
+that the mechanical noise variance is suppressed below
+the vacuum noise level once C > ¯nth/2.
+10−3
+10−2
+10−1
+100
+101
+C/nth
+10−2
+100
+102
+Variance
+(a)
+0
+0.5
+1
+F
+(b)
+Coherent→
+(c)
+↑→
+P
+Q
+Cat→
+Fock→
+0.0 0.2 0.4 0.6 0.8 1.0
+η
+0.0
+0.5
+1.0
+F
+−0.30
+−0.15
+0.00
+0.15
+0.30
+Wtransferred
+FIG. 3.
+(a) Contributions to the variance as a function of
+interaction strength of mechanical noise (blue), optical signal
+(yellow), and two contributions of optical noise: mode mis-
+match on Q (black), and inefficiency (red). The size of the
+markers correspond to the inefficiency (η = 0.9, 0.75, 0.5) for
+decreasing size respectively. (b) The transfer fidelity (F) as a
+function of interaction strength for a coherent state (black),
+cat state (green) (α = 2) and single photon Fock state (dark
+blue). Inset shows F as a function of η for the coherent state,
+at a fixed value of C/nth = 10. (c) Corresponding plots of the
+Wigner distributions for a coherent state (top row), cat state
+(middle row) and Fock state (bottom row) at the interaction
+strengths indicated by the grey lines connected to subplot (b).
+The black dotted circle in the top right indicates the length
+scale of the contour of the ground state. The orientation of
+the plots is indicated by the black arrows in the top right plot.
+Since the feedback process is linear and all noise
+sources are Gaussian, it is straightforward to extend our
+analysis beyond the transfer of vacuum states, to more
+elaborate states such as Schr¨odinger cat states. This can
+be achieved using Wigner functions (Supplementary Ma-
+terial, Section IV [56]). Imperfections introduced by the
+thermal noise, mode mismatch, and inefficiency tend to
+‘smear out’ quantum features of the transferred optical
+mode’s Wigner function. Mathematically, this is repre-
+sented by convolving the signal’s Wigner function with a
+
+5
+Gaussian noise kernel G(r) (with r = (Q P)T ) [57]:
+Wtransferred(r) = (W ⊛ G) (r).
+(18)
+In the regime relevant to this paper, G is typically close to
+symmetric, with a slight wider spread in the Q direction
+due to mode mismatch. The transfer fidelity can then be
+determined for any pure input state as
+F = 2π
+� ∞
+−∞
+� ∞
+−∞
+W(r)Wtransferred(r)d2r.
+(19)
+We plot the fidelity for input coherent, Fock, and cat
+states in Fig. 3 (b) as a function of C/nth and assum-
+ing that η = 1. The coherent state fidelity exceeds the
+classical limit of 1/2 at C/nth = 0.25 and the no-cloning
+bound of 2/3 at C/nth = 0.50. The fidelity for the non-
+Gaussian states also reach fidelities greater than 0.5 at
+similar, experimentally accessible [33, 43, 58] coopera-
+tivities.
+For the chosen experimental parameters, the
+maximum achievable fidelities are 0.98, 0.93, and 0.82
+for coherent, Fock, and cat states, respectively, and are
+limited by the mode-mismatch noise. The fidelity is ro-
+bust against measurement inefficiencies as visible in the
+inset of Fig. 3 (b), which shows that the coherent state
+fidelity can exceed 1/2 even with a detection efficiency as
+low as η = 0.2. Fig. 3 (c) plots the Wigner distributions
+of transferred coherent, Fock, and cat states at three dif-
+ferent values of C/nth, showing that the negativity of
+the Fock and cat states can be transferred, and therefore
+non-classical properties of the input state preserved.
+In conclusion, we have identified that feedback can be
+used to achieve continuous optical-to-mechanical state
+transfer in the unresolved sideband regime. We predict
+that state transfer can be achieved with high fidelity and
+whilst preserving non-classical features such as Wigner
+negativity.
+The ability to implement continuous state
+transfer in the unresolved sideband regime significantly
+widens the class of optomechanical systems that can be
+used as interfaces in quantum networks.
+ACKNOWLEDGEMENTS
+The authors thank Mr S. Khademi and Dr C. Meng
+for useful discussions. This research was primarily sup-
+ported by the Australian Research Council Centre of
+Excellence for Engineered Quantum Systems (EQUS,
+CE170100009). Support was also provided by the by the
+Air Force Office of Scientific Research under award num-
+ber FA9550-20-1-0391.
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+
+1
+Supplemental Material: Continuous optical-to-mechanical quantum state transfer in
+the unresolved sideband regime
+I.
+THEORETICAL MODEL
+The Quantum Langevin equation used to derive the equation of motion for an arbitrary quadrature (O) is [S1]:
+dO
+dt = 1
+iℏ [O, H] −
+�
+O, a†� �γ
+2 a − √γain(t)
+�
++
+�γ
+2 a† − √γa†
+in(t)
+�
+[O, a] ,
+(S1)
+with γ the linewidth and a† (a) the creation (annihilation) operator associated with the operator O.
+The full set of equations of motion (EOM) in frequency space, after including adiabatic elimination on the optical
+quadratures, is:
+−iωP = −ΩQ − Γ
+2 P +
+√
+ΓPin − 2αgom,0X
+−iωQ = ΩP − Γ
+2 Q +
+√
+ΓQin
+0 = −κ
+2 X + √κXin
+0 = −κ
+2 Y − √κYin − 2αgom,0Q,
+(S2)
+where Γ (κ) is the mechanical (optical) linewidth, and gom,0 the single-photon optomechanical interaction strength,
+which is boosted by α from the intracavity photon amplitude, P(in) and Q(in) are the (intracavity) momentum and
+position quadratures of the mechanical oscillator, respectively, Ω is the mechanical resonance frequency and X(in) and
+Y(in) are the (intracavity) amplitude and phase quadratures of the optical cavity respectively.
+Using input-output relations (Oout = Oin − √γO), we can determine an estimate for the mechanical position
+quadrature, by rescaling the detected photocurrent:
+Yout = −√ηYin + 2
+�
+ηΓCQ +
+�
+1 − ηYv,
+(S3)
+such that
+Qest = Q −
+√κ
+4αgom,0
+�
+Yin −
+�1 − η
+η
+Yv
+�
+,
+(S4)
+which includes the added noise (Yv) from detection inefficiencies (η). After convolution with an arbitrary causal
+real-valued filter function (f(t) ⊛ Qest) this is equivalent to an estimate of the momentum quadrature (Pest). To
+model applying a feedback force we subtract G Γ
+2 Pest from the equation of motion for P, where G is the strength of
+the feedback force.
+The steady state solutions for the mechanical quadratures are then as included in the main text (reproduced here
+for ease of reading):
+Q (ω) =
+√
+Γχ(ω)
+�
+Qin + φ(ω)Pin − 2
+√
+Cφ(ω)Xin + Gf(ω)φ(ω)
+4
+√
+C
+�
+Yin −
+�1 − η
+η
+Yv
+� �
+,
+(S5)
+and
+P (ω) =
+√
+Γχ(ω)
+�
+Pin −
+�Gf(ω)Γ
+2Ω
++ 1
+�
+φ(ω)Qin − 2
+√
+CXin + Gf(ω)
+4
+√
+C
+�
+Yin −
+�1 − η
+η
+Yv
+� �
+,
+(S6)
+with
+φ(ω) =
+Ω
+Γ/2 − iω ,
+(S7)
+
+2
+and
+χ(ω) =
+1
+Ωφ(ω)−1 + (Ω + GΓ f(ω)
+2 )φ(ω)
+.
+(S8)
+Following the main text, we re-write Eqs. (S5) and (S6) in the form:
+Q = gXXtrans + Qnoise, optical + Qnoise, mechanical
+(S9)
+P = gY Ytrans + Pnoise, optical + Pnoise, mechanical.
+(S10)
+Here, Xtrans and Ytrans are the quadratures of the transferred optical mode that are imprinted on the position and
+momentum, respectively, and gX and gY are the transfer gains allowing for possible differences in gain between
+position and momentum. Terms labelled with a subscript ‘noise’ encompass the residual mechanical noise remaining
+after feedback, and the optical noise imprinted on the mechanical oscillator by both feedback and radiation pressure.
+Following the main text, we can derive the susceptibility of the transferred mode, by using the relations Xtrans =
+(a†
+trans + atrans)/
+√
+2 and Ytrans = i(a†
+trans − atrans)/
+√
+2 to find:
+u(ω) = 2
+√
+ΓC
+gY
+χ(ω)
+�Gf(ω)
+8C
+− i
+�
+(S11)
+or in terms of Xtrans and Ytrans:
+Xtrans = 2
+√
+ΓC
+gY
+χ(ω)
+�Gf(ω)
+8C
+Xin + Yin
+�
+(S12)
+Ytrans = −2
+√
+ΓC
+gY
+χ(ω)
+�
+Xin − Gf(ω)
+8C
+Yin
+�
+.
+(S13)
+The gain parameters are found through the method described in the main text:
+gY =
+�4ΓC
+2π
+� ∞
+−∞
+|χ(ω)|2 �
+|f(ω)|2 + 1
+�
+dω
+�1/2
+(S14)
+gX = − 1
+gY
+8ΓC
+2π
+� ∞
+−∞
+|χ(ω)|2ℑ(φ(ω))ℑ(f(ω))dω.
+(S15)
+II.
+MECHANICAL POWER SPECTRAL DENSITY
+We construct the symmetrised power spectral densities for all separate contributions listed in Eqs. (S9) and (S10)
+using the following relations:
+¯SOO = SOO(ω) + SOO(−ω)
+2
+,
+(S16)
+and
+⟨Qin (t)Qin (t′)⟩ = ⟨Pin (t) Pin (t′)⟩ = (¯nth + 1/2) δ (t − t′) ,
+(S17)
+where ¯nth ≈ kBT/ℏΩ is the mean thermal occupancy of the mechanical resonator and T is its temperature. Due to
+the typically high frequency of the optical field we approximate the optical field to have no thermal occupancy, with
+⟨Xin (t)Xin (t′)⟩ = ⟨Yin (t) Yin (t′)⟩ = δ (t − t′) /2.
+(S18)
+
+3
+10−12
+10−10
+10−8
+10−6
+10−4
+PSD (Hz−1)
+(a)
+C/nth = 0.01
+10−1
+100
+101
+ω/Ω
+10−6
+10−10
+10−8
+PSD (Hz−1)
+C/nth = 10.0
+(b)
+10−1
+100
+101
+|ω/Ω|
+10−14
+10−9
+10−4
+|u(ω)|2 (Hz−1)
+10−1
+100
+101
+|ω/Ω|
+10−10
+10−6
+10−8
+|u(ω)|2 (Hz−1)
+FIG. S1. (a) Four separate contributions to the power spectral density (PSD) with C/nth = 0.01 of the mechanical quadratures:
+transferred optical mode (yellow), mechanical noise (blue), optical noise from mode mismatch on Q (black) and optical noise
+from inefficiencies (η) (red). The width of the red lines correspond to η = 0.9, 0.75, and 0.5 (decreasing width respectively).
+The inset shows the spectral mode of the signal for ω < 0 (green) and ω > 0 (purple). (b) Same as in (a) but with a larger
+interaction strength, C/nth = 10.
+Using these, we find:
+SQQ, optical signal =g2
+Y
+1
+2π
+1
+24Γ4g2
+om
+Γκ |χ(ω)|2 �
+|f(ω)|2 + 1
+�
+SQQ, optical noise =1
+24Γ4g2
+om
+Γκ |χ(ω)|2
+� �1 − η
+η
+�
+|f(ω)φ(ω)|2 + |φ(ω)f(ω) − gXgY |2 + |−φ(ω) − gY gXf(ω)|2
+�
+SQQ, mechanical noise =Γ|χ(ω)|2 �
+1 + |φ(ω)|2�
+(¯nth + 1
+2)
+SPP, optical signal =g2
+Y
+1
+24Γ4g2
+om
+Γκ |χ(ω)|2 �
+|f(ω)|2 + 1
+�
+SPP, optical noise =1
+24Γ4g2
+om
+Γκ |χ(ω)|2
+�1 − η
+η
+� �
+|f(ω)|2 + 1
+�
+SPP, mechanical noise =Γ|χ(ω)|2
+�
+1 +
+����4φ(ω)f(ω) 1
+Ω
+4g2
+om
+κΓ + 1
+����
+2� �
+¯nth + 1
+2
+�
+SPQ =4g2
+om
+πκ |χ(ω)|2
+�
+1 + |f(ω)|2
+4
+�
+1 −
+�1 − η
+η
+�2�
+ℜ(φ(ω))+
+Γ
+2π |χ(ω)|2 8g2
+om
+κΩ2 (ℑ (f(ω))ℑ (φ(ω)) − ℜ(f(ω))ℜ(φ(ω)))
+�
+¯nth + 1
+2
+�
+,
+(S19)
+Each contribution is plotted for two different optomechanical cooperativities in Fig. S1 (a) and (b), including
+inefficiency noise for three different detection efficiencies. Apart from the additional mode mismatch noise on Q, the
+contributions are near identical for Q and P in the limit that C ≪ Ω/Γ. We therefore only plot them for Q.
+Apart from the mode-mismatch noise, all contributions to the power spectral density are peaked at the mechanical
+
+4
+resonance frequency and have a roughly Lorenzian shape. The mismatch-noise, by contrast, is peaked at ω = 0 and
+approximates the shape of a low pass filter. At low cooperativity (Fig. S1 (a)), the noise dominates the optical signal,
+preventing an effective quantum state transfer. At higher cooperativity (Fig. S1 (b)), the optical signal rises above
+the noise contributions, suggesting that quantum state transfer is possible. The inset in both figures plots the spectral
+modeshape of the signal mode (u(ω)). The optical signal peaks at −Ω (blue) and is suppressed relative to this peak
+by several orders of magnitude at Ω (green), evidencing that the chosen filter enables a single-sideband transfer.
+III.
+VARIANCES
+We obtain numerical values for the variances of each component of the power spectral density by integrating over
+frequency. To obtain the variance plot of the main text we sweep the interaction strength gom, while keeping the
+other system parameters constant.
+We also determine analytical approximations in the limit of a large bandwidth filter and high mechanical quality
+factor (Q = Ω
+Γ ≫ 1). In that limit, and with our chosen filter, the expressions for gY and gX can be approximated as:
+gY =
+�
+�
+�
+�2C
+�
+1 +
+G2
+64C2
+1 + G/2
+�
+,
+(S20)
+and
+gX =
+1
+gY(1 + 2/G).
+(S21)
+Using these expressions and approximating the spectral densities in the high quality limit, we calculate analytic
+expressions for the variance of each component through the residue theorem. The results are:
+VXtrans = 1
+2gX
+2,
+(S22)
+VQmech =
+1
+1 + G/2
+�1
+2 + nth
+�
+,
+(S23)
+VQmismatch = 0,
+(S24)
+VQ η = 1
+4
+�1 − η
+η
+�
+gX
+2,
+(S25)
+VQmismatchQsignal = 0,
+(S26)
+and
+VPQ = VQP = 0.
+(S27)
+Eq. S24 only occurs in the variance for Q.
+The analytic variances for P are similar, with the subscript Q (X)
+substituted for P (Y).
+
+5
+IV.
+WIGNER FUNCTIONS
+The Wigner functions used as optical input states are given by the following equations in Wigner space (r = (QP)T ):
+WCat (Q=2,P=0)(r) = exp(−r · r)
+�
+exp(−2α · α) cosh
+�
+2
+√
+2r · α
+�
++ cos
+�
+2
+√
+2r · (ϖα)
+��
+π(exp(−2α · α) + 1)
+WFock (n=1)(r) = e−r·r (2r · r − 1)
+π
+WCoherent, vacuum(r) = e− r·r
+2
+2π ,
+(S28)
+where α = (αr, αi), and we use αr = 2, and αi = 0 for the calculation of the fidelity in the main text, and ϖ =
+((0, 1), (−1, 0)) is a symplectic matrix.
+As stated in the main text, the interactions are Gaussian and we can construct a Wigner function associated with
+the noise in the system:
+Wnoise(r) = 1
+2π
+1
+�
+det (Vnoise)
+exp
+�Q2V11 − 2QPV12 + P 2V22
+2V 2
+12 − 2V11V22
+�
+,
+(S29)
+where Vij are the elements of the correlation matrix that only contains all the noise contributions. Specifically:
+Vnoise =
+�
+VQQ, optical noise + VQQ, mechanical noise
+VQnoise Pnoise
+VPnoise Qnoise
+VPP, optical noise + VPP, mechanical noise
+�
+.
+(S30)
+The transferred Wigner functions Wtransferred are found by:
+Wtransferred(r′) = (Wtarget ⊛ Wnoise) (r),
+(S31)
+which we use directly for the calculation of the transfer fidelity:
+F = 2π
+� ∞
+−∞
+Wtarget(r)Wtransferred(r)dr.
+(S32)
+REFERENCES
+[S1] W. P. Bowen and G. J. Milburn, Quantum optomechanics (CRC Press, Boca Raton London New York, 2015), ISBN
+978-0-367-57519-9.
+
diff --git a/XdE2T4oBgHgl3EQfYgeB/content/tmp_files/load_file.txt b/XdE2T4oBgHgl3EQfYgeB/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d894db62a9770c865948d2a558c17b6185ba99b9
--- /dev/null
+++ b/XdE2T4oBgHgl3EQfYgeB/content/tmp_files/load_file.txt
@@ -0,0 +1,962 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf,len=961
+page_content='Continuous optical-to-mechanical quantum state transfer in the unresolved sideband  regime  Amy Navarathna1,2, James S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Bennett1,2,3, and Warwick P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Bowen1,2∗  1 ARC Centre of Excellence for Engineered Quantum Systems, St Lucia, Queensland 4072, Australia  2 School of Mathematics and Physics, University of Queensland, St Lucia, Queensland 4072, Australia  3 Centre for Quantum Dynamics, Griffith University, Nathan, QLD 4222, Australia  (Dated: January 11, 2023)  Optical-to-mechanical quantum state transfer is an important capability for future quantum net-  works, quantum communication, and distributed quantum sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' However, existing continuous  state transfer protocols operate in the resolved sideband regime, necessitating a high-quality optical  cavity and a high mechanical resonance frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Here, we propose a continuous protocol that  operates in the unresolved sideband regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The protocol is based on feedback cooling, can be im-  plemented with current technology, and is able to transfer non-Gaussian quantum states with high  fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Our protocol significantly expands the kinds of optomechanical devices for which continu-  ous optical-to-mechanical state transfer is possible, paving the way towards quantum technological  applications and the preparation of macroscopic superpositions to test the fundamentals of quantum  science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  The ability to transfer quantum states between op-  tical communication channels and quantum computing  nodes is a necessary ingredient of the emerging quan-  tum internet [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Quantum state transfer also has im-  portant applications in quantum-enhanced sensing [2, 3],  quantum-secure communications [4], and fundamental  tests of macroscopic quantum mechanics [5–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' A lead-  ing approach is to mediate the transfer using an optome-  chanical resonator [11–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' This is attractive because me-  chanical resonators interact via radiation pressure with  electromagnetic fields of all frequencies [1] and can also  be functionalized to interact with most quantum com-  puting nodes, such as spins [18–20], superconducting de-  vices [21–23] and atomic ensembles [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  The first step in the transfer process is an optical-  to-mechanical state transfer, with a subsequent transfer  to the final computing node [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  An optical cav-  ity is employed to enhance the radiation pressure during  the optical-to-mechanical state transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  Leading pro-  posals work only in the resolved sideband regime, where  the decay rate of this cavity is lower than the mechani-  cal resonance frequency [12, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' By contrast, most op-  tomechanical systems operate in the unresolved sideband  regime [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  In many cases this is due to the benefits  that low mechanical frequencies convey for applications,  for instance in precision sensing [30–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  In others, it  is because of the difficulty of simultaneously achieving a  low decay rate, a high resonance frequency, and sufficient  radiation pressure coupling [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  To date, the only proposals for optical-to-mechanical  state transfer in the unresolved sideband regime have  used pulsed, rather than continuous, optomechanical in-  teractions [34–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  This narrows the range of applica-  tions, introduces significant technical challenges due to  the additional timing and phase accuracy required [36–  ∗ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='bowen@uq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='au  38], and involves large radiation pressure impulse forces  that can be problematic [35, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  It is well known that a mechanical resonator can be  feedback cooled close to its motional ground state in  the unresolved sideband regime [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Here we propose  a continuous optical-to-mechanical state transfer proto-  col based on the same concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' By modelling the open  quantum system dynamics, we show that feedback cool-  ing can be understood as the transfer of a vacuum state  of light onto the mechanical resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' We find that ap-  propriate choice of the feedback parameters allows the  transfer of arbitrary quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The requirements  for successful transfer closely match those for ground-  state cooling – once the optomechanical cooperativity ex-  ceeds the thermal occupancy of the mechanical resonator,  a coherent state can be transferred with near unity fi-  delity and the Wigner-negativity of non-Gaussian states  can be preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  Moreover, the feedback parameters  can be used to phase-sensitively amplify (or squeeze) the  transferred state, to engineer its temporal profile, and –  in direct analogy to state-transfer via resolved sideband  cooling [42] – to achieve the transfer of a single optical  sideband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  Our work extends continuous optomechanical state  transfer beyond the resolved sideband limit, to low-  quality optical cavities and low frequency mechanical res-  onators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Feedback cooling of a mechanical resonator to  near its motional ground state has recently been demon-  strated, both in cryogenic [43] and room temperature en-  vironments [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' As such, our proposal can be directly  implemented with existing technology, providing a new  tool for quantum networks and opening a new pathway  to create and study macroscopic quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Our  work also provides new insights into feedback cooling,  showing that the process is in fact a quantum state trans-  fer from light to mechanical motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  We consider an optomechanical system in the unre-  solved sideband, high mechanical quality regime (κ ≫  Ω ≫ Γ) with resonant optical driving, where κ (Γ) is  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='03855v1  [quant-ph]  10 Jan 2023  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  Schematic optomechanical system with feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  Light is coupled into an optomechanical cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The reflected  light is measured through homodyne detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The detected  photocurrent (Yout(t)) is convolved with a filter f(t) and di-  rectly fed back to the momentum of the mechanical resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  the optical (mechanical) energy decay rate, and Ω the  mechanical resonance frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  In this scenario, the  amplitude quadrature of the input optical field Xin is di-  rectly imprinted on the mechanical motion via radiation  pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The phase quadrature Yin is not,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' but is encoded  on the phase quadrature of the output optical field as [1]:  Yout = −√ηYin + 2  �  ηΓCQ +  �  1 − ηYv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (1)  where η is the detection efficiency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' C  =  4g2  om/Γκ  is the optomechanical cooperativity with gom being  the  coherent-amplitude-boosted  optomechanical  cou-  pling rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Yv is the vacuum noise introduced due to  detection loss,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Q (P) is the dimensionless mechanical  position (momentum) operator with [Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' P] = i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' and all  optical quadrature operators are normalised such that  [X(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Y (t′)] = iδ(t−t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' We propose to detect the output  phase quadrature and use continuous feedback to trans-  fer it to the mechanical resonator, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' We  note that feed-forward, similar to our feedback, has been  applied to improve microwave-to-optical state transfer in  the resolved sideband regime [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' In contrast, the feed-  forward functioned in that experiment to suppress cor-  related noise terms, while both optical quadratures were  transferred by radiation pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  Our scheme is analogous to feedback cooling [41, 43,  44, 46–50], with the detected signal applied as a force  onto the mechanical resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Using quantum Langevin  equations, we find that it is described by the following  equations of motion:  ˙Q = ΩP − Γ  2 Q +  √  ΓQin,  (2)  and  ˙P = − ΩQ − Γ  2 P +  √  ΓPin − 2  √  ΓCXin  (3)  − ΓG  2 f(t) ⊛  �  −  �  Yin −  �1 − η  η  Yv  �  1  2  √  ΓC  + Q  �  ,  where Pin and Qin are white thermal noise operators that  satisfy [Qin(t), Pin(t′)] = iδ(t−t′), and we have made the  rotating wave approximation (RWA) with respect to the  mechanical bath [1, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The last term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' (3) repre-  sents the feedback force, where the measured photocur-  rent is convolved with an arbitrary causal filter function  f(t) ∈ R and amplified by the gain factor G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The fil-  ter function is normalised so that |f(Ω)| = 1, where  f(ω) =  � ∞  −∞ f (t) eiωtdt is the Fourier transform of f(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  The steady-state solutions to Eqs (2) and (3) are found  by moving into frequency space and adiabatically elimi-  nating the dynamics of the optical cavity field (Supple-  mentary Material, Section I [56]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  This results in the  quadratures  Q (ω) =  √  Γχ(ω)  �  Qin + φ(ω)Pin − 2  √  Cφ(ω)Xin + Gf(ω)φ(ω)  4  √  C  �  Yin −  �1 − η  η  Yv  � �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (4)  P (ω) =  √  Γχ(ω)  �  Pin −  �Gf(ω)Γ  2Ω  + 1  �  φ(ω)Qin − 2  √  CXin + Gf(ω)  4  √  C  �  Yin −  �1 − η  η  Yv  � �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (5)  where  φ(ω) =  Ω  Γ/2 − iω ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (6)  the feedback-broadened mechanical susceptibility is  χ(ω) =  1  Ωφ(ω)−1 + (Ω + GΓ f(ω)  2 )φ(ω)  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (7)  and the adiabatic elimination is valid in the unresolved  sideband regime ({Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' CΓ} ≪ κ) taken throughout this  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' (7), we see that the mechanical suscep-  tibility decreases as G increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' This suppresses most  of the mechanical terms in Eqs (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The only  term that remains is Qin in P(ω), but this is suppressed  by the large mechanical quality factor (Ω/Γ ≫ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' It is  this combined suppression of all mechanical terms that  enables optical state transfer with high fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  The optical input field consists of a continuum of op-  tical modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' To build insight into which of these modes  is best transferred to the single mechanical mode, as well  as the gain and noise of the transfer process, we re-write  3  Eqs (4) and (5) as:  Q = gXXtrans + Qnoise,optical + Qnoise,mechanical  (8)  P = gY Ytrans + Pnoise,optical + Pnoise,mechanical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (9)  Here, Xtrans and Ytrans are the optical quadratures trans-  ferred to position and momentum, respectively, and gX  and gY are the transfer gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  Terms labelled with a  subscript ‘noise’ encompass the residual thermal variance  remaining after feedback, and any optical terms not aris-  ing from the temporal mode of interest (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=', inefficient  detection, mode mismatch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  The input optical quadratures transferred to Q and P  in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' (4) and (5) are not perfectly conjugate observ-  ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The difference is embodied in φ, and is a result  of the retarded response of the mechanical position to an  applied force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The imperfection introduces an ambigu-  ity in the optical mode that is optimally transferred –  a different mode is best transferred to P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Here,  we choose to assess the transfer of the mode that is op-  timally transferred to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' This mode is described by the  annihilation operator  atrans(ω) = u(ω)ain(ω)  (10)  and spectral modeshape  u(ω) = 2  √  ΓC  gY  χ(ω)  �Gf(ω)  8C  − i  �  ,  (11)  where ain(ω) = (Xin(ω) + iYin(ω))/  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  Using the  relations Xtrans = (a†  trans + atrans)/  √  2 and Ytrans =  i(a†  trans − atrans)/  √  2, its amplitude and phase quadra-  tures are found to be  Xtrans = 2  √  ΓC  gY  χ(ω)  �Gf(ω)  8C  Xin + Yin  �  (12)  Ytrans = 2  √  ΓC  gY  χ(ω)  �  −Xin + Gf(ω)  8C  Yin  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (13)  Comparison of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' (13) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' (5) confirms that Ytrans  is reproduced exactly in P(ω), scaled by the momentum  gain gY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  The phase quadrature transfer gain, gY , can be de-  termined by enforcing the boson commutation relation  [atrans(t), a†  trans(t)] = 1 on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' (10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' while that for  the amplitude quadrature, gX, can be found by requir-  ing that the optical noise on position commutes with  both Xtrans and Ytrans, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=', [Qnoise,optical(t), Xtrans(t)] =  [Qnoise,optical(t), Ytrans(t)] = 0, where Qnoise,optical is ob-  tained by rearranging Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Together, these give  gY =  �4ΓC  2π  � ∞  −∞  |χ(ω)|2 �  |f(ω)|2 + 1  �  dω  �1/2  (14)  gX = − 1  gY  8ΓC  2π  � ∞  −∞  |χ(ω)|2ℑ(φ(ω))ℑ(f(ω))dω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (15)  The spectral modeshape and quadratures of the trans-  ferred mode depend on both the feedback-broadened me-  chanical susceptibility χ(ω) and the feedback filter func-  tion f(ω), so that the transferred state can be controlled  through appropriate choice of the filter properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Thus  far our results are valid for an arbitrary real-valued causal  filter function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' In the remainder of the paper we choose  the generalized-Lorentzian filter  f(ω) =  Γ′Ω  ω2 − Ω2 + iΓ′ω ,  (16)  where Γ′ is the filter bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' This filter is commonly  used for feedback cooling [41, 48, 50, 52] and is close  to the known optimal filter for both momentum estima-  tion [53] and feedback cooling [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Γ′ is chosen to be  much larger than Ω, so that the filter acts as an integra-  tor near the mechanical resonance frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The gain  factor G can then be understood as the fractional increase  in the mechanical decay rate due to the feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  With the filter in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' (16) and in the limit of large filter  bandwidth and mechanical quality factor (Ω/Γ ≫ 1), the  amplitude and phase quadrature transfer gains can be  approximated as  gY = 2  �  �  �  �C  �  1 +  G2  64C2  2 + G  �  and gX =  1  gY (1 + 2/G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (17)  We define the overall gain of the transfer process as  √gXgY , so that it is independent of unitary squeezing  operations on the transferred state [55], and define the  level of squeezing applied during the transfer as gX/gY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  The overall gain and squeezing level are plotted as a  function of the feedback gain factor G in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' 2 using  both numerical calculations and the analytic approxima-  tions of Eqs (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' For these plots and throughout the  paper we use the system parameters Ω/2π = 1 MHz,  Γ/2π = 1 Hz, Γ′/2π = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='59 MHz, κ/2π = 100 MHz, and  gom/2π = 395 kHz, T = 30 mK which have been achieved  in a range of optomechanical experiments [33, 43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  The overall transfer gain approaches unity for G ≫ 1,  and the transfer generally involves amplitude quadrature  squeezing (gX/gY < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  Only at G = 8C do we find  that the input state is transferred without any squeezing  (gX/gY = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Comparison of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' (12) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' (4) shows  that, in the high quality limit for which f(ω) can be sub-  stituted with f(±Ω) = ∓i, this choice of gain also results  in near-agreement between Xtrans and the optical input  terms in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The remaining discrepancy arises from the  retardation factor φ(ω), and this discrepancy approaches  zero in the high-quality-factor limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' We therefore select  G = 8C for the remainder of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  It is illustrative to consider how our choice of filter  function and gain factor influences the spectral mode-  shape u(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The frequency dependence of the prefactor  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' (11) depends only on χ(ω), and is sharply peaked  at both ±Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' However, since f(±Ω) = ∓i, for G = 8C the  term in parentheses is precisely zero at −Ω and equals  4  10−6  10−5  10−4  10−3  10−2  10−1  100  101  G/C  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='0  √gXgY  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='5  0  gX/gY  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  Transfer  gain  (√gXgY , red)  and  squeezing  (gX/gY , blue) as a function of the feedback strength by co-  operativity (G/C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The dashed line indicates G = 1 and the  full grey line indicates the optimal gain (G = 8C), where  gX/gY = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The dots are numerically obtained, and the lines  are using the analytic expressions derived in the high quality  factor limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  −2i at Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  Our particular choice, therefore, enables a  single-sideband state transfer, transferring only the lower  optical sideband and doing this with a modeshape given  approximately by χ(ω) (see also Supplementary Mate-  rial, Section II [56]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  To quantitatively assess the quality of transfer we  first consider an input vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' We calculate the  contributions to the position and momentum variances  from this input and from the noise sources specified in  Eqs (8) and (9) (see Supplementary Material, Sections  II & III [56]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' We separate the optical noise into con-  tributions arising from inefficiences and mode mismatch,  so that the non-ideality of the transfer that arises due  to φ(ω) can be assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  The results are plotted in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' 3 (a) as a function of C/nth (with G = 8C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The  variance of the transferred optical mode increases with C,  asymptoting to the vacuum variance of 1/2 once C ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  Conversely, the mechanical noise contribution decreases,  dropping below the vacuum level for C ≫ nth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The vari-  ance of the optical inefficiency noise has a cooperativity  dependence that is similar to the optical signal, increas-  ing with C and asymptoting to a constant value once  C ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' As expected, this noise increases as the detec-  tion efficiency degrades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' However, even for η as low as  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='5 the transferred signal variance still dominates ineffi-  ciency noise for the whole range of C/nth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The mode-  mismatch noise on Q is very low for small C, increases  approximately linearly with C, and eventually exceeds  the signal variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Thus, the mode-mismatch ultimately  constrains the performance of the state transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  Using the analytic expressions for the gains in Eqs (17),  we derive analytic expressions for the different variance  contributions that are valid in the same high-quality,  high-bandwidth limit (see Supplementary Material [56],  Section III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' With the exception of the mismatch noise,  which is zero in the limit of high quality, these expres-  sions agree well with the numerical results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  From them, we find that when C ≫ 1 the noise variance  introduced by optical inefficiency is Vη = (1−η)/4η, and  that the mechanical noise variance is suppressed below  the vacuum noise level once C > ¯nth/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  10−3  10−2  10−1  100  101  C/nth  10−2  100  102  Variance  (a)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='5  1  F  (b)  Coherent→  (c)  ↑→  P  Q  Cat→  Fock→  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='0  η  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='0  F  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='30  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='30  Wtransferred  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (a) Contributions to the variance as a function of  interaction strength of mechanical noise (blue), optical signal  (yellow), and two contributions of optical noise: mode mis-  match on Q (black), and inefficiency (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The size of the  markers correspond to the inefficiency (η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='5) for  decreasing size respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' (b) The transfer fidelity (F) as a  function of interaction strength for a coherent state (black),  cat state (green) (α = 2) and single photon Fock state (dark  blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Inset shows F as a function of η for the coherent state,  at a fixed value of C/nth = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' (c) Corresponding plots of the  Wigner distributions for a coherent state (top row), cat state  (middle row) and Fock state (bottom row) at the interaction  strengths indicated by the grey lines connected to subplot (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  The black dotted circle in the top right indicates the length  scale of the contour of the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The orientation of  the plots is indicated by the black arrows in the top right plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  Since the feedback process is linear and all noise  sources are Gaussian, it is straightforward to extend our  analysis beyond the transfer of vacuum states, to more  elaborate states such as Schr¨odinger cat states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' This can  be achieved using Wigner functions (Supplementary Ma-  terial, Section IV [56]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Imperfections introduced by the  thermal noise, mode mismatch, and inefficiency tend to  ‘smear out’ quantum features of the transferred optical  mode’s Wigner function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Mathematically, this is repre-  sented by convolving the signal’s Wigner function with a  5  Gaussian noise kernel G(r) (with r = (Q P)T ) [57]:  Wtransferred(r) = (W ⊛ G) (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (18)  In the regime relevant to this paper, G is typically close to  symmetric, with a slight wider spread in the Q direction  due to mode mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The transfer fidelity can then be  determined for any pure input state as  F = 2π  � ∞  −∞  � ∞  −∞  W(r)Wtransferred(r)d2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (19)  We plot the fidelity for input coherent, Fock, and cat  states in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' 3 (b) as a function of C/nth and assum-  ing that η = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The coherent state fidelity exceeds the  classical limit of 1/2 at C/nth = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='25 and the no-cloning  bound of 2/3 at C/nth = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The fidelity for the non-  Gaussian states also reach fidelities greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='5 at  similar, experimentally accessible [33, 43, 58] coopera-  tivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  For the chosen experimental parameters, the  maximum achievable fidelities are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='98, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='93, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='82  for coherent, Fock, and cat states, respectively, and are  limited by the mode-mismatch noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The fidelity is ro-  bust against measurement inefficiencies as visible in the  inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' 3 (b), which shows that the coherent state  fidelity can exceed 1/2 even with a detection efficiency as  low as η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' 3 (c) plots the Wigner distributions  of transferred coherent, Fock, and cat states at three dif-  ferent values of C/nth, showing that the negativity of  the Fock and cat states can be transferred, and therefore  non-classical properties of the input state preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  In conclusion, we have identified that feedback can be  used to achieve continuous optical-to-mechanical state  transfer in the unresolved sideband regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' We predict  that state transfer can be achieved with high fidelity and  whilst preserving non-classical features such as Wigner  negativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  The ability to implement continuous state  transfer in the unresolved sideband regime significantly  widens the class of optomechanical systems that can be  used as interfaces in quantum networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  The authors thank Mr S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Khademi and Dr C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Meng  for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' This research was primarily sup-  ported by the Australian Research Council Centre of  Excellence for Engineered Quantum Systems (EQUS,  CE170100009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Support was also provided by the by the  Air Force Office of Scientific Research under award num-  ber FA9550-20-1-0391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
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+page_content=' Schnabel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Ralph,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Symul, and Ping Koy Lam, IEEE Journal of Se-  lected Topics in Quantum Electronics 9, 1519 (2003),  ISSN 1077-260X, URL http://ieeexplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='org/  document/1263784/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  [56] See Supplemental Material (below) for more details to  reproduce the work, a figure of the contributions to the  power spectral density of the mechanical quadratures, and  analytic expressions for the variance contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
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+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
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+page_content='tandfonline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='com/doi/abs/  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='1080/09500349908231397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  [58] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
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+page_content='  Wilson,  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  Sudhir,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  Piro,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  Schilling,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Ghadimi, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Kippenberg, Nature 524, 325  (2015), ISSN 0028-0836, 1476-4687, URL http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='com/articles/nature14672.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  1  Supplemental Material: Continuous optical-to-mechanical quantum state transfer in  the unresolved sideband regime  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  THEORETICAL MODEL  The Quantum Langevin equation used to derive the equation of motion for an arbitrary quadrature (O) is [S1]:  dO  dt = 1  iℏ [O, H] −  �  O, a†� �γ  2 a − √γain(t)  �  +  �γ  2 a† − √γa†  in(t)  �  [O, a] ,  (S1)  with γ the linewidth and a† (a) the creation (annihilation) operator associated with the operator O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  The full set of equations of motion (EOM) in frequency space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' after including adiabatic elimination on the optical  quadratures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' is:  −iωP = −ΩQ − Γ  2 P +  √  ΓPin − 2αgom,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='0X  −iωQ = ΩP − Γ  2 Q +  √  ΓQin  0 = −κ  2 X + √κXin  0 = −κ  2 Y − √κYin − 2αgom,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='0Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (S2)  where Γ (κ) is the mechanical (optical) linewidth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' and gom,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='0 the single-photon optomechanical interaction strength,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  which is boosted by α from the intracavity photon amplitude,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' P(in) and Q(in) are the (intracavity) momentum and  position quadratures of the mechanical oscillator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Ω is the mechanical resonance frequency and X(in) and  Y(in) are the (intracavity) amplitude and phase quadratures of the optical cavity respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  Using input-output relations (Oout = Oin − √γO), we can determine an estimate for the mechanical position  quadrature, by rescaling the detected photocurrent:  Yout = −√ηYin + 2  �  ηΓCQ +  �  1 − ηYv,  (S3)  such that  Qest = Q −  √κ  4αgom,0  �  Yin −  �1 − η  η  Yv  �  ,  (S4)  which includes the added noise (Yv) from detection inefficiencies (η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' After convolution with an arbitrary causal  real-valued filter function (f(t) ⊛ Qest) this is equivalent to an estimate of the momentum quadrature (Pest).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' To  model applying a feedback force we subtract G Γ  2 Pest from the equation of motion for P, where G is the strength of  the feedback force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  The steady state solutions for the mechanical quadratures are then as included in the main text (reproduced here  for ease of reading):  Q (ω) =  √  Γχ(ω)  �  Qin + φ(ω)Pin − 2  √  Cφ(ω)Xin + Gf(ω)φ(ω)  4  √  C  �  Yin −  �1 − η  η  Yv  � �  ,  (S5)  and  P (ω) =  √  Γχ(ω)  �  Pin −  �Gf(ω)Γ  2Ω  + 1  �  φ(ω)Qin − 2  √  CXin + Gf(ω)  4  √  C  �  Yin −  �1 − η  η  Yv  � �  ,  (S6)  with  φ(ω) =  Ω  Γ/2 − iω ,  (S7)  2  and  χ(ω) =  1  Ωφ(ω)−1 + (Ω + GΓ f(ω)  2 )φ(ω)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (S8)  Following the main text, we re-write Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' (S5) and (S6) in the form:  Q = gXXtrans + Qnoise, optical + Qnoise, mechanical  (S9)  P = gY Ytrans + Pnoise, optical + Pnoise, mechanical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (S10)  Here, Xtrans and Ytrans are the quadratures of the transferred optical mode that are imprinted on the position and  momentum, respectively, and gX and gY are the transfer gains allowing for possible differences in gain between  position and momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Terms labelled with a subscript ‘noise’ encompass the residual mechanical noise remaining  after feedback, and the optical noise imprinted on the mechanical oscillator by both feedback and radiation pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  Following the main text, we can derive the susceptibility of the transferred mode, by using the relations Xtrans =  (a†  trans + atrans)/  √  2 and Ytrans = i(a†  trans − atrans)/  √  2 to find:  u(ω) = 2  √  ΓC  gY  χ(ω)  �Gf(ω)  8C  − i  �  (S11)  or in terms of Xtrans and Ytrans:  Xtrans = 2  √  ΓC  gY  χ(ω)  �Gf(ω)  8C  Xin + Yin  �  (S12)  Ytrans = −2  √  ΓC  gY  χ(ω)  �  Xin − Gf(ω)  8C  Yin  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (S13)  The gain parameters are found through the method described in the main text:  gY =  �4ΓC  2π  � ∞  −∞  |χ(ω)|2 �  |f(ω)|2 + 1  �  dω  �1/2  (S14)  gX = − 1  gY  8ΓC  2π  � ∞  −∞  |χ(ω)|2ℑ(φ(ω))ℑ(f(ω))dω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (S15)  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  MECHANICAL POWER SPECTRAL DENSITY  We construct the symmetrised power spectral densities for all separate contributions listed in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' (S9) and (S10)  using the following relations:  ¯SOO = SOO(ω) + SOO(−ω)  2  ,  (S16)  and  ⟨Qin (t)Qin (t′)⟩ = ⟨Pin (t) Pin (t′)⟩ = (¯nth + 1/2) δ (t − t′) ,  (S17)  where ¯nth ≈ kBT/ℏΩ is the mean thermal occupancy of the mechanical resonator and T is its temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Due to  the typically high frequency of the optical field we approximate the optical field to have no thermal occupancy, with  ⟨Xin (t)Xin (t′)⟩ = ⟨Yin (t) Yin (t′)⟩ = δ (t − t′) /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (S18)  3  10−12  10−10  10−8  10−6  10−4  PSD (Hz−1)  (a)  C/nth = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='01  10−1  100  101  ω/Ω  10−6  10−10  10−8  PSD (Hz−1)  C/nth = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='0  (b)  10−1  100  101  |ω/Ω|  10−14  10−9  10−4  |u(ω)|2 (Hz−1)  10−1  100  101  |ω/Ω|  10−10  10−6  10−8  |u(ω)|2 (Hz−1)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' (a) Four separate contributions to the power spectral density (PSD) with C/nth = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='01 of the mechanical quadratures:  transferred optical mode (yellow), mechanical noise (blue), optical noise from mode mismatch on Q (black) and optical noise  from inefficiencies (η) (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The width of the red lines correspond to η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='75, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='5 (decreasing width respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  The inset shows the spectral mode of the signal for ω < 0 (green) and ω > 0 (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' (b) Same as in (a) but with a larger  interaction strength, C/nth = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  Using these,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' we find:  SQQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' optical signal =g2  Y  1  2π  1  24Γ4g2  om  Γκ |χ(ω)|2 �  |f(ω)|2 + 1  �  SQQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' optical noise =1  24Γ4g2  om  Γκ |χ(ω)|2  � �1 − η  η  �  |f(ω)φ(ω)|2 + |φ(ω)f(ω) − gXgY |2 + |−φ(ω) − gY gXf(ω)|2  �  SQQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' mechanical noise =Γ|χ(ω)|2 �  1 + |φ(ω)|2�  (¯nth + 1  2)  SPP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' optical signal =g2  Y  1  24Γ4g2  om  Γκ |χ(ω)|2 �  |f(ω)|2 + 1  �  SPP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' optical noise =1  24Γ4g2  om  Γκ |χ(ω)|2  �1 − η  η  � �  |f(ω)|2 + 1  �  SPP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' mechanical noise =Γ|χ(ω)|2  �  1 +  ����4φ(ω)f(ω) 1  Ω  4g2  om  κΓ + 1  ����  2� �  ¯nth + 1  2  �  SPQ =4g2  om  πκ |χ(ω)|2  �  1 + |f(ω)|2  4  �  1 −  �1 − η  η  �2�  ℜ(φ(ω))+  Γ  2π |χ(ω)|2 8g2  om  κΩ2 (ℑ (f(ω))ℑ (φ(ω)) − ℜ(f(ω))ℜ(φ(ω)))  �  ¯nth + 1  2  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (S19)  Each contribution is plotted for two different optomechanical cooperativities in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' S1 (a) and (b), including  inefficiency noise for three different detection efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Apart from the additional mode mismatch noise on Q, the  contributions are near identical for Q and P in the limit that C ≪ Ω/Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' We therefore only plot them for Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  Apart from the mode-mismatch noise, all contributions to the power spectral density are peaked at the mechanical  4  resonance frequency and have a roughly Lorenzian shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The mismatch-noise, by contrast, is peaked at ω = 0 and  approximates the shape of a low pass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' At low cooperativity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' S1 (a)), the noise dominates the optical signal,  preventing an effective quantum state transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' At higher cooperativity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' S1 (b)), the optical signal rises above  the noise contributions, suggesting that quantum state transfer is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The inset in both figures plots the spectral  modeshape of the signal mode (u(ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The optical signal peaks at −Ω (blue) and is suppressed relative to this peak  by several orders of magnitude at Ω (green), evidencing that the chosen filter enables a single-sideband transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  VARIANCES  We obtain numerical values for the variances of each component of the power spectral density by integrating over  frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' To obtain the variance plot of the main text we sweep the interaction strength gom, while keeping the  other system parameters constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  We also determine analytical approximations in the limit of a large bandwidth filter and high mechanical quality  factor (Q = Ω  Γ ≫ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' In that limit, and with our chosen filter, the expressions for gY and gX can be approximated as:  gY =  �  �  �  �2C  �  1 +  G2  64C2  1 + G/2  �  ,  (S20)  and  gX =  1  gY(1 + 2/G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (S21)  Using these expressions and approximating the spectral densities in the high quality limit, we calculate analytic  expressions for the variance of each component through the residue theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' The results are:  VXtrans = 1  2gX  2,  (S22)  VQmech =  1  1 + G/2  �1  2 + nth  �  ,  (S23)  VQmismatch = 0,  (S24)  VQ η = 1  4  �1 − η  η  �  gX  2,  (S25)  VQmismatchQsignal = 0,  (S26)  and  VPQ = VQP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (S27)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' S24 only occurs in the variance for Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  The analytic variances for P are similar, with the subscript Q (X)  substituted for P (Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  5  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  WIGNER FUNCTIONS  The Wigner functions used as optical input states are given by the following equations in Wigner space (r = (QP)T ):  WCat (Q=2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='P=0)(r) = exp(−r · r)  �  exp(−2α · α) cosh  �  2  √  2r · α  �  + cos  �  2  √  2r · (ϖα)  ��  π(exp(−2α · α) + 1)  WFock (n=1)(r) = e−r·r (2r · r − 1)  π  WCoherent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' vacuum(r) = e− r·r  2  2π ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (S28)  where α = (αr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' αi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' and we use αr = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' and αi = 0 for the calculation of the fidelity in the main text,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' and ϖ =  ((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' (−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' 0)) is a symplectic matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  As stated in the main text, the interactions are Gaussian and we can construct a Wigner function associated with  the noise in the system:  Wnoise(r) = 1  2π  1  �  det (Vnoise)  exp  �Q2V11 − 2QPV12 + P 2V22  2V 2  12 − 2V11V22  �  ,  (S29)  where Vij are the elements of the correlation matrix that only contains all the noise contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Specifically:  Vnoise =  �  VQQ, optical noise + VQQ, mechanical noise  VQnoise Pnoise  VPnoise Qnoise  VPP, optical noise + VPP, mechanical noise  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (S30)  The transferred Wigner functions Wtransferred are found by:  Wtransferred(r′) = (Wtarget ⊛ Wnoise) (r),  (S31)  which we use directly for the calculation of the transfer fidelity:  F = 2π  � ∞  −∞  Wtarget(r)Wtransferred(r)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content='  (S32)  REFERENCES  [S1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Bowen and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
+page_content=' Milburn, Quantum optomechanics (CRC Press, Boca Raton London New York, 2015), ISBN  978-0-367-57519-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE2T4oBgHgl3EQfYgeB/content/2301.03855v1.pdf'}
diff --git a/YtE4T4oBgHgl3EQfOAxw/content/tmp_files/2301.04961v1.pdf.txt b/YtE4T4oBgHgl3EQfOAxw/content/tmp_files/2301.04961v1.pdf.txt
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+Silicon Carbide Metasurfaces for Controlling the
+Spontaneous Emission of Embedded Color
+Centers
+Mohammed Ashahar Ahamad and Faraz Ahmed Inam
+Department of Physics, Aligarh Muslim University, Aligarh, Uttar Pradesh 202002, India
+faraz.inam.phy@amu.ac.in
+Stefania Castelletto
+School of Engineering, RMIT University, Melbourne, Victoria 3001, Australia
+Stefania.castelletto@rmit.edu.au
+Abstract:
+Nanopillars fabricated in diamond or silicon-carbide (SiC) have been used to
+enhance the light harvesting or absorption or to increase the collection efficiency of embed-
+ded single photon emission in the visible or near infrared for their detection using confocal
+microscopy. While electric and magnetic dipolar resonances in SiC have been studied in
+the far-infrared, they have not been studied in the near infrared. Here we show for the first
+time that electromagnetic Mie-scattering moments within SiC metasurfaces can control the
+spontaneous emission process of point defects in the near infrared. Using SiC nanopillars
+based metasurfaces, we theoretically demonstrate a control over the spontaneous emission
+rate of embedded color-centers by using the coherent superposition of the electric dipolar
+and magnetic quadrupolar electromagnetic Mie-scattering moments of the structure. More
+than an order of magnitude emission/decay rate enhancement is obtained with the maximum
+enhancement close to 30. We also demonstrate that the relative phase of the Mie-scattering
+moments helps in controlling the emission directionality. SiC metasurfaces in the spectral
+range of color centres, from the visible to the near infrared, can be used to control the con-
+finement and directionality of their spontaneous emission, increasing the opportunities to
+study light-matter interaction and to advance quantum photonic and quantum sensing de-
+vice integration.
+Keywords: Mie-scattering moments, silicon carbide nanopillars metasurface, emission en-
+hancements, radiation directionality, color centres
+© 2022 The Author(s)
+1.
+Introduction
+Color-centers in silicon-carbide (SiC) are example of emitters that possess single photon emission [1], optical spin
+read out and control, and have been amongst the most studied for optical coherence spin control and spin-photon
+interface [2,3] due to their very long coherence time [4,5] and photo-stability. SiC is quite distinguished from the
+other material platforms as it possesses color centre with optical-spin properties combined with advanced material
+fabrication methods, metal-oxide-semiconductor functionalities [6] and nonlinear second and third order optical
+properties. Due to its wide electronic bandgap which leads to broad optical transparency, photostable color centres
+emission [7] which extend to the near infrared, CMOS compatibility [8] and availability of quantum-grade wafer-
+scale SiC on insulator, it has emerged as one of the most a promising material for integrated quantum photonic
+applications [9,10].
+In particular, SiC can host a wide range of point defects/color centers including silicon vacancy VSi (V1, V2,
+V3), divacancies VSiVC and carbon antisite vacancy pair CSiVC [11]. The VSi in SiC is a promising single photon
+source (SPS) for spin-photon photon interface in the near infrared region around 917nm [12, 13]. At present
+the main challenge in applying SiC for quantum networks is to significantly enhance the rate of single photon
+generation and collection from embedded color-centers. Photonics is mainly used to enhance the properties of
+these systems.
+So far in SiC bulk material, nanopillars have been fabricated to enhance the light harvesting or collection
+efficiency of embedded single photon emission for their detection using confocal microscopy [12,14]; while meta-
+lenses [15] are used to modify the phase front of the emitted light, achieving high focusing and large emission
+directionality of the color-centers emitting below these meta-lenses. Currently metasurfaces used to excite/enhance
+arXiv:2301.04961v1  [physics.optics]  12 Jan 2023
+
+the magnetic and electric resonances in SiC have been investigated only in the far infrared [16] and in the context
+of surface phonon polaritons studies [17]. Recently it has been shown that metamaterial/metasurface light matter
+interaction can be used to control, enhance and tune the quantum properties of bulk materials [18,19]. In particular
+all dielectric metasurfaces due to their zero absorption losses have emerged as the preferred platform compared to
+plasmonics in photo-luminescence enhancement [20]. When a dielectric structure is placed under electromagnetic
+excitation, various charge and current distributions are excited in it. These distributions results in multi-polar Mie-
+scattering resonances being excited in structures with dimensions of the order of the excitation wavelength [21]. A
+coherent superposition of these resonances leads to many interesting phenomena like, bound states in continuum
+(BIC) [22], tuning of the radiation directionality in the lateral or transverse directions [23] and tuning of the local
+optical density of states (LDOS) to achieve emission rate enhancement for emitters embedded in the metasurfaces
+[24].
+Here for the first time we study the electric dipolar and magnetic quadrupolar resonances in the near infrared
+in SiC for controlling the spontaneous emission rate of the embedded color centers in the dielectric nanopillars
+forming Mie resonators.
+In this study, using the coherent superposition of Mie-scattering resonances in SiC pillars based metasurface, we
+theoretically demonstrate that it is possible to control the spectral spontaneous emission process of the embedded
+color-centers.We first optimise the scattering efficiency of the SiC metasurface when excited by a plane wave and
+then by a dipole emitter. In the case the light source is a dipole, namely the VSi embedded in the SiC metamaterial,
+we study the effects of the metasurface based on array of nanopillars Mie resonances on the LDOS and emission
+directionality. In particular, we study the effect of the periodicity of the nanopillars array to increase the emission
+rate and maintain high directionality compared to the case of a single pillar.
+2.
+Theoretical background
+Scattering is the phenomenon of re-emission of radiation by a particle after undergoing interaction with radiation
+[25]. When a plane electromagnetic wave is incident on a particle, charge distribution and displacement currents
+J(r) = −iωε0(εr − 1)E(r) (here E(r) is the field at the position vector r, ω = 2πr is the angular frequency, εr
+and ε0 are the permittivity of the particle and surrounding medium) are excited within it. When the particle’s
+dimensions are of the order of the excitation wavelength, the excited charge and current distributions leads to the
+development of multipolar Mie-scattering modes [26, 27]. The amplitude and phase of excitation of the electric
+and magnetic resonances or multi-polar Mie-scattering moments inside the scatterer are totally governed by its
+size, shape and surrounding electromagnetic environment [28]. These multi-polar Mie resonances in the visible
+spectral range have been demonstrated experimentally in the last decade using a silicon spherical nanopartciles
+and nanodiamonds [29,30].
+The total scattering efficiency (SE) Ctotal
+sca
+is calculated by normalizing the total far field scattered power to the
+energy flux of the incident wave on the scatterer [31]. The total SE Ct
+sca is the sum of partial SE from different mul-
+tipoles: Cp
+sca, Cm
+sca, CQ
+sca and CM
+sca represents contributions from electric dipole, magnetic dipole, electric quadrupole
+and magnetic quadrupole respectively [32].
+Ctotal
+sca
+=
+Cp
+sca +Cm
+sca +CQ
+sca +CM
+sca
+(1)
+Ctotal
+sca
+=
+k4
+6πε2
+0|Einc|2
+�
+�∑
+�
+|pα|2 +
+���mα
+c
+���
+2�
++ 1
+120 ∑
+�
+�|kQe
+αβ|2 +
+�����
+kQm
+αβ
+c
+�����
+2�
+�
+�
+�
+(2)
+where, pα and mα are the electric and magnetic dipole moments with Qe
+αβ and Qm
+αβ being the corresponding
+quadrupole moments. |Einc| is the amplitude of the incident electric field, k is the wave-vector and c is the speed
+of light. They are mathematically expressed as [32]:
+ED : pα
+=
+− 1
+iω
+��
+d3rJω
+α j0(kr)+ k2
+2
+�
+d3r
+�
+3(r.Jω)rα −r2Jω
+α
+� j2(kr)
+(kr)2
+�
+(3)
+MD : mα
+=
+3
+2
+�
+d3r(r×Jω)α
+j1(kr)
+kr
+(4)
+EQ : Qe
+αβ
+=
+− 3
+iω
+��
+d3r[3(rβJω
+α +rαJω
+β )−2(r.Jω)δαβ] j1(kr)
+(kr)
++
+2k2
+�
+d3[5rαrβ(r.Jω)−(rαJβ +rβJα)r2 −r2(r.Jω)δαβ] j3(kr)
+(kr)3
+�
+(5)
+MQ : Qm
+αβ
+=
+15
+�
+d3r
+�
+rα(r×Jω)β +rβ(r×Jω)α
+� j2(kr)
+(kr)2
+(6)
+
+Fig. 1. (a) Schematic of the metasurface with a 2D periodic lattice of SiC pillars under plane-wave
+excitation at 917 nm with wave-vector along the +z-direction. (b) Schematic of the unit cell. Each
+pillar has a length L = 2 µm and diameter D = 424 nm with a dipole emitter located at the center
+of each pillar. (c) The SE of the individual multipolar Mie-scattering moments as a function of the
+lattice periodicity, P, under plane-wave excitation at 917 nm. P is varied from 450 nm to 2500 nm.
+(d) The corresponding phase of the individual multipolar Mie-scattering moments as a function of
+the lattice periodicity, P. The dotted black lines corresponds to overlapping ED and MQ resonances
+with P = 915 nm, 1095 nm, 1500 nm and 2315 nm.
+The Mie-resonances control the electromagnetic field amplitudes within the scatterer and therefore contribute
+in tuning the local electromagnetic density of states (LDOS). The LDOS due to the local electromagnetic environ-
+ment around a point dipole emitter is defined as [33]
+ρ(ω,r) = ∑
+k,σ
+| ˆd ·Ek,σ(r)|2δ(ω −ωk,σ).
+(7)
+Here, ˆd is the unit vector specifying the direction of the transition dipole moment with ω being the transition
+frequency. The summation is over all wavevectors (k) and polarizations (σ). E is the total electric field at the
+source position resulting from the superposition of the fields directly radiated by the dipole emitter embedded
+inside the scatterer with the fields reflected and scattered back from the surroundings. The LDOS govern the
+complete radiation process of a dipole emitter. Hence the Mie-scattering modes play a vital role in tuning the
+spontaneous emission process of the emitter by controlling the scattered electric field at the source point.
+The balancing of the electric and magnetic Mie-scattering moments leads to the directionality of the scattered
+radiation pattern [32]. The radiation pattern is controlled by the relative phase of the balanced electric and magnetic
+multi-polar moments [34]. When the electric and magnetic dipolar moments are balanced and in phase, |ED| =
+|MD|, arg(ED) = arg(MD), this leads to a completely forward radiation directionality, known as the Kerker
+condition [32]. When these dipolar moments are out-of-phase, |ED| = |MD|, arg(ED) = arg(MD)+π, it results in
+a completely backward directionality, known as the anti-Kerker condition [34]. When the superposition of dipolar
+
+a
+L=2um
+Unit cell
+D = 424 nm
+X
+P
+10
+！
+(c)
+ED
+！！
+MD
+Phase(rad)
+EQ
+MQ
+0
+L
+S
+T
+业
+0
+500
+1000
+1500
+2000
+2500
+1000
+1500
+2000
+P(nm)
+P(nm)Fig. 2. The 2D electric field norm within (a) a single SiC pillar and the SiC pillar metasurface with
+P = (b) 2315 nm, (c) 1500 nm and (d) 1095 nm under (i) plane wave excitation with wave-vector
+along the +z-direction and electric field polarized along the +x-direction and (ii) dipole excitation
+with dipole emitter placed at the center of the SiC pillar with orientation along the x-direction.
+as well as the quadrapolar moments are balanced and are in phase, |ED + MD| = |EQ + MQ| with arg(ED +
+MD) = arg(ED + MD), the radiation pattern is highly directional along the forward direction, known as the
+generalised Kerker condition [34]. However, when these superpositions are out-of-phase, this leads to a complete
+transverse scattering [34]. The phase of the Mie-scattering moments therefore controls the far-field scattering
+radiation pattern of the scatterer. Under a point dipole emitter excitation of the structure, the far-field scattering
+pattern of the structures also influences the radiation pattern of the dipole emitter placed in the vicinity of the
+scatterer [35].
+In the following we investigate these effects in SiC nanopillars array under plane wave-excitation and under a
+single dipole excitation simulating the VSi.
+3.
+Results and discussion
+3.1.
+Scattering efficiency and decay-rate enhancement
+We have computationally optimised the SiC metasurface, shown in Fig. 1(a) with unit cell in Fig. 1(b), to achieve
+the generalised Kerker’s condition in SiC for the specific color centre of interest. The metasurface consists of a
+periodic 2D lattice of SiC pillars, each of length, L = 2µm. The electrodynamics calculations are performed using
+the commercial Comsol Multiphysics RF module. The details of the calculations are presented in the Methods
+sections. The metasurface is excited by a plane wave with wavelength, λexc, travelling along the +z-direction
+(arrow symbol in Fig. 1(a)) with the electric field polarized along the +x-direction. Under the influence of the
+plane electromagnetic wave, Mie scattering moments are excited within the SiC pillars. We first optimised the
+diameter, D, of the SiC pillars for the maxima in the SE at λexc = 917 nm corresponding to the zero phonon line
+(ZPL) of the silicon vacancy, VSi in SiC [36,37]. The optimised D value was found to be around 424 nm. We then
+study the coherent superposition of the Mie-scattering modes of the individual SiC pillars by varying the lattice
+periodicity, P. For P ≫ λexc, the structure is expected to behave as a single isolated pillar. With decreasing P, the
+interactions between the Mie-scattering modes of the individual pillars will increase. When these modes will be
+in phase, their coherent superposition will lead to a maxima for the total SE of the 2D SiC pillar lattice. Fig. 1(c)
+and (d) show the amplitude and the phase of the individual Mie-scattering moments of the SiC pillar metasurface
+as a function of P. Sharp resonance peaks are observed in the amplitude of the individual Mie-scattering moments
+
+(iD)
+(a) Single pillar
+(a)Single pillar (b)P: 2315 nm
+(b) P: 2315 nm
+X
+3
+2.5
+('n'e)
+1.5
+(c) P: 1500 nm
+(d) P: 1095 nm
+(c)P: 1500 nm
+(d)P: 1095 nm
+0.5Fig. 3. The SE of the individual excited multipolar Mie-scattering moments and the emitter’s (VSi
+color-center) relative decay rate in the SiC pillar metasurface as a function of the lattice periodicity,
+P. The dipole emitter is placed at the center of the SiC pillars with dipole orientation along the
+horizontal plane. The γ∞ is the emitter’s decay rate in the bulk SiC. (b) The schematic representation
+of the tuning of the embedded color-center’s emission with the lattice periodicity, P set to (i) off-
+resonant P1 and (ii) resonant P2 values.
+(Fig. 1(c)). At these sharp resonances, a sharp jump in the phase of the corresponding Mie-scattering moment is
+observed (Fig. 1(d)). We will now focus on the local maxima arising due to the ED and the MQ moments (under
+dipole excitation of the structure only these two resonances were excited and were observed to have an influence
+on the dipole emitter’s decay rates). These local maxima are observed for P =915 nm, 1095 nm, 1500 nm and
+2315 nm (black dotted lines in Fig. 1(c) and (d)). At P =915 nm, the ED resonance peak is much greater than the
+MQ resonance with the phase of these two resonances being equal. For P =1095 nm, 1500 nm and 2315 nm, the
+ED and MQ moments are nearly balanced and a sharp jump is also observed in their phase. We will now examine
+the balanced superposition of these two moments at P =1095 nm, 1500 nm and 2315 nm.
+Figure 2(i) shows the normalised electric field distribution within a (a) single SiC pillar and the SiC pillar meta-
+surface with P = (b) 2315 nm, (c) 1500 nm and (d) 1095 nm under plane wave excitation with wave-vector along
+the +z-direction and electric field polarized along the +x-direction. Strong confinement of the electric field within
+the SiC cylinder is observed at these P values corresponding to the balanced superposition of the ED and MQ
+resonances, with the maximum field confinement observed for P = 1500 nm (the SE was also observed to be max-
+imum at this P value). The field confinement will in-turn lead to LDOS enhancement within the SiC pillar (Eq 7).
+For a dipole emitter placed at the field maxima points, the LDOS enhancement will lead to its decay rate enhance-
+ment. Figure Fig. 2(ii) shows the normalised electric field distribution within (a) a single SiC pillar and the SiC
+pillar metasurface with P = (b) 2315 nm, (c) 1500 nm and (d) 1095 nm under dipole excitation with dipole emitter
+placed at the center of the SiC pillars with orientation along the x-direction. Large field confinement/enhancement
+which will lead to large LDOS enhancement can be observed here.
+We now study the influence of the above LDOS enhancement on the spontaneous emission rates of a dipole
+emitter, the VSi color-center embedded at the center of each SiC pillar. Figure 3(a) shows the emitter’s (VSi color-
+center) relative decay rate together with the SE of the individual Mie-scattering moments in the SiC pillar meta-
+surface as a function of the lattice periodicity, P. The decay rates of the VSi emitter in the SiC pillar metasurface,
+γ are scaled relative to its decay rates in a bulk SiC crystal, γ∞. The influence of the LDOS enhancement arising
+from the electric field confinement (Fig. 2) in tuning the emitter’s decay rate can be clearly observed here. Also, it
+can be observed that the relative decay rates ( γ
+γ∞ ) (dash-dotted red curve) only tunes with the local maxima which
+are dominated by ED and MQ resonances. These resonances corresponds to P = 1095 nm, 1500 nm and 2315
+nm, respectively. A schematic representation of the embedded dipole emitter’s radiation tuning with the SiC pillar
+lattice periodicity, P at an off-resonant (i) and resonant (ii) value is presented in Fig. 3.
+In Fig. 4, we study the SE spectral response due to all the excited Mie-scattering moments and the effect
+on the relative decay rates of a horizontally oriented (along x-direction) dipole source for the above resonant
+
+20
+(a)
+ED
+(b)
+30
+MD
+15
+EQ
+SE (a.u.)
+-MQ
+20
+8
+10
+-8
+10
+(i)
+(iD)
+500
+1000
+1500 2000
+2500
+P (nm)Fig. 4. The spectral response of the SE with the individual excited multipolar Mie-scattering mo-
+ments under horizontal dipole excitation and the emitter’s (VSi color-center with dipole orientation
+along the horizontal plane) relative decay rate in a (a) single SiC pillar; SiC pillar metastuface with
+P = (b) 2315nm, (c) 1500 nm and (d) 1095 nm, respectively.
+periodicity values (P = 1095 nm, 1500 nm and 2315 nm) of the metasurface. Here, the Mie-scattering moments
+of the metasurface are excited by the dipole source itself. For an isolated SiC pillar (Fig. 4(a)), all the studied
+Mie-scattering moments are observed to be weakly excited with no clear resonances. The relative decay rate
+(dash-dotted red curve) is observed to be around 2.7 at 917 nm. However, for all resonant P values (P = 1095
+nm, 1500 nm and 2315 nm), significant contributions are observed only from the ED and MQ moments. Their
+superposition is controlling the behaviour of the SE and the relative decay rate. The maximum relative decay rate
+enhancement is close to 30 at 917 nm for P = 1500 nm and 1095 nm. For P = 2315 nm the enhancement is about
+20. Therefore, it can be concluded that the coherent superposition of the ED and MQ Mie-scattering moments
+of the individual pillars are enhancing the decay rates of an embedded dipole emitter by more than an order of
+magnitude.
+We now study the role of the phase of the excited Mie-scattering moments (ED and MQ) of the SiC pillar
+metasurface on the far field radiation pattern of an embedded dipole emitter.
+3.2.
+Phase analysis and Radiation pattern
+Figure 5 shows a narrow range of values for both the relative amplitudes and phase of the ED and MQ moments
+at the resonant P values. At the VSi color-center’s peak emission wavelength of 917 nm, the MQ moment appears
+to be slightly larger than the ED moment. For P = 1500 nm, the ED
+MQ = 0.7 and for P = 1095 nm and 2315 nm, the
+ED
+MQ = 0.81. The corresponding phase difference between ED and MQ moments is π at 917 nm for all resonant P
+values.
+In Fig. 6(i) we show the in-phase and out-of-phase superposition of the balanced ED and MQ moments. The
+influence of this superposition on both the in-plane (X-Z plane, blue curve) and out-of-plane (Y-Z plane, red
+curve) far-field scattering patterns is shown here. The in-phase (ED + MQ) superposition results in longitudinal
+(along top and bottom directions) scattering and the out-of-phase (ED − MQ) superposition results in transverse
+scattering along the out-of-plane direction (red curve).
+We now study the influence of the ED and MQ moments superpositions on the embedded dipole emitter radi-
+ation pattern for a single nanopillar and SiC metasurface of array of interacting nanopillars. The emitter’s dipole
+orientation is along the X-direction. Fig. 6(ii) shows the far-field emission patterns of the embedded VSi color-
+
+1.5
+1.5
+Single Pillar
+ED
+P = 2315nm
+-ED
+(b)
+30
+30
+MD
+MD
+-EQ
+-EQ
+1
+SE(a.u.)
+20
+-MQ
+20.8
+8
+-MQ
+a
+.... TOTAL
+..... TOTAL
+0.5
+10
+10
+8
+8
+0
+0
+0
+900
+910
+920
+930
+940
+950
+900
+910
+930
+940
+920
+950
+入(nm)
+入(nm)
+1.5
+1.5
+= 1500 nm
+P = 1095nm
+(d)
+D
+(c)
+ED
+ED 
+30
+30
+MD
+-MD
+-EQ
+-EQ
+L
+(a.u.)
+20_8
+一MQ
+20.8
+一MQ
+....TOTAL
+..... TOTAL
+10
+10
+8
+0
+0
+0
+900
+910
+920
+930
+940
+950
+900
+910
+920
+930
+940
+950
+入(nm)
+入(nm)Fig. 5. The spectral response of the (a) relative amplitudes and (b) phase difference of the elec-
+tric dipolar (ED) and magnetic quadrapular (MQ) Mie-scattering moments excited under horizontal
+dipole excitation.
+center at 917 nm for different resonant P values. In a single pillar, the majority of the emission is observed to be
+directed towards the bottom surface and lies mainly along the longitudinal direction (bottom and top). However,
+in the SiC metasurface at the resonant P values corresponding to ED and MQ moments, the out-of-phase super-
+position of the ED and MQ moments (the phase difference between these moments was observed to be close to π
+at 917 nm in Fig. 5) directs the embedded emitter’s radiation pattern along the transverse direction, especially in
+the out-of-plane (Y-Z plane, red curve).
+4.
+Conclusion
+We studied for the first time the coherent superposition of the eletric and magnetic dipolar and quadrupolar Mie-
+scattering moments of SiC metamaterial nanopillars array in the near infrared emission (917 nm). We first de-
+termined the design of the metasurface periodicity to induce sharp resonances in the amplitude and phase of the
+Mie-scattering moments. Strong electric field confinement was observed within the SiC pillar when the periodicity
+of the lattice matched with the resonance of the ED and MQ modes. The field confinement leads to large LDOS
+and subsequently decay rate enhancement for a color-center dipole embedded at the center of the SiC pillar. Under
+a point dipole emitter excitation within SiC, it was determined that only the ED and MQ moments are contribut-
+ing to the electromagnetic scattering in the SiC nanopillars metasurface. Both these moments were observed to
+be nicely coupled and were showing collective resonance at the optimised wavelength (917 nm). The coherent
+superposition of these two moments controls the complete spontaneous emission process of the embedded color
+center. At the collective resonance point of these two moments (λ = 917 nm), we determined more than an order
+of magnitude decay rate enhancement with the maximum enhancement reaching 30. Such an enhancement has
+never been reported in dielectric neither in metal-dielectric individual nanopillars [38], thus paving the way for
+the use SiC metasurfaces to to enhance and control light extraction from quantum emitters, to study light matter
+interaction effects in integrated quantum photonics and for applications in quantum sensing. We also observed that
+by designing specific resonant structures, the coherent superposition of the ED and MQ moments can be used to
+better control the radiation/emission pattern and hence the emission directionality of the embedded dipole emitter
+compared to a single nanopillar. Specifically the embedded emitter’s radiation pattern can be more confined along
+the transverse direction, especially in the out-of-plane (Y-Z plane, red curve), thus facilitating the planar emission
+propagation. Such result is relevant for applications of SiC metasurfaces for planar integrated photonics. Our study
+can prompt further studies on SiC quantum states of light based on multi-emitters-resonators coupling enabled by
+metamaterials such as superradiance [39].
+5.
+Materials and Methods
+All the electrodynamics calculations have been performed using the commercial COMSOL Multiphysics Radio
+Frequency (RF) module. The periodic boundary conditions are applied to all the horizontal planes defining the 2D
+lattice of the SiC substrate to build an array of dielectric pillars metasurface. The Scattering boundary conditions
+are applied at the top and bottom boundaries of the computational domain. The optical constant for the SiC has
+been extracted from the experimentally reported values by Singh et.al. [40]. During the entire calculation the
+minimum meshing size was 1 nm with the maximum being λ/7.
+
+(a)
+(b)
+- Single pillar
+Phase difference (rad)
+T
+ED
+.....P: 1095 nm
+e
+arg(ED) -
+1.5
+MQ
+-P: 1500 nm
+arg (MQ)
+- P: 2315 nm
+1
+0.5
+-T
+0
+913
+915
+917
+919
+921
+913
+915
+917
+919
+921
+入 (nm)
+入(nm)Fig. 6. (i) In-plane (X-Z plane, blue curves) and out-of-plane (Y-Z plane, red curves) far-field scat-
+tering pattern corresponding to the in-phase (ED+MD) and out-of-phase (ED-MQ) superposition
+of the electric dipolar (ED) and magnetic quadrupolar (MQ) Mie-scattering moments. (ii) Farfield
+radiation patterns in-plane (X-Z plane, blue curves) and out-of-plane (Y-Z plane, red curves) of a
+dipole emitter (orientation along the X-direction, same as that of electric field in (i)) placed at the
+center of (a) single pillar; SiC pillar metastuface with P = (b) 2315nm, (c) 1500 nm and (d) 1095
+nm, respectively.
+5.1.
+Scattering efficiency calculation
+The scattering cross section is defined as the amount of power scattered by the scatterer to the amount of power
+per unit area carried by the incident wave. The SE is obtained just by dividing the scattering cross-section by
+the geometrical cross-section. Mathematically it is expressed as SE = σs/G [41]. Here σs is the scattering cross
+section and G is geometrical cross section.
+The SE is calculated semi-analytically using electric field values at each mesh point in the computational grid
+under plane-wave excitation using Comsol Multi-physics module. Using these field values and permittivity profile
+at each mesh points, current density is calculated as: Jω(r) = iωε0(εr −1)Eω(r). Here ε0 and εr are permittivity of
+free space and SiC medium, respectively. The computationally obtained values of E(r), Jω(r) and ε(r) are used to
+calculate individual multipolar Mie-scattering moments, pα, mα, Qe
+αβ and Qm
+αβ described in Eq. 2. The integration
+referred in Eq. 2 is carried on the domain of the SiC pillar.
+5.2.
+Relative decay rate calculations
+In these calculations, the SiV center is treated as a classical radiating point dipole source. In the computational
+domain, it is modelled as a point current source driven at the emission frequency, ν = c
+λ [42]. Scattering/perfectly
+matched layer (PML) boundary conditions are applied on the exterior boundaries of the computational domain.
+The total power radiated by the dipole is integrated over a closed surface enclosing the current source. The relative
+decay-rate is calculated as Γrel = γ/γ∞ = P/P∞ [42], where P∞ is the power corresponding to the point dipole’s
+emission in the bulk SiC. The permittivity of SiC is taken from [43].
+Acknowledgement The authors would like to acknowledge the financial support from the Department of Sci-
+ence and Technology (DST), India (CRG/2021/001167). The authors thank Dr Nadeem Ahmed for his help regard-
+ing the plotting of the figures and the formatting of the manuscript. MA and FAI also thank Dr Ahmed Mekawy
+for his help regarding multipole decomposition of the Mie-scattering moments.
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+spontaneous emission lifetime in a microcavity,” J. Opt. Soc. Am. B 16, 465 (1999).
+43. P. T. B. Shaffer, “Refractive Index, Dispersion, and Birefringence of Silicon Carbide Polytypes,” Appl. Opt. Vol. 10,
+Issue 5, pp. 1034-1036 10, 1034–1036 (1971).
+
diff --git a/YtE4T4oBgHgl3EQfOAxw/content/tmp_files/load_file.txt b/YtE4T4oBgHgl3EQfOAxw/content/tmp_files/load_file.txt
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@@ -0,0 +1,779 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf,len=778
+page_content='Silicon Carbide Metasurfaces for Controlling the  Spontaneous Emission of Embedded Color  Centers  Mohammed Ashahar Ahamad and Faraz Ahmed Inam  Department of Physics, Aligarh Muslim University, Aligarh, Uttar Pradesh 202002, India  faraz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='inam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='phy@amu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='in  Stefania Castelletto  School of Engineering, RMIT University, Melbourne, Victoria 3001, Australia  Stefania.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='castelletto@rmit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='au  Abstract:  Nanopillars fabricated in diamond or silicon-carbide (SiC) have been used to  enhance the light harvesting or absorption or to increase the collection efficiency of embed-  ded single photon emission in the visible or near infrared for their detection using confocal  microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' While electric and magnetic dipolar resonances in SiC have been studied in  the far-infrared, they have not been studied in the near infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Here we show for the first  time that electromagnetic Mie-scattering moments within SiC metasurfaces can control the  spontaneous emission process of point defects in the near infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Using SiC nanopillars  based metasurfaces, we theoretically demonstrate a control over the spontaneous emission  rate of embedded color-centers by using the coherent superposition of the electric dipolar  and magnetic quadrupolar electromagnetic Mie-scattering moments of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' More  than an order of magnitude emission/decay rate enhancement is obtained with the maximum  enhancement close to 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' We also demonstrate that the relative phase of the Mie-scattering  moments helps in controlling the emission directionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' SiC metasurfaces in the spectral  range of color centres, from the visible to the near infrared, can be used to control the con-  finement and directionality of their spontaneous emission, increasing the opportunities to  study light-matter interaction and to advance quantum photonic and quantum sensing de-  vice integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  Keywords: Mie-scattering moments, silicon carbide nanopillars metasurface, emission en-  hancements, radiation directionality, color centres  © 2022 The Author(s)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  Introduction  Color-centers in silicon-carbide (SiC) are example of emitters that possess single photon emission [1], optical spin  read out and control, and have been amongst the most studied for optical coherence spin control and spin-photon  interface [2,3] due to their very long coherence time [4,5] and photo-stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' SiC is quite distinguished from the  other material platforms as it possesses color centre with optical-spin properties combined with advanced material  fabrication methods, metal-oxide-semiconductor functionalities [6] and nonlinear second and third order optical  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Due to its wide electronic bandgap which leads to broad optical transparency, photostable color centres  emission [7] which extend to the near infrared, CMOS compatibility [8] and availability of quantum-grade wafer-  scale SiC on insulator, it has emerged as one of the most a promising material for integrated quantum photonic  applications [9,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  In particular, SiC can host a wide range of point defects/color centers including silicon vacancy VSi (V1, V2,  V3), divacancies VSiVC and carbon antisite vacancy pair CSiVC [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The VSi in SiC is a promising single photon  source (SPS) for spin-photon photon interface in the near infrared region around 917nm [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' At present  the main challenge in applying SiC for quantum networks is to significantly enhance the rate of single photon  generation and collection from embedded color-centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Photonics is mainly used to enhance the properties of  these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  So far in SiC bulk material, nanopillars have been fabricated to enhance the light harvesting or collection  efficiency of embedded single photon emission for their detection using confocal microscopy [12,14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' while meta-  lenses [15] are used to modify the phase front of the emitted light, achieving high focusing and large emission  directionality of the color-centers emitting below these meta-lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Currently metasurfaces used to excite/enhance  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='04961v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='optics]  12 Jan 2023  the magnetic and electric resonances in SiC have been investigated only in the far infrared [16] and in the context  of surface phonon polaritons studies [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Recently it has been shown that metamaterial/metasurface light matter  interaction can be used to control, enhance and tune the quantum properties of bulk materials [18,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' In particular  all dielectric metasurfaces due to their zero absorption losses have emerged as the preferred platform compared to  plasmonics in photo-luminescence enhancement [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' When a dielectric structure is placed under electromagnetic  excitation, various charge and current distributions are excited in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' These distributions results in multi-polar Mie-  scattering resonances being excited in structures with dimensions of the order of the excitation wavelength [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' A  coherent superposition of these resonances leads to many interesting phenomena like, bound states in continuum  (BIC) [22], tuning of the radiation directionality in the lateral or transverse directions [23] and tuning of the local  optical density of states (LDOS) to achieve emission rate enhancement for emitters embedded in the metasurfaces  [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  Here for the first time we study the electric dipolar and magnetic quadrupolar resonances in the near infrared  in SiC for controlling the spontaneous emission rate of the embedded color centers in the dielectric nanopillars  forming Mie resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  In this study, using the coherent superposition of Mie-scattering resonances in SiC pillars based metasurface, we  theoretically demonstrate that it is possible to control the spectral spontaneous emission process of the embedded  color-centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='We first optimise the scattering efficiency of the SiC metasurface when excited by a plane wave and  then by a dipole emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' In the case the light source is a dipole, namely the VSi embedded in the SiC metamaterial,  we study the effects of the metasurface based on array of nanopillars Mie resonances on the LDOS and emission  directionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' In particular, we study the effect of the periodicity of the nanopillars array to increase the emission  rate and maintain high directionality compared to the case of a single pillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  Theoretical background  Scattering is the phenomenon of re-emission of radiation by a particle after undergoing interaction with radiation  [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' When a plane electromagnetic wave is incident on a particle, charge distribution and displacement currents  J(r) = −iωε0(εr − 1)E(r) (here E(r) is the field at the position vector r, ω = 2πr is the angular frequency, εr  and ε0 are the permittivity of the particle and surrounding medium) are excited within it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' When the particle’s  dimensions are of the order of the excitation wavelength, the excited charge and current distributions leads to the  development of multipolar Mie-scattering modes [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The amplitude and phase of excitation of the electric  and magnetic resonances or multi-polar Mie-scattering moments inside the scatterer are totally governed by its  size, shape and surrounding electromagnetic environment [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' These multi-polar Mie resonances in the visible  spectral range have been demonstrated experimentally in the last decade using a silicon spherical nanopartciles  and nanodiamonds [29,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  The total scattering efficiency (SE) Ctotal  sca  is calculated by normalizing the total far field scattered power to the  energy flux of the incident wave on the scatterer [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The total SE Ct  sca is the sum of partial SE from different mul-  tipoles: Cp  sca, Cm  sca, CQ  sca and CM  sca represents contributions from electric dipole, magnetic dipole, electric quadrupole  and magnetic quadrupole respectively [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  Ctotal  sca  =  Cp  sca +Cm  sca +CQ  sca +CM  sca  (1)  Ctotal  sca  =  k4  6πε2  0|Einc|2  �  �∑  �  |pα|2 +  ���mα  c  ���  2�  + 1  120 ∑  �  �|kQe  αβ|2 +  �����  kQm  αβ  c  �����  2�  �  �  �  (2)  where, pα and mα are the electric and magnetic dipole moments with Qe  αβ and Qm  αβ being the corresponding  quadrupole moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' |Einc| is the amplitude of the incident electric field, k is the wave-vector and c is the speed  of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' They are mathematically expressed as [32]:  ED : pα  =  − 1  iω  ��  d3rJω  α j0(kr)+ k2  2  �  d3r  �  3(r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='Jω)rα −r2Jω  α  � j2(kr)  (kr)2  �  (3)  MD : mα  =  3  2  �  d3r(r×Jω)α  j1(kr)  kr  (4)  EQ : Qe  αβ  =  − 3  iω  ��  d3r[3(rβJω  α +rαJω  β )−2(r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='Jω)δαβ] j1(kr)  (kr)  +  2k2  �  d3[5rαrβ(r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='Jω)−(rαJβ +rβJα)r2 −r2(r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='Jω)δαβ] j3(kr)  (kr)3  �  (5)  MQ : Qm  αβ  =  15  �  d3r  �  rα(r×Jω)β +rβ(r×Jω)α  � j2(kr)  (kr)2  (6)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' (a) Schematic of the metasurface with a 2D periodic lattice of SiC pillars under plane-wave  excitation at 917 nm with wave-vector along the +z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' (b) Schematic of the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Each  pillar has a length L = 2 µm and diameter D = 424 nm with a dipole emitter located at the center  of each pillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' (c) The SE of the individual multipolar Mie-scattering moments as a function of the  lattice periodicity, P, under plane-wave excitation at 917 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' P is varied from 450 nm to 2500 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  (d) The corresponding phase of the individual multipolar Mie-scattering moments as a function of  the lattice periodicity, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The dotted black lines corresponds to overlapping ED and MQ resonances  with P = 915 nm, 1095 nm, 1500 nm and 2315 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  The Mie-resonances control the electromagnetic field amplitudes within the scatterer and therefore contribute  in tuning the local electromagnetic density of states (LDOS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The LDOS due to the local electromagnetic environ-  ment around a point dipole emitter is defined as [33]  ρ(ω,r) = ∑  k,σ  | ˆd ·Ek,σ(r)|2δ(ω −ωk,σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  (7)  Here, ˆd is the unit vector specifying the direction of the transition dipole moment with ω being the transition  frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The summation is over all wavevectors (k) and polarizations (σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' E is the total electric field at the  source position resulting from the superposition of the fields directly radiated by the dipole emitter embedded  inside the scatterer with the fields reflected and scattered back from the surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The LDOS govern the  complete radiation process of a dipole emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Hence the Mie-scattering modes play a vital role in tuning the  spontaneous emission process of the emitter by controlling the scattered electric field at the source point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  The balancing of the electric and magnetic Mie-scattering moments leads to the directionality of the scattered  radiation pattern [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The radiation pattern is controlled by the relative phase of the balanced electric and magnetic  multi-polar moments [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' When the electric and magnetic dipolar moments are balanced and in phase, |ED| =  |MD|, arg(ED) = arg(MD), this leads to a completely forward radiation directionality, known as the Kerker  condition [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' When these dipolar moments are out-of-phase, |ED| = |MD|, arg(ED) = arg(MD)+π, it results in  a completely backward directionality, known as the anti-Kerker condition [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' When the superposition of dipolar  a  L=2um  Unit cell  D = 424 nm  X  P  10  ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  (c)  ED  ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  MD  Phase(rad)  EQ  MQ  0  L  S  T  业  0  500  1000  1500  2000  2500  1000  1500  2000  P(nm)  P(nm)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The 2D electric field norm within (a) a single SiC pillar and the SiC pillar metasurface with  P = (b) 2315 nm, (c) 1500 nm and (d) 1095 nm under (i) plane wave excitation with wave-vector  along the +z-direction and electric field polarized along the +x-direction and (ii) dipole excitation  with dipole emitter placed at the center of the SiC pillar with orientation along the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  as well as the quadrapolar moments are balanced and are in phase, |ED + MD| = |EQ + MQ| with arg(ED +  MD) = arg(ED + MD), the radiation pattern is highly directional along the forward direction, known as the  generalised Kerker condition [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' However, when these superpositions are out-of-phase, this leads to a complete  transverse scattering [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The phase of the Mie-scattering moments therefore controls the far-field scattering  radiation pattern of the scatterer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Under a point dipole emitter excitation of the structure, the far-field scattering  pattern of the structures also influences the radiation pattern of the dipole emitter placed in the vicinity of the  scatterer [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  In the following we investigate these effects in SiC nanopillars array under plane wave-excitation and under a  single dipole excitation simulating the VSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  Results and discussion  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  Scattering efficiency and decay-rate enhancement  We have computationally optimised the SiC metasurface, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 1(a) with unit cell in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 1(b), to achieve  the generalised Kerker’s condition in SiC for the specific color centre of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The metasurface consists of a  periodic 2D lattice of SiC pillars, each of length, L = 2µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The electrodynamics calculations are performed using  the commercial Comsol Multiphysics RF module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The details of the calculations are presented in the Methods  sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The metasurface is excited by a plane wave with wavelength, λexc, travelling along the +z-direction  (arrow symbol in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 1(a)) with the electric field polarized along the +x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Under the influence of the  plane electromagnetic wave, Mie scattering moments are excited within the SiC pillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' We first optimised the  diameter, D, of the SiC pillars for the maxima in the SE at λexc = 917 nm corresponding to the zero phonon line  (ZPL) of the silicon vacancy, VSi in SiC [36,37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The optimised D value was found to be around 424 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' We then  study the coherent superposition of the Mie-scattering modes of the individual SiC pillars by varying the lattice  periodicity, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' For P ≫ λexc, the structure is expected to behave as a single isolated pillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' With decreasing P, the  interactions between the Mie-scattering modes of the individual pillars will increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' When these modes will be  in phase, their coherent superposition will lead to a maxima for the total SE of the 2D SiC pillar lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 1(c)  and (d) show the amplitude and the phase of the individual Mie-scattering moments of the SiC pillar metasurface  as a function of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Sharp resonance peaks are observed in the amplitude of the individual Mie-scattering moments  (iD)  (a) Single pillar  (a)Single pillar (b)P: 2315 nm  (b) P: 2315 nm  X  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content="5  ('n'e)  1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='5  (c) P: 1500 nm  (d) P: 1095 nm  (c)P: 1500 nm  (d)P: 1095 nm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='5Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The SE of the individual excited multipolar Mie-scattering moments and the emitter’s (VSi  color-center) relative decay rate in the SiC pillar metasurface as a function of the lattice periodicity,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The dipole emitter is placed at the center of the SiC pillars with dipole orientation along the  horizontal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The γ∞ is the emitter’s decay rate in the bulk SiC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' (b) The schematic representation  of the tuning of the embedded color-center’s emission with the lattice periodicity, P set to (i) off-  resonant P1 and (ii) resonant P2 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' At these sharp resonances, a sharp jump in the phase of the corresponding Mie-scattering moment is  observed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 1(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' We will now focus on the local maxima arising due to the ED and the MQ moments (under  dipole excitation of the structure only these two resonances were excited and were observed to have an influence  on the dipole emitter’s decay rates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' These local maxima are observed for P =915 nm, 1095 nm, 1500 nm and  2315 nm (black dotted lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 1(c) and (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' At P =915 nm, the ED resonance peak is much greater than the  MQ resonance with the phase of these two resonances being equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' For P =1095 nm, 1500 nm and 2315 nm, the  ED and MQ moments are nearly balanced and a sharp jump is also observed in their phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' We will now examine  the balanced superposition of these two moments at P =1095 nm, 1500 nm and 2315 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  Figure 2(i) shows the normalised electric field distribution within a (a) single SiC pillar and the SiC pillar meta-  surface with P = (b) 2315 nm, (c) 1500 nm and (d) 1095 nm under plane wave excitation with wave-vector along  the +z-direction and electric field polarized along the +x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Strong confinement of the electric field within  the SiC cylinder is observed at these P values corresponding to the balanced superposition of the ED and MQ  resonances, with the maximum field confinement observed for P = 1500 nm (the SE was also observed to be max-  imum at this P value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The field confinement will in-turn lead to LDOS enhancement within the SiC pillar (Eq 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  For a dipole emitter placed at the field maxima points, the LDOS enhancement will lead to its decay rate enhance-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Figure Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 2(ii) shows the normalised electric field distribution within (a) a single SiC pillar and the SiC  pillar metasurface with P = (b) 2315 nm, (c) 1500 nm and (d) 1095 nm under dipole excitation with dipole emitter  placed at the center of the SiC pillars with orientation along the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Large field confinement/enhancement  which will lead to large LDOS enhancement can be observed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  We now study the influence of the above LDOS enhancement on the spontaneous emission rates of a dipole  emitter, the VSi color-center embedded at the center of each SiC pillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Figure 3(a) shows the emitter’s (VSi color-  center) relative decay rate together with the SE of the individual Mie-scattering moments in the SiC pillar meta-  surface as a function of the lattice periodicity, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The decay rates of the VSi emitter in the SiC pillar metasurface,  γ are scaled relative to its decay rates in a bulk SiC crystal, γ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The influence of the LDOS enhancement arising  from the electric field confinement (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 2) in tuning the emitter’s decay rate can be clearly observed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Also, it  can be observed that the relative decay rates ( γ  γ∞ ) (dash-dotted red curve) only tunes with the local maxima which  are dominated by ED and MQ resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' These resonances corresponds to P = 1095 nm, 1500 nm and 2315  nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' A schematic representation of the embedded dipole emitter’s radiation tuning with the SiC pillar  lattice periodicity, P at an off-resonant (i) and resonant (ii) value is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 4, we study the SE spectral response due to all the excited Mie-scattering moments and the effect  on the relative decay rates of a horizontally oriented (along x-direction) dipole source for the above resonant  20  (a)  ED  (b)  30  MD  15  EQ  SE (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=')  MQ  20  8  10  8  10  (i)  (iD)  500  1000  1500 2000  2500  P (nm)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The spectral response of the SE with the individual excited multipolar Mie-scattering mo-  ments under horizontal dipole excitation and the emitter’s (VSi color-center with dipole orientation  along the horizontal plane) relative decay rate in a (a) single SiC pillar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' SiC pillar metastuface with  P = (b) 2315nm, (c) 1500 nm and (d) 1095 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  periodicity values (P = 1095 nm, 1500 nm and 2315 nm) of the metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Here, the Mie-scattering moments  of the metasurface are excited by the dipole source itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' For an isolated SiC pillar (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 4(a)), all the studied  Mie-scattering moments are observed to be weakly excited with no clear resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The relative decay rate  (dash-dotted red curve) is observed to be around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='7 at 917 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' However, for all resonant P values (P = 1095  nm, 1500 nm and 2315 nm), significant contributions are observed only from the ED and MQ moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Their  superposition is controlling the behaviour of the SE and the relative decay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The maximum relative decay rate  enhancement is close to 30 at 917 nm for P = 1500 nm and 1095 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' For P = 2315 nm the enhancement is about  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Therefore, it can be concluded that the coherent superposition of the ED and MQ Mie-scattering moments  of the individual pillars are enhancing the decay rates of an embedded dipole emitter by more than an order of  magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  We now study the role of the phase of the excited Mie-scattering moments (ED and MQ) of the SiC pillar  metasurface on the far field radiation pattern of an embedded dipole emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  Phase analysis and Radiation pattern  Figure 5 shows a narrow range of values for both the relative amplitudes and phase of the ED and MQ moments  at the resonant P values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' At the VSi color-center’s peak emission wavelength of 917 nm, the MQ moment appears  to be slightly larger than the ED moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' For P = 1500 nm, the ED  MQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='7 and for P = 1095 nm and 2315 nm, the  ED  MQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The corresponding phase difference between ED and MQ moments is π at 917 nm for all resonant P  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 6(i) we show the in-phase and out-of-phase superposition of the balanced ED and MQ moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The  influence of this superposition on both the in-plane (X-Z plane, blue curve) and out-of-plane (Y-Z plane, red  curve) far-field scattering patterns is shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The in-phase (ED + MQ) superposition results in longitudinal  (along top and bottom directions) scattering and the out-of-phase (ED − MQ) superposition results in transverse  scattering along the out-of-plane direction (red curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  We now study the influence of the ED and MQ moments superpositions on the embedded dipole emitter radi-  ation pattern for a single nanopillar and SiC metasurface of array of interacting nanopillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The emitter’s dipole  orientation is along the X-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 6(ii) shows the far-field emission patterns of the embedded VSi color-  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='5  Single Pillar  ED  P = 2315nm  ED  (b)  30  30  MD  MD  EQ  EQ  1  SE(a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=')  20  MQ  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='8  8  MQ  a  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='. TOTAL  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' TOTAL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='5  10  10  8  8  0  0  0  900  910  920  930  940  950  900  910  930  940  920  950  入(nm)  入(nm)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='5  = 1500 nm  P = 1095nm  (d)  D  (c)  ED  ED  30  30  MD  MD  EQ  EQ  L  (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=')  20_8  一MQ  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='8  一MQ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='.TOTAL  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' TOTAL  10  10  8  0  0  0  900  910  920  930  940  950  900  910  920  930  940  950  入(nm)  入(nm)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The spectral response of the (a) relative amplitudes and (b) phase difference of the elec-  tric dipolar (ED) and magnetic quadrapular (MQ) Mie-scattering moments excited under horizontal  dipole excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  center at 917 nm for different resonant P values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' In a single pillar, the majority of the emission is observed to be  directed towards the bottom surface and lies mainly along the longitudinal direction (bottom and top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' However,  in the SiC metasurface at the resonant P values corresponding to ED and MQ moments, the out-of-phase super-  position of the ED and MQ moments (the phase difference between these moments was observed to be close to π  at 917 nm in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 5) directs the embedded emitter’s radiation pattern along the transverse direction, especially in  the out-of-plane (Y-Z plane, red curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  Conclusion  We studied for the first time the coherent superposition of the eletric and magnetic dipolar and quadrupolar Mie-  scattering moments of SiC metamaterial nanopillars array in the near infrared emission (917 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' We first de-  termined the design of the metasurface periodicity to induce sharp resonances in the amplitude and phase of the  Mie-scattering moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Strong electric field confinement was observed within the SiC pillar when the periodicity  of the lattice matched with the resonance of the ED and MQ modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The field confinement leads to large LDOS  and subsequently decay rate enhancement for a color-center dipole embedded at the center of the SiC pillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Under  a point dipole emitter excitation within SiC, it was determined that only the ED and MQ moments are contribut-  ing to the electromagnetic scattering in the SiC nanopillars metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Both these moments were observed to  be nicely coupled and were showing collective resonance at the optimised wavelength (917 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The coherent  superposition of these two moments controls the complete spontaneous emission process of the embedded color  center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' At the collective resonance point of these two moments (λ = 917 nm), we determined more than an order  of magnitude decay rate enhancement with the maximum enhancement reaching 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Such an enhancement has  never been reported in dielectric neither in metal-dielectric individual nanopillars [38], thus paving the way for  the use SiC metasurfaces to to enhance and control light extraction from quantum emitters, to study light matter  interaction effects in integrated quantum photonics and for applications in quantum sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' We also observed that  by designing specific resonant structures, the coherent superposition of the ED and MQ moments can be used to  better control the radiation/emission pattern and hence the emission directionality of the embedded dipole emitter  compared to a single nanopillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Specifically the embedded emitter’s radiation pattern can be more confined along  the transverse direction, especially in the out-of-plane (Y-Z plane, red curve), thus facilitating the planar emission  propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Such result is relevant for applications of SiC metasurfaces for planar integrated photonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Our study  can prompt further studies on SiC quantum states of light based on multi-emitters-resonators coupling enabled by  metamaterials such as superradiance [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  Materials and Methods  All the electrodynamics calculations have been performed using the commercial COMSOL Multiphysics Radio  Frequency (RF) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The periodic boundary conditions are applied to all the horizontal planes defining the 2D  lattice of the SiC substrate to build an array of dielectric pillars metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The Scattering boundary conditions  are applied at the top and bottom boundaries of the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The optical constant for the SiC has  been extracted from the experimentally reported values by Singh et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' During the entire calculation the  minimum meshing size was 1 nm with the maximum being λ/7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  (a)  (b)  Single pillar  Phase difference (rad)  T  ED  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='P: 1095 nm  e  arg(ED) -  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='5  MQ  P: 1500 nm  arg (MQ)  P: 2315 nm  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='5  T  0  913  915  917  919  921  913  915  917  919  921  入 (nm)  入(nm)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' (i) In-plane (X-Z plane, blue curves) and out-of-plane (Y-Z plane, red curves) far-field scat-  tering pattern corresponding to the in-phase (ED+MD) and out-of-phase (ED-MQ) superposition  of the electric dipolar (ED) and magnetic quadrupolar (MQ) Mie-scattering moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' (ii) Farfield  radiation patterns in-plane (X-Z plane, blue curves) and out-of-plane (Y-Z plane, red curves) of a  dipole emitter (orientation along the X-direction, same as that of electric field in (i)) placed at the  center of (a) single pillar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' SiC pillar metastuface with P = (b) 2315nm, (c) 1500 nm and (d) 1095  nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  Scattering efficiency calculation  The scattering cross section is defined as the amount of power scattered by the scatterer to the amount of power  per unit area carried by the incident wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The SE is obtained just by dividing the scattering cross-section by  the geometrical cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Mathematically it is expressed as SE = σs/G [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Here σs is the scattering cross  section and G is geometrical cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  The SE is calculated semi-analytically using electric field values at each mesh point in the computational grid  under plane-wave excitation using Comsol Multi-physics module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Using these field values and permittivity profile  at each mesh points, current density is calculated as: Jω(r) = iωε0(εr −1)Eω(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Here ε0 and εr are permittivity of  free space and SiC medium, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The computationally obtained values of E(r), Jω(r) and ε(r) are used to  calculate individual multipolar Mie-scattering moments, pα, mα, Qe  αβ and Qm  αβ described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The integration  referred in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 2 is carried on the domain of the SiC pillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  Relative decay rate calculations  In these calculations, the SiV center is treated as a classical radiating point dipole source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' In the computational  domain, it is modelled as a point current source driven at the emission frequency, ν = c  λ [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' Scattering/perfectly  matched layer (PML) boundary conditions are applied on the exterior boundaries of the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  The total power radiated by the dipole is integrated over a closed surface enclosing the current source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The relative  decay-rate is calculated as Γrel = γ/γ∞ = P/P∞ [42], where P∞ is the power corresponding to the point dipole’s  emission in the bulk SiC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The permittivity of SiC is taken from [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  Acknowledgement The authors would like to acknowledge the financial support from the Department of Sci-  ence and Technology (DST), India (CRG/2021/001167).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' The authors thank Dr Nadeem Ahmed for his help regard-  ing the plotting of the figures and the formatting of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' MA and FAI also thank Dr Ahmed Mekawy  for his help regarding multipole decomposition of the Mie-scattering moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
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+page_content=' Scherer, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
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+page_content=' 10,  Issue 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
+page_content=' 1034-1036 10, 1034–1036 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE4T4oBgHgl3EQfOAxw/content/2301.04961v1.pdf'}
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+ 
+1 
+ 
+ 
+Designing an Improved Deep Learning-based Model for 
+COVID-19 Recognition in Chest X-ray Images: A Knowledge 
+Distillation Approach 
+AmirReza BabaAhmadi1, Sahar Khalafi2, Masoud ShariatPanahi3 , Moosa Ayati4 
+ 
+Abstract 
+Background and Objectives: COVID-19 has adversely affected humans and societies in different aspects. Numerous 
+people have perished due to inaccurate COVID-19 identification and, consequently, a lack of appropriate medical 
+treatment. Numerous solutions based on manual and automatic feature extraction techniques have been investigated 
+to address this issue by researchers worldwide. Typically, automatic feature extraction methods, particularly deep 
+learning models, necessitate a powerful hardware system to perform the necessary computations. Unfortunately, many 
+institutions and societies cannot benefit from these advancements due to the prohibitively high cost of high-quality 
+hardware equipment.  As a result, this study focused on two primary goals: first, lowering the computational costs 
+associated with running the proposed model on embedded devices, mobile devices, and conventional computers; and 
+second, improving the model's performance in comparison to previously published methods (at least performs on par 
+with state of the art models) in order to ensure its performance and accuracy for the medical recognition task. 
+Methods: This study used two neural networks to improve feature extraction from our dataset: VGG19 and 
+ResNet50V2. Both of these networks are capable of providing semantic features from the nominated dataset. 
+Streaming in a fully connected classifier layer that feeds richer features, therefore feature vectors of these networks 
+have been merged, and this action resulted in satisfactory classification results for normal and COVID-19 cases. On 
+the other hand, these two networks have many layers and require a significant amount of computation.  To this end, 
+An alternative network was considered, namely MobileNetV2, which excels at extracting semantic features while 
+requiring minimal computation on mobile and embedded devices. Knowledge distillation (KD) was used to transfer 
+knowledge from the teacher network (concatenated ResNet50V2 and VGG19) to the student network (MobileNetV2) 
+to improve MobileNetV2 performance and to achieve a robust and accurate model for the COVID-19 identification 
+task from chest X-ray images. 
+Results: Pre-trained networks were used to provide a more useful starting point for the COVID-19 detection task. 
+Additionally, a 5-fold cross-validation technique was used on both the teacher and student networks to evaluate the 
+ 
+1 School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran (email: babaahmadi.amir@ut.ac.ir) 
+2 Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan, Iran 
+3 School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran 
+4 School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran 
+
+ 
+2 
+ 
+ 
+proposed method's performance.  Finally, the proposed model achieved 98.8% accuracy in detecting infectious and 
+normal cases. 
+Conclusion: The study results demonstrate the proposed method's superior performance. With the student model 
+achieving acceptable accuracy and F1-score using cross-validation technique, it can be concluded that this network is 
+well-suited for conventional computers, embedded systems, and clinical experts' cell phones. 
+Keywords: COVID-19, Deep Learning, Medical Image Analysis, Knowledge Distillation, Chest X-ray images, 
+Teacher-Student Model 
+ 
+ 
+1. Background and Objectives 
+COVID-19 has been a significant threat to human life and health in recent years. It has delivered a detrimental effect 
+on healthcare systems worldwide. Unfortunately, COVID-19 is highly contagious and transmissible. As a result, it is 
+inevitable to develop prediction systems capable of rapidly diagnosing it and averting its adverse effects. Numerous 
+scientists have conducted multiple studies to develop human-level prediction and diagnosis systems to aid 
+communities in combating this disease. 
+These methods are typically effective at detecting COVID-19 using CT and chest X-ray images. However, the fatal 
+flaw is that automatic feature extraction techniques, particularly deep learning models, require a significant amount of 
+computation to perform a specific task. Numerous hospitals and institutions are unable to procure expensive hardware 
+systems capable of performing these algorithms. 
+Additionally, in the absence of medical experts, the health of the population in many developing countries may be 
+jeopardized. In some developing countries, the scarcity of human experts capable of identifying medical diseases from 
+medical images is a stark reality. As a result, an effort is made to offer an alternate solution to remedy this issue in 
+this study.  
+This paper focuses on two primary objectives and charts the course accordingly: first, a fast algorithm capable of being 
+deployed in conventional computers and embedded in mobile devices and tablets, and other embedded systems 
+requires development. The second objective, which prompted developing a novel deep learning model, was to deliver 
+more accurate and reliable results than previously published methods to ensure the algorithm's performance for 
+medical diagnosis. 
+Recognizing COVID-19 is the first step toward treatment. The main objective of this paper is an attempt to address 
+this step. COVID-19 is typically accompanied by symptoms including coughing, colds, and shortness of breath, among 
+others. According to the WHO, respiratory problems are the fatal symptoms of COVID-19 and can be identified using 
+CT scans or chest X-rays obtained in hospitals and medical clinics. 
+
+ 
+3 
+ 
+ 
+Deep learning and its associated techniques have been widely applied to computer vision tasks such as medical image 
+analysis in recent years. Examples include tumor detection [1], diabetic retinopathy [2], among others. 
+Considering that medical image analysis is one of the most exciting applications of computer vision and deep learning, 
+automatic feature extractor algorithms can be critical in diagnosing diseases at levels comparable to human experts. 
+Due to the critical nature of the initial diagnosis, researchers have identified this step's accuracy as a significant 
+challenge. The following section examines related works and studies in order to create a solid framework for our 
+research. In [3], a model based on stacked convolutional neural networks with multiple pre-trained networks was 
+proposed. In [4], a residual-layer-based neural network with a modified kernel was used to prognosticate the presence 
+of COVID-19 in normal and pneumonia chest X-ray images. 
+[5] proposes a model based on an ensemble learning classifier to address the imbalance dataset issue. Another paper 
+[6] introduced a combination of LSTM5 (for detection) and a convolutional neural network (for recognition). This 
+method produced a 99.4% accuracy metric. [7] describes the development of a model using LSTM and GAN 
+architectures. This method does not require the addition of a network for feature extraction for the binary classification 
+task. [8] investigated the presence of COVID-19 in medical images using a deep neural network based on a CNN with 
+additional components in different layers. 
+In [9], CNN and SVM6 were combined to analyze COVID-19 CXR images. [10] proposes a new framework based on 
+the use of DarkNet53 to perform chest X-ray images. [11] conducted a comparative study to determine which model 
+performs the best among pre-trained deep learning architectures. Fifteen models were examined in order to determine 
+the optimal architecture. According to this study, VGG19 was the most accurate pre-trained model. Another 
+comparative study [12] classified COVID-19 from normal cases and pneumonia using novel pre-trained architectures 
+such as DenseNet 201, Inception V3, Resnet50, and a few other networks. [13] introduced a new model based on 
+GAN and CNN. Multiple classifiers were used to perform the classification task in [13]. Furthermore, this paper 
+developed a COVID-19 segmentation model using a unique dataset.  
+[14] compared pre-trained architectures and then modified those networks to reduce trainable parameters and enable 
+faster algorithm training. [15] examined a variety of pre-trained architectures, including Alex-Net, VGG16, and the 
+ResNet family. Moreover, this article concentrated on evaluating freezing layers and other configurations used in 
+transfer learning methods. A study was conducted in [16] to determine the best candidate for COVID-19 recognition. 
+Based on this research, the outstanding architecture for COVID-19 detection is achieved through the Inception 
+Network. Another comparative study was conducted in [17] using DenseNet121, ResNet50, VGG16, and VGG19. 
+In [18], a binary classification task has been performed. Fine-tuning was performed on the final layer of the pre-trained 
+SqueezeNet, DenseNet121, ResNet50, and ResNet18 networks. [19] depicts a voting-based procedure for evaluating 
+ 
+5 Long Short-Term Memory (LSTM) 
+6 Support Vector Machine (SVM) 
+
+ 
+4 
+ 
+ 
+VGG16, InceptionV3, and ResNet50 predictions. The final layer (output) compares the performance of all of the 
+networks mentioned previously in a simultaneous manner and selects the best result. 
+In [20], an assessment of the effect of preprocessing on the results of CNN-based networks is made. [21] describes 
+the development of a customized network dubbed COVID-Net for CXR and chest X-ray medical images. Another 
+study [22] compared COVID-19, bacterial pneumonia, and viral-pneumonia classification results using state-of-the-
+art algorithms and CNN architecture. In another study [23], PCA7 was used to increase the efficiency of feature 
+extraction. YOLO8 Networks with additional convolution layers involving modified kernels were used in [24] to detect 
+COVID-19 from chest X-ray images. 
+[25] examined the performance of the Efficient-Net family in detecting COVID-19 and pneumonia in normal cases. 
+In [26], a network, termed CoroNet, was introduced based on the Xception network to perform multi-class 
+classification from normal category images, including pneumonia-viral, pneumonia-bacterial, and COVID-19. 
+This paper is structured as follows: The second section provides an overview of previous publications on the task. The 
+third section describes the knowledge distillation method and explains why it was incorporated into the algorithm. 
+Section four discusses the dataset that was used to conduct this research. Section five describes the training phase and 
+the evaluation metrics. Finally, section six presents the proposed model's evaluation results, and section seven 
+summarizes our paper's findings. 
+ 
+2. Methods 
+Nowadays, Deep Neural Networks9 have revolutionized the field of computer vision. Their applications have been 
+extensively investigated in a variety of fields, including self-driving cars [27], medical image analysis [28][29], and 
+agriculture [30], among others. CNNs10 have established themselves as the most effective tools for automatic feature 
+extraction in computer vision and NLP11, speech processing, and video classification tasks. 
+This paper demonstrates how CNN-based neural networks can improve semantic feature extraction for binary 
+classification tasks involving COVID-19 and normal cases. DenseNet [31], ResNet [32], VGG [33], Xception [34], 
+and Mobile-NetV2 [35] are some of the most powerful pre-trained networks. VGG19 is an extension of VGG16; it 
+features sixteen convolutional layers and three fully connected layers. Five MaxPool layers are used, and the final 
+layer is a SoftMax layer. ResNet50V2 is an enhanced version of ResNet50 that outperforms previous versions such 
+as ResNet101. 
+ 
+7 Principal Component Analysis (PCA) 
+8 You Only Look Once (YOLO) 
+9 Deep Neural Networks (DNN) 
+10 Convolutional Neural Networks (CNN) 
+11 Natural Language Processing (NLP) 
+
+ 
+5 
+ 
+ 
+ResNet50V2 has been equipped with several new connections between different blocks, allowing this network to 
+achieve a high level of accuracy in the ImageNet competition. MobileNetV2 is a convolutional network designed with 
+depth-wise convolution layers to improve accuracy while lowering computational costs. This reduction is caused by 
+decreasing trainable parameters. Due to the use of inverted residual blocks, this network can be embedded in mobile 
+phones, tablets, and other conventional embedded devices. 
+Since both ResNet50V2 and VGG19 generate the same size output layer (feature vector), their feature vectors were 
+combined to produce a richer semantic feature set for the specified task. Afterward, a CNN layer was added to this 
+architecture (kernel size 1 and 1024 filters). The network's output was then flattened and streamed into the fully 
+connected layers. Notably, no activation function in the final CNN layer was used.  The classification layer was 
+constructed using 64 neurons in fully connected layers, with a dropout rate of 0.5. Another fully connected layer has 
+been added to this layer, the final layer, whose neurons count is equal to the size of the problem's classes. 
+The entire architecture is termed the teacher network, with MobileNetV2 serving as the student network in a broader 
+context referred to as the teacher-student model or knowledge distillation framework. The structure of the teacher 
+model is depicted in Figure 1. The tensors' feature size for both ResNet50V2 and VGG19 is 10*10*2048, and the size 
+of the concatenated feature is 10*10*4096. 
+ 
+ 
+Figure 1. Teacher Model Architecture 
+ 
+
+Teacher Model Architecture
+ResNet50V2
+10*10*2048
+CNNlayer
+Filteri024
+Dropout(50%)
+VGG19
+Kernel(1,l)
+10*10+2048
+10*10*4096
+n
+Classifier
+Merged Features 
+6 
+ 
+ 
+ 
+ 
+3. Knowledge Distillation 
+Hinton et al. pioneered  Knowledge Distillation12 in [36]. KD is a process that involves training a smaller network to 
+imitate the behavior of a more extensive network.  The purpose of designing a complex network as a teacher is to 
+learn more sophisticated features and deliver better results. However, we typically want to run our network on a 
+standard computer or embedded device. 
+Due to the limitation of memory size and computational cost, frequent issues arise. As a result, a solution is required 
+to address these issues. A weighted average (mean) is necessary to distill knowledge from teacher to student. Cross-
+Entropy with soft targets is the initial objective function. Through the softmax function in the smaller network, this 
+objective function is calculated based on high temperature. A more significant architecture (network) must be used to 
+generate soft targets. Cross-Entropy with valid labels is the second objective function. This function is calculated 
+using the softmax output from the student model by setting the temperature to zero. 
+The teacher network and the student network begin receiving training data in parallel. The teacher model contains a 
+softmax with temperature in its output.  By contrast, the student model generates two different outputs. The first output 
+is softmax with temperature, while the second output contains standard softmax. The student model is intended to 
+produce softened probabilities (the output of the teacher model). The following formula is used to calculate the loss 
+of knowledge distillation:  
+2
+( . )
+(1
+) (
+. )
+KD
+s
+L
+KL p q T
+L W x
+
+
+=
++
+−
+     
+(1) 
+Where p and q denote the probabilities generated by student and teacher networks in a specific temperature (T), 
+respectively, and KL denotes the Kullback-Leibler divergence, which measures the level of distinction between two 
+probabilistic distributions. The Cross-Entropy of the student model with T=1 is (LWs.x). According to [36],  and T  
+are hyperparameters where the greater the value of  , the better the learning experience for the student model. 
+Back-propagation must be performed only in the student network during the distillation phase to add a significant 
+element to this description since the teacher has already tuned its parameters. The teacher's knowledge is then 
+transferred to the student model throughout the distillation procedure. Notably, the student model can be trained at a 
+faster rate than the teacher model. For more details regarding the distillation procedure, please refer to [37]. The 
+procedure for knowledge distillation is depicted in Figure 1. 
+ 
+12 Knowledge Distillation (KD) 
+
+ 
+7 
+ 
+ 
+ 
+Figure 2.  Knowledge Distillation for COVID-19 detection 
+4.  Dataset 
+Two public datasets were used to train the proposed deep learning model to build the required dataset. First, a public 
+dataset available at (https://github.com/ieee8023/covid-chestxray-dataset) was used for positive samples of COVID-
+19. Afterward, the dataset available at (https://www.kaggle.com/c/rsna-pneumonia-detection-challenge) was used to 
+collect negative samples (normal cases).  
+Following the two datasets being merged, 118 COVID-19 cases and 8851 normal cases were established. It became 
+clear that an unbalanced dataset was created due to the number of positive cases (COVID-19) being significantly 
+lower than the number of normal chest X-ray medical images. As a result, the issue was mitigated through the use of 
+sampling techniques. The central concept is to select an equal number of items from each category for the binary 
+classification task. The oversampling method was used to increase COVID-19 (positive cases) samples to ensure that 
+both positive and negative classes had an equal number of samples. 
+The number of positive cases increased to 8851 following the oversampling technique, while the number of negative 
+(normal) samples remained unchanged. It should be noted that no images of pneumonia were used in this study. 
+Pneumonia is classified into several different classes, including SARS, Streptococcus, ARDS, and Pneumocystis. In 
+this respect, treating all of these categories as a single class was deemed as impractical. This cannot be very clear in 
+terms of interpreting recognition task results as distinct pneumonia types require a unique type of treatment. As a 
+result, developing a pneumonia-type classifier was deferred to a later date. Figure 3  illustrates a selection of patients 
+with COVID-19 and normal images from the dataset. 
+
+Generating soft targets
+Teacher
+Network(ResNet50V2/VGG19)
+Chest X-ray Dataset
+Backpropagation
+Loss
+Student
+Trainingthe student
+Network(MobileNetV2) 
+8 
+ 
+ 
+ 
+ 
+Figure 3(a): Healthy person  
+ 
+Figure 3(b): Patient with COVID-19 
+4.1 Data Augmentation 
+Before data augmentation, the images were normalized to avoid issues with vanishing and exploding gradients. After 
+that, the image was resized to 224x224. The model was then enhanced with data augmentation techniques to make it 
+more responsive to variations within the medical images. During data augmentation, it was assumed that variations in 
+the images did not affect the label's (ground truth) definition. Random rotation was the only data augmentation 
+
+ 
+9 
+ 
+ 
+approach considered for the dataset, between (0-200). This can prevent overfitting and also helps the learning curve 
+converge more quickly. 
+5. Training Phase 
+The training phase consumed approximately 80% of the dataset, with the remaining 20% used for the test phase. The 
+K-fold Cross-Validation technique was used to ensure the accuracy of the performance evaluation. The loss function 
+was binary Cross-Entropy, and the optimizer was Adam (with a learning rate=1e-5). The number of folds was 5 (k=5). 
+The batch size was 32, and the epoch number was set to 120. 
+5.1 Evaluation Metrics 
+Several specific criteria need to be established for evaluating the model's performance on the test dataset. True Positive 
+refers to correctly identifying COVID-19 positives among both positive and normal cases. In the conceptualization, 
+true negativity entails accurately identifying normal cases. False Positive refers to the practice of misdiagnosing 
+COVID-19 cases as normal. False-negative prediction is misclassifying normal cases as COVID-19 cases. 
+Precision is defined as the ratio of True Positives over the sum of True Positives and False Positives. 
+ 
+TP
+Precision
+TP
+FP
+=
++
+                  (1) 
+ 
+Sensitivity is the ratio of True Positives over to the sum of True Positives and False Negatives. 
+ 
+(
+)
+TP
+Sensitivity recall
+TP
+FN
+=
++
+ 
+(2) 
+ 
+Specificity is the ratio of True Negatives over the sum of False Positives and True Negatives. 
+ 
+TN
+Specificity
+TN
+FP
+=
++
+ 
+(3) 
+ 
+
+ 
+10 
+ 
+ 
+Finally, three additional criteria were used to measure the performance of the proposed model on the study's dataset 
+(F1 Score and Balanced-Accuracy in addition to conventional accuracy).  
+2*
+*
+1
+(
+)
+precision recall
+F
+Score
+precision
+recall
+−
+=
++
+ 
+(4) 
+ 
+2
+specificity
+sensitivity
+Balanced Accuracy
++
+=
+   (5) 
+ 
+TP
+TN
+Accuracy
+Positive
+Negative
++
+=
++
+ 
+(6) 
+ 
+6. Results and Discussion 
+In this section, we present the results of our method applied to the dataset mentioned above. The images which have 
+been used for the test have not been seen before by the algorithm. Therefore, these results are approvable that the 
+proposed algorithm can perform very well on unseen medical images. Table 1 presents the results of the training 
+teacher model. It is conspicuous that the teacher's performance is enough to detect chest X-ray images with good 
+accuracy and F1-score. Table 2 shows the results of MobileNetV2 (student model). Its results are satisfactory for the 
+classification task. Despite presenting good results of the student network, we decided to improve its classification 
+ability via the knowledge distillation approach. Table 3 is presenting the results of the student network after 
+knowledge distillation is completed. Eventually, the student network performs  on par with the existing methods in 
+the literature and sometimes achieves better accuracy and F1-score in comparison with previous publications. 
+ 
+Table 1: Evaluation results of the Teacher Network (ResNet50V2/VGG19) 
+Fold No.  
+Acc 
+Precision 
+Specificity 
+Recall(sensitivity) 
+F1 score 
+Balanced- 
+Acc 
+1 
+0.978 
+0.978 
+0.978 
+0.978 
+0.978 
+0.978 
+2 
+0.989 
+0.978 
+1.000 
+1.000 
+0.988 
+1.000 
+3 
+1.000 
+1.000 
+1.000 
+1.000 
+1.000 
+1.000 
+4 
+0.989 
+0.978 
+1.000 
+1.000 
+0.988 
+1.000 
+5 
+1.000 
+1.000 
+1.000 
+1.000 
+1.000 
+1.000 
+Average 
+0.992 
+0.987 
+0.996 
+0.996 
+0.991 
+0.996 
+ 
+
+ 
+11 
+ 
+ 
+ 
+ 
+Table 2: Evaluation results of the Student Network (MobileNetV2) Before Knowledge-Distillation 
+Fold No.  
+Acc 
+Precision 
+Specificity 
+Recall(sensitivity) 
+F1 score 
+Balanced- 
+Acc 
+1 
+0.987 
+0.978 
+1.000 
+1.000 
+0.992 
+1.000 
+2 
+0.974 
+0.976 
+0.976 
+0.976 
+0.972 
+0.976 
+3 
+0.974 
+0.976 
+0.976 
+0.976 
+0.972 
+0.976 
+4 
+0.987 
+0.978 
+1.000 
+1.000 
+0.992 
+1.000 
+5 
+0.987 
+0.978 
+1.000 
+1.000 
+0.992 
+1.000 
+Average 
+0.980 
+0.977 
+0.990 
+0.990 
+0.984 
+0.990 
+ 
+Table 3: Evaluation results of the Student Network (MobileNetV2) After Knowledge-Distillation 
+Fold No.  
+Acc 
+Precision 
+Specificity 
+Recall(sensitivity) 
+F1 score 
+Balanced- 
+Acc 
+1 
+0.974 
+0.976 
+0.976 
+0.976 
+0.972 
+0.976 
+2 
+1.000 
+1.000 
+1.000 
+1.000 
+1.000 
+1.000 
+3 
+0.978 
+0.978 
+1.000 
+1.000 
+0.992 
+1.000 
+4 
+1.000 
+1.000 
+1.000 
+1.000 
+1.000 
+1.000 
+5 
+0.989 
+0.978 
+1.000 
+1.000 
+0.988 
+1.000 
+Average 
+0.988 
+0.987 
+0.996 
+0.996 
+0.991 
+0.996 
+%Improvement 
+0.8 % 
+1.0 % 
+0.6 % 
+0.6 % 
+0.7 % 
+0.6 % 
+ 
+ 
+Table 4: Number of total parameters in each architecture 
+Teacher Network (ResNet50V2/VGG19) 
+49,222,390 
+Student Network (MobileNetV2) 
+2,334,966 
+Number of Parameters Reduction (%) 
+-95.3  %  
+ 
+ 
+It is noteworthy that the student network has outperformed its previous version(without KD) in terms of evaluation 
+metrics performance, and this is because knowledge distillation improves MobileNetV2's performance to some extent. 
+Meanwhile, KD aids the network in mitigating common neural network forgetting problems. Thus, the student model's 
+performance demonstrates that it can be used for medical recognition tasks in embedding systems while requiring 
+minimal computation, owing to the use of depthwise convolutional layers and knowledge distillation. Table 4 shows 
+the number of parameters in each architecture. It’s conspicuous that not only KD can improve student model’s 
+performance, but also it reduces  the number of parameters about 95.3% while maintainging perfromance. Therfore, 
+student model can be an eligible candidate for COVID-19 recognition task. 
+
+ 
+12 
+ 
+ 
+ 
+7. Conclusion 
+In this paper, a novel method was developed for identifying COVID-19 medical chest X-ray images. Due to 
+encountering an unbalanced dataset, this issue was resolved using oversampling and data augmentation techniques. In 
+evaluating the algorithm, the fivefold cross-validation method was used to ensure the proposed model's performance. 
+After performing knowledge distillation, an accuracy and F1-score of 98.8% and 99.1% were achieved respectively. 
+The proposed method, we believe, is an excellent choice for the COVID-19 recognition task. However, adding more 
+diverse datasets from different countries will help improve the algorithm. For future works on medical image datasets, 
+incremental learning techniques, self-supervised deep learning methods, vision transformer architectures, knowledge 
+distillation under adversarial attacks are proposed. 
+ 
+Declaration 
+Conflict of interest 
+The corresponding author declares that there are no conflicts of interest on behalf of all authors. 
+ 
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+
diff --git a/Z9E0T4oBgHgl3EQf4QJC/content/tmp_files/load_file.txt b/Z9E0T4oBgHgl3EQf4QJC/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf,len=789
+page_content='1  Designing an Improved Deep Learning-based Model for COVID-19 Recognition in Chest X-ray Images: A Knowledge Distillation Approach AmirReza BabaAhmadi1, Sahar Khalafi2, Masoud ShariatPanahi3 , Moosa Ayati4  Abstract Background and Objectives: COVID-19 has adversely affected humans and societies in different aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Numerous people have perished due to inaccurate COVID-19 identification and, consequently, a lack of appropriate medical treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Numerous solutions based on manual and automatic feature extraction techniques have been investigated to address this issue by researchers worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Typically, automatic feature extraction methods, particularly deep learning models, necessitate a powerful hardware system to perform the necessary computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Unfortunately, many institutions and societies cannot benefit from these advancements due to the prohibitively high cost of high-quality hardware equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' As a result, this study focused on two primary goals: first, lowering the computational costs associated with running the proposed model on embedded devices, mobile devices, and conventional computers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=" and second, improving the model's performance in comparison to previously published methods (at least performs on par with state of the art models) in order to ensure its performance and accuracy for the medical recognition task." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Methods: This study used two neural networks to improve feature extraction from our dataset: VGG19 and ResNet50V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Both of these networks are capable of providing semantic features from the nominated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='  Streaming in a fully connected classifier layer that feeds richer features, therefore feature vectors of these networks have been merged, and this action resulted in satisfactory classification results for normal and COVID-19 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' On the other hand, these two networks have many layers and require a significant amount of computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' To this end, An alternative network was considered, namely MobileNetV2, which excels at extracting semantic features while requiring minimal computation on mobile and embedded devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Knowledge distillation (KD) was used to transfer knowledge from the teacher network (concatenated ResNet50V2 and VGG19) to the student network (MobileNetV2) to improve MobileNetV2 performance and to achieve a robust and accurate model for the COVID-19 identification task from chest X-ray images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Results: Pre-trained networks were used to provide a more useful starting point for the COVID-19 detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Additionally, a 5-fold cross-validation technique was used on both the teacher and student networks to evaluate the  1 School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran (email: babaahmadi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='amir@ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content="ir) 2 Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan, Iran 3 School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran 4 School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran  2  proposed method's performance." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Finally, the proposed model achieved 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='8% accuracy in detecting infectious and normal cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=" Conclusion: The study results demonstrate the proposed method's superior performance." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=" With the student model achieving acceptable accuracy and F1-score using cross-validation technique, it can be concluded that this network is well-suited for conventional computers, embedded systems, and clinical experts' cell phones." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Keywords: COVID-19, Deep Learning, Medical Image Analysis, Knowledge Distillation, Chest X-ray images, Teacher-Student Model  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Background and Objectives COVID-19 has been a significant threat to human life and health in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' It has delivered a detrimental effect on healthcare systems worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Unfortunately, COVID-19 is highly contagious and transmissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' As a result, it is inevitable to develop prediction systems capable of rapidly diagnosing it and averting its adverse effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Numerous scientists have conducted multiple studies to develop human-level prediction and diagnosis systems to aid communities in combating this disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' These methods are typically effective at detecting COVID-19 using CT and chest X-ray images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' However, the fatal flaw is that automatic feature extraction techniques, particularly deep learning models, require a significant amount of computation to perform a specific task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Numerous hospitals and institutions are unable to procure expensive hardware systems capable of performing these algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Additionally, in the absence of medical experts, the health of the population in many developing countries may be jeopardized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' In some developing countries, the scarcity of human experts capable of identifying medical diseases from medical images is a stark reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' As a result, an effort is made to offer an alternate solution to remedy this issue in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='  This paper focuses on two primary objectives and charts the course accordingly: first, a fast algorithm capable of being deployed in conventional computers and embedded in mobile devices and tablets, and other embedded systems requires development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=" The second objective, which prompted developing a novel deep learning model, was to deliver more accurate and reliable results than previously published methods to ensure the algorithm's performance for medical diagnosis." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Recognizing COVID-19 is the first step toward treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The main objective of this paper is an attempt to address this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' COVID-19 is typically accompanied by symptoms including coughing, colds, and shortness of breath, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' According to the WHO, respiratory problems are the fatal symptoms of COVID-19 and can be identified using CT scans or chest X-rays obtained in hospitals and medical clinics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='  3  Deep learning and its associated techniques have been widely applied to computer vision tasks such as medical image analysis in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Examples include tumor detection [1], diabetic retinopathy [2], among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Considering that medical image analysis is one of the most exciting applications of computer vision and deep learning, automatic feature extractor algorithms can be critical in diagnosing diseases at levels comparable to human experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=" Due to the critical nature of the initial diagnosis, researchers have identified this step's accuracy as a significant challenge." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The following section examines related works and studies in order to create a solid framework for our research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' In [3], a model based on stacked convolutional neural networks with multiple pre-trained networks was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' In [4], a residual-layer-based neural network with a modified kernel was used to prognosticate the presence of COVID-19 in normal and pneumonia chest X-ray images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' [5] proposes a model based on an ensemble learning classifier to address the imbalance dataset issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Another paper [6] introduced a combination of LSTM5 (for detection) and a convolutional neural network (for recognition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' This method produced a 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='4% accuracy metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' [7] describes the development of a model using LSTM and GAN architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' This method does not require the addition of a network for feature extraction for the binary classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='  [8] investigated the presence of COVID-19 in medical images using a deep neural network based on a CNN with additional components in different layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' In [9], CNN and SVM6 were combined to analyze COVID-19 CXR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' [10] proposes a new framework based on the use of DarkNet53 to perform chest X-ray images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' [11] conducted a comparative study to determine which model performs the best among pre-trained deep learning architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Fifteen models were examined in order to determine the optimal architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' According to this study, VGG19 was the most accurate pre-trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Another comparative study [12] classified COVID-19 from normal cases and pneumonia using novel pre-trained architectures such as DenseNet 201, Inception V3, Resnet50, and a few other networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' [13] introduced a new model based on GAN and CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Multiple classifiers were used to perform the classification task in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Furthermore, this paper developed a COVID-19 segmentation model using a unique dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' [14] compared pre-trained architectures and then modified those networks to reduce trainable parameters and enable faster algorithm training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' [15] examined a variety of pre-trained architectures, including Alex-Net, VGG16, and the ResNet family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Moreover, this article concentrated on evaluating freezing layers and other configurations used in transfer learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' A study was conducted in [16] to determine the best candidate for COVID-19 recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='  Based on this research, the outstanding architecture for COVID-19 detection is achieved through the Inception Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Another comparative study was conducted in [17] using DenseNet121, ResNet50, VGG16, and VGG19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' In [18], a binary classification task has been performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Fine-tuning was performed on the final layer of the pre-trained SqueezeNet, DenseNet121, ResNet50, and ResNet18 networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' [19] depicts a voting-based procedure for evaluating  5 Long Short-Term Memory (LSTM) 6 Support Vector Machine (SVM)  4  VGG16, InceptionV3, and ResNet50 predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The final layer (output) compares the performance of all of the networks mentioned previously in a simultaneous manner and selects the best result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' In [20], an assessment of the effect of preprocessing on the results of CNN-based networks is made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' [21] describes the development of a customized network dubbed COVID-Net for CXR and chest X-ray medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Another study [22] compared COVID-19, bacterial pneumonia, and viral-pneumonia classification results using state-of-the- art algorithms and CNN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' In another study [23], PCA7 was used to increase the efficiency of feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' YOLO8 Networks with additional convolution layers involving modified kernels were used in [24] to detect COVID-19 from chest X-ray images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' [25] examined the performance of the Efficient-Net family in detecting COVID-19 and pneumonia in normal cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' In [26], a network, termed CoroNet, was introduced based on the Xception network to perform multi-class classification from normal category images, including pneumonia-viral, pneumonia-bacterial, and COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' This paper is structured as follows: The second section provides an overview of previous publications on the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The third section describes the knowledge distillation method and explains why it was incorporated into the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Section four discusses the dataset that was used to conduct this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Section five describes the training phase and the evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content="  Finally, section six presents the proposed model's evaluation results, and section seven summarizes our paper's findings." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Methods Nowadays, Deep Neural Networks9 have revolutionized the field of computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Their applications have been extensively investigated in a variety of fields, including self-driving cars [27], medical image analysis [28][29], and agriculture [30], among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' CNNs10 have established themselves as the most effective tools for automatic feature extraction in computer vision and NLP11, speech processing, and video classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' This paper demonstrates how CNN-based neural networks can improve semantic feature extraction for binary classification tasks involving COVID-19 and normal cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' DenseNet [31], ResNet [32], VGG [33], Xception [34], and Mobile-NetV2 [35] are some of the most powerful pre-trained networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' VGG19 is an extension of VGG16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' it features sixteen convolutional layers and three fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Five MaxPool layers are used, and the final layer is a SoftMax layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' ResNet50V2 is an enhanced version of ResNet50 that outperforms previous versions such as ResNet101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='  7 Principal Component Analysis (PCA) 8 You Only Look Once (YOLO) 9 Deep Neural Networks (DNN) 10 Convolutional Neural Networks (CNN) 11 Natural Language Processing (NLP)  5  ResNet50V2 has been equipped with several new connections between different blocks, allowing this network to achieve a high level of accuracy in the ImageNet competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' MobileNetV2 is a convolutional network designed with depth-wise convolution layers to improve accuracy while lowering computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' This reduction is caused by decreasing trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Due to the use of inverted residual blocks, this network can be embedded in mobile phones, tablets, and other conventional embedded devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Since both ResNet50V2 and VGG19 generate the same size output layer (feature vector), their feature vectors were combined to produce a richer semantic feature set for the specified task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Afterward, a CNN layer was added to this architecture (kernel size 1 and 1024 filters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=" The network's output was then flattened and streamed into the fully connected layers." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Notably, no activation function in the final CNN layer was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The classification layer was constructed using 64 neurons in fully connected layers, with a dropout rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=" Another fully connected layer has been added to this layer, the final layer, whose neurons count is equal to the size of the problem's classes." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The entire architecture is termed the teacher network, with MobileNetV2 serving as the student network in a broader context referred to as the teacher-student model or knowledge distillation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The structure of the teacher model is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content="  The tensors' feature size for both ResNet50V2 and VGG19 is 10*10*2048, and the size of the concatenated feature is 10*10*4096." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Teacher Model Architecture  Teacher Model Architecture  ResNet50V2  10 10 2048  CNNlayer  Filteri024  Dropout(50%)  VGG19  Kernel(1,l)  10 10+2048  10 10 4096  n  Classifier  Merged Features  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Knowledge Distillation Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' pioneered  Knowledge Distillation12 in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' KD is a process that involves training a smaller network to imitate the behavior of a more extensive network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The purpose of designing a complex network as a teacher is to learn more sophisticated features and deliver better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' However, we typically want to run our network on a standard computer or embedded device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Due to the limitation of memory size and computational cost, frequent issues arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' As a result, a solution is required to address these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' A weighted average (mean) is necessary to distill knowledge from teacher to student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Cross- Entropy with soft targets is the initial objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Through the softmax function in the smaller network, this objective function is calculated based on high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' A more significant architecture (network) must be used to generate soft targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Cross-Entropy with valid labels is the second objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' This function is calculated using the softmax output from the student model by setting the temperature to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The teacher network and the student network begin receiving training data in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The teacher model contains a softmax with temperature in its output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' By contrast, the student model generates two different outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The first output is softmax with temperature, while the second output contains standard softmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The student model is intended to produce softened probabilities (the output of the teacher model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='  The following formula is used to calculate the loss of knowledge distillation: 2 ( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' ) (1 ) ( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' ) KD s L KL p q T L W x \uf061 \uf061 = + −  (1) Where p and q denote the probabilities generated by student and teacher networks in a specific temperature (T), respectively, and KL denotes the Kullback-Leibler divergence, which measures the level of distinction between two probabilistic distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The Cross-Entropy of the student model with T=1 is (LWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' According to [36], \uf061 and T are hyperparameters where the greater the value of \uf061 , the better the learning experience for the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Back-propagation must be performed only in the student network during the distillation phase to add a significant element to this description since the teacher has already tuned its parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=" The teacher's knowledge is then transferred to the student model throughout the distillation procedure." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Notably, the student model can be trained at a faster rate than the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' For more details regarding the distillation procedure, please refer to [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The procedure for knowledge distillation is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='  12 Knowledge Distillation (KD)  7  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Knowledge Distillation for COVID-19 detection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Dataset Two public datasets were used to train the proposed deep learning model to build the required dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' First, a public dataset available at (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='com/ieee8023/covid-chestxray-dataset) was used for positive samples of COVID- 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Afterward, the dataset available at (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='com/c/rsna-pneumonia-detection-challenge) was used to collect negative samples (normal cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Following the two datasets being merged, 118 COVID-19 cases and 8851 normal cases were established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' It became clear that an unbalanced dataset was created due to the number of positive cases (COVID-19) being significantly lower than the number of normal chest X-ray medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' As a result, the issue was mitigated through the use of sampling techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The central concept is to select an equal number of items from each category for the binary classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The oversampling method was used to increase COVID-19 (positive cases) samples to ensure that both positive and negative classes had an equal number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The number of positive cases increased to 8851 following the oversampling technique, while the number of negative (normal) samples remained unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' It should be noted that no images of pneumonia were used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Pneumonia is classified into several different classes, including SARS, Streptococcus, ARDS, and Pneumocystis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='  In this respect, treating all of these categories as a single class was deemed as impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' This cannot be very clear in terms of interpreting recognition task results as distinct pneumonia types require a unique type of treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' As a result, developing a pneumonia-type classifier was deferred to a later date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Figure 3  illustrates a selection of patients with COVID-19 and normal images from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='  Generating soft targets  Teacher  Network(ResNet50V2/VGG19)  Chest X ray Dataset  Backpropagation  Loss  Student  Trainingthe student  Network(MobileNetV2)  8  Figure 3(a): Healthy person  Figure 3(b): Patient with COVID-19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='1 Data Augmentation Before data augmentation, the images were normalized to avoid issues with vanishing and exploding gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' After that, the image was resized to 224x224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The model was then enhanced with data augmentation techniques to make it more responsive to variations within the medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=" During data augmentation, it was assumed that variations in the images did not affect the label's (ground truth) definition." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Random rotation was the only data augmentation  9  approach considered for the dataset, between (0-200).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' This can prevent overfitting and also helps the learning curve converge more quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Training Phase The training phase consumed approximately 80% of the dataset, with the remaining 20% used for the test phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The K-fold Cross-Validation technique was used to ensure the accuracy of the performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The loss function was binary Cross-Entropy, and the optimizer was Adam (with a learning rate=1e-5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The number of folds was 5 (k=5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The batch size was 32, and the epoch number was set to 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content="1 Evaluation Metrics Several specific criteria need to be established for evaluating the model's performance on the test dataset." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' True Positive refers to correctly identifying COVID-19 positives among both positive and normal cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' In the conceptualization, true negativity entails accurately identifying normal cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' False Positive refers to the practice of misdiagnosing COVID-19 cases as normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' False-negative prediction is misclassifying normal cases as COVID-19 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Precision is defined as the ratio of True Positives over the sum of True Positives and False Positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='  TP  Precision  TP  FP  =  +  (1)  Sensitivity is the ratio of True Positives over to the sum of True Positives and False Negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='  (  )  TP  Sensitivity recall  TP  FN  =  +  (2)  Specificity is the ratio of True Negatives over the sum of False Positives and True Negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content="  TN  Specificity  TN  FP  =  +  (3)  10  Finally, three additional criteria were used to measure the performance of the proposed model on the study's dataset (F1 Score and Balanced-Accuracy in addition to conventional accuracy)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' 2* * 1 ( ) precision recall F Score precision recall − = +  (4)  2  specificity  sensitivity  Balanced Accuracy  +  =  (5)  TP  TN  Accuracy  Positive  Negative  +  =  +  (6)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Results and Discussion In this section, we present the results of our method applied to the dataset mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The images which have been used for the test have not been seen before by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Therefore, these results are approvable that the proposed algorithm can perform very well on unseen medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Table 1 presents the results of the training teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=" It is conspicuous that the teacher's performance is enough to detect chest X-ray images with good accuracy and F1-score." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Table 2 shows the results of MobileNetV2 (student model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Its results are satisfactory for the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Despite presenting good results of the student network, we decided to improve its classification ability via the knowledge distillation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Table 3 is presenting the results of the student network after knowledge distillation is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Eventually, the student network performs  on par with the existing methods in the literature and sometimes achieves better accuracy and F1-score in comparison with previous publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='  Table 1: Evaluation results of the Teacher Network (ResNet50V2/VGG19) Fold No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
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+page_content='6 %  Table 4: Number of total parameters in each architecture Teacher Network (ResNet50V2/VGG19) 49,222,390 Student Network (MobileNetV2) 2,334,966 Number of Parameters Reduction (%) -95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content="3  %  It is noteworthy that the student network has outperformed its previous version(without KD) in terms of evaluation metrics performance, and this is because knowledge distillation improves MobileNetV2's performance to some extent." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Meanwhile, KD aids the network in mitigating common neural network forgetting problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=" Thus, the student model's performance demonstrates that it can be used for medical recognition tasks in embedding systems while requiring minimal computation, owing to the use of depthwise convolutional layers and knowledge distillation." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Table 4 shows the number of parameters in each architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' It’s conspicuous that not only KD can improve student model’s performance, but also it reduces  the number of parameters about 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='3% while maintainging perfromance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Therfore, student model can be an eligible candidate for COVID-19 recognition task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='  12  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Conclusion In this paper, a novel method was developed for identifying COVID-19 medical chest X-ray images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' Due to encountering an unbalanced dataset, this issue was resolved using oversampling and data augmentation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=" In evaluating the algorithm, the fivefold cross-validation method was used to ensure the proposed model's performance." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' After performing knowledge distillation, an accuracy and F1-score of 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='8% and 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='1% were achieved respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' The proposed method, we believe, is an excellent choice for the COVID-19 recognition task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' However, adding more diverse datasets from different countries will help improve the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content=' For future works on medical image datasets, incremental learning techniques, self-supervised deep learning methods, vision transformer architectures, knowledge distillation under adversarial attacks are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='  Declaration Conflict of interest The corresponding author declares that there are no conflicts of interest on behalf of all authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
+page_content='  References [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
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+page_content=' Valls, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E0T4oBgHgl3EQf4QJC/content/2301.02735v1.pdf'}
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diff --git a/cdFST4oBgHgl3EQfDjjb/content/tmp_files/2301.13711v1.pdf.txt b/cdFST4oBgHgl3EQfDjjb/content/tmp_files/2301.13711v1.pdf.txt
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+arXiv:2301.13711v1  [astro-ph.HE]  31 Jan 2023
+MNRAS 000, 1–14 (2022)
+Preprint 1 February 2023
+Compiled using MNRAS LATEX style file v3.0
+Wideband Study of the Brightest Black Hole X-ray Binary 4U 1543−47 in
+the 2021 Outburst: Signature of Disk-Wind Regulated Accretion
+Geethu Prabhakar1⋆, Samir Mandal1†, Bhuvana G. R.2, and Anuj Nandi3
+1 Department of Earth and Space Sciences, Indian Institute of Space Science and Technology (IIST), Trivandrum - 695547, India
+2 Department of Physics, Dayananda Sagar University, Bengaluru - 560068, India
+3 Space Astronomy Group, ISITE Campus, U R Rao Satellite Centre, Bengaluru - 560037, India
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+A comprehensive wideband spectral analysis of the brightest black hole X-ray binary 4U 1543 − 47 during its 2021 outburst
+is carried out for the first time using NICER, NuSTAR, and AstroSat observations by phenomenological and reflection mod-
+elling. The source attains a super-Eddington peak luminosity and remains in the soft state, with a small fraction (< 3%) of
+the inverse-Comptonized photons. The spectral modelling reveals a steep photon index (Γ ∼ 2 − 2.6) and relatively high inner
+disk temperature (Tin ∼ 0.9 − 1.27 keV). The line-of-sight column density varies between (0.45 − 0.54)×1022 cm−2. Reflection
+modelling using the RELXILL model suggests that 4U 1543 − 47 is a low-inclination system (θ ∼ 32◦ − 40◦). The accretion
+disk is highly ionized (log ξ > 3) and has super solar abundance (3.6−10 AFe,⊙) over the entire period of study. We detected a
+prominent dynamic absorption feature between ∼ 8 − 11 keV in the spectra throughout the outburst. This detection is the first
+of its kind for X-ray binaries. We infer that the absorption of the primary X-ray photons by the highly ionized, fast-moving
+disk-winds can produce the observed absorption feature. The phenomenological spectral modelling also shows the presence of
+a neutral absorption feature ∼ 7.1 −7.4 keV, and both ionized and neutral absorption components follow each other with a delay
+of a typical viscous timescale of 10 − 15 days.
+Key words: accretion, accretion disc - black hole physics - X-rays: binaries - stars: individual: 4U 1543 − 47
+1 INTRODUCTION
+X-ray spectroscopy of black hole X-ray binaries (BH-XRBs)
+holds the key to unveil the geometry of the system and the
+dynamics of the accretion process. The spectrum of BH-XRBs
+mainly consists of a hard powerlaw and a soft thermal compo-
+nent. The soft component, which is a multi-temperature black-
+body, is assumed to be originated from an optically thick, ge-
+ometrically thin accretion disk (Shakura & Sunyaev 1973). The
+hard powerlaw component is generally believed to be emitted
+from an optically thin, hot electron cloud called ‘corona’ by the
+Comptonization of the soft disk photons (Sunyaev & Titarchuk
+1980, 1985; Zdziarski et al. 1994; Chakrabarti & Titarchuk 1995;
+Poutanen & Coppi 1998; Chakrabarti & Mandal 2006; Iyer et al.
+2015; Poutanen et al. 2018). The relative strength of these com-
+ponents leads to different states in outbursting BH-XRBs. In the
+Low/Hard State (LHS), the non-thermal component dominates and
+in the High/Soft State (HSS), the disk emission dominates. There
+are short-lived intermediate states also, namely, the Hard Interme-
+diate State (HIMS) and Soft Intermediate State (SIMS), lying be-
+tween LHS and HSS (Homan et al. 2001; Homan & Belloni 2005;
+Remillard & McClintock 2006; Nandi et al. 2012; Sreehari et al.
+2019; Aneesha et al. 2019; Bhuvana et al. 2021; Prabhakar et al.
+2022). A typical outburst starts with the LHS and proceeds through
+⋆ geethuprabhakar.17@res.iist.ac.in
+† samir@iist.ac.in
+intermediate states to HSS then back to LHS again and finally reach
+quiescence. However, it does not always have to go through all the
+states mentioned above (Debnath et al. 2015; Radhika et al. 2016;
+García et al. 2019; Baby et al. 2020, 2021; Prabhakar et al. 2022).
+The advent of high resolution spectroscopy reveals the presence
+of reflection features in the spectra of many BH-XRBs. Irrespective
+of the geometry of the corona, it is believed that the photons upscat-
+terd by the corona, the primary photons interact with the disc mate-
+rial and a part of which produces the reflection features (Basko et al.
+1974). The reprocessed X-ray spectrum consists of fluorescent line
+emission from various elements, a soft thermal continuum and a
+Compton hump peaked at ∼ 20 − 30 keV. The most prominent fea-
+ture among the fluorescent emission lines is the iron K-edge at ∼ 7.1
+keV (Ueda et al. 1998) and Kα line at ∼ 6 − 7 keV (White et al.
+1986; Barret & Olive 2002; Di Salvo et al. 2005). This is because
+the fluorescence yield increases with the atomic number (Burhop
+1952). For a distant observer, these reflection features appeared to
+be diluted/broadened and distorted (asymmetric) due to relativistic
+effects of the strong gravity region in the close vicinity of the BH
+(Fabian et al. 1989, 2000). Spectral modelling using relativistic re-
+flection models can address the effect of blurring of the spectral fea-
+tures and helps to probe the physics of the strong gravity region at
+the inner disk. The accretion disk characteristics, such as the ion-
+ization of the disk material, the iron abundance, inclination of the
+system, spin of the BH etc., can also be obtained from reflection
+modelling. The line broadening can also be due to Comptonization
+in a highly ionized, optically thick cloud, and the resultant feature
+© 2022 The Authors
+
+2
+Prabhakar et al.
+is broad and symmetric. However, this mechanism is important for
+high inclination systems only (Petrucci et al. 2001).
+The Fe−K band (5 − 8 keV) is the energy range where most of
+the emission/absorption features appear. The first observational ev-
+idence of the Fe−K absorption lines was provided by Ueda et al.
+(1998) with ASCA in the spectra of galactic superluminal BH source
+GRO J1655 − 40. Kotani et al. (2000) and Lee et al. (2002) also de-
+tected similar features in the superluminal jet source GRS 1915+105
+with ASCA. Later, it is revealed that the absorption features are
+very common in the spectra of BH-XRBs (Shidatsu et al. 2013;
+King et al. 2014; Xu et al. 2018). Photon interaction with neutral
+and static material produces sharp fluorescent lines at their corre-
+sponding transition energy levels. In case of ionized absorbers, there
+would be an increase in the transition line energy compared to their
+neutral ones. The absorption lines from highly ionized ions give an
+insight into the highly ionized plasma around the compact object.
+The process of accretion in XRBs is usually accompanied with
+outflows and/or jets (Fender et al. 1999, 2004, 2010; Miller et al.
+2012, 2013; Radhika & Nandi 2014; Radhika et al. 2016). The per-
+sistent jets are present in the LHS of the system, and it gets turned
+off in HSS. Accretion disk-wind is generally observed in the disk-
+dominated HSS, though it can exist in other spectral states as well
+(Lee et al. 2002; Miller et al. 2008; Neilsen & Lee 2009; Neilsen
+2013). The disk-winds carry a sufficient amount of matter which
+suppresses the launch of radio jets (Neilsen & Lee 2009) in HSS.
+The disk-wind can also be highly ionized and their presence can be
+inferred by the blue-shifted absorption features in the X-ray spec-
+trum (Ebisawa 1997; Kotani et al. 1997). In general, it seems that
+the absorption lines are absent in the LHS, which is still a matter
+of debate. Neilsen & Lee (2009) suggests that the wind gets pho-
+toionized completely in LHS, and the medium becomes transparent;
+this could be a possible reason for the absence of absorption lines in
+the spectra. Usually, the disk-winds are observed in high inclination
+systems (Ponti et al. 2012). Such systems may show intensity ‘dips’
+in their X-ray spectra, for e.g., GRS 1915 + 105, 4U 1630 − 47, H
+1743 − 322, MAXI J1305 − 704, GRO J1655 − 40 (Leahy 1997;
+Kuulkers et al. 1998; Shidatsu et al. 2013). The dips are believed to
+be caused by obscuring material associated with the accretion disk
+(Frank et al. 1987) and are visible for highly inclined systems with
+inclination angle 60◦ ≲ θ ≲ 80◦ (Frank et al. 1987). The disk-winds
+play a major role in regulating the accretion scenario of BH-XRBs.
+For example, Muñoz-Darias et al. (2016) showed how winds control
+the violent outburst of V404 Cygni by diminishing a significant frac-
+tion of the outer disk. Disk-wind studies in BH-XRBs can provide
+great insights into the physical mechanisms involved in the accretion
+process.
+4U 1543 − 47 is a BH-XRB, discovered by Uhuru satellite in
+1971 (Matilsky et al. 1972). Since the discovery, it has undergone
+five outbursts; the first four are in an interval of ∼ 10 years, in
+1984 (Kitamoto et al. 1984), 1992 (Harmon et al. 1992) and 2002
+(Park et al. 2004). After a gap of 19 years, the fifth outburst hap-
+pened in 2021 (Negoro et al. 2021a), which marks the source as
+the brightest BH-XRB with a peak X-ray intensity of 11 Crab in
+2 − 4 keV with MAXI/GSC (Negoro et al. 2021b). The 2002 out-
+burst was also brighter (4 Crab in 2 − 12 keV), while the previ-
+ous three outbursts have comparable intensities (Park et al. 2004).
+Its optical counterpart, IL Lupi, was discovered by Pedersen (1983).
+The central engine is a dynamically confirmed BH with a mass of
+9.4 ± 1.0 M⊙, and the companion is an A2V star of mass 2.45 ±
+0.15 M⊙ (Russell et al. 2006). It is located at RA = 15h47m8s.27,
+Dec = −47◦40
+′10
+′′.8 (J2000) (Park et al. 2004) at a distance of
+7.5 ± 0.5 kpc (Jonker & Nelemans 2004). Orosz (2003) estimated
+the orbital inclination of the system as 20.7◦ ± 1.5◦. There were
+multiple attempts to estimate the spin (a∗, dimensionless spin pa-
+rameter) of the BH in 4U 1543 − 47 using RXTE observations of
+the 2002 outburst. Shafee et al. (2006) estimated a spin of ∼ 0.75 −
+0.85 using continuum-fitting of RXTE data. Miller et al. (2009) and
+Morningstar & Miller (2014) estimated the spin value as 0.3 ± 0.1
+and 0.43+0.22
+−0.31 respectively using relativistic disk reflection and disk
+continuum modelling. These three estimations are based on a BH
+mass of 9.4 ± 1.0 M⊙ and a distance of 7.5 ± 0.5 kpc. Shafee et al.
+(2006) and Morningstar & Miller (2014) used the binary inclination
+(θ) of 20.7 ± 1.5 degree, while Miller et al. (2009) used a θ of 32+3
+−4
+degree for the spin estimation. Dong et al. (2020) reported a spin of
+0.67+0.15
+−0.08 and θ of 36.3+5.3
+−3.4 degree by reflection modelling of RXTE
+data using the model RELXILL.
+The Giant Metrewave Radio Telescope (GMRT) detected radio
+flares from the source in 2002 outburst (Park et al. 2004). Multi-
+ple flaring occasions are reported at different phases of the outburst.
+Russell et al. (2020) reported the presence of a compact jet in the
+SIMS of the 2002 outburst of 4U 1543 − 47 using multiwavelength
+observations (X-ray, optical, IR, and radio). Since the system has
+a low inclination, the jet angle and axis of rotation may coincide.
+Russell et al. (2020) tested the chances of jet contribution to the lu-
+minosity of the system and renounced that possibility.
+Until now, there is no study in literature based on the 2021 out-
+burst of 4U 1543 − 47. We aim for a detailed analysis of the wide-
+band spectral characteristics of the 2021 outburst using three dif-
+ferent instrument data from NICER (Neutron star Interior Compo-
+sition ExploreR), NuSTAR (Nuclear Spectroscopic Telescope Array)
+and AstroSat during outburst decay. The evolution of spectral pa-
+rameters is investigated using phenomenological and reflection mod-
+elling. Even though the reflection modelling of RXTE data of 2002
+outburst (Miller et al. 2009; Morningstar & Miller 2014; Dong et al.
+2020) unveil the fundamental quantities of the system like a∗ and
+θ, data from much better spectral resolution instruments like NuS-
+TAR (Harrison et al. 2013) are highly promising. It can also provide
+outburst specific quantities like the iron abundance and ionization
+of the accretion disk. We report the presence of strong and dynamic
+absorption features in the 2021 outburst spectra, which has not been
+observed in any previous outbursts of 4U 1543 − 47. We examine
+these features quantitatively using phenomenological modelling of
+NuSTAR data.
+This paper is structured as follows: The observations and the data
+reduction procedure are discussed in §2. The evolution of the out-
+burst lightcurve and hardness ratio are examined in §3. The spectral
+modelling and parameter evolution are presented in §4. Phenomeno-
+logical and reflection modelling of different epochs are discussed in
+§4.1 and §4.2, respectively. The detailed study of the absorption fea-
+tures in the spectra of 4U 1543 − 47 is carried out in §4.3. We dis-
+cussed our overall findings in §5. Finally, we summarise the results
+in §6 and then conclude.
+2 OBSERVATIONS AND DATA REDUCTION
+We perform the present study based on the 2021 outburst of 4U
+1543 − 47 using NuSTAR, NICER and AstroSat observations over
+a period from 17 June 2021 (MJD 59382) to 14 September 2021
+(MJD 59471). We considered all the NuSTAR and AstroSat obser-
+vations in this period and used the NICER observations which are
+simultaneous with NuSTAR. The list of observations considered for
+this study is given in Table 1. There are a total of 16 epochs of ob-
+MNRAS 000, 1–14 (2022)
+
+Disk-wind regulated accretion in 4U 1543−47
+3
+Table 1. The list of observations of the source 4U 1543 − 47 considered for the study. There are 16 epochs consisting of ten NuSTAR and six AstroSat
+observations. Seven NuSTAR epochs have simultaneous NICER coverage also.
+Epoch
+Obs. ID (MJD)
+Remarks
+NuSTAR
+NICER
+AstroSat
+1
+80702317002 (59382.42)
+4655060101 (59382.44)
+2
+80702317004 (59389.47)
+4655060201 (59389.47)
+3
+T04_018T01_9000004494 (59396.04)
+Offset
+4
+80702317006 (59396.18)
+4655060301 (59396.19)
+5
+80702317008 (59403.02)
+4655060401 (59403.04)
+6
+T04_021T01_9000004526 (59405.36)
+Offset
+7
+T04_030T01_9000004588 (59421.19)
+Pointed
+8
+90702326002 (59421.67)
+9
+90702326004 (59428.18)
+4202230143 (59428.13)
+10
+T04_035T01_9000004622 (59430.59)
+Pointed
+11
+90702326006 (59450.19)
+12
+90702326008 (59455.55)
+13
+T04_046T01_9000004680 (59457.06)
+Pointed
+14
+T04_051T01_9000004686 (59461.05)
+Pointed
+15
+90702326010 (59465.67)
+4202230166 (59466.07)
+16
+90702326012 (59471.51)
+4202230171 (59471.43)
+servations consisting of ten NuSTAR and six AstroSat observations.
+Seven NuSTAR epochs have simultaneous NICER coverage also.
+2.1 NuSTAR Data Reduction
+NuSTAR (Harrison et al. 2013) has observed the source several
+times in the 2021 outburst. NuSTAR is devoid of pile-up issues
+and moreover, its good energy resolution in the energy cover-
+age (3 − 79 keV) makes it suitable for the study of enormously
+bright sources like 4U 1543 − 47. NuSTAR consists two focal
+plane module telescopes (FPMA and FPMB), both are operat-
+ing in 3 − 78 keV band. The NuSTAR data for the 2021 out-
+burst is reduced using HEASOFT v.6.29, NUSTARDAS pipeline
+v.2.1.1 and CALDB v.20211115. For extremely bright sources,
+we set statusexpr="STATUS==b0000xxx00xxxx000"1 and set
+saamode to strict and tentacle to yes. A circular region of
+radius 35 pixels centered on the brightest pixel is extracted as the
+source region and as the background region, we also choose a 35
+pixel circular region away from this. These files are used for gener-
+ating science products such as the spectrum, background, lightcurve,
+Auxiliary Response File (ARF) and Response Matrix File (RMF),
+through the NUPRODUCTS task, independently for both FPMA and
+FPMB. The spectra are grouped with a minimum of 50 counts per
+bin without any systematics.
+2.2 NICER Data Reduction
+The
+X-ray
+Timing
+Instrument
+(XTI)
+onboard
+NICER
+(Gendreau et al. 2016) operates in 0.2 − 12 keV band. NICER
+has observed the source 4U 1543 − 47 in almost every day during
+the 2021 outburst. We analyse NICER data of the source between
+MJD 59382 and MJD 59471 which is simultaneous with the
+NuSTAR observations (Table 1). The data is reduced using the
+tool NICERDAS2 in HEASOFT v.6.29 with the 20210707 caldb
+version. There are 56 focal plane modules (FPMs) of NICER/XTI.
+1 https://heasarc.gsfc.nasa.gov/docs/nustar/analysis/
+2 https://heasarc.gsfc.nasa.gov/docs/nicer/nicer_analysis.html
+We excluded FPM-14 and 34 in addition to the non-functional
+FPMs (FPM-11, 20, 22, and 60) due to increased noise levels.
+Since 4U 1543 − 47 is extremely bright at the beginning of the
+outburst, the initial epochs (till ∼ MJD 59425) are affected by
+telemetry saturation. For such observations, a lower number of
+FPMs were kept active by the instrument team and we considered
+only the active detectors in the data reduction. Level-2 standard
+calibration and filtering are done using nicerl2 task and applied
+barycenter corrections through barycorr with refframe="ICRS".
+Spectra is generated using XSELECT (V2.4m). Lightcurve of
+the NICER observation on MJD 59428.18 shows a flaring in the
+high energy band; therefore, the corresponding GTIs are excluded
+from the extraction. The ARF and RMF files are generated for
+each observation based on the number of active detectors. The
+task nibackgen3C503 (Remillard et al. 2021) is used for creating
+background files. Finally, the source spectra are grouped with 25
+photons per bin and a systematic uncertainty of 1 % is added to the
+spectra.
+2.3 AstroSat Data Reduction
+The Soft X-ray Telescope (SXT) and Large Area X-ray Propor-
+tional Counter (LAXPC) on-board AstroSat (Yadav et al. 2016;
+Agrawal et al. 2017) together observes the astronomical sources in
+wideband energy range (0.3−80 keV). AstroSat has observed the
+2021 outburst of 4U 1543-47 during 6 different epochs. The first
+two of these observations are carried out with an offset of 40′ since
+the source was too bright to have pointed observation (Garg et al.
+2021). We obtain Level-1 LAXPC and Level-2 SXT data of all six
+observations available at data archive hosted by ISSDC4.
+LAXPC consists of three identical proportional counts namely
+LAXPC10, LAXPC20 and LAXPC30. However, for our analysis, we
+have used data from LAXPC20 alone because of its steady gain (see
+also Bhuvana et al. 2021; Baby et al. 2021; Bhuvana et al. 2022;
+Prabhakar et al. 2022). To extract the Level-2 LAXPC data file i.e.,
+source spectrum, lightcurve, RMF and background spectrum and
+3 https://heasarc.gsfc.nasa.gov/docs/nicer/tools/nicer_bkg_est_tools.html
+4 https://astrobrowse.issdc.gov.in/astro_archive/archive/Home.jsp
+MNRAS 000, 1–14 (2022)
+
+4
+Prabhakar et al.
+0
+5
+10
+0
+25
+50
+75
+100
+125
+150
+175
+0
+20
+(a)
+NICER
+NuSTAR
+AstroSat
+0.0
+0.5
+0
+25
+50
+75
+100
+125
+150
+175
+0.0
+0.2
+(b)
+0
+25
+50
+75
+100
+125
+150
+175
+Time (days since MJD 59370)
+0.0
+0.2
+0.4
+(c)
+Flux (photons/sec/cm2)
+Ratio (4-10/2-4 keV)
+10-20 keV
+2-10 keV
+Flux (Crab)
+Figure 1. MAXI/GSC daily average lightcurve in the energy bands (a) 2 − 10 keV and (b) 10 − 20 keV with flux in units of photons/sec/cm2 and Crab in the left
+and right Y-axes respectively. The hardness ratio (c) is defined by the flux in 4 − 10 keV to 2 − 4 keV. The NICER, NuSTAR and AstroSat observations for this
+study are marked using lines with the colours cyan, blue and red respectively.
+lightcurve, we make use of latest version of single routine LAXPC
+software LaxpcSoftversion3.4.35 (Antia et al. 2022). Level-2
+files are extracted from a single event and the top layer of LAXPC
+unit to avoid the instrument effects at high energy. While the soft-
+ware generated LAXPC response files are used for pointed obser-
+vations, a 40′ offset LAXPC response file provided by the instru-
+ment is used for the offset observations (see also Baby et al. 2020;
+Katoch et al. 2021).
+SXT has observed the source in Photon Counting (PC) mode dur-
+ing all the epochs. The orbit-wise SXT cleaned Level-2 event files
+are merged to get single event file for each observation using event
+merger python routine6 based on Julia v 1.1.1. The merged event file
+is then loaded into XSELECT, where we select single-pixel events
+by applying grade 0 filter to avoid optical data leakage (Singh et al.
+2021; Prabhakar et al. 2022). From the XSELECT images, we find
+that the first two offset observations have count rate < 40 counts s−1
+and hence the corresponding spectra wouldn’t have pileup issues.
+We select a circular region of radius 10′ in the image to extract the
+source spectrum and lightcurve files. In all the pointed observations
+(see Table 1), we find the central region of the image to be very
+bright which could cause a pile-up effect. Therefore, source files are
+extracted from an annular region of the outer radius of 15′ and inner
+radius of 2′ for these observations. The standard SXT background
+spectrum and instrument response file provided by the instrument
+team7 are used. ARF for the selected region is obtained from python-
+based tool sxtarfmodule provided by the SXT team. Extracted SXT
+and LAXPC spectra are grouped to have 30 counts per bin in the first
+5 http://www.tifr.res.in/~astrosat_laxpc/LaxpcSoft.html
+6 https://www.tifr.res.in/~astrosat_sxt/dataanalysis.html
+7 https://www.tifr.res.in/~astrosat_sxt/dataanalysis.html
+two observations and 20 counts per bin in the rest of the observations
+based on the source brightness. A systematics of 2% (Sreehari et al.
+2019; Athulya et al. 2022) is applied for both SXT and LAXPC spec-
+tra.
+3 OUTBURST PROFILE AND HARDNESS RATIO
+During the 2021 outburst of 4U 1543 − 47, the flux reached the
+peak value within a few days of the commencement of the out-
+burst. The outburst is monitored by multiple X-ray instruments. The
+MAXI/GSC8 daily lightcurve of the source is generated for two dif-
+ferent energy bands, 2 − 10 keV and 10 − 20 keV and are plotted in
+Fig. 1. The MJD 59370 (05 June 2021) is defined as day 0 through-
+out the study and according to this, the outburst continues over ∼175
+days. The lightcurve reveals that the source is extremely luminous in
+low energies with a very high count rate (Fig. 1a), while the contri-
+bution to the luminosity in the high energy band (Fig. 1b) is an order
+of magnitude lower. The highest value of flux in 2 − 10 keV band is
+32.67 photons/sec/cm2 (∼ 10 Crab) on day 9, whereas the same in
+10 − 20 keV is just 0.19 photons/sec/cm2 (∼ 0.5 Crab). The source
+flux reached 11.65 Crab in 2 − 4 keV, which is the highest value ob-
+served among the BH-XRBs. We define hardness ratio (HR) as the
+ratio of flux in 4 − 10 keV to 2 − 4 keV since beyond 10 keV the
+contribution is significantly low. The HR evolution (Fig. 1c) shows
+that the source mostly remains in the soft state during the outburst.
+The NICER, NuSTAR and AstroSat observations used in this study
+are marked by cyan, blue and red dashed lines, respectively. There
+is no simultaneous broadband observation in the rising phase of the
+outburst.
+8 http://maxi.riken.jp/mxondem/
+MNRAS 000, 1–14 (2022)
+
+Disk-wind regulated accretion in 4U 1543−47
+5
+10
+5
+0.8
+1
+1.2
+Ratio
+Energy (keV)
+Figure 2. The ratios (model to data) of the spectral fitting of NuSTAR obser-
+vations (Table 1) using tbabs(diskbb+powerlaw) model. The colours black,
+red, green, blue, cyan, pink, magenta, orange, yellow and grey represent the
+NuSTAR epochs in ascending order (Table 1). A strong absorption feature ex-
+ists between ∼ 8 − 11 keV for all the epochs. The absorption depth increases
+up to Epoch 9 (pink in colour) and then decreases as the outburst progresses.
+The figure is zoomed around the absorption feature for better clarity.
+4 SPECTRAL MODELLING AND RESULTS
+We studied the spectral properties of the 2021 outburst of 4U 1543−
+47 from MJD 59382 (17 June 2021) to MJD 59471 (14 September
+2021). All the three instruments, NICER, NuSTAR and AstroSat have
+good coverage over this period. Table 1 summarises the list of obser-
+vations used in this work. In total, there are 16 epochs, comprising
+ten NuSTAR and six AstroSat (SXT-LAXPC) observations. In addi-
+tion, there are seven NICER observations which are simultaneous
+with that of NuSTAR and we used these pairs for NICER-NuSTAR
+wideband spectral analysis. We carried out phenomenological and
+reflection modelling of each NuSTAR observations and extended that
+to wideband NICER-NuSTAR and AstroSat observations.
+4.1 Phenomenological Spectral Modelling
+We used HEASOFT v.6.29 and XSPEC V12.12.0 package for the
+spectral modelling of NICER, NuSTAR and AstroSat data. We have
+done spectral modelling of NICER-NuSTAR and AstroSat data to
+see the nature of the broad-band spectrum. The NICER data below
+0.8 keV shows large residuals; therefore, we used 0.8 − 10 keV for
+NICER spectra and 4 − 60 keV for NuSTAR spectra. Spectra from
+both FPMA and FPMB telescopes of NuSTAR give similar results.
+We present only FPMA spectra throughout the study. We used 0.5−7
+keV for SXT and 3 − 30 keV for LAXPC as significant data is not
+available beyond this range.
+To accommodate the interstellar absorption, we used the tbabs
+model which uses an equivalent hydrogen column density nH
+through the solar abundance table provided by Wilms et al.
+(2000). We initially modelled the NuSTAR observations with
+tbabs(diskbb+powerlaw) model. Here, diskbb (Mitsuda et al. 1984)
+represents the multicolor blackbody spectrum from the accretion
+disc and powerlaw employs the inverse Comptonization of the soft
+blackbody photons. We detected a broad absorption feature at ∼
+8−11 keV in all epochs. The ratio of data to model of all the NuSTAR
+observations are shown in Fig. 2. The different NuSTAR epochs are
+shown in black, red, green, blue, cyan, pink, magenta, orange, yel-
+low and grey colours in ascending order. It shows that the depth of
+absorption feature starts with a low value (black in colour) and then
+keeps on increasing as the outburst progress, reaching the maximum
+on Epoch 9 (pink in colour). Finally, the absorption depth decreases
+towards the end (grey in colour) of our study. We found a similar
+absorption feature in the AstroSat/LAXPC data as well.
+We used a partial covering fraction absorption model pcfabs9 in
+XSPEC to check if this strong absorption feature at ∼ 8−11 keV can
+be due to an intervening absorber, but it did not improve the fitting
+and the low energy residuals were high. We also tried several other
+models, for example, the thermal Comptonization model nthcomp
+(Zdziarski et al. 1996) or thcomp (Zdziarski et al. 2020) in-place of
+powerlaw, and diskbb was replaced by kerrbb (Li et al. 2005). Var-
+ious combinations of these models, fit to the data also showed the
+presence of the absorption feature at ∼ 8 − 11 keV. Model combina-
+tion with kerrbb as the seed photon source did not provide a good
+fit; moreover, it failed to constrain the BH mass and spin. So, we
+prefer to use diskbb model in combination with thcomp which is an
+improved version of nthcomp. The thcomp is a convolution model
+which allows a variable fraction (parameter cov_ f rac) of seed pho-
+ton to Comptonize both up-scattering and down-scattering. Other
+parameters are the photon index (Γ) and electron temperature (kTe)
+of the corona.
+The observed absorption feature has a symmetrical profile and in-
+clusion of a Gaussian absorption model gabs10 fits the absorption
+feature well. The parameters of gabs component are line energy
+(line E, Eg), line width (σ) and line depth (strength). However, if
+we use a smeared absorption edge model, smedge (Ebisawa et al.
+1994) in XSPEC to compensate for the absorption feature, it re-
+sults in an abnormally high value of absorption width due to the
+asymmetric nature of the model. In addition, the NuSTAR data show
+a weak presence of a Fe Kα absorption edge. Therefore, we also
+used an edge component to improve the fit residual for NuSTAR.
+The model parameters for the edge component are the threshold
+energy of the absorption edge (edge E, Ee) and the correspond-
+ing absorption depth (D). Therefore, the final model for NICER-
+NuSTAR data is tbabs(thcomp × diskbb)edge × gabs (model M1,
+hereafter). In contrast, the absorption edge feature was not visible
+in the AstroSat/LAXPC data, possibly due to the low spectral resolu-
+tion of LAXPC. Therefore, the model M1 for AstroSat data becomes
+tbabs(thcomp×diskbb)×gabs. However, the AstroSat data of Epoch
+3 & 6 show the presence of a weak Fe Kα emission line feature in
+the residual, and we include a gauss model component for these two
+epochs of AstroSat data.
+All the seven NICER-NuSTAR simultaneous pairs and six AstroSat
+observations are fitted with the model M1. The Fig. 3a and Fig. 3b
+show the wideband spectra of NICER-NuSTAR (Epoch 2 & 9) and
+AstroSat (Epoch 3 & 7) respectively modelled using M1. In each
+case, the spectrum in the red colour is relatively softer than that of
+the black colour; therefore, both figures illustrate moderate spectral
+changes during the outburst decay.
+The goodness of the fit is determined using χ2 statistics. The re-
+duced χ2 (χ2
+red) varies between 0.9 − 1.3. All the parameters es-
+timated from the wideband phenomenological modelling are pre-
+sented in Table 2. The parameter uncertainties are calculated within
+the 90% confidence range. Note that the NuSTAR observations on
+Epoch 8, 11 & 12 (Table 1) are not included here as no simultane-
+ous NICER observations available. The nH is left free and it varies
+between (0.45 − 0.54) × 1022 cm−2. We aim to find out the evolution
+9 https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/XSmodelPcfabs.html
+10 https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/node246.html
+MNRAS 000, 1–14 (2022)
+
+6
+Prabhakar et al.
+10−4
+10−3
+0.01
+0.1
+1
+10
+Photons cm−2 s−1 keV−1
+1
+10
+2
+5
+20
+50
+1
+1.5
+Ratio
+Energy (keV)
+(a)
+Epoch 2
+Epoch 9
+10−3
+0.01
+0.1
+1
+10
+Photons cm−2 s−1 keV−1
+1
+10
+0.5
+2
+5
+20
+1
+1.5
+Ratio
+Energy (keV)
+(b)
+Epoch 3
+Epoch 7
+Figure 3. (a) Simultaneous NICER-NuSTAR pair on Epoch 2 (black in colour) and Epoch 9 (red in colour) fitted with the model tbabs(thcomp × diskbb)edge ×
+gabs. Epoch 2 spectrum is harder compared to that of Epoch 9. (b) The AstroSat spectra on Epoch 3 (black) and 7 (red) respectively modelled using
+tbabs(thcomp × diskbb) × gabs. We include an additional gauss component for AstroSat (Epoch 3) data. Both instruments show spectral changes during
+the outburst.
+Table 2. Wideband NICER-NuSTAR simultaneous pairs and AstroSat observations (highlighted with grey colour) using the model tbabs(thcomp×diskbb)edge×
+gabs and tbabs(thcomp×diskbb)×gabs respectively. The error values represent 90% confidence interval. The NuSTAR data on Epoch 8, 11 & 12 are not included
+here as no simultaneous NICER observations available. The bolometric (0.5 − 100 keV) observed flux and estimated luminosity for each epoch are also shown.
+Epoch
+nH
+Model
+χ2
+red
+Fbol
+Lbol
+diskbb
+thcomp
+edge or gauss∗
+gabs
+(×1022
+Tin
+norm
+Γ
+kTe
+cov_frac
+line E
+D or σ
+line E
+σ
+strength
+(×10−8
+cm−2)
+(keV)
+(×103)
+(keV)
+(×10−2)
+(keV)
+(keV)
+(keV)
+(keV)
+(keV)
+erg cm−2 s−1)
+(LEdd)
+1
+0.470+0.002
+−0.002 1.272+0.002
+−0.002 7.96+0.05
+−0.05 2.44+0.06
+−0.05
+20f
+2.3+0.2
+−0.2
+7.16+0.05
+−0.05 0.04+0.01
+−0.01
+10.0+0.1
+−0.1
+1.60+0.09
+−0.09 0.84+0.08
+−0.08 0.90
+27.64+0.04
+−0.04
+1.52+0.14
+−0.14
+2
+0.458+0.002
+−0.002 1.159+0.002
+−0.002 6.07+0.04
+−0.04
+2.0+0.1
+−0.1
+11.6+3.9
+−1.9
+0.9+0.2
+−0.1
+7.19+0.07
+−0.07 0.03+0.01
+−0.01
+9.9+0.1
+−0.1
+1.65+0.07
+−0.08
+1.2+0.1
+−0.1
+1.00
+17.22+0.02
+−0.03
+0.95+0.09
+−0.09
+3
+0.52+0.02
+−0.02
+0.99+0.01
+−0.01
+15.4+1.5
+−1.4
+1.84+0.14
+−0.12
+20f
+0.39+0.07
+−0.06 6.77+0.15
+−0.15
+0.6f
+7.35+0.34
+−0.39
+2f
+1.73+0.47
+−0.47 1.17
+19.5+0.1
+−0.1
+1.07+0.001
+−0.001
+4
+0.461+0.002
+−0.002 1.094+0.002
+−0.002 5.82+0.05
+−0.05
+2.2+0.1
+−0.1
+20f
+0.3+0.1
+−0.1
+7.27+0.07
+−0.07 0.05+0.01
+−0.01
+10.4+0.1
+−0.1
+1.94+0.08
+−0.07
+2.2+0.2
+−0.1
+0.95
+12.38+0.02
+−0.03
+0.68+0.06
+−0.06
+5
+0.458+0.002
+−0.002 1.050+0.002
+−0.002 5.77+0.04
+−0.04
+2.3+0.2
+−0.2
+20f
+0.2+0.1
+−0.1
+7.28+0.05
+−0.05 0.06+0.01
+−0.01
+10.7+0.1
+−0.1
+1.97+0.08
+−0.07
+2.7+0.2
+−0.2
+1.02
+10.10+0.01
+−0.02
+0.56+0.05
+−0.05
+6
+0.50+0.02
+−0.02
+0.97+0.02
+−0.02
+12.1+1.5
+−1.4
+2.3f
+20f
+0.3+0.02
+−0.02
+6.77+0.18
+−0.18
+0.6f
+9.05+0.51
+−0.55
+2.5f
+2.27+0.66
+−0.68 1.27
+13.9+0.06
+−0.05
+0.76+0.001
+−0.001
+7
+0.54+0.01
+−0.01
+0.98+0.01
+−0.01
+7.64+0.33
+−0.33 2.26+0.06
+−0.05
+20f
+0.2f
+-
+-
+10.7+0.2
+−0.2
+1.46+0.24
+−0.18 1.56+0.25
+−0.33 1.29
+9.36+0.21
+−0.20
+0.51+0.004
+−0.004
+9
+0.451+0.003
+−0.003 0.966+0.002
+−0.002 5.60+0.05
+−0.05
+2.6+0.5
+−0.4
+20f
+0.1+0.2
+−0.1
+7.44+0.08
+−0.08 0.08+0.02
+−0.02
+10.5+0.2
+−0.1
+2.02+0.09
+−0.09
+3.3+0.3
+−0.2
+1.05
+6.88+0.02
+−0.02
+0.38+0.04
+−0.04
+10
+0.53+0.01
+−0.01
+0.96+0.01
+−0.01
+5.63+0.35
+−0.31 1.92+0.30
+−0.25
+20f
+0.11+0.05
+−0.03
+-
+-
+10.1+0.2
+−0.2
+1.56+0.20
+−0.18 2.13+0.39
+−0.32 1.28
+7.59+0.04
+−0.03
+0.40+0.001
+−0.001
+13
+0.52+0.01
+−0.01
+0.91+0.01
+−0.01
+7.48+0.24
+−0.23 1.77+0.03
+−0.03
+20f
+0.12+0.05
+−0.05
+-
+-
+9.42+0.27
+−0.27 0.75+0.30
+−0.28 0.47+0.13
+−0.11 1.10
+6.59+0.06
+−0.07
+0.36+0.002
+−0.003
+14
+0.48+0.01
+−0.01
+0.91+0.01
+−0.01
+7.92+0.58
+−0.53 1.94+0.02
+−0.02
+20f
+0.4f
+-
+-
+10.7+0.24
+−0.21 1.34+0.18
+−0.17 1.47+0.18
+−0.18 1.31
+6.22+1.89
+−6.56
+0.34+0.08
+−0.29
+15
+0.465+0.004
+−0.010 0.904+0.030
+−0.004 6.10+0.09
+−0.69 2.09+0.04
+−0.04
+20f
+1.6+0.1
+−0.1
+7.31+0.08
+−0.14 0.10+0.05
+−0.03
+9.6+0.3
+−0.3
+2.1+1.2
+−0.4
+1.1+1.1
+−0.3
+1.27
+5.66+0.02
+−0.02
+0.31+0.03
+−0.03
+16
+0.471+0.002
+−0.002 0.897+0.002
+−0.001 6.22+0.05
+−0.05 2.04+0.04
+−0.06
+20f
+1.6+0.1
+−0.1
+7.35+0.06
+−0.03 0.12+0.02
+−0.03
+9.4+0.4
+−0.3
+1.5+0.4
+−0.3
+0.5+0.1
+−0.1
+1.00
+5.39+0.01
+−0.01
+0.30+0.03
+−0.03
+∗ edge is used for NICER-NuSTAR pairs whereas gauss component is used only for Epoch 3 & 6 of AstroSat data.
+f Frozen parameters
+MNRAS 000, 1–14 (2022)
+
+Disk-wind regulated accretion in 4U 1543−47
+7
+20
+40
+60
+80
+100
+1.0
+1.2
+Tin (keV)
+(a)
+20
+40
+60
+80
+100
+2.0
+2.5
+3.0
+Γ
+(b)
+20
+40
+60
+80
+100
+Time (days since MJD 59370)
+0.25
+0.50
+τ0
+(c)
+Figure 4. Evolution of (a) Tin and (b) Γ from simultaneous NICER-NuSTAR
+pairs (green in colour) and AstroSat observations (red in colour) fitted using
+the model M1. We calculate the (c) optical depth (τ0) of the absorber from
+gabs components for each epoch. The Γ of Epoch 13 marked with a blue
+circle carries a special signature that is discussed in §5.
+of various parameters with the progress of the outburst. Fig. 4 gives
+the variation of the inner disk temperature Tin, photon index Γ and
+optical depth (τ0) with time. Here, points in green and red colour
+represent the parameter value estimated from NICER-NuSTAR and
+AstroSat spectral modelling respectively.
+We find that the inner disk temperature (Fig. 4a and Table 2)
+monotonically decreases throughout the outburst decay. The evolu-
+tion of the diskbb norm (Table 2) estimated using NICER-NuSTAR
+decreases till Epoch 9 and a reverse trend is observed for later
+epochs. The diskbb norm from the AstroSat data also shows a sim-
+ilar pattern though AstroSat values are higher than that of NICER-
+NuSTAR. The variation of photon index Γ is presented in Fig. 4b. The
+value of Γ estimated from AstroSat data (red square) differs from that
+of NICER-NuSTAR pairs (green square). This can be due to the non-
+availability of the high energy contribution (beyond 30 keV) in the
+LAXPC data. From NICER-NuSTAR fitting, Γ varies between 2−2.6,
+and it shows spectral softening till Epoch 9. We wanted to estimate
+the electron temperature (kTe) of the corona using the thcomp model.
+But the broadband spectral fitting could not constrain the value of
+kTe, except for Epoch 2, for which we obtained kTe = 11.6+3.9
+−1.9 keV
+(Table 2). For all the remaining epochs, we freeze kTe at 20 keV.
+Only a tiny fraction, cov_frac < 3 % (Table 2), of the soft photons
+Comptonized in the corona. It gradually decreases till Epoch 10 and
+increases afterwards. This behaviour is consistent with the spectral
+softening trend shown in Fig. 4b.
+The broad absorption feature at ∼ 8−11 keV in the spectrum is
+well represented by the gabs model. The strength shows an increas-
+ing trend and reaches the maximum in Epoch 9 and declines beyond
+that. We calculate the optical depth (τ0) associated with the gabs
+component using gabs strength and σ as τ0 = strength/σ
+√
+2π. The
+evolution of τ0 is shown in Fig. 4c, which shows that the absorption
+optical depth increases and reaches a maximum on Epoch 9 and then
+decreases. The dynamic behaviour of the absorption strength seems
+10−4
+10−3
+10−2
+10−1
+100
+101
+Photons cm−2 s−1 keV−1
+1.00
+1.25
+(a)
+χ2
+red =2.46
+tbabs(diskbb+relxill)
+0.75
+1.00
+1.25
+(b)
+χ2
+red =1.60
+tbabs(diskbb+relxill)gabs
+0.8
+1.0
+1.2
+(c)
+χ2
+red =1.63
+tbabs(diskbb+relxillCp)gabs
+0.75
+1.00
+1.25
+(d)
+χ2
+red =1.79
+tbabs(diskbb+relxilllpCp)gabs
+100
+101
+Energy (keV)
+1.0
+1.2
+(e)
+χ2
+red =0.91
+tbabs(diskbb+relxilllp)gabs
+Ratio
+Figure 5. Unfolded spectrum (top panel) of simultaneous NICER-NuSTAR
+pair on Epoch 2 and ratio of the model to the data using various re-
+flection models (a) tbabs(diskbb+relxill), (b) tbabs(diskbb+relxill)gabs,
+(c) tbabs(diskbb+relxillCp)gabs, (d) tbabs(diskbb+relxilllpCp)gabs and (e)
+tbabs(diskbb+relxilllp)gabs. The model and the value of χ2
+red are mentioned
+at the top left and bottom left corners, respectively, in each case.
+interesting. We attempt to characterize the strong and dynamic ab-
+sorption features in §4.3. The evolution of the edge component is
+listed in Table 2. We discuss a possible connection between edge
+and gabs components in §5.
+We estimate the observed bolometric flux (Fbol) in 0.5 − 100
+keV with uncertainty in 90% confidence interval from the wide-
+band simultaneous spectral data, which is also shown in Table 2.
+Corresponding bolometric luminosity (Lbol) of the source is also
+calculated by assuming the distance to the source as 7.5 ± 0.5 kpc
+(Jonker & Nelemans 2004). The Eddington luminosity of the source
+is LEdd = 1.22±1×1039 ergs s−1 with an assumed BH mass of 9.4±1
+M⊙ (Russell et al. 2006). It can be seen from Table 2 that the lumi-
+nosity of the source exceeds the LEdd at the peak (Epoch 1 is close
+to the peak) of the outburst, and the luminosity decreases gradually
+with the decay of the outburst.
+MNRAS 000, 1–14 (2022)
+
+8
+Prabhakar et al.
+4.2 Spectral Modelling for Reflection Studies
+To understand the reflection features in the spectra of 4U 1543 − 47,
+we use the relativistic reflection model RELXILL11v1.4.3. Dif-
+ferent flavours of the RELXILL model are tried. The unfolded
+NICER-NuSTAR spectra and data-to-model ratios using various re-
+flection models are shown in Fig. 5 for Epoch 2. We started with
+the model tbabs(diskbb+relxill) which gives a χ2
+red of 2.46. The
+data-to-model ratio (Fig. 5a) shows that the absorption feature at
+∼ 8 − 11 keV in the spectrum cannot be fitted by the reflection
+model. We added the absorption model gabs with this, and the model
+tbabs(diskbb+relxill)gabs (Fig. 5b) improves the residual but still
+the χ2
+red for this combination is 1.6. Then, we replaced the model
+relxill with relxillCp (Fig. 5c) where a thermal Comptonizing con-
+tinuum is assumed for the illuminating flux. This combination has
+χ2
+red = 1.63, which is also unacceptable. Then, we used relxilllpCp
+(Fig. 5d) as the reflection model, where a lamp-post (lp) geome-
+try is assumed for the corona. In all the ‘lp’ flavours of the RELX-
+ILL model, the inner disk is illuminated by a point-like corona sit-
+uated at a height ‘h’ from the disk surface on the axis of rotation.
+The modelling with relxilllpCp also resulted in a large residual, with
+χ2
+red = 1.79. We replaced relxilllpCp with relxilllp (Fig. 5e) where
+the illuminating flux is modelled as a powerlaw with a high-energy
+cutoff just like the relxill. For this trial, we got a reasonable fit with
+χ2
+red = 0.91, and we decided to proceed with the model combina-
+tion, tbabs(diskbb+relxilllp)gabs (model M2, hereafter) as the final
+model to study the reflection features in the spectra. Note that, no ad-
+ditional edge or gauss component is required in the reflection mod-
+elling.
+Using the model M2, we did the spectral fitting of the simultane-
+ous NICER-NuSTAR pairs and AstroSat observations. Here, nH is a
+free parameter. Inner and outer disk radii are frozen at the innermost
+stable circular orbit RISCO and 400 rg (where rg ≡ GM/c2, the gravi-
+tational radius of the BH), respectively. The powerlaw cutoff energy,
+Ecut, is fixed at 60 keV since it is hitting the upper limit. All other
+parameters of relxilllp are kept free; the lamp-post height h (in units
+of rg), inclination angle of the system θ (in degree), Γ of the incident
+radiation, ionization parameter log ξ (erg cm s−1), iron abundance
+(AFe) of the accretion disk in terms of the solar abundance AFe,⊙, and
+the reflection fraction Rf. Here, Rf is defined as the ratio of the pri-
+mary photon flux illuminating the disk to that reach the observer at
+infinity (Dauser et al. 2016). We wish to estimate the spin parameter,
+a∗, of the system, but it is found to hit the upper limit for almost all
+the epochs. Based on previous studies (Morningstar & Miller 2014;
+Dong et al. 2020), we freeze a∗ = 0.4 for this study.
+The estimated reflection model (M2) parameters are listed in Ta-
+ble 3. The errors represent 90% confidence interval. The evolution
+of Tin follows the same trend as that observed in the phenomeno-
+logical spectral modelling (Table 2). The value of Γ varies between
+2 − 3.3 and appears slightly steeper than that of phenomenologi-
+cal modelling (Table 2). It may be due to the additional low energy
+contribution from the reflection component over the diskbb compo-
+nent. However, reflection modelling also shows spectral softening
+till Epoch 9. The NICER-NuSTAR results suggest that 4U 1543 − 47
+is a low inclination system with θ varies between ∼ 32◦ − 40◦. We
+could not constrain the inclination angle from AstroSat data and
+freeze it to 40◦.
+The evolution of few important model parameters (h, log ξ
+and Rf ) are shown in Fig. 6 for better presentation. Simultane-
+ous NICER-NuSTAR pairs and AstroSat observations are marked in
+11 http://www.sternwarte.uni-erlangen.de/~dauser/research/relxill/index.html
+20
+40
+60
+80
+100
+0
+50
+100
+h
+(a)
+20
+40
+60
+80
+100
+2
+4
+log ξ
+(b)
+20
+40
+60
+80
+100
+Time (days since MJD 59370)
+1
+5
+10
+Rf
+(c)
+Figure 6. Evolution of (a) h, (b) log ξ and (c) Rf from reflection modelling
+of simultaneous NICER-NuSTAR pairs (green in colour) and AstroSat obser-
+vations (red in colour) using the model tbabs(diskbb + relxilllp)gabs. See
+text for details.
+green and red colour, respectively. We could not estimate the un-
+certainty of h in most of the epochs. The evolution of h (Fig. 6a)
+indicates that the primary source is moving away from the BH till
+Epoch 6 and then gradually coming closer to the central object. The
+ionization structure of the disk is established through the parameter
+log ξ. Its value gradually increases (Fig. 6b) and reaches the maxi-
+mum around Epoch 10 and then gradually decreases. The high value
+of log ξ (>3) suggests a highly ionized disc material throughout the
+outburst. We could estimate iron abundance AFe for Epoch 1, 2, 4
+and 5, and it hits the upper limit during the rest of the epochs. Our
+study reveals an overabundance (3.6−10 AFe,⊙) of iron in the disk.
+The reflection fraction Rf is estimated well at the first three and
+last two epochs only. Fig. 6c suggests that the fraction of primary
+photons reaching the disk increases till Epoch 9, and it decreases af-
+terwards. The gabs strength shows a similar behaviour found in the
+phenomenological modelling. We have discussed more on this result
+in §4.3.
+Very recently, the RELXILL model (version v2.2) has undergone
+some modifications by considering the effect of returning radiation
+in the calculation of reflected flux. Particularly in relxilllpCp, where
+the effects of returning radiation, the density profile and ionization
+gradient of the disk and the velocity of the primary source are also
+included. However, the velocity of the primary source and the effects
+of returning radiation are the new parameters added to the relxilllp
+model. We applied this modified relxilllp model to the broadband
+NICER-NuSTAR data but could not constrain the source velocity.
+Also, we tried to estimate the parameters using the v2.2 flavour of
+relxilllpCp. We fitted all the wideband NICER-NuSTAR observations
+using the model tbabs(diskbb+relxilllpCp)gabs. But, we could not
+constrain most of the parameters since the number of free parameters
+is very large. Therefore, we need to essentially freeze all the new
+parameters introduced in the updated version, and RELXILL v2.2
+does not bring any improvement in the result.
+MNRAS 000, 1–14 (2022)
+
+Disk-wind regulated accretion in 4U 1543−47
+9
+Table 3. Reflection modelling of NICER-NuSTAR simultaneous pairs and AstroSat observations (highlighted with grey colour) using the model
+tbabs(diskbb+relxilllp)gabs. The error values represent 90% confidence interval. The NuSTAR data on Epoch 8, 11 & 12 are not included here as no si-
+multaneous NICER observations available.
+Epoch
+nH
+Model
+χ2
+red
+diskbb
+relxilllp
+gabs
+(×1022
+Tin
+norm
+h
+θ
+Γ
+log ξ
+AFe
+Rf
+norm
+line E
+σ
+strength
+cm−2)
+(keV)
+(×103)
+(GM/c2)
+(deg)
+(erg cm s−1)
+(AFe,⊙)
+(×10−3)
+(keV)
+(keV)
+(keV)
+1
+0.49+0.01
+−0.01
+1.276+0.001
+−0.001 6.31+0.03
+−0.03
+25.50a
+32.7+3.4
+−2.9
+2.81+0.03
+−0.03
+3.58+0.09
+−0.07
+8.5+1.2
+−1.3
+1.4+0.1
+−0.2
+282.7+84.3
+−46.1
+9.9+0.1
+−0.1
+2.05+0.1
+−0.03
+1.1+0.1
+−0.1
+0.80
+2
+0.70+0.04
+−0.05
+1.151+0.006
+−0.005
+6.3+0.2
+−0.2
+9.4a
+36.3+1.6
+−1.8
+2.94+0.04
+−0.08
+3.65+0.05
+−0.08
+7.7+1.2
+−1.1
+3.5+0.9
+−0.7
+200.4+47.8
+−53.9
+9.8+0.1
+−0.1
+1.94+0.09
+−0.09
+1.3+0.2
+−0.2
+0.91
+3
+0.46+0.02
+−0.01
+1.01+0.006
+−0.006
+11.7+0.76
+−0.44
+30f
+40f
+3.0f
+4.7b
+9.37b
+−0.40 5.59+1.85
+−3.48 0.32+0.08
+−0.08
+9.37+0.27
+−0.13 0.75+0.37
+−0.64 0.48+0.03
+−0.19 1.37
+4
+0.683+0.004
+−0.03
+1.081+0.001
+−0.001 6.14+0.008
+−0.04
+30.5+24.3
+−2.0
+35.7+2.2
+−2.7
+3.23+0.01
+−0.06
+4.23+0.05
+−0.04
+5.8+0.8
+−0.7
+10.0a
+41.1+20.3
+−2.0
+10.04+0.04
+−0.07 1.92+0.04
+−0.06 1.99+0.2
+−0.08 0.83
+5
+0.67+0.01
+−0.03
+1.044+0.001
+−0.001 5.83+0.01
+−0.04 84.0+56.6
+−18.3 39.6+2.6
+−3.1 3.385+0.006
+−0.05
+4.7b
+7.1+0.6
+−0.7
+10.0a
+33.6+3.7
+−2.5
+10.33+0.03
+−0.03 2.06+0.03
+−0.05 2.76+0.02
+−0.06 0.83
+6
+0.49+0.01
+−0.02
+0.99+0.07
+−0.01
+9.23+1.46
+−0.19
+100f
+40f
+3f
+4.30+0.21
+−0.21
+10b
+7.18a
+7.11+1.95
+−4.1
+10.0+0.13
+−0.14 1.30+0.13
+−0.13 1.39+0.28
+−0.11 1.30
+7
+0.53+0.03
+−0.01
+0.99+0.003
+−0.003
+6.94+0.16
+−0.27
+8.26a
+40f
+2.46+0.36
+−0.25
+3.70+0.75
+−0.68
+10b
+9.99a
+5.75+0.55
+−0.69
+10.5+0.16
+−0.14 1.45+0.20
+−0.17 1.65+0.30
+−0.25 1.44
+9
+0.496+0.01
+−0.007 0.958+0.001
+−0.001 5.85+0.04
+−0.04
+40.29a
+40f
+3.17+0.1
+−0.06
+4.7b
+10b
+10f
+4.0+2.3
+−1.0
+10.11+0.08
+−0.08 1.86+0.06
+−0.06
+3.1+0.2
+−0.2
+1.10
+10
+0.44+0.03
+−0.01
+0.98+0.004
+−0.004
+4.95+0.10
+−0.90
+70f
+40f
+2.87+0.33
+−0.24
+4.7b
+10b
+8.10a
+1.33+0.28
+−0.12
+10.2+0.13
+−0.12 1.78+0.16
+−0.19 2.76+0.36
+−0.38 1.19
+13
+0.47+0.01
+−0.01
+0.94+0.002
+−0.003
+5.90+0.08
+−0.09
+44.3a
+40f
+2.25f
+3.64+0.09
+−0.18
+3.56+0.56
+−0.46
+8.10a
+1.11+0.07
+−0.16
+9.22+0.21
+−0.18 0.84+0.20
+−0.21 0.74+0.10
+−0.13 1.11
+14
+0.48+0.01
+−0.01
+0.91+0.002
+−0.002
+7.95+0.21
+−0.12
+6.69a
+40f
+2.61+0.08
+−0.19
+3.96+0.07
+−0.26
+10b
+10.6a
+0.14+0.01
+−0.01
+10.1+0.13
+−0.13 1.03+0.12
+−0.13 1.25+0.14
+−0.14 1.24
+15
+0.475+0.004
+−0.003 0.906+0.001
+−0.001 5.91+0.04
+−0.04
+33.74a
+38.0+8.5
+−4.4
+2.07+0.06
+−0.06
+2.7+0.1
+−0.3
+10.0b
+0.9+0.2
+−0.2
+4.9+1.5
+−0.7
+9.1+0.1
+−0.1
+1.5+0.1
+−0.2
+1.0+0.1
+−0.1
+1.12
+16
+0.482+0.004
+−0.004 0.904+0.001
+−0.001 5.63+0.05
+−0.05
+50.01a
+36.7+9.4
+−5.5
+2.05+0.05
+−0.05
+3.7+0.1
+−0.2
+10.0b
+0.9+0.2
+−0.1
+4.2+0.8
+−0.6
+8.7+0.1
+−0.1
+1.49+0.08
+−0.08
+1.2+0.2
+−0.2
+0.87
+a Parameter uncertainty can’t be estimated.
+b Parameter hits the boundary.
+f Frozen parameters.
+4.3 Absorption Features in the Spectra of 4U 1543−47
+The wideband spectral analysis of 4U 1543 − 47 reveals the pres-
+ence of a very strong absorption feature (Fig. 2, Table 2, Table 3)
+whose strength changes throughout the outburst. We use gabs model
+to characterize the absorption feature in phenomenological and re-
+flection modelling. The gabs strength estimated from both methods
+follow the same trend; getting more stronger as the outburst pro-
+gresses and reaches the maximum value on Epoch 9, then declines
+gradually.
+In general, the absorption features in the spectrum can be due to
+multiple reasons like the presence of obscuring cloud in the line-
+of-sight, occultation due to the companion star, strong accretion
+disk-wind and/or the stellar wind from the companion (Miller et al.
+2008; Szostek & Zdziarski 2008; Koljonen & Tomsick 2020). We
+have discarded the chances of absorption due to obscuring cloud in
+the line-of-sight by fitting the data with the partial covering fraction
+model pcfabs and found no improvement in the fitting. If the ab-
+sorption feature is produced by the occultation or stellar wind of the
+binary companion, the features must show some orbital variations.
+Precise diagnostic of the orbital variations provide significant insight
+into the understanding of the nature and origin of the absorption fea-
+tures. Since 4U 1543−47 is a low inclination system (θ ∼ 32◦ −40◦),
+the expected orbital variation of the absorption feature, if any, will be
+weak. Therefore, we avoid using multi-instrument data to check the
+orbital variation of the absorption feature, as the differences in the
+estimated parameters between instruments may screw up the varia-
+tion. We use only the NuSTAR observations for this purpose.
+We extracted the spectrum from different patches of GTIs of each
+NuSTAR observation epoch. Since the GTI-patches have low expo-
+sure time, we grouped the spectrum with only 30 counts per bin.
+Patches with an exposure time less than 500 seconds are merged to-
+gether before extracting the spectrum. We did a simultaneous joint
+fitting of all the GTI-patches under each epoch using the model M1.
+In the joint fitting, all parameters are tied between the patches ex-
+cept gabs strength. The line-of-sight column density, nH, is frozen
+at 0.45 × 1022 cm−2 found from broadband NICER-NuSTAR spec-
+tral modelling. In Fig. 7, we plot the simultaneous joint fitting of the
+spectra for Epoch 9, which has 7 patches of GTIs. The black, red,
+green, blue, cyan, magenta and yellow colours represent them in the
+ascending order of time.
+We also fitted all the 10 epochs (Table 1) of NuSTAR observations
+using the models M1 and M2 discussed before. We calculated the
+gabs strength (Si) for each NuSTAR epoch using both models. The
+evolution of Si is shown in Fig. 8a. The colours black, red, green,
+blue, cyan, pink, magenta, orange, yellow and grey indicate the NuS-
+TAR epochs in chronological order. The gabs strength of NuSTAR
+data is showing the same trend of wideband spectral data; reach-
+ing maximum on Epoch 9 (pink in colour). The absorption strength
+MNRAS 000, 1–14 (2022)
+
+10
+Prabhakar et al.
+10−5
+10−4
+10−3
+0.01
+0.1
+1
+Photons cm−2 s−1 keV−1
+10
+5
+20
+1
+1.2
+Ratio
+Energy (keV)
+Figure 7. Folded spectra of different patches in NuSTAR observation of
+Epoch 9 using the model tbabs(thcomp × diskbb)edge × gabs. The param-
+eters, except the gabs strength, are tied between the patches. See text for
+details.
+0
+20
+40
+60
+80
+100
+Time (days since MJD 59370)
+1
+2
+(a)
+0
+100
+200
+300
+Phase (degree)
+−0.5
+0.0
+0.5
+1.0
+Residual Strength (keV)
+(b)
+Strength (keV)
+Figure 8. (a) Evolution of gabs strength (Si) estimated from NuSTAR us-
+ing the models tbabs(thcomp × diskbb)edge × gabs (square symbol) and
+tbabs(diskbb + relxilllp) × gabs (star symbol). (b) Residual strength (Sp -
+Si) for different patches in each NuSTAR epoch with the orbital phase. The
+colours black, red, green, blue, cyan, pink, magenta, orange, yellow and grey
+represent the NuSTAR epochs in chronological order. See text for details.
+represented by the square symbol is estimated using the model M1,
+whereas the same using M2 are denoted by the star symbol. We no-
+tice that the value estimated using M2 are marginally higher than the
+same from M1. This can be the effect of an additional edge compo-
+nent used in the M1 model. Similarly, we estimated the gabs strength
+(Sp) corresponding to each GTI-patch for a given epoch from the
+patch-spectra modelling using M1. The residual strength (Sp - Si) is
+measured for each patches inside an epoch and are plotted against
+the orbital phase in Fig. 8b. The binary orbital period (P) of 4U
+1543 − 47 is 26.79377 ± 0.00007 hours (Orosz et al. 1998; Orosz
+2003). The orbital position of each NuSTAR patch has been identi-
+fied based on the start time (MJD 59382.42) of Epoch 1 as the refer-
+ence time. For Epoch 12 (data in orange colour in Fig. 8a), the value
+of Si is unusually low, and the estimated Sp from the patch-spectra
+modelling of this epoch is not reliable. Therefore, we ignore Epoch
+12 from Fig. 8b. The residual varies within ±0.5 keV (except one
+patch), and we see only a marginal variation (within uncertainties)
+in strength within an orbit. This implies that the orbital position of
+the BH and the companion is not responsible for the dynamic nature
+of the absorption features. In fact, we do not expect such behaviour
+for a low inclination system.
+The X-ray luminosity of the source at the peak (see Epoch 1 in
+Table 2) of the outburst is extremely high, and it may irradiate (see
+Lasota (2001) for a review) the outer accretion disk and the compan-
+ion star. If the irradiation affects the companion star, either a fresh
+accretion of matter starts at the hot spot or enhances the stellar wind
+in the companion star. The former may produce multiple trigger-
+ing in the same outburst event, which has been observed, for exam-
+ple, in GX 339 − 4 (Aneesha et al. 2019). If the latter happens, the
+highly ionized wind material may absorb the X-ray emission from
+the primary to produce the broad absorption feature. The compan-
+ion of 4U 1543 − 47 is an A2V type star with a mass Mc=2.45 M⊙
+(Russell et al. 2006) and radius Rc=2.84 R⊙ (Orosz et al. 1998). The
+escape velocity (ve) of the stellar material from the surface of the
+companion is calculated as 573 km s−1. The binary separation (a)
+between the BH and the companion is estimated as 7.18 ×1011 cm
+by considering a BH mass, MBH=9.4 M⊙ (Russell et al. 2006) using
+the relation P2/a3 = 4π2/ G(MBH + Mc), where G is the Gravitational
+constant. We observe that the stellar wind takes only a few hours to
+reach the primary. The column density of stellar wind and the ion-
+ization state should reduce with the decrease of the X-ray luminosity
+of the primary. Therefore, we expect that the strength of the broad
+absorption feature should reduce along with the progress of the out-
+burst. Instead, we observe that the absorption strength enhances and
+becomes strongest during Epoch 9. Moreover, the estimated stellar
+wind speed is not sufficient to blue shift the highest ionized lines
+of Fe XXVI, to produce the observed absorption feature. Therefore,
+the stellar wind has no role in the dynamic absorption features in the
+spectra of 4U 1543 − 47.
+The irradiation of the outer accretion disk enhances the accre-
+tion rate; therefore, the outburst source stays in the high luminosity
+state for a longer duration (King 1998; Lasota 2001; Aneesha et al.
+2019; Aneesha & Mandal 2020). This is possibly causing 2021 out-
+burst of 4U 1543 − 47 to decline very slowly (over ∼ 175 days,
+see Fig. 1). The super Eddington peak luminosity (Epoch 1 in Ta-
+ble 2) of the source can launch strong disk-wind (e.g., King et al.
+(2015); Muñoz-Darias et al. (2018)). The presence of the accre-
+tion disk-wind is more prominent in the soft state of X-ray bi-
+naries (Miller et al. 2008; Neilsen & Lee 2009; Ponti et al. 2012),
+though disk-winds are not exclusively confined to soft spectral state
+(Lee et al. 2002). Spectral analysis of 4U 1543 − 47 (Table 2) sug-
+gests that the source was in the HSS during our study. Also, we no-
+tice spectral softening happens till Epoch 9 (∼ day 60) and beyond
+which spectra gradually become harder (Table 2 & Table 3). If the
+disk-ionized winds are responsible for the absorption features, then
+the strength of the features would be maximum when the source is
+softer. Surprisingly the strength of the absorption feature is maxi-
+mum on ∼ day 60 as per our analysis (Fig. 8a), and it is keeps-on
+decreasing further. The optical depth (τ0) evolution (Fig. 4c) also
+suggests that the absorption column is maximally populated on ∼
+day 60.
+The transition energy of the most ionized line with the highest
+absorption yield, i.e., Fe XXV and Fe XXVI, are 6.68 keV and 6.97
+MNRAS 000, 1–14 (2022)
+
+Disk-wind regulated accretion in 4U 1543−47
+11
+keV respectively (provided by XSTAR line finding list12). Assuming
+the absorption feature (with line E ∼ 10 keV) in the NuSTAR spectra
+is produced due to the absorption of the accretion disk photons by
+the highly ionized blue shifted disk-wind, the estimated wind speed
+is reaching 30% of the speed of light to blue shift the Fe XXVI line
+energy to 10 keV. Such a fast disk-wind has never been observed in
+X-ray binary systems. In fact, highly ionized wind (say, Fe XXVI)
+is never detected (Ponti et al. 2012) in BH-XRB systems with low
+inclination angle; for example, GX 339 − 4, XTE J1817 − 330, 4U
+1957 + 115, XTE J1650 − 500, GRS 1758 − 258 etc. Therefore,
+this detection is the first of its kind for X-ray binaries. However,
+mildly relativistic disk-wind is not uncommon in quasars and AGNs
+(Reeves et al. 2009; Tombesi et al. 2015; Hagino et al. 2017). The
+other difficulty is the width (σ) of the absorption feature, which is
+as broad as 2 keV on Epoch 9 (Table 2). Known line-broadening
+processes due to turbulence or scattering will face serious challenges
+in explaining the line width if it is from a single line. Instead, it is
+more likely that the broad feature can be produced by combining
+multiple lines of various ionization states of iron.
+The phenomenological spectral fitting of NuSTAR data with
+model M1 reveals the presence of neutral Fe K−α absorption edge
+and the broad ionized absorption features. We calculate the equiv-
+alent width (EW), which is a measure of the strength of an ab-
+sorption line, of both absorption features to find if there exists any
+connection between these two components. The EW is defined as
+(Arumugasamy et al. 2018),
+EW =
+� ∞
+0
+[1 − F(E)] dE.
+(1)
+The energy dependent function F(E) for the gabs component is
+given by,
+F(E) = exp(−τ);
+τ = τ0 exp
+�
+−(E − Eg)2/2σ2�
+,
+(2)
+where τ0, Eg and σ are the optical depth, line energy and line width
+respectively. Similarly, F(E) corresponds to the edge component in
+XSPEC is given by,
+I(E) =
+
+1
+if E ≤ Ee
+exp[−D (E/Ee)−3]
+if E ≥ Ee,
+(3)
+where Ee and D are the threshold energy and absorption depth, re-
+spectively.
+The evolution of EW calculated based on NuSTAR phenomeno-
+logical modelling is shown in Fig. 9 for both edge (red in colour)
+and gabs (blue in colour) components. The gabs EW increases till
+Epoch 9 and then gradually decline, except for Epoch 12 (Fig. 8a)
+& 13 (Table 2). The implication of this result and the connection
+between both components (gabs and edge) are discussed in §5.
+5 DISCUSSION
+The wideband spectral modelling of NICER-NuSTAR and AstroSat
+data reveals that the inner disc temperature Tin is highest (1.27 keV)
+on Epoch 1, and it keeps on decreasing during the decay of the out-
+burst (Fig. 4a). The estimated diskbb norm (Table 2, Table 3) sug-
+gests a marginal inward movement of the inner disk radius rin since
+diskbb norm ∝ r2
+in. However, the decrease in rin could not prevent
+the drop in Tin due to the gradual decline in ˙M as Tin ∝ ˙M1/4 r−3/4
+in
+12 https://heasarc.gsfc.nasa.gov/docs/software/xstar/xstar.html
+20
+40
+60
+80
+100
+Time (days since MJD 59370)
+0.1
+0.5
+1
+2
+3
+EW (keV)
+A
+B C
+D
+E
+Figure 9. Evolution of equivalent width of gabs (blue in colour) and edge
+(red in colour) components from the NuSTAR epochs. The vertical lines (A-
+E) are used to explain the figure in the description.
+(Frank et al. 2002). The extreme luminosity in the inner disk may
+slow down the accretion of matter to the BH. On the other hand,
+the high accretion disk luminosity (Table 2) can irradiate the outer
+accretion disk, enhancing the accretion of matter. If most of the ac-
+creted matter is released as the disk-wind, the amount of matter ac-
+tually transfers to the inner disk for falling onto the BH is much less.
+Therefore, the gradual decline of Tin is due to the reduction of effec-
+tive infall of matter onto the BH through the inner disk. The source
+luminosity is completely soft-photons-dominated due to very little
+fractional Comptonization (cov_frac in Table 2), low corona tem-
+perature, and steeper photon index Γ (Fig. 4b). Therefore, the source
+was in the high/soft spectral state during our study, and it was the
+softest on Epoch 9.
+The important parameters for reflection modelling are shown in
+Table 3 and in Fig. 6. The lamp-post height comes closer to the
+central object as the source becomes softer. The reflection fraction
+(Rf) increases and hits the boundary when the source is softest.
+If the value of ionization parameter, log ξ ≳ 3, the fluorescence
+yield of the highly ionized Fe line (more ionized than Fe XXIII)
+increases (Matt et al. 1993). The high value of log ξ (Table 3) ob-
+tained from reflection modelling suggests a highly ionized accretion
+disk throughout the outburst. Combining all the factors like extreme
+luminosity, high log ξ, and an overabundance of Fe (parameter AFe
+in Table 3) refer to a highly ionized disk-wind having a significant
+yield of Fe XXV, Fe XXVI etc.
+An important characteristic of this source is the presence of a
+broad, symmetric and dynamic absorption feature in the spectrum
+∼ 8−11 keV. We presented various possibilities regarding the origin
+of this absorption feature in §4.3. Finally, we concluded that the fast
+moving ionized disk-wind could absorb the primary X-ray photons
+and produce this broad absorption feature. The phenomenological
+modelling shows the presence of the neutral Fe Kα absorption (edge)
+at ∼ 7.1 − 7.4 keV, originating from the outer part of the accretion
+disk. The initial steep rise of the EW of the neutral component (red
+star in Fig. 9) indicates the enhancement of disk matter due to irra-
+diation of the outer disk. Due to the availability of more matter, the
+radiation pressure of the highly luminous inner part could release
+more ionized matter. The source enters to the HSS, and the disk-
+wind gradually becomes very active till Epoch 9 (possibly Epoch
+MNRAS 000, 1–14 (2022)
+
+12
+Prabhakar et al.
+10 also) where the gabs EW is maximum (marked A in Fig. 9) and
+the disk spectrum is the softest (Fig. 4b). Therefore, the evolution
+of the ionized EW (blue square in Fig. 9) followed the neutral com-
+ponent till day 60 (Epoch 9). After that, EW of the ionized compo-
+nent declines gradually (AB in Fig. 9) because the disk luminosity
+has reduced significantly, and a good fraction of inner disk matter
+has already been lost in the form of wind. The neutral component
+remains unaffected as it takes a viscous timescale to propagate the
+same to the outer disk. The NuSTAR data on Epoch 12 reveals a sud-
+den drop of the strength (orange points in Fig. 8a) and EW (marked
+‘C’ in Fig. 9) of gabs component. We also notice the same signature
+in the AstroSat data on Epoch 13, observed after 2 days of the NuS-
+TAR’s Epoch 12 observation; the gabs strength (Table 2) on Epoch
+13 is smaller by a few factor compared to the nearby observations.
+We identify this sudden drop of ionized EW can be due to the evacu-
+ation of the inner disk. The huge central luminosity may slow down
+the accretion onto the central object, and most of the accreted mat-
+ter is released through disk-wind. Once the disk luminosity reduces,
+there is a sudden infall of matter onto the BH, leading to an evacua-
+tion of the inner disk.
+If this interpretation is correct, we expect a relatively harder spec-
+trum due to a significant drop of soft photon flux during Epochs 12
+& 13. To characterise this, we calculate the observed flux in 0.5 − 7
+keV (soft) and 7−20 keV (hard) band for NuSTAR and AstroSat data.
+The AstroSat soft and hard fluxes on Epoch 10 are 7.45 × 10−8 erg
+cm−2 s−1 and 1.23 × 10−9 erg cm−2 s−1 respectively. The drop of As-
+troSat soft flux on the next observation (Epoch 13) is 14%, whereas
+the hard flux increases by 5%. This resulted in a relatively harder
+spectral index (marked by a blue circle in Fig. 4b) on Epoch 13. Sim-
+ilarly, the NuSTAR soft and hard fluxes on Epoch 11 are 6.48 × 10−8
+erg cm−2 s−1 and 1.1 × 10−9 erg cm−2 s−1 respectively. The drop of
+NuSTAR soft flux on Epoch 12 is 10%, whereas the hard flux in-
+creases by a factor of 2. Therefore, the suddenly enhanced accretion
+at the inner disk resulted in these dramatic changes in the EW and
+spectral properties. However, the inner accretion disk recovers over
+the next 10 days (marked CD) due to the transfer of matter from
+the outer disk, and the ionized component returns back to a gradual
+declination. Interestingly, the neutral component (or the outer disk)
+follows the same trend as the ionized component, namely the de-
+cline and refilling signature (red stars between BE), with a delay of
+the typical viscous timescale of 10 − 15 days.
+6 SUMMARY AND CONCLUSION
+We study the wideband spectral properties of the 2021 outburst of
+4U 1543 − 47. The MAXI/GSC lightcurve (Fig. 1) shows that the
+outburst rises over 9 days followed by a slow decay over ∼ 175 days.
+We use multi-instruments data (NICER, NuSTAR and AstroSat) for
+simultaneous broadband spectral study over a period of 100 days
+from MJD 59370. We have performed the spectral study using the
+phenomenological model M1 and reflection model M2. The major
+findings from our study are summarized below:
+• The source generally remains very bright during this outburst
+with a super Eddington peak luminosity on Epoch 1 (Table 2).
+• The source was in the HSS during our study, with a steep pho-
+ton index (Fig. 4b) due to a very small fraction (< 3%) of inverse-
+Comptonized photons and low corona temperature.
+• The reflection modelling reveals that the inclination of the sys-
+tem is between 32◦−40◦.
+• The extreme luminosity, high ionization (log ξ > 3) and over-
+abundance of iron (3.6−10 AFe,⊙) indicate the presence of disk-wind
+with a significant yield of highly ionized iron species.
+• Presence of a broad, dynamic absorption feature at ∼ 8 − 11
+keV is observed throughout our study. This detection is the first of
+its kind for X-ray binaries. We propose that this feature is due to
+the absorption of the accretion disk photons by the highly ionized,
+blue shifted disk-wind. The strength of the ionized absorption fea-
+ture (Table 2 & Table 3) increases between Epoch 1 to Epoch 9 as the
+disk-wind column density is expected to increase with the spectral
+softening of the source.
+• The observed line energy of the absorption feature suggests an
+estimated wind speed of nearly 30% of the speed of light to blue
+shift the most ionized line with the highest absorption yield like Fe
+XXVI. Hence it would become the first X-ray binary source to show
+a highly relativistic disk-wind.
+• The initial steep rise of the neutral component EW (red star
+in Fig. 9) is an indication of the enhancement of disk matter due
+to irradiation of the outer disk. It enhances the accretion rate and
+hence the source remains in the high luminosity state and decays
+very slowly.
+• The evolution of EW (Fig. 9) of the neutral absorption compo-
+nent (edge) and the same of the ionized component (gabs), follow
+each other with a delay of the typical viscous timescale of 10 − 15
+days.
+• An evacuation of the inner accretion disk is observed during
+Epoch 12 − 13. This event leaves a signature of the drop in the soft
+photon flux and an enhancement of hard flux. Therefore, the spec-
+trum becomes relatively harder (blue circle in Fig. 4b).
+Finally, this study suggests that accretion dynamics of 4U 1543 −
+47 during 2021 outburst is regulated by the disk-wind.
+ACKNOWLEDGEMENTS
+The authors wish to thank the anonymous reviewer for the insightful
+suggestions which significantly improved the quality of the publi-
+cation. This work uses data from the NICER and NuSTAR mission
+by the National Aeronautics and Space Administration. This work
+also has used data from the AstroSat mission of the ISRO archived
+at the Indian Space Science Data Centre (ISSDC). The work has
+been performed utilizing the calibration databases, and auxiliary
+analysis tools developed, maintained and distributed by AstroSat-
+SXT team with members from various institutions in India and
+abroad. The High Energy Astrophysics Science Archive Research
+Center (HEASARC), which provides the software and NASA’s As-
+trophysics Data System Bibliographic Services are also acknowl-
+edged. BGR acknowledges the financial support of ISRO under As-
+troSat archival data utilization program Sanction order No. DS-2B-
+13013(2)/13/2019-Sec.2. AN thanks GH, SAG, DD, PDMSA, and
+Director, URSC for the support to carry out this research.
+DATA AVAILABILITY
+The
+data
+from
+NICER
+and
+NuSTAR
+underly-
+ing
+this
+article
+are
+available
+in
+HEASARC,
+at
+https://heasarc.gsfc.nasa.gov/docs/archive.html.
+AstroSat
+data
+archive
+is
+available
+at
+https://astrobrowse.issdc.gov.in/astro_archive/archive/Home.jsp.
+MNRAS 000, 1–14 (2022)
+
+Disk-wind regulated accretion in 4U 1543−47
+13
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf,len=2397
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='13711v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='HE]  31 Jan 2023  MNRAS 000, 1–14 (2022)  Preprint 1 February 2023  Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='0  Wideband Study of the Brightest Black Hole X-ray Binary 4U 1543−47 in  the 2021 Outburst: Signature of Disk-Wind Regulated Accretion  Geethu Prabhakar1⋆, Samir Mandal1†, Bhuvana G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='2, and Anuj Nandi3  1 Department of Earth and Space Sciences, Indian Institute of Space Science and Technology (IIST), Trivandrum - 695547, India  2 Department of Physics, Dayananda Sagar University, Bengaluru - 560068, India  3 Space Astronomy Group, ISITE Campus, U R Rao Satellite Centre, Bengaluru - 560037, India  Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' in original form ZZZ  ABSTRACT  A comprehensive wideband spectral analysis of the brightest black hole X-ray binary 4U 1543 − 47 during its 2021 outburst  is carried out for the first time using NICER, NuSTAR, and AstroSat observations by phenomenological and reflection mod-  elling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The source attains a super-Eddington peak luminosity and remains in the soft state, with a small fraction (< 3%) of  the inverse-Comptonized photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The spectral modelling reveals a steep photon index (Γ ∼ 2 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='6) and relatively high inner  disk temperature (Tin ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='9 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='27 keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The line-of-sight column density varies between (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='45 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='54)×1022 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Reflection  modelling using the RELXILL model suggests that 4U 1543 − 47 is a low-inclination system (θ ∼ 32◦ − 40◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The accretion  disk is highly ionized (log ξ > 3) and has super solar abundance (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='6−10 AFe,⊙) over the entire period of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We detected a  prominent dynamic absorption feature between ∼ 8 − 11 keV in the spectra throughout the outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' This detection is the first  of its kind for X-ray binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We infer that the absorption of the primary X-ray photons by the highly ionized, fast-moving  disk-winds can produce the observed absorption feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The phenomenological spectral modelling also shows the presence of  a neutral absorption feature ∼ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='1 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='4 keV, and both ionized and neutral absorption components follow each other with a delay  of a typical viscous timescale of 10 − 15 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Key words: accretion, accretion disc - black hole physics - X-rays: binaries - stars: individual: 4U 1543 − 47  1 INTRODUCTION  X-ray spectroscopy of black hole X-ray binaries (BH-XRBs)  holds the key to unveil the geometry of the system and the  dynamics of the accretion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The spectrum of BH-XRBs  mainly consists of a hard powerlaw and a soft thermal compo-  nent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The soft component, which is a multi-temperature black-  body, is assumed to be originated from an optically thick, ge-  ometrically thin accretion disk (Shakura & Sunyaev 1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The  hard powerlaw component is generally believed to be emitted  from an optically thin, hot electron cloud called ‘corona’ by the  Comptonization of the soft disk photons (Sunyaev & Titarchuk  1980, 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Zdziarski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Chakrabarti & Titarchuk 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Poutanen & Coppi 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Chakrabarti & Mandal 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Iyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Poutanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The relative strength of these com-  ponents leads to different states in outbursting BH-XRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' In the  Low/Hard State (LHS), the non-thermal component dominates and  in the High/Soft State (HSS), the disk emission dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' There  are short-lived intermediate states also, namely, the Hard Interme-  diate State (HIMS) and Soft Intermediate State (SIMS), lying be-  tween LHS and HSS (Homan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Homan & Belloni 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Remillard & McClintock 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Nandi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Sreehari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Aneesha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Bhuvana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Prabhakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' A typical outburst starts with the LHS and proceeds through  ⋆ geethuprabhakar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='17@res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='iist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='in  † samir@iist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='in  intermediate states to HSS then back to LHS again and finally reach  quiescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' However, it does not always have to go through all the  states mentioned above (Debnath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Radhika et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Baby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Prabhakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The advent of high resolution spectroscopy reveals the presence  of reflection features in the spectra of many BH-XRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Irrespective  of the geometry of the corona, it is believed that the photons upscat-  terd by the corona, the primary photons interact with the disc mate-  rial and a part of which produces the reflection features (Basko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The reprocessed X-ray spectrum consists of fluorescent line  emission from various elements, a soft thermal continuum and a  Compton hump peaked at ∼ 20 − 30 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The most prominent fea-  ture among the fluorescent emission lines is the iron K-edge at ∼ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='1  keV (Ueda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1998) and Kα line at ∼ 6 − 7 keV (White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Barret & Olive 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Di Salvo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' This is because  the fluorescence yield increases with the atomic number (Burhop  1952).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' For a distant observer, these reflection features appeared to  be diluted/broadened and distorted (asymmetric) due to relativistic  effects of the strong gravity region in the close vicinity of the BH  (Fabian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1989, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Spectral modelling using relativistic re-  flection models can address the effect of blurring of the spectral fea-  tures and helps to probe the physics of the strong gravity region at  the inner disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The accretion disk characteristics, such as the ion-  ization of the disk material, the iron abundance, inclination of the  system, spin of the BH etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=', can also be obtained from reflection  modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The line broadening can also be due to Comptonization  in a highly ionized, optically thick cloud, and the resultant feature  © 2022 The Authors  2  Prabhakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  is broad and symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' However, this mechanism is important for  high inclination systems only (Petrucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The Fe−K band (5 − 8 keV) is the energy range where most of  the emission/absorption features appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The first observational ev-  idence of the Fe−K absorption lines was provided by Ueda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  (1998) with ASCA in the spectra of galactic superluminal BH source  GRO J1655 − 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Kotani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' (2000) and Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' (2002) also de-  tected similar features in the superluminal jet source GRS 1915+105  with ASCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Later, it is revealed that the absorption features are  very common in the spectra of BH-XRBs (Shidatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Photon interaction with neutral  and static material produces sharp fluorescent lines at their corre-  sponding transition energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' In case of ionized absorbers, there  would be an increase in the transition line energy compared to their  neutral ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The absorption lines from highly ionized ions give an  insight into the highly ionized plasma around the compact object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The process of accretion in XRBs is usually accompanied with  outflows and/or jets (Fender et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1999, 2004, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  2012, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Radhika & Nandi 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Radhika et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The per-  sistent jets are present in the LHS of the system, and it gets turned  off in HSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Accretion disk-wind is generally observed in the disk-  dominated HSS, though it can exist in other spectral states as well  (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Neilsen & Lee 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Neilsen  2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The disk-winds carry a sufficient amount of matter which  suppresses the launch of radio jets (Neilsen & Lee 2009) in HSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The disk-wind can also be highly ionized and their presence can be  inferred by the blue-shifted absorption features in the X-ray spec-  trum (Ebisawa 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Kotani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' In general, it seems that  the absorption lines are absent in the LHS, which is still a matter  of debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Neilsen & Lee (2009) suggests that the wind gets pho-  toionized completely in LHS, and the medium becomes transparent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  this could be a possible reason for the absence of absorption lines in  the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Usually, the disk-winds are observed in high inclination  systems (Ponti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Such systems may show intensity ‘dips’  in their X-ray spectra, for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=', GRS 1915 + 105, 4U 1630 − 47, H  1743 − 322, MAXI J1305 − 704, GRO J1655 − 40 (Leahy 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Kuulkers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Shidatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The dips are believed to  be caused by obscuring material associated with the accretion disk  (Frank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1987) and are visible for highly inclined systems with  inclination angle 60◦ ≲ θ ≲ 80◦ (Frank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The disk-winds  play a major role in regulating the accretion scenario of BH-XRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  For example, Muñoz-Darias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' (2016) showed how winds control  the violent outburst of V404 Cygni by diminishing a significant frac-  tion of the outer disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Disk-wind studies in BH-XRBs can provide  great insights into the physical mechanisms involved in the accretion  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  4U 1543 − 47 is a BH-XRB, discovered by Uhuru satellite in  1971 (Matilsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Since the discovery, it has undergone  five outbursts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' the first four are in an interval of ∼ 10 years, in  1984 (Kitamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1984), 1992 (Harmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1992) and 2002  (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' After a gap of 19 years, the fifth outburst hap-  pened in 2021 (Negoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2021a), which marks the source as  the brightest BH-XRB with a peak X-ray intensity of 11 Crab in  2 − 4 keV with MAXI/GSC (Negoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The 2002 out-  burst was also brighter (4 Crab in 2 − 12 keV), while the previ-  ous three outbursts have comparable intensities (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Its optical counterpart, IL Lupi, was discovered by Pedersen (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The central engine is a dynamically confirmed BH with a mass of  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='0 M⊙, and the companion is an A2V star of mass 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='45 ±  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='15 M⊙ (Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' It is located at RA = 15h47m8s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='27,  Dec = −47◦40  ′10  ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='8 (J2000) (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2004) at a distance of  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='5 kpc (Jonker & Nelemans 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Orosz (2003) estimated  the orbital inclination of the system as 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='7◦ ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' There were  multiple attempts to estimate the spin (a∗, dimensionless spin pa-  rameter) of the BH in 4U 1543 − 47 using RXTE observations of  the 2002 outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Shafee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' (2006) estimated a spin of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='75 −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='85 using continuum-fitting of RXTE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' (2009) and  Morningstar & Miller (2014) estimated the spin value as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='1  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='43+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='22  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='31 respectively using relativistic disk reflection and disk  continuum modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' These three estimations are based on a BH  mass of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='0 M⊙ and a distance of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='5 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Shafee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  (2006) and Morningstar & Miller (2014) used the binary inclination  (θ) of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='5 degree, while Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' (2009) used a θ of 32+3  −4  degree for the spin estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' (2020) reported a spin of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='67+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='15  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='08 and θ of 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='3+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='3  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='4 degree by reflection modelling of RXTE  data using the model RELXILL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The Giant Metrewave Radio Telescope (GMRT) detected radio  flares from the source in 2002 outburst (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Multi-  ple flaring occasions are reported at different phases of the outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' (2020) reported the presence of a compact jet in the  SIMS of the 2002 outburst of 4U 1543 − 47 using multiwavelength  observations (X-ray, optical, IR, and radio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Since the system has  a low inclination, the jet angle and axis of rotation may coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' (2020) tested the chances of jet contribution to the lu-  minosity of the system and renounced that possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Until now, there is no study in literature based on the 2021 out-  burst of 4U 1543 − 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We aim for a detailed analysis of the wide-  band spectral characteristics of the 2021 outburst using three dif-  ferent instrument data from NICER (Neutron star Interior Compo-  sition ExploreR), NuSTAR (Nuclear Spectroscopic Telescope Array)  and AstroSat during outburst decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The evolution of spectral pa-  rameters is investigated using phenomenological and reflection mod-  elling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Even though the reflection modelling of RXTE data of 2002  outburst (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Morningstar & Miller 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  2020) unveil the fundamental quantities of the system like a∗ and  θ, data from much better spectral resolution instruments like NuS-  TAR (Harrison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2013) are highly promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' It can also provide  outburst specific quantities like the iron abundance and ionization  of the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We report the presence of strong and dynamic  absorption features in the 2021 outburst spectra, which has not been  observed in any previous outbursts of 4U 1543 − 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We examine  these features quantitatively using phenomenological modelling of  NuSTAR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  This paper is structured as follows: The observations and the data  reduction procedure are discussed in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The evolution of the out-  burst lightcurve and hardness ratio are examined in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The spectral  modelling and parameter evolution are presented in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Phenomeno-  logical and reflection modelling of different epochs are discussed in  §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='1 and §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The detailed study of the absorption fea-  tures in the spectra of 4U 1543 − 47 is carried out in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We dis-  cussed our overall findings in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Finally, we summarise the results  in §6 and then conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  2 OBSERVATIONS AND DATA REDUCTION  We perform the present study based on the 2021 outburst of 4U  1543 − 47 using NuSTAR, NICER and AstroSat observations over  a period from 17 June 2021 (MJD 59382) to 14 September 2021  (MJD 59471).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We considered all the NuSTAR and AstroSat obser-  vations in this period and used the NICER observations which are  simultaneous with NuSTAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The list of observations considered for  this study is given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' There are a total of 16 epochs of ob-  MNRAS 000, 1–14 (2022)  Disk-wind regulated accretion in 4U 1543−47  3  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The list of observations of the source 4U 1543 − 47 considered for the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' There are 16 epochs consisting of ten NuSTAR and six AstroSat  observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Seven NuSTAR epochs have simultaneous NICER coverage also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Epoch  Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' ID (MJD)  Remarks  NuSTAR  NICER  AstroSat  1  80702317002 (59382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='42)  4655060101 (59382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='44)  2  80702317004 (59389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='47)  4655060201 (59389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='47)  3  T04_018T01_9000004494 (59396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='04)  Offset  4  80702317006 (59396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='18)  4655060301 (59396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='19)  5  80702317008 (59403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='02)  4655060401 (59403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='04)  6  T04_021T01_9000004526 (59405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='36)  Offset  7  T04_030T01_9000004588 (59421.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='19)  Pointed  8  90702326002 (59421.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='67)  9  90702326004 (59428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='18)  4202230143 (59428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='13)  10  T04_035T01_9000004622 (59430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='59)  Pointed  11  90702326006 (59450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='19)  12  90702326008 (59455.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='55)  13  T04_046T01_9000004680 (59457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='06)  Pointed  14  T04_051T01_9000004686 (59461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='05)  Pointed  15  90702326010 (59465.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='67)  4202230166 (59466.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='07)  16  90702326012 (59471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='51)  4202230171 (59471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='43)  servations consisting of ten NuSTAR and six AstroSat observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Seven NuSTAR epochs have simultaneous NICER coverage also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='1 NuSTAR Data Reduction  NuSTAR (Harrison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2013) has observed the source several  times in the 2021 outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' NuSTAR is devoid of pile-up issues  and moreover, its good energy resolution in the energy cover-  age (3 − 79 keV) makes it suitable for the study of enormously  bright sources like 4U 1543 − 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' NuSTAR consists two focal  plane module telescopes (FPMA and FPMB), both are operat-  ing in 3 − 78 keV band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The NuSTAR data for the 2021 out-  burst is reduced using HEASOFT v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='29, NUSTARDAS pipeline  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='1 and CALDB v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='20211115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' For extremely bright sources,  we set statusexpr="STATUS==b0000xxx00xxxx000"1 and set  saamode to strict and tentacle to yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' A circular region of  radius 35 pixels centered on the brightest pixel is extracted as the  source region and as the background region, we also choose a 35  pixel circular region away from this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' These files are used for gener-  ating science products such as the spectrum, background, lightcurve,  Auxiliary Response File (ARF) and Response Matrix File (RMF),  through the NUPRODUCTS task, independently for both FPMA and  FPMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The spectra are grouped with a minimum of 50 counts per  bin without any systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='2 NICER Data Reduction  The  X-ray  Timing  Instrument  (XTI)  onboard  NICER  (Gendreau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2016) operates in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='2 − 12 keV band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' NICER  has observed the source 4U 1543 − 47 in almost every day during  the 2021 outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We analyse NICER data of the source between  MJD 59382 and MJD 59471 which is simultaneous with the  NuSTAR observations (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The data is reduced using the  tool NICERDAS2 in HEASOFT v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='29 with the 20210707 caldb  version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' There are 56 focal plane modules (FPMs) of NICER/XTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  1 https://heasarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='gov/docs/nustar/analysis/  2 https://heasarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='gov/docs/nicer/nicer_analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='html  We excluded FPM-14 and 34 in addition to the non-functional  FPMs (FPM-11, 20, 22, and 60) due to increased noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Since 4U 1543 − 47 is extremely bright at the beginning of the  outburst, the initial epochs (till ∼ MJD 59425) are affected by  telemetry saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' For such observations, a lower number of  FPMs were kept active by the instrument team and we considered  only the active detectors in the data reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Level-2 standard  calibration and filtering are done using nicerl2 task and applied  barycenter corrections through barycorr with refframe="ICRS".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Spectra is generated using XSELECT (V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='4m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Lightcurve of  the NICER observation on MJD 59428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='18 shows a flaring in the  high energy band;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' therefore, the corresponding GTIs are excluded  from the extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The ARF and RMF files are generated for  each observation based on the number of active detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The  task nibackgen3C503 (Remillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2021) is used for creating  background files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Finally, the source spectra are grouped with 25  photons per bin and a systematic uncertainty of 1 % is added to the  spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='3 AstroSat Data Reduction  The Soft X-ray Telescope (SXT) and Large Area X-ray Propor-  tional Counter (LAXPC) on-board AstroSat (Yadav et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Agrawal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2017) together observes the astronomical sources in  wideband energy range (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='3−80 keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' AstroSat has observed the  2021 outburst of 4U 1543-47 during 6 different epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The first  two of these observations are carried out with an offset of 40′ since  the source was too bright to have pointed observation (Garg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We obtain Level-1 LAXPC and Level-2 SXT data of all six  observations available at data archive hosted by ISSDC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  LAXPC consists of three identical proportional counts namely  LAXPC10, LAXPC20 and LAXPC30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' However, for our analysis, we  have used data from LAXPC20 alone because of its steady gain (see  also Bhuvana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Baby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Bhuvana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Prabhakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' To extract the Level-2 LAXPC data file i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=',  source spectrum, lightcurve, RMF and background spectrum and  3 https://heasarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
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+page_content='gov/docs/nicer/tools/nicer_bkg_est_tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='html  4 https://astrobrowse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='issdc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='in/astro_archive/archive/Home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='jsp  MNRAS 000, 1–14 (2022)  4  Prabhakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  0  5  10  0  25  50  75  100  125  150  175  0  20  (a)  NICER  NuSTAR  AstroSat  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
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+page_content='5  0  25  50  75  100  125  150  175  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='2  (b)  0  25  50  75  100  125  150  175  Time (days since MJD 59370)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='4  (c)  Flux (photons/sec/cm2)  Ratio (4-10/2-4 keV)  10-20 keV  2-10 keV  Flux (Crab)  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' MAXI/GSC daily average lightcurve in the energy bands (a) 2 − 10 keV and (b) 10 − 20 keV with flux in units of photons/sec/cm2 and Crab in the left  and right Y-axes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The hardness ratio (c) is defined by the flux in 4 − 10 keV to 2 − 4 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The NICER, NuSTAR and AstroSat observations for this  study are marked using lines with the colours cyan, blue and red respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  lightcurve, we make use of latest version of single routine LAXPC  software LaxpcSoftversion3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='35 (Antia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Level-2  files are extracted from a single event and the top layer of LAXPC  unit to avoid the instrument effects at high energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' While the soft-  ware generated LAXPC response files are used for pointed obser-  vations, a 40′ offset LAXPC response file provided by the instru-  ment is used for the offset observations (see also Baby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Katoch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  SXT has observed the source in Photon Counting (PC) mode dur-  ing all the epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The orbit-wise SXT cleaned Level-2 event files  are merged to get single event file for each observation using event  merger python routine6 based on Julia v 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The merged event file  is then loaded into XSELECT, where we select single-pixel events  by applying grade 0 filter to avoid optical data leakage (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Prabhakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' From the XSELECT images, we find  that the first two offset observations have count rate < 40 counts s−1  and hence the corresponding spectra wouldn’t have pileup issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  We select a circular region of radius 10′ in the image to extract the  source spectrum and lightcurve files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' In all the pointed observations  (see Table 1), we find the central region of the image to be very  bright which could cause a pile-up effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Therefore, source files are  extracted from an annular region of the outer radius of 15′ and inner  radius of 2′ for these observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The standard SXT background  spectrum and instrument response file provided by the instrument  team7 are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' ARF for the selected region is obtained from python-  based tool sxtarfmodule provided by the SXT team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Extracted SXT  and LAXPC spectra are grouped to have 30 counts per bin in the first  5 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
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+page_content='html  7 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='tifr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='in/~astrosat_sxt/dataanalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='html  two observations and 20 counts per bin in the rest of the observations  based on the source brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' A systematics of 2% (Sreehari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Athulya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2022) is applied for both SXT and LAXPC spec-  tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  3 OUTBURST PROFILE AND HARDNESS RATIO  During the 2021 outburst of 4U 1543 − 47, the flux reached the  peak value within a few days of the commencement of the out-  burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The outburst is monitored by multiple X-ray instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The  MAXI/GSC8 daily lightcurve of the source is generated for two dif-  ferent energy bands, 2 − 10 keV and 10 − 20 keV and are plotted in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The MJD 59370 (05 June 2021) is defined as day 0 through-  out the study and according to this, the outburst continues over ∼175  days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The lightcurve reveals that the source is extremely luminous in  low energies with a very high count rate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1a), while the contri-  bution to the luminosity in the high energy band (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1b) is an order  of magnitude lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The highest value of flux in 2 − 10 keV band is  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='67 photons/sec/cm2 (∼ 10 Crab) on day 9, whereas the same in  10 − 20 keV is just 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='19 photons/sec/cm2 (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='5 Crab).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The source  flux reached 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='65 Crab in 2 − 4 keV, which is the highest value ob-  served among the BH-XRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We define hardness ratio (HR) as the  ratio of flux in 4 − 10 keV to 2 − 4 keV since beyond 10 keV the  contribution is significantly low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The HR evolution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1c) shows  that the source mostly remains in the soft state during the outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The NICER, NuSTAR and AstroSat observations used in this study  are marked by cyan, blue and red dashed lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' There  is no simultaneous broadband observation in the rising phase of the  outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  8 http://maxi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='riken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='jp/mxondem/  MNRAS 000, 1–14 (2022)  Disk-wind regulated accretion in 4U 1543−47  5  10  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='2  Ratio  Energy (keV)  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The ratios (model to data) of the spectral fitting of NuSTAR obser-  vations (Table 1) using tbabs(diskbb+powerlaw) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The colours black,  red, green, blue, cyan, pink, magenta, orange, yellow and grey represent the  NuSTAR epochs in ascending order (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' A strong absorption feature ex-  ists between ∼ 8 − 11 keV for all the epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The absorption depth increases  up to Epoch 9 (pink in colour) and then decreases as the outburst progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The figure is zoomed around the absorption feature for better clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  4 SPECTRAL MODELLING AND RESULTS  We studied the spectral properties of the 2021 outburst of 4U 1543−  47 from MJD 59382 (17 June 2021) to MJD 59471 (14 September  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' All the three instruments, NICER, NuSTAR and AstroSat have  good coverage over this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Table 1 summarises the list of obser-  vations used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' In total, there are 16 epochs, comprising  ten NuSTAR and six AstroSat (SXT-LAXPC) observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' In addi-  tion, there are seven NICER observations which are simultaneous  with that of NuSTAR and we used these pairs for NICER-NuSTAR  wideband spectral analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We carried out phenomenological and  reflection modelling of each NuSTAR observations and extended that  to wideband NICER-NuSTAR and AstroSat observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='1 Phenomenological Spectral Modelling  We used HEASOFT v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='29 and XSPEC V12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='0 package for the  spectral modelling of NICER, NuSTAR and AstroSat data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We have  done spectral modelling of NICER-NuSTAR and AstroSat data to  see the nature of the broad-band spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The NICER data below  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='8 keV shows large residuals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' therefore, we used 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='8 − 10 keV for  NICER spectra and 4 − 60 keV for NuSTAR spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Spectra from  both FPMA and FPMB telescopes of NuSTAR give similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  We present only FPMA spectra throughout the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We used 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='5−7  keV for SXT and 3 − 30 keV for LAXPC as significant data is not  available beyond this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  To accommodate the interstellar absorption, we used the tbabs  model which uses an equivalent hydrogen column density nH  through the solar abundance table provided by Wilms et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We initially modelled the NuSTAR observations with  tbabs(diskbb+powerlaw) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Here, diskbb (Mitsuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1984)  represents the multicolor blackbody spectrum from the accretion  disc and powerlaw employs the inverse Comptonization of the soft  blackbody photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We detected a broad absorption feature at ∼  8−11 keV in all epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The ratio of data to model of all the NuSTAR  observations are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The different NuSTAR epochs are  shown in black, red, green, blue, cyan, pink, magenta, orange, yel-  low and grey colours in ascending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' It shows that the depth of  absorption feature starts with a low value (black in colour) and then  keeps on increasing as the outburst progress, reaching the maximum  on Epoch 9 (pink in colour).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Finally, the absorption depth decreases  towards the end (grey in colour) of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We found a similar  absorption feature in the AstroSat/LAXPC data as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  We used a partial covering fraction absorption model pcfabs9 in  XSPEC to check if this strong absorption feature at ∼ 8−11 keV can  be due to an intervening absorber, but it did not improve the fitting  and the low energy residuals were high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We also tried several other  models, for example, the thermal Comptonization model nthcomp  (Zdziarski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1996) or thcomp (Zdziarski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2020) in-place of  powerlaw, and diskbb was replaced by kerrbb (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Var-  ious combinations of these models, fit to the data also showed the  presence of the absorption feature at ∼ 8 − 11 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Model combina-  tion with kerrbb as the seed photon source did not provide a good  fit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' moreover, it failed to constrain the BH mass and spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' So, we  prefer to use diskbb model in combination with thcomp which is an  improved version of nthcomp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The thcomp is a convolution model  which allows a variable fraction (parameter cov_ f rac) of seed pho-  ton to Comptonize both up-scattering and down-scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Other  parameters are the photon index (Γ) and electron temperature (kTe)  of the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The observed absorption feature has a symmetrical profile and in-  clusion of a Gaussian absorption model gabs10 fits the absorption  feature well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The parameters of gabs component are line energy  (line E, Eg), line width (σ) and line depth (strength).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' However, if  we use a smeared absorption edge model, smedge (Ebisawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  1994) in XSPEC to compensate for the absorption feature, it re-  sults in an abnormally high value of absorption width due to the  asymmetric nature of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' In addition, the NuSTAR data show  a weak presence of a Fe Kα absorption edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Therefore, we also  used an edge component to improve the fit residual for NuSTAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The model parameters for the edge component are the threshold  energy of the absorption edge (edge E, Ee) and the correspond-  ing absorption depth (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Therefore, the final model for NICER-  NuSTAR data is tbabs(thcomp × diskbb)edge × gabs (model M1,  hereafter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' In contrast, the absorption edge feature was not visible  in the AstroSat/LAXPC data, possibly due to the low spectral resolu-  tion of LAXPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Therefore, the model M1 for AstroSat data becomes  tbabs(thcomp×diskbb)×gabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' However, the AstroSat data of Epoch  3 & 6 show the presence of a weak Fe Kα emission line feature in  the residual, and we include a gauss model component for these two  epochs of AstroSat data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  All the seven NICER-NuSTAR simultaneous pairs and six AstroSat  observations are fitted with the model M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 3a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 3b  show the wideband spectra of NICER-NuSTAR (Epoch 2 & 9) and  AstroSat (Epoch 3 & 7) respectively modelled using M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' In each  case, the spectrum in the red colour is relatively softer than that of  the black colour;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' therefore, both figures illustrate moderate spectral  changes during the outburst decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The goodness of the fit is determined using χ2 statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The re-  duced χ2 (χ2  red) varies between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='9 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' All the parameters es-  timated from the wideband phenomenological modelling are pre-  sented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
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+page_content=' (a) Simultaneous NICER-NuSTAR pair on Epoch 2 (black in colour) and Epoch 9 (red in colour) fitted with the model tbabs(thcomp × diskbb)edge ×  gabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Epoch 2 spectrum is harder compared to that of Epoch 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' (b) The AstroSat spectra on Epoch 3 (black) and 7 (red) respectively modelled using  tbabs(thcomp × diskbb) × gabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
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+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Wideband NICER-NuSTAR simultaneous pairs and AstroSat observations (highlighted with grey colour) using the model tbabs(thcomp×diskbb)edge×  gabs and tbabs(thcomp×diskbb)×gabs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
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+page_content='  f Frozen parameters  MNRAS 000, 1–14 (2022)  Disk-wind regulated accretion in 4U 1543−47  7  20  40  60  80  100  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
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+page_content='0  Γ  (b)  20  40  60  80  100  Time (days since MJD 59370)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
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+page_content='50  τ0  (c)  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Evolution of (a) Tin and (b) Γ from simultaneous NICER-NuSTAR  pairs (green in colour) and AstroSat observations (red in colour) fitted using  the model M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We calculate the (c) optical depth (τ0) of the absorber from  gabs components for each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The Γ of Epoch 13 marked with a blue  circle carries a special signature that is discussed in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  of various parameters with the progress of the outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 4 gives  the variation of the inner disk temperature Tin, photon index Γ and  optical depth (τ0) with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Here, points in green and red colour  represent the parameter value estimated from NICER-NuSTAR and  AstroSat spectral modelling respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  We find that the inner disk temperature (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 4a and Table 2)  monotonically decreases throughout the outburst decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The evolu-  tion of the diskbb norm (Table 2) estimated using NICER-NuSTAR  decreases till Epoch 9 and a reverse trend is observed for later  epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The diskbb norm from the AstroSat data also shows a sim-  ilar pattern though AstroSat values are higher than that of NICER-  NuSTAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The variation of photon index Γ is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The  value of Γ estimated from AstroSat data (red square) differs from that  of NICER-NuSTAR pairs (green square).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' This can be due to the non-  availability of the high energy contribution (beyond 30 keV) in the  LAXPC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' From NICER-NuSTAR fitting, Γ varies between 2−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='6,  and it shows spectral softening till Epoch 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We wanted to estimate  the electron temperature (kTe) of the corona using the thcomp model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  But the broadband spectral fitting could not constrain the value of  kTe, except for Epoch 2, for which we obtained kTe = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='6+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='9  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='9 keV  (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' For all the remaining epochs, we freeze kTe at 20 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Only a tiny fraction, cov_frac < 3 % (Table 2), of the soft photons  Comptonized in the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' It gradually decreases till Epoch 10 and  increases afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' This behaviour is consistent with the spectral  softening trend shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The broad absorption feature at ∼ 8−11 keV in the spectrum is  well represented by the gabs model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The strength shows an increas-  ing trend and reaches the maximum in Epoch 9 and declines beyond  that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We calculate the optical depth (τ0) associated with the gabs  component using gabs strength and σ as τ0 = strength/σ  √  2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The  evolution of τ0 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 4c, which shows that the absorption  optical depth increases and reaches a maximum on Epoch 9 and then  decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The dynamic behaviour of the absorption strength seems  10−4  10−3  10−2  10−1  100  101  Photons cm−2 s−1 keV−1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='25  (a)  χ2  red =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='46  tbabs(diskbb+relxill)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='25  (b)  χ2  red =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='60  tbabs(diskbb+relxill)gabs  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='2  (c)  χ2  red =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='63  tbabs(diskbb+relxillCp)gabs  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='25  (d)  χ2  red =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='79  tbabs(diskbb+relxilllpCp)gabs  100  101  Energy (keV)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='2  (e)  χ2  red =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='91  tbabs(diskbb+relxilllp)gabs  Ratio  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Unfolded spectrum (top panel) of simultaneous NICER-NuSTAR  pair on Epoch 2 and ratio of the model to the data using various re-  flection models (a) tbabs(diskbb+relxill), (b) tbabs(diskbb+relxill)gabs,  (c) tbabs(diskbb+relxillCp)gabs, (d) tbabs(diskbb+relxilllpCp)gabs and (e)  tbabs(diskbb+relxilllp)gabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The model and the value of χ2  red are mentioned  at the top left and bottom left corners, respectively, in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We attempt to characterize the strong and dynamic ab-  sorption features in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The evolution of the edge component is  listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We discuss a possible connection between edge  and gabs components in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  We estimate the observed bolometric flux (Fbol) in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='5 − 100  keV with uncertainty in 90% confidence interval from the wide-  band simultaneous spectral data, which is also shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Corresponding bolometric luminosity (Lbol) of the source is also  calculated by assuming the distance to the source as 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='5 kpc  (Jonker & Nelemans 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The Eddington luminosity of the source  is LEdd = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='22±1×1039 ergs s−1 with an assumed BH mass of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='4±1  M⊙ (Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' It can be seen from Table 2 that the lumi-  nosity of the source exceeds the LEdd at the peak (Epoch 1 is close  to the peak) of the outburst, and the luminosity decreases gradually  with the decay of the outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  MNRAS 000, 1–14 (2022)  8  Prabhakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='2 Spectral Modelling for Reflection Studies  To understand the reflection features in the spectra of 4U 1543 − 47,  we use the relativistic reflection model RELXILL11v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Dif-  ferent flavours of the RELXILL model are tried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The unfolded  NICER-NuSTAR spectra and data-to-model ratios using various re-  flection models are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 5 for Epoch 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We started with  the model tbabs(diskbb+relxill) which gives a χ2  red of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The  data-to-model ratio (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 5a) shows that the absorption feature at  ∼ 8 − 11 keV in the spectrum cannot be fitted by the reflection  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We added the absorption model gabs with this, and the model  tbabs(diskbb+relxill)gabs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 5b) improves the residual but still  the χ2  red for this combination is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Then, we replaced the model  relxill with relxillCp (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 5c) where a thermal Comptonizing con-  tinuum is assumed for the illuminating flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' This combination has  χ2  red = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='63, which is also unacceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Then, we used relxilllpCp  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 5d) as the reflection model, where a lamp-post (lp) geome-  try is assumed for the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' In all the ‘lp’ flavours of the RELX-  ILL model, the inner disk is illuminated by a point-like corona sit-  uated at a height ‘h’ from the disk surface on the axis of rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The modelling with relxilllpCp also resulted in a large residual, with  χ2  red = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We replaced relxilllpCp with relxilllp (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 5e) where  the illuminating flux is modelled as a powerlaw with a high-energy  cutoff just like the relxill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' For this trial, we got a reasonable fit with  χ2  red = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='91, and we decided to proceed with the model combina-  tion, tbabs(diskbb+relxilllp)gabs (model M2, hereafter) as the final  model to study the reflection features in the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Note that, no ad-  ditional edge or gauss component is required in the reflection mod-  elling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Using the model M2, we did the spectral fitting of the simultane-  ous NICER-NuSTAR pairs and AstroSat observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Here, nH is a  free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Inner and outer disk radii are frozen at the innermost  stable circular orbit RISCO and 400 rg (where rg ≡ GM/c2, the gravi-  tational radius of the BH), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The powerlaw cutoff energy,  Ecut, is fixed at 60 keV since it is hitting the upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' All other  parameters of relxilllp are kept free;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' the lamp-post height h (in units  of rg), inclination angle of the system θ (in degree), Γ of the incident  radiation, ionization parameter log ξ (erg cm s−1), iron abundance  (AFe) of the accretion disk in terms of the solar abundance AFe,⊙, and  the reflection fraction Rf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Here, Rf is defined as the ratio of the pri-  mary photon flux illuminating the disk to that reach the observer at  infinity (Dauser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We wish to estimate the spin parameter,  a∗, of the system, but it is found to hit the upper limit for almost all  the epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Based on previous studies (Morningstar & Miller 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2020), we freeze a∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='4 for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The estimated reflection model (M2) parameters are listed in Ta-  ble 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The errors represent 90% confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The evolution  of Tin follows the same trend as that observed in the phenomeno-  logical spectral modelling (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The value of Γ varies between  2 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='3 and appears slightly steeper than that of phenomenologi-  cal modelling (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' It may be due to the additional low energy  contribution from the reflection component over the diskbb compo-  nent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' However, reflection modelling also shows spectral softening  till Epoch 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The NICER-NuSTAR results suggest that 4U 1543 − 47  is a low inclination system with θ varies between ∼ 32◦ − 40◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We  could not constrain the inclination angle from AstroSat data and  freeze it to 40◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The evolution of few important model parameters (h, log ξ  and Rf ) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 6 for better presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Simultane-  ous NICER-NuSTAR pairs and AstroSat observations are marked in  11 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='sternwarte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='uni-erlangen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='de/~dauser/research/relxill/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='html  20  40  60  80  100  0  50  100  h  (a)  20  40  60  80  100  2  4  log ξ  (b)  20  40  60  80  100  Time (days since MJD 59370)  1  5  10  Rf  (c)  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Evolution of (a) h, (b) log ξ and (c) Rf from reflection modelling  of simultaneous NICER-NuSTAR pairs (green in colour) and AstroSat obser-  vations (red in colour) using the model tbabs(diskbb + relxilllp)gabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' See  text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  green and red colour, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We could not estimate the un-  certainty of h in most of the epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The evolution of h (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 6a)  indicates that the primary source is moving away from the BH till  Epoch 6 and then gradually coming closer to the central object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The  ionization structure of the disk is established through the parameter  log ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Its value gradually increases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 6b) and reaches the maxi-  mum around Epoch 10 and then gradually decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The high value  of log ξ (>3) suggests a highly ionized disc material throughout the  outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We could estimate iron abundance AFe for Epoch 1, 2, 4  and 5, and it hits the upper limit during the rest of the epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Our  study reveals an overabundance (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='6−10 AFe,⊙) of iron in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The reflection fraction Rf is estimated well at the first three and  last two epochs only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 6c suggests that the fraction of primary  photons reaching the disk increases till Epoch 9, and it decreases af-  terwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The gabs strength shows a similar behaviour found in the  phenomenological modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We have discussed more on this result  in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Very recently, the RELXILL model (version v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='2) has undergone  some modifications by considering the effect of returning radiation  in the calculation of reflected flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Particularly in relxilllpCp, where  the effects of returning radiation, the density profile and ionization  gradient of the disk and the velocity of the primary source are also  included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' However, the velocity of the primary source and the effects  of returning radiation are the new parameters added to the relxilllp  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We applied this modified relxilllp model to the broadband  NICER-NuSTAR data but could not constrain the source velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Also, we tried to estimate the parameters using the v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='2 flavour of  relxilllpCp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We fitted all the wideband NICER-NuSTAR observations  using the model tbabs(diskbb+relxilllpCp)gabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' But, we could not  constrain most of the parameters since the number of free parameters  is very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Therefore, we need to essentially freeze all the new  parameters introduced in the updated version, and RELXILL v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='2  does not bring any improvement in the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  MNRAS 000, 1–14 (2022)  Disk-wind regulated accretion in 4U 1543−47  9  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Reflection modelling of NICER-NuSTAR simultaneous pairs and AstroSat observations (highlighted with grey colour) using the model  tbabs(diskbb+relxilllp)gabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The error values represent 90% confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The NuSTAR data on Epoch 8, 11 & 12 are not included here as no si-  multaneous NICER observations available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Epoch  nH  Model  χ2  red  diskbb  relxilllp  gabs  (×1022  Tin  norm  h  θ  Γ  log ξ  AFe  Rf  norm  line E  σ  strength  cm−2)  (keV)  (×103)  (GM/c2)  (deg)  (erg cm s−1)  (AFe,⊙)  (×10−3)  (keV)  (keV)  (keV)  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
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+page_content='87  a Parameter uncertainty can’t be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  b Parameter hits the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  f Frozen parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='3 Absorption Features in the Spectra of 4U 1543−47  The wideband spectral analysis of 4U 1543 − 47 reveals the pres-  ence of a very strong absorption feature (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2, Table 2, Table 3)  whose strength changes throughout the outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We use gabs model  to characterize the absorption feature in phenomenological and re-  flection modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The gabs strength estimated from both methods  follow the same trend;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' getting more stronger as the outburst pro-  gresses and reaches the maximum value on Epoch 9, then declines  gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  In general, the absorption features in the spectrum can be due to  multiple reasons like the presence of obscuring cloud in the line-  of-sight, occultation due to the companion star, strong accretion  disk-wind and/or the stellar wind from the companion (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Szostek & Zdziarski 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Koljonen & Tomsick 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We  have discarded the chances of absorption due to obscuring cloud in  the line-of-sight by fitting the data with the partial covering fraction  model pcfabs and found no improvement in the fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' If the ab-  sorption feature is produced by the occultation or stellar wind of the  binary companion, the features must show some orbital variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Precise diagnostic of the orbital variations provide significant insight  into the understanding of the nature and origin of the absorption fea-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Since 4U 1543−47 is a low inclination system (θ ∼ 32◦ −40◦),  the expected orbital variation of the absorption feature, if any, will be  weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Therefore, we avoid using multi-instrument data to check the  orbital variation of the absorption feature, as the differences in the  estimated parameters between instruments may screw up the varia-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We use only the NuSTAR observations for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  We extracted the spectrum from different patches of GTIs of each  NuSTAR observation epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Since the GTI-patches have low expo-  sure time, we grouped the spectrum with only 30 counts per bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Patches with an exposure time less than 500 seconds are merged to-  gether before extracting the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We did a simultaneous joint  fitting of all the GTI-patches under each epoch using the model M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  In the joint fitting, all parameters are tied between the patches ex-  cept gabs strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The line-of-sight column density, nH, is frozen  at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='45 × 1022 cm−2 found from broadband NICER-NuSTAR spec-  tral modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 7, we plot the simultaneous joint fitting of the  spectra for Epoch 9, which has 7 patches of GTIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The black, red,  green, blue, cyan, magenta and yellow colours represent them in the  ascending order of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  We also fitted all the 10 epochs (Table 1) of NuSTAR observations  using the models M1 and M2 discussed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We calculated the  gabs strength (Si) for each NuSTAR epoch using both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The  evolution of Si is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 8a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The colours black, red, green,  blue, cyan, pink, magenta, orange, yellow and grey indicate the NuS-  TAR epochs in chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The gabs strength of NuSTAR  data is showing the same trend of wideband spectral data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' reach-  ing maximum on Epoch 9 (pink in colour).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The absorption strength  MNRAS 000, 1–14 (2022)  10  Prabhakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  10−5  10−4  10−3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='1  1  Photons cm−2 s−1 keV−1  10  5  20  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='2  Ratio  Energy (keV)  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Folded spectra of different patches in NuSTAR observation of  Epoch 9 using the model tbabs(thcomp × diskbb)edge × gabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The param-  eters, except the gabs strength, are tied between the patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' See text for  details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  0  20  40  60  80  100  Time (days since MJD 59370)  1  2  (a)  0  100  200  300  Phase (degree)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='0  Residual Strength (keV)  (b)  Strength (keV)  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' (a) Evolution of gabs strength (Si) estimated from NuSTAR us-  ing the models tbabs(thcomp × diskbb)edge × gabs (square symbol) and  tbabs(diskbb + relxilllp) × gabs (star symbol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' (b) Residual strength (Sp -  Si) for different patches in each NuSTAR epoch with the orbital phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The  colours black, red, green, blue, cyan, pink, magenta, orange, yellow and grey  represent the NuSTAR epochs in chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  represented by the square symbol is estimated using the model M1,  whereas the same using M2 are denoted by the star symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We no-  tice that the value estimated using M2 are marginally higher than the  same from M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' This can be the effect of an additional edge compo-  nent used in the M1 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Similarly, we estimated the gabs strength  (Sp) corresponding to each GTI-patch for a given epoch from the  patch-spectra modelling using M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The residual strength (Sp - Si) is  measured for each patches inside an epoch and are plotted against  the orbital phase in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 8b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The binary orbital period (P) of 4U  1543 − 47 is 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='79377 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='00007 hours (Orosz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Orosz  2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The orbital position of each NuSTAR patch has been identi-  fied based on the start time (MJD 59382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='42) of Epoch 1 as the refer-  ence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' For Epoch 12 (data in orange colour in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 8a), the value  of Si is unusually low, and the estimated Sp from the patch-spectra  modelling of this epoch is not reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Therefore, we ignore Epoch  12 from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 8b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The residual varies within ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='5 keV (except one  patch), and we see only a marginal variation (within uncertainties)  in strength within an orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' This implies that the orbital position of  the BH and the companion is not responsible for the dynamic nature  of the absorption features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' In fact, we do not expect such behaviour  for a low inclination system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The X-ray luminosity of the source at the peak (see Epoch 1 in  Table 2) of the outburst is extremely high, and it may irradiate (see  Lasota (2001) for a review) the outer accretion disk and the compan-  ion star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' If the irradiation affects the companion star, either a fresh  accretion of matter starts at the hot spot or enhances the stellar wind  in the companion star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The former may produce multiple trigger-  ing in the same outburst event, which has been observed, for exam-  ple, in GX 339 − 4 (Aneesha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' If the latter happens, the  highly ionized wind material may absorb the X-ray emission from  the primary to produce the broad absorption feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The compan-  ion of 4U 1543 − 47 is an A2V type star with a mass Mc=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='45 M⊙  (Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2006) and radius Rc=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='84 R⊙ (Orosz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The  escape velocity (ve) of the stellar material from the surface of the  companion is calculated as 573 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The binary separation (a)  between the BH and the companion is estimated as 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='18 ×1011 cm  by considering a BH mass, MBH=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='4 M⊙ (Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2006) using  the relation P2/a3 = 4π2/ G(MBH + Mc), where G is the Gravitational  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We observe that the stellar wind takes only a few hours to  reach the primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The column density of stellar wind and the ion-  ization state should reduce with the decrease of the X-ray luminosity  of the primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Therefore, we expect that the strength of the broad  absorption feature should reduce along with the progress of the out-  burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Instead, we observe that the absorption strength enhances and  becomes strongest during Epoch 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Moreover, the estimated stellar  wind speed is not sufficient to blue shift the highest ionized lines  of Fe XXVI, to produce the observed absorption feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Therefore,  the stellar wind has no role in the dynamic absorption features in the  spectra of 4U 1543 − 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The irradiation of the outer accretion disk enhances the accre-  tion rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' therefore, the outburst source stays in the high luminosity  state for a longer duration (King 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Lasota 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Aneesha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Aneesha & Mandal 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' This is possibly causing 2021 out-  burst of 4U 1543 − 47 to decline very slowly (over ∼ 175 days,  see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The super Eddington peak luminosity (Epoch 1 in Ta-  ble 2) of the source can launch strong disk-wind (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=', King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Muñoz-Darias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The presence of the accre-  tion disk-wind is more prominent in the soft state of X-ray bi-  naries (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Neilsen & Lee 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Ponti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2012),  though disk-winds are not exclusively confined to soft spectral state  (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Spectral analysis of 4U 1543 − 47 (Table 2) sug-  gests that the source was in the HSS during our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Also, we no-  tice spectral softening happens till Epoch 9 (∼ day 60) and beyond  which spectra gradually become harder (Table 2 & Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' If the  disk-ionized winds are responsible for the absorption features, then  the strength of the features would be maximum when the source is  softer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Surprisingly the strength of the absorption feature is maxi-  mum on ∼ day 60 as per our analysis (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 8a), and it is keeps-on  decreasing further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The optical depth (τ0) evolution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 4c) also  suggests that the absorption column is maximally populated on ∼  day 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The transition energy of the most ionized line with the highest  absorption yield, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=', Fe XXV and Fe XXVI, are 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='68 keV and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='97  MNRAS 000, 1–14 (2022)  Disk-wind regulated accretion in 4U 1543−47  11  keV respectively (provided by XSTAR line finding list12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Assuming  the absorption feature (with line E ∼ 10 keV) in the NuSTAR spectra  is produced due to the absorption of the accretion disk photons by  the highly ionized blue shifted disk-wind, the estimated wind speed  is reaching 30% of the speed of light to blue shift the Fe XXVI line  energy to 10 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Such a fast disk-wind has never been observed in  X-ray binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' In fact, highly ionized wind (say, Fe XXVI)  is never detected (Ponti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2012) in BH-XRB systems with low  inclination angle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' for example, GX 339 − 4, XTE J1817 − 330, 4U  1957 + 115, XTE J1650 − 500, GRS 1758 − 258 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Therefore,  this detection is the first of its kind for X-ray binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' However,  mildly relativistic disk-wind is not uncommon in quasars and AGNs  (Reeves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Tombesi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Hagino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The  other difficulty is the width (σ) of the absorption feature, which is  as broad as 2 keV on Epoch 9 (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Known line-broadening  processes due to turbulence or scattering will face serious challenges  in explaining the line width if it is from a single line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Instead, it is  more likely that the broad feature can be produced by combining  multiple lines of various ionization states of iron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The phenomenological spectral fitting of NuSTAR data with  model M1 reveals the presence of neutral Fe K−α absorption edge  and the broad ionized absorption features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We calculate the equiv-  alent width (EW), which is a measure of the strength of an ab-  sorption line, of both absorption features to find if there exists any  connection between these two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The EW is defined as  (Arumugasamy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2018),  EW =  � ∞  0  [1 − F(E)] dE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  (1)  The energy dependent function F(E) for the gabs component is  given by,  F(E) = exp(−τ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  τ = τ0 exp  �  −(E − Eg)2/2σ2�  ,  (2)  where τ0, Eg and σ are the optical depth, line energy and line width  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Similarly, F(E) corresponds to the edge component in  XSPEC is given by,  I(E) =  \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3  1  if E ≤ Ee  exp[−D (E/Ee)−3]  if E ≥ Ee,  (3)  where Ee and D are the threshold energy and absorption depth, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The evolution of EW calculated based on NuSTAR phenomeno-  logical modelling is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 9 for both edge (red in colour)  and gabs (blue in colour) components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The gabs EW increases till  Epoch 9 and then gradually decline, except for Epoch 12 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 8a)  & 13 (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The implication of this result and the connection  between both components (gabs and edge) are discussed in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  5 DISCUSSION  The wideband spectral modelling of NICER-NuSTAR and AstroSat  data reveals that the inner disc temperature Tin is highest (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='27 keV)  on Epoch 1, and it keeps on decreasing during the decay of the out-  burst (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The estimated diskbb norm (Table 2, Table 3) sug-  gests a marginal inward movement of the inner disk radius rin since  diskbb norm ∝ r2  in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' However, the decrease in rin could not prevent  the drop in Tin due to the gradual decline in ˙M as Tin ∝ ˙M1/4 r−3/4  in  12 https://heasarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='gov/docs/software/xstar/xstar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='html  20  40  60  80  100  Time (days since MJD 59370)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='5  1  2  3  EW (keV)  A  B C  D  E  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Evolution of equivalent width of gabs (blue in colour) and edge  (red in colour) components from the NuSTAR epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The vertical lines (A-  E) are used to explain the figure in the description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  (Frank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The extreme luminosity in the inner disk may  slow down the accretion of matter to the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' On the other hand,  the high accretion disk luminosity (Table 2) can irradiate the outer  accretion disk, enhancing the accretion of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' If most of the ac-  creted matter is released as the disk-wind, the amount of matter ac-  tually transfers to the inner disk for falling onto the BH is much less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Therefore, the gradual decline of Tin is due to the reduction of effec-  tive infall of matter onto the BH through the inner disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The source  luminosity is completely soft-photons-dominated due to very little  fractional Comptonization (cov_frac in Table 2), low corona tem-  perature, and steeper photon index Γ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Therefore, the source  was in the high/soft spectral state during our study, and it was the  softest on Epoch 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The important parameters for reflection modelling are shown in  Table 3 and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The lamp-post height comes closer to the  central object as the source becomes softer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The reflection fraction  (Rf) increases and hits the boundary when the source is softest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  If the value of ionization parameter, log ξ ≳ 3, the fluorescence  yield of the highly ionized Fe line (more ionized than Fe XXIII)  increases (Matt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The high value of log ξ (Table 3) ob-  tained from reflection modelling suggests a highly ionized accretion  disk throughout the outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Combining all the factors like extreme  luminosity, high log ξ, and an overabundance of Fe (parameter AFe  in Table 3) refer to a highly ionized disk-wind having a significant  yield of Fe XXV, Fe XXVI etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  An important characteristic of this source is the presence of a  broad, symmetric and dynamic absorption feature in the spectrum  ∼ 8−11 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We presented various possibilities regarding the origin  of this absorption feature in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Finally, we concluded that the fast  moving ionized disk-wind could absorb the primary X-ray photons  and produce this broad absorption feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The phenomenological  modelling shows the presence of the neutral Fe Kα absorption (edge)  at ∼ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='1 − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='4 keV, originating from the outer part of the accretion  disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The initial steep rise of the EW of the neutral component (red  star in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 9) indicates the enhancement of disk matter due to irra-  diation of the outer disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Due to the availability of more matter, the  radiation pressure of the highly luminous inner part could release  more ionized matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The source enters to the HSS, and the disk-  wind gradually becomes very active till Epoch 9 (possibly Epoch  MNRAS 000, 1–14 (2022)  12  Prabhakar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  10 also) where the gabs EW is maximum (marked A in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 9) and  the disk spectrum is the softest (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Therefore, the evolution  of the ionized EW (blue square in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 9) followed the neutral com-  ponent till day 60 (Epoch 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' After that, EW of the ionized compo-  nent declines gradually (AB in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 9) because the disk luminosity  has reduced significantly, and a good fraction of inner disk matter  has already been lost in the form of wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The neutral component  remains unaffected as it takes a viscous timescale to propagate the  same to the outer disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The NuSTAR data on Epoch 12 reveals a sud-  den drop of the strength (orange points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 8a) and EW (marked  ‘C’ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 9) of gabs component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We also notice the same signature  in the AstroSat data on Epoch 13, observed after 2 days of the NuS-  TAR’s Epoch 12 observation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' the gabs strength (Table 2) on Epoch  13 is smaller by a few factor compared to the nearby observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  We identify this sudden drop of ionized EW can be due to the evacu-  ation of the inner disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The huge central luminosity may slow down  the accretion onto the central object, and most of the accreted mat-  ter is released through disk-wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Once the disk luminosity reduces,  there is a sudden infall of matter onto the BH, leading to an evacua-  tion of the inner disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  If this interpretation is correct, we expect a relatively harder spec-  trum due to a significant drop of soft photon flux during Epochs 12  & 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' To characterise this, we calculate the observed flux in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='5 − 7  keV (soft) and 7−20 keV (hard) band for NuSTAR and AstroSat data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The AstroSat soft and hard fluxes on Epoch 10 are 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='45 × 10−8 erg  cm−2 s−1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='23 × 10−9 erg cm−2 s−1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The drop of As-  troSat soft flux on the next observation (Epoch 13) is 14%, whereas  the hard flux increases by 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' This resulted in a relatively harder  spectral index (marked by a blue circle in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 4b) on Epoch 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Sim-  ilarly, the NuSTAR soft and hard fluxes on Epoch 11 are 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='48 × 10−8  erg cm−2 s−1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='1 × 10−9 erg cm−2 s−1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The drop of  NuSTAR soft flux on Epoch 12 is 10%, whereas the hard flux in-  creases by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Therefore, the suddenly enhanced accretion  at the inner disk resulted in these dramatic changes in the EW and  spectral properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' However, the inner accretion disk recovers over  the next 10 days (marked CD) due to the transfer of matter from  the outer disk, and the ionized component returns back to a gradual  declination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Interestingly, the neutral component (or the outer disk)  follows the same trend as the ionized component, namely the de-  cline and refilling signature (red stars between BE), with a delay of  the typical viscous timescale of 10 − 15 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  6 SUMMARY AND CONCLUSION  We study the wideband spectral properties of the 2021 outburst of  4U 1543 − 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The MAXI/GSC lightcurve (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 1) shows that the  outburst rises over 9 days followed by a slow decay over ∼ 175 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  We use multi-instruments data (NICER, NuSTAR and AstroSat) for  simultaneous broadband spectral study over a period of 100 days  from MJD 59370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We have performed the spectral study using the  phenomenological model M1 and reflection model M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The major  findings from our study are summarized below:  The source generally remains very bright during this outburst  with a super Eddington peak luminosity on Epoch 1 (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The source was in the HSS during our study, with a steep pho-  ton index (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 4b) due to a very small fraction (< 3%) of inverse-  Comptonized photons and low corona temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The reflection modelling reveals that the inclination of the sys-  tem is between 32◦−40◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The extreme luminosity, high ionization (log ξ > 3) and over-  abundance of iron (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='6−10 AFe,⊙) indicate the presence of disk-wind  with a significant yield of highly ionized iron species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Presence of a broad, dynamic absorption feature at ∼ 8 − 11  keV is observed throughout our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' This detection is the first of  its kind for X-ray binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' We propose that this feature is due to  the absorption of the accretion disk photons by the highly ionized,  blue shifted disk-wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The strength of the ionized absorption fea-  ture (Table 2 & Table 3) increases between Epoch 1 to Epoch 9 as the  disk-wind column density is expected to increase with the spectral  softening of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The observed line energy of the absorption feature suggests an  estimated wind speed of nearly 30% of the speed of light to blue  shift the most ionized line with the highest absorption yield like Fe  XXVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Hence it would become the first X-ray binary source to show  a highly relativistic disk-wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The initial steep rise of the neutral component EW (red star  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 9) is an indication of the enhancement of disk matter due  to irradiation of the outer disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' It enhances the accretion rate and  hence the source remains in the high luminosity state and decays  very slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  The evolution of EW (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 9) of the neutral absorption compo-  nent (edge) and the same of the ionized component (gabs), follow  each other with a delay of the typical viscous timescale of 10 − 15  days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  An evacuation of the inner accretion disk is observed during  Epoch 12 − 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' This event leaves a signature of the drop in the soft  photon flux and an enhancement of hard flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' Therefore, the spec-  trum becomes relatively harder (blue circle in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  Finally, this study suggests that accretion dynamics of 4U 1543 −  47 during 2021 outburst is regulated by the disk-wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  The authors wish to thank the anonymous reviewer for the insightful  suggestions which significantly improved the quality of the publi-  cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' This work uses data from the NICER and NuSTAR mission  by the National Aeronautics and Space Administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' This work  also has used data from the AstroSat mission of the ISRO archived  at the Indian Space Science Data Centre (ISSDC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The work has  been performed utilizing the calibration databases, and auxiliary  analysis tools developed, maintained and distributed by AstroSat-  SXT team with members from various institutions in India and  abroad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' The High Energy Astrophysics Science Archive Research  Center (HEASARC), which provides the software and NASA’s As-  trophysics Data System Bibliographic Services are also acknowl-  edged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' BGR acknowledges the financial support of ISRO under As-  troSat archival data utilization program Sanction order No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' DS-2B-  13013(2)/13/2019-Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content=' AN thanks GH, SAG, DD, PDMSA, and  Director, URSC for the support to carry out this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  DATA AVAILABILITY  The  data  from  NICER  and  NuSTAR  underly-  ing  this  article  are  available  in  HEASARC,  at  https://heasarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='gov/docs/archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='  AstroSat  data  archive  is  available  at  https://astrobrowse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='issdc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='in/astro_archive/archive/Home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
+page_content='jsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
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+page_content='  MNRAS 000, 1–14 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf'}
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+Optical solitons in curved spacetime
+Felix Spengler,1 Alessio Belenchia,1, 2 Dennis R¨atzel,3, 4 and Daniel Braun1
+1Institut f¨ur Theoretische Physik, Eberhard-Karls-Universit¨at T¨ubingen, 72076 T¨ubingen, Germany
+2Centre for Theoretical Atomic, Molecular, and Optical Physics,
+School of Mathematics and Physics, Queens University, Belfast BT7 1NN, United Kingdom
+3ZARM, University of Bremen, Am Fallturm 2, 28359 Bremen, Germany
+4Humboldt Universit¨at zu Berlin, Institut f¨ur Physik, Newtonstraße 15, 12489 Berlin, Germany
+(Dated: January 13, 2023)
+Light propagation in curved spacetime is at the basis of some of the most stringent tests of Einstein’s general
+relativity. At the same time, light propagation in media is at the basis of several communication systems. Given
+the ubiquity of the gravitational field, and the exquisite level of sensitivity of optical measurements, the time is
+ripe for investigations combining these two aspects and studying light propagation in media located in curved
+spacetime. In this work, we focus on the effect of a weak gravitational field on the propagation of optical
+solitons in non-linear optical media. We derive a non-linear Schr¨odinger equation describing the propagation of
+an optical pulse in an effective, gradient-index medium in flat spacetime, encoding both the material properties
+and curved spacetime effects. In analyzing the special case of propagation in a 1D optical fiber, we also include
+the effect of mechanical deformations and show it to be the dominant effect for a fiber oriented in the radial
+direction in Schwarzschild spacetime.
+INTRODUCTION
+The properties of light propagating in optical media is a
+subject as old as optics itself. In recent years, the possibility
+to engineer novel metamaterials has opened the door to the
+so-called transformation optics [1], a field promising to en-
+hance existing devices and create novel ones. At the basis of
+this revolution is the fact that, in the geometric optics limit
+– and neglecting dispersion –, light rays propagate in media
+following the geodesics of an effective Lorentzian metric, the
+so-called optical metric [2]. This has also led to the investi-
+gation of light in optical media as an analogue gravity model,
+i.e., a model in which field perturbations propagate as if in a
+curved spacetime background, particularly useful in the inves-
+tigation of kinematic effects of quantum field theory in curved
+spacetime, like the Hawking radiation and cosmological par-
+ticle production [3–5]. When also the effect of dispersion is
+considered, the metric description can be cast aside for a more
+powerful Hamiltonian formalism, giving rise to the so-called
+ray-optical structures [6, 7].
+This analogy between optical media and curved space-
+times can be pushed even further by showing that Maxwell
+equations in vacuum, curved spacetime are equivalent to
+flat-spacetime Maxwell equations in the presence of a bi-
+anisotropic moving medium whose dielectric permittivity and
+magnetic permeability are determined entirely by the space-
+time metric [8]. Spacetime itself can then be described as an
+optical medium at the level of full electromagnetism. It is then
+natural to wonder what would happen if light were to propa-
+gate in an optical medium placed in a curved spacetime. Far
+from being a far-fetched situation, this is exactly the case for
+light propagating in media on Earth due to the non-vanishing,
+albeit weak, gravitational field of our planet. In this work, we
+are interested in exactly this situation. In particular, while at
+the geometric optics level the formalism of ray-optical struc-
+tures can be used, we aim here at a description, analogous to
+the one in [8], at the level of full Maxwell equations. Indeed,
+such a description allows for the modelling of the propagation
+of intense pulses in situations of physical interest, like soliton
+propagation in optical fibers, taking into account the effect of
+a weak gravitational field.
+We show that light propagation in a medium in curved
+spacetime is equivalent to propagation in an effective medium
+in flat spacetime. We then use this formalism to investigate the
+propagation of intense light pulses in non-linear media, giving
+rise to optical solitons. Solitons, and more in general propa-
+gating pulses, in optical fibers are at the basis of several com-
+munication protocols. Given that fibers on Earth are de facto
+in a curved spacetime due to our planet’s gravitational field,
+it is relevant to analyze how gravity influences light-pulses
+propagation. Our result allows us to set up a framework for
+the analysis of the effect of acceleration and curvature on the
+propagation of pulses in optical fibers in curved spacetimes.
+We numerically investigate some of these effects for the sim-
+ple case of 1D propagation in the weak-field limit.
+AN EFFECTIVE “SPACETIME MEDIUM”
+While light in media can propagate as in a curved space-
+time, curved spacetime can also be seen as an effective
+medium with non-trivial permeability and permittivity [8, 9].
+It is not difficult to generalize the derivations in [8, 9] to the
+case in which light propagates in an optical medium placed
+in curved spacetime. Also in this case it can be shown that
+Maxwell’s equations are equivalent to Maxwell’s equations
+in flat spacetime for an effective medium whose properties
+encode both the ones of the physical medium and of curved
+spacetime.
+Indeed, consider a dielectric and permeable medium in
+curved spacetime characterized by a Lorentzian metric gµν
+with mostly plus signature. We follow here the notation of [7],
+arXiv:2301.04986v1  [gr-qc]  12 Jan 2023
+
+2
+also reported in the Supplemental Material [10]. Maxwell’s
+equations in the absence of free charges and currents are given
+by
+∇kF∗ ik = 0
+(1)
+∇kGik = 0,
+(2)
+where F∗ is the Hodge dual of the electromagnetic tensor F,
+and G and F are related by the constitutive equations of the
+material. Choosing an observer field ui, the electric and mag-
+netic field strengths can be defined with respect to it as
+Ba = −1
+2ηabcdubFcd; Ei = Fijuj
+(3)
+Ha = −1
+2ηabcdubGcd; Di = Giju j
+(4)
+Fab = −ηcd
+abudBc + 2u[aEb]
+(5)
+Gab = −ηcd
+abudHc + 2u[aDb],
+(6)
+in the reference frame of the observer in which the medium
+is assumed to be at rest. Here ηijkl = √−gδijkl is the Levi-
+Civita tensor and T[abc... ] denotes the antisymmetrization of
+the tensor with respect to the indices in square brackets.
+As discussed in [10], choosing ui = δi
+0/ √−g00, the projec-
+tion of Maxwell’s equations in 3-dimensional form leads to
+δαβγ∂βHγ − ∂0Dα = 0; ∂lDl = 0
+(7)
+δαβγ∂βEγ + ∂0Bα = 0; ∂lBl = 0,
+(8)
+where Eα = √−g00Eα, Hα = √−g00Hα, and
+Dα = − √−ggαβ
+g00
+Dβ − δαβγ g0γ
+g00
+Hβ
+(9)
+Bα = − √−ggαβ
+g00
+Bβ + δαβγ g0γ
+g00
+Eβ,
+(10)
+with Bα = √−g00Bα, and Dα = √−g00Dα. These expres-
+sions are equivalent to Maxwell’s equations in flat space-
+time in the presence of an optical medium.
+In particular,
+for a non-dispersive medium characterized by constitutive re-
+lations Da = εb
+aEb, and Ba = µb
+aHb, the effective medium
+will be characterized by a dielectric and magnetic permeabil-
+ity given by the product of the material ones and the ones
+characterizing the curved spacetime [8, 9]. Indeed, expressing
+Dα = ˜εαβEβ+ ˜γβ
+αHβ and correspondingly Bα = ˜µαβHβ− ˜γβ
+αEβ,
+where ˜γβ
+α encode magnetoelectric effects, we see that
+˜µαβ = − √−ggαγ
+g00
+µ β
+γ
+(11)
+˜εαβ = − √−ggαγ
+g00
+ε β
+γ ,
+(12)
+and ˜γαβ = −δαβγg0γ/g001. As a direct consequence, whenever
+the refractive index of the effective medium can be defined, it
+1 Note that, in the case the material itself possesses magnetoelectric terms in
+the constitutive equations, i.e., Da = εb
+aEb + γb
+aHb, and Ba = µb
+aHb − γb
+aEb
+then ˜γαβ = −δαβγ g0γ
+g00 − √−g gαδ
+g00 γβ
+δ
+will also be the product of the material refractive index times
+the vacuum spacetime effective one. The same result can be
+easily obtained at the level of geometric optics.
+Finally, we make two observations relevant for the study of
+the propagation of light pulses. Firstly, a non-magnetic mate-
+rial in curved spacetime corresponds to a magnetic effective
+medium in Minkowski due to the “magnetic permeability”
+of the background spacetime. Secondly, when considering
+a non-linear material, we see that the non-linearity will also
+be affected by the curvature of spacetime as well as the linear
+polarizability.
+PULSE PROPAGATION: NON-LINEAR SCHR ¨ODINGER
+EQUATION
+We next consider the propagation of light pulses in a Kerr
+non-linear, non-magnetic material in curved spacetime.
+In
+particular, we focus on the case in which the material is in a
+stationary orbit of Schwarzschild spacetime and use isotropic
+coordinates. This situation well-captures the cases of interest
+for optical communication and laboratory experiments like,
+e.g., optical fibers hanging still above Earth’s surface.
+In flat spacetime, the non-linear Schr¨odinger equation
+(NLSE) is often used when considering the propagation of
+light pulses whose amplitude is well-described by a scalar
+envelope slowly varying with respect to the light period and
+wavelength [11, 12]. In the case of a medium stationary in
+Schwarzschild’ spacetime, by employing the correspondence
+with an effective medium in flat spacetime as described in the
+previous section, the usual derivation of the NLSE can be car-
+ried out. However, the effective medium will be inhomoge-
+neous due to the curved spacetime contribution to the polar-
+izability and permeability of the material medium. This gives
+rise to extra terms in the NLSE which are of purely gravita-
+tional origin. Furthermore, another source of inhomogeneity
+in the medium can be included when considering the effect of
+tidal forces on the material that, through photoelasticity, ren-
+der the refractive index position-dependent.
+Neglecting for the moment photoelasticity, i.e., consider-
+ing a rigid dielectric, we can write Maxwell’s equation in flat
+spacetime for the effective medium in the familiar notation,
+using the fields and field strengths that we indicate with plain
+capital letters from now on,
+∇ · B = 0, ∇ · D = 0
+(13)
+∇ × E = −∂tB, ∇ × H = ∂tD,
+(14)
+where D
+=
+˜εE and H
+=
+B/˜µ.
+Here ˜µ
+=
+˜µ(r) and
+˜ε = ˜ε(E, r, ω) in frequency space, allowing us to account
+for the effect of material dispersion, are the permeability
+and permittivity of the effective medium.
+Expressing the
+Schwarzschild’ spacetime metric in isotropic coordinates as
+ds2 = − (B(t, r)/A(t, r))2 dt2 + A4(t, r)δαβdxαdxβ, with A(r) =
+1 + rS /4r and B(r) = 1 − rS /4r, with rS the Schwarzschild
+
+3
+radius, we have
+˜ε(E, r, ω) = ε0εspε = ε0
+A(r)3
+B(r)
+�
+1 + χ(1)(ω) + 3χ(3) |E|2
+Ω
+�
+,
+(15)
+˜µ = ˜µ(r) = µ0µsp = µ0A(r)3B(r)−1,
+(16)
+with Ω = A(r)−4 the conformal factor relating the spacial part
+of the metric with the flat, Euclidean one2. The explicit ra-
+dial dependence in the linear part of these effective quantities
+comes from the curved spacetime optical properties encoded
+in the diagonal terms √−ggαα/g00 (cf. eq.(11)) that we define
+as εsp = µsp = A(r)3B(r)−1. The field dependency of ˜ε takes
+into account the non-linearity of the physical medium. Note
+also that dispersion implies that the dielectric permeability is
+a function of the physical frequency ω defined with respect to
+our stationary observer uµ.
+From eq. (13), and writing D = ˜εℓE + PNL, where ˜εℓ =
+ε0εsp(1 + χ(1)(ω)) is the linear part of the dielectric permeabil-
+ity in eq. (15) and PNL is the non-linear polarization, we can
+then obtain the wave equation, in frequency space,
+∇2E − ∇(∇ · E) + ˜µ˜εℓν2E = −˜µν2PNL − (∇ log(µsp)) × (∇ × E) .
+(17)
+Here we indicate with ν the conjugate variable to the coordi-
+nate time t in the flat spacetime of the effective medium. Note
+that the homogeneous Maxwell equations imply that
+∇ · E = −(∇ log ˜εℓ) · E − 1
+˜εℓ
+∇ · PNL,
+(18)
+and thus
+−∇(∇ · E) = (E · ∇)∇ log ˜εℓ + �(∇ log ˜εℓ) · ∇� E
+(19)
++ (∇ log ˜εℓ) × (∇ × E) + ∇
+� 1
+˜εℓ
+(∇ · PNL)
+�
+.
+Eq. (18) makes evident that ∇ · E is of the same order as the
+non-linearities and inhomogeneities in the electric permittiv-
+ity, which is also why it is usually safely neglected in deriva-
+tions of the NLSE.
+The wave equation in eq. (17) is equivalent to Maxwell
+equations and, as such, presents the same level of complexity
+if analytical or numerical solutions are attempted. The NLSE
+is a scalar propagation equation for the electric field’s slowly
+varying amplitude that allows one to numerically simulate the
+pulse propagation. We thus want to write the electric field as
+the product of a slowly varying amplitude times a phase prop-
+agating along the propagation direction, that we will identify
+with the z direction in the following. In this context, notice
+that the dispersion relation of the physical medium, in its rest
+2 This conformal factor arises due to the fact that EaEa in curved space-
+time corresponds to |E|2/Ω with |E|2 = EaEbδab the flat spacetime norm
+squared of the electric strength field.
+frame, is given simply by n(ω) = cκ/ω, with κ the modulus of
+the spatial projection of the wave 4-vector. For the effective
+medium, this relation reads ˜n = c˜κ/ν, where ˜n = √εspµspn
+is the product of the material refractive index and the “space-
+time refractive index” nsp = √εspµsp. Moreover, since ν is
+the frequency defined with respect to Minkowski coordinate
+time, i.e., the conjugate Fourier variable to t, it is related to
+the physical frequency, i.e., the one measured by a physical
+observer in curved spacetime, by the gravitational redshift
+ν = ω √−g00. From the equivalence of the dispersion rela-
+tions, we see that ˜κ(r) = κnsp(r)
+�
+−g00(r). We will thus write
+E(r, t) ∝ E(r)ei(˜κ0z−ν0t) + cc., with ˜κ0 = ˜κ(r, ν0) evaluated at a
+central frequency ν0.
+In order to proceed with the derivation of the NLSE, and to
+further simplify our equations, we consider two separate sit-
+uations of physical interest: (i) pulse propagation at approxi-
+mately constant radius; (ii) pulse propagating radially.
+Horizontal motion at (almost) constant radius
+We assume the propagation direction of the light pulse to
+be the z axis taken to be perpendicular to the radial direction
+for horizontal motion, and consider linearly polarized light
+propagating in a medium stationary on Earth for concreteness.
+Then, for propagation distances much smaller than Earth’s ra-
+dius (r⊕), i.e., z ≪ r⊕, the horizontal motion can be considered
+as happening at constant radius. With these approximations,
+the spacetime permeability and permittivity are constant func-
+tions of r⊕, µsp = εsp = A(r⊕)3B(r⊕)−1 and also the physical
+frequency is not changing with z. Thus, we see that in eq. (17)
+the last term on the right-hand side vanishes.
+We follow the derivation in [4] where the pulse propaga-
+tion in a single-mode optical fiber was considered. Indeed,
+for µsp εsp constant, eq. (17) is formally equivalent to eq. (S1)
+of [4] in frequency space. We thus end up with an effective
+one dimensional problem for the slowly varying envelope, and
+the derivation of the NLSE is the textbook one [10, 12]. In
+particular, recall that the slowly varying envelope approxima-
+tion(s) (SVEA) consists in neglecting terms ∂2
+zE ≪ ˜κ0∂zE and
+(˜κ1/˜κ0)∂t ≪ 1 on the basis that the envelope will contain many
+wavelengths and optical cycles. If we apply now the SVEA
+we end up with, in the time domain,
+i(∂z + ˜κ1∂t)E − ˜κ2
+2 ∂2
+t E = −n2ν0nsp(r⊕)ε0
+|E|2
+Ω E,
+(20)
+where ˜κi(ν0) are the coefficients of the power series expansion
+˜κ(ν) = �
+n ˜κn(ν0)/n! (ν − ν0)n in ν − ν0 and we are considering
+Kerr non-linear media for which the nonlinear index is n2 =
+3χ(3)/(2n(ω0)cε0).
+Considering
+an
+anomalous
+dispersive
+material,
+i.e.,
+κ2(ν0) < 0, an analytical solution of the NLSE can be found
+(see, e.g., [4]) and reads
+E(t, z) =
+�
+Ω|˜κ2|
+ν0n2nspε0T 2
+0
+cosh
+�t − ˜κ1z
+T0
+�−1
+exp
+������
+iz|˜κ2|
+2T 2
+0
+������ , (21)
+
+4
+z (m)
+rS/r⊕ = 10−3
+rS/r⊕ = 10−2
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+1.9487 ×108
+1.9488 ×108
+1.9489 ×108
+1.9490 ×108
+1.9491 ×108
+1.9492 ×108
+1.9493 ×108
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+1.950× 108
+1.952× 108
+1.954× 108
+1.956× 108
+1.958× 108
+(m/s)
+FIG. 1: Velocity of the soliton along the fiber, with respect to
+an observer comoving with the segment of the dielectric
+material where the (peak of the) soliton is located, for
+L = 0.1 m, rs = 10−3r⊕, and including photoelasticity. The
+red, dashed and blue, solid curves represent the analytical
+expression in eq. (25) including or in the absence,
+respectively, of photoelasticity. The red points and blue
+squares are obtained by numerical simulations and agrees
+perfectly with the analytical formula of eq. (25). The inset
+shows the case with photoelasticity in which rs = 10−2r⊕.
+This shows a deviation from a purely linear relation between
+the velocity and the propagation distance.
+where T0 is the pulse length, and 1/˜κ1 is its speed of prop-
+agation. This reduces to the result from Philbin et al.[4] –
+eq.(S74) of the supplementary material in [4] – in the limit
+of rS → 0. From this expression, combined with the fact
+that ˜κ1(ν0) = nspκ1(ω0), we can conclude that the velocity of
+the horizontally propagating soliton in curved spacetime with
+respect to an observer comoving with the segment of the di-
+electric material3 is given simply by κ1(ω0)−1.
+Radial motion
+Let us now consider the case in which the light pulse propa-
+gates radially along the z direction. Care is in order here, since
+now all the quantities appearing in the wave equation will
+change along the propagation direction, including the physical
+frequency that will be subject to gravitational redshift. Moti-
+vated by the symmetry of the problem, and in order to obtain a
+scalar, one-dimensional equation whose solution can be sim-
+ulated, we assume that all the quantities entering the wave
+equation depend solely on z. This is tantamount to identifying
+the radial direction with the z-axis and work close to x = y = 0
+so that r = r⊕ + z, which is a reasonable assumption since we
+are considering the vertical propagation of a well localized
+pulse. With this approximation, the wave equations (17) re-
+duce to a system of three decoupled equations [13]
+∂2
+z Ex(y) + ˜µ˜εℓν2Ex(y) = − ˜µν2PNL,x(y) + (∂z(ln ˜µ)) ∂zEx(y)
+(22)
+∂2
+z Ez + ˜µ˜εℓν2Ez = − ˜µν2PNL,z − ∂z
+� 1
+˜εℓ
+∂zPNL,z
+�
+(23)
+− 2(∂z ln ˜εℓ)∂zEz − Ez∂2
+z ln ˜εℓ
+It is immediate to realize that Ez = 0 is a solution of the cor-
+responding equation so that we can consider the propagation
+of linearly polarized light (in a direction orthogonal to z) and
+we end up with a single equation of the form of eq. (22).
+Proceeding as before with substituting the ansatz E(z, t) ∝
+E(z, t)ei(˜κ0(z)z−ν0t) + cc., expanding ˜κ(z, ν) around ν0, and using
+the SVEA approximation(s) we obtain the NLSE given by
+i(∂z + ˜κ1∂t)E − ˜κ2
+2 ∂2
+t E + 2i∂z˜κ0
+2˜κ0
+E + 2iz∂z˜κ0
+2˜κ0
+∂zE + iz∂2
+z ˜κ0
+2˜κ0
+E − z∂z˜κ0E − z2 (∂z˜κ0)2
+2˜κ0
+E = −n2ν0nsp(r)ε0|E|2E/Ω + ∂z ln nsp
+2˜κ0
+(i˜κ0E + ∂zE + iz(∂z˜κ0)E) .
+(24)
+Eq. (24) contains several additional terms with respect to the
+equation for the horizontal propagation due to the fact that
+now the wavevector ˜κ0 depends explicitly on the coordinate
+along the propagation direction and so does the refractive
+index, i.e., we are propagating in a gradient-index medium
+(GRIN)4. All geometrical quantities appearing in the equation
+3 Indeed note that proper length and proper time for an observer comoving
+with the segment of the dielectric material and in connection with coor-
+dinate quantities are given by ℓ = A2 z and τ = t B/A so that � ≡ ℓ/τ =
+A3B−1z/t = nsp˜�.
+4 See also [14–16] for early studies of soliton propagation in inhomogeneous
+are evaluated at r⊕ + z. Finally, consistently with the horizon-
+tal propagation case, upon setting ˜κ0 constant, we return to
+eq. (20).
+INCLUDING PHOTOELASTICITY
+Up until now, we have considered rigid dielectrics, i.e., di-
+electric media in which the speed of sound is infinite. For
+media.
+
+U5
+realistic materials, this is of course never the case and the di-
+electric gets deformed by the action of forces, including the
+tidal ones in our set-up. Let us consider an optical fiber as a
+paradigmatic example. In this case, the deformation due to the
+action of gravity will be relevant only for the case of vertical
+propagation.
+Deformations of a dielectric lead to a change in the relative
+permeability of the material, and thus of the refractive index,
+a phenomenon known as photoelasticity [17]. The contribu-
+tions to this effect coming from the curvature of spacetime
+and the inertial acceleration of the fiber can be separately ac-
+counted for following the discussion in [18]. Consider a fiber
+of length L hanging from at support located at r⊕ + L. As
+far as the strain is within the elastic limit of the material, we
+can relate it with the stresses through a linear relation, i.e.,
+Hooke’s law. Thus, we write the strain tensor as Skl = 1
+Y σkl,
+where Y is the Young’s modulus of the material and σkl = Fk
+Al
+is the stress tensor given by the ratio between the force Fk in
+direction ˆek and the cross-sectional area Al normal to ˆel upon
+which the force acts. The photoelastic (or acousto-optic) ef-
+fect consists in the change of the relative electric permeability
+by ∆(εr)−1
+kl = Pkl mnSmn, where P is the photoelastic tensor.
+In the following, we limit ourselves to the case of isotropic
+materials and a diagonal stress tensor (see [10] for the de-
+tails of the computation). It should be noted that photoelastic-
+ity is far from negligible in the case under investigation and
+becomes the dominant effect in the vertical propagation sce-
+nario, overwhelming the effect related to the optical properties
+of the background spacetime.
+While photoelasticity introduces a further radial depen-
+dence in the optical properties of the effective medium, this
+does not affect the form of eq. (24), which remains valid. The
+only difference is in the expressions for the quantities ˜κi and
+their derivatives, due to the fact that now the refractive index
+of the medium is given by n(ω) =
+�
+1 + χ1(ω) + ∆εr(ω) [10].
+NUMERICAL RESULTS
+While the wave equation in eq. (17) gives us the full
+Maxwell equations, including possibly interesting effects re-
+lated to the vectorial nature of the electric field, and thus to the
+interplay between gravity and the light polarization, its numer-
+ical investigation is beyond the scope of the current work, and
+it is left for future investigations. Here, we focus on the prop-
+agation of light pulses as described by the simplified eq.(24),
+motivated by light propagation in optical fibers [4]. Note that
+in the case of eq. (20) an analytical solution was presented in
+eq. (21).
+Equation (24) for the vertical propagation is solved numeri-
+cally – being a non-linear PDE with coordinate dependent co-
+efficients – using the split-step Fourier (SSF) method [11] and
+taking into account also the effect of the fiber deformation.
+For this purpose, we utilize the same fiber parameters as in [4]
+(see also table I in [10]) and initialize the temporal profile at
+0
+20
+40
+60
+80
+100
+0
+5.×10-14
+1.×10-13
+1.5×10-13
+2.×10-13
+2.5×10-13
+δτ (s)
+× 10−13
+z (m)
+0
+20
+40
+60
+80
+100
+0
+1.×10-21
+2.×10-21
+3.×10-21
+4.×10-21
+5.×10-21
+6.×10-21
+FIG. 2: Time of arrival of the soliton for the case of
+propagation in the gravitational field of Earth for which we
+assume rS = 9 × 10−3 m. The main figure shows the
+difference in time of arrival, with respect to an observer
+comoving with the segment of the dielectric material where
+the soliton is located, between vertically and horizontally
+propagating solitons over the propagation coordinate length
+z. The inset shows the same in the case photoelasticity is
+neglected.
+z = 0 as the one of the input pulse in the same reference.
+The intuition based on the SSF method– where the propa-
+gation equation (24) is rewritten in the form ∂zE =
+� ˆD + ˆN
+�
+E
+with the diffusive dynamics enclosed in the operator ˆD =
+ˆD(z, ∂t) [10] – allows us to formulate the educated guess that
+the propagation speed of the soliton, in the effective flat space-
+time, is given by
+˜� = 1 + z ˜κ′
+0(z)/˜κ0(z)
+˜κ1(z)
+.
+(25)
+Indeed, this appears as (the real part of) the inverse of the
+coefficient of the time derivative in ˆD(z, ∂t). Then, in order to
+translate this result into the speed measured by an observer co-
+moving with the segment of the dielectric material where the
+soliton peak is located, we need to just multiply eq. (25) by the
+spacetime refractive index. That this intuition is indeed cor-
+rect is verified by the numerical simulations reported in Fig. 1.
+We see that the z-dependence of the propagation velocity is
+strongly enhanced by the effects of mechanical deformation
+of the fiber with respect to the case in which photoelasticity is
+ignored. The z-dependence of the vertical propagation veloc-
+ity without photoelasticity is weak, and the velocity is close to
+the one of the horizontal case. To quantify the latter statement,
+in Fig. 2 we show the difference in the (proper) time of arrival
+of the soliton for the case of propagation in the gravitational
+field of Earth, corresponding to a Schwarzschild radius that
+
+2.5 × 10-13
+2. × 10-13
+1.5× 10-13
+1. × 10-13
+5. × 10-14
+0
+0
+20
+40
+60
+80
+1006
+we take as rS = 9 × 10−3 m. The main figure shows
+δτ = |z(
+�
+−g00(r⊕ + z)˜�−1
+↑ ) −
+�
+−g00(r⊕)˜�−1
+→ )|,
+(26)
+with ˜�↑ and ˜�→ the propagation velocities, in the effective flat
+spacetime, for vertical and horizontal propagation. The inset
+shows instead the case in which for the vertical propagation
+the photoelasticity is neglected, showing a much weaker de-
+pendence.
+Finally, in Fig. 3 we show the deviation of the average ve-
+locity along the vertical direction �a�(rS ) with respect to the
+constant velocity at rS = 0 as a function of the dimension-
+less ratio rS /r⊕. The average velocity is obtained numeri-
+cally from the simulations as the ratio of the total length L
+and the propagation time of the soliton and transformed into
+the frame of the observer comoving with the fiber at its upper
+end-point – i.e., multiplied by nsp(r⊕ + L). Analytically, we
+use �a� = (
+� L
+0 � dz)/L with � = nsp˜� and ˜� given in eq. (25).
+Fig. 3 shows once again the agreement between the simulated
+data and our analytical ansatz and it also shows that the pho-
+toelasticity is the main effect that allows one to have a sizable
+difference between the flat and curved spacetime propagation.
+Another quantity characterizing the propagating pulse is its
+temporal width. In the horizontal propagation case, the du-
+ration of the pulse is constant. The same is not, in general,
+true when considering the vertical propagation. In the Supple-
+mental Material [10], we report the evolution of the temporal
+width along the fiber. In particular, our simulations show a
+focusing of the pulse which is however sizable only in the
+presence of photoelasticity.
+CONCLUSIONS
+We have considered the propagation of light pulses in non-
+linear, non-magnetic media stationary in curved spacetime.
+Taking some intuition from the seminal work of Plebanski [8],
+we showed that light propagation in such media can be equiv-
+alently described as the propagation in an effective medium
+in flat spacetime whose electric and magnetic properties ac-
+quire a multiplicative factor encoding the spacetime structure.
+Having done that, eq. (17) describes the propagation of light
+in the effective medium. It is interesting to note, even though
+we did not investigate it in this work, that the vectorial na-
+ture of this equation encodes the interplay between the light
+polarization and the gravitational field. Such interplay should
+be expected on the basis of the fact that the effective medium
+is an inhomogeneous, gradient-index medium for which it is
+well known that the propagation of light is influenced by its
+own polarization [19–21]. Furthermore, the effect of polariza-
+tion on the propagation of light in curved, vacuum spacetime
+has been extensively considered in the literature and shown to
+take place also for static spacetimes [22, 23].
+Neglecting the aforementioned effects, which would be un-
+doubtedly small, by virtue of approximations we have been
+able to derive a scalar NLSE describing the propagation of
+rS/r⊕
+L = 0.1 m
+L = 0.1 m, P1122 = 0
+L = 1 m
+0.0000
+0.0002
+0.0004
+0.0006
+0.0008
+0.0010
+0.0000
+0.0005
+0.0010
+0.0015
+0.0020
+(rS)/
+(rS = 0) − 1
+FIG. 3: Change in average velocity (�a�) of the soliton in the
+fiber – with respect to the observer comoving with the
+dielectric – compared to the case with rS = 0. Orange, square
+points corresponds to the case of a L = 1 m propagation with
+photoelasticity. Blue, round points correspond to the case of
+a L = 0.1 m propagation with photoelasticity. Green,
+diamonds correspond to the case of a L = 0.1 m propagation
+without photoelasticity. The lines correspond to the
+analytical result that fits perfectly the different sets of data.
+a light pulse. It is important to notice that, when solving the
+NLSE employing the SSF method, we are implicitly consider-
+ing a unidirectional equation and ignoring any possible back-
+propagating field in the boundary conditions imposed, for all
+times, at z = 0. This means that backscattered light from the
+pulse is assumed negligible relative to the pulse itself, a condi-
+tion common to all unidirectional envelope propagation equa-
+tions [24]. While this is not a problem for the horizontal prop-
+agation, in which case only the weak non-linearity could give
+rise to back-reflection, in the case of the vertical propagation
+light is effectively propagating in a gradient-index medium
+with the refracting index slowly varying in the propagation
+direction.
+This by itself can give rise to back-propagating
+fields, and effectively limits the validity of our treatment to
+regimes in which the photoelasticity allows to employ a uni-
+directional equation. Luckily, the regime of validity of the
+equation – which depends on the parameter chosen for the
+physical medium – can be readily estimated by following the
+discussion in [25] as we detail in [10].
+Given these caveats, the NLSE that we have derived shows
+that an optical pulse propagating radially in a Kerr non-liner
+medium stationary in Schwarzschild spacetime experiences a
+change in its propagation velocity captured by eq. (25). This
+effect is mostly due to photoelasticity which overwhelms the
+purely spatiotemporal effects encoded in nsp. The difference
+in propagation velocity between the vertically and horizon-
+
+UVav7
+tally propagating pulses results, in turn, in a difference of the
+time of arrival of two pulses of the order of hundreds of fem-
+toseconds in Earth gravitational field, a fact that puts this dif-
+ference in the reach of current technologies (see [26–28] and
+references therein).
+ACKNOWLEDGEMENTS
+The authors thank Francesco Marino for interesting dis-
+cussions. A. Belenchia and D. Braun acknowledge support
+from the Deutsche Forschungsgemeinschaft (DFG, German
+Research Foundation) project number BR 5221/4-1. D. R¨atzel
+acknowledges funding by the Federal Ministry of Education
+and Research of Germany in the project “Open6GHub” (grant
+number:
+16KISK016) and support through the Deutsche
+Forschungsgemeinschaft (DFG, German Research Founda-
+tion) under Germany’s Excellence Strategy – EXC-2123
+QuantumFrontiers – 390837967, the Research Training Group
+1620 “Models of Gravity” and the TerraQ initiative from the
+Deutsche Forschungsgemeinschaft (DFG, German Research
+Foundation) – Project-ID 434617780 – SFB 1464.
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+1
+Supplemental Material: Optical solitons in curved spacetime
+Felix Spengler1, Alessio Belenchia1,2, Dennis R¨atzel3, Daniel Braun1
+1Institut f¨ur Theoretische Physik, Eberhard-Karls-Universit¨at T¨ubingen, 72076 T¨ubingen, Germany
+2Centre for Theoretical Atomic, Molecular, and Optical Physics,
+School of Mathematics and Physics, Queen’s University, Belfast BT7 1NN, United Kingdom
+3Humboldt Universit¨at zu Berlin, Institut f¨ur Physik, Newtonstraße 15, 12489 Berlin, Germany
+In this supplemental material, we collect the detailed derivations of the results in the main text.
+VACUUM SPACETIME AS AN OPTICAL MEDIUM & THE EFFECTIVE MEDIUM DESCRIPTION
+Thanks to the seminal work of Plebanski in the ’60s [8], it is well known that electromagnetism in curved spacetime is
+equivalent to propagation in an optical medium. Following the derivation presented in [9], Maxwell vacuum equations in curved
+spacetime are written as
+∇kF∗ ik = 0
+(S1)
+∇kFik = 0,
+(S2)
+where F∗ is the Hodge dual of the e.m. tensor, Latin indices run from 0 to 3, and the metric gi j has mostly plus signature. As in
+the main text, we consider the case with no currents.
+Choosing an observer field ui, the electric and magnetic field strength can be defined with respect to it as
+Hi = F∗ i juj, Ei = Fi juj
+(S3)
+Fi j = ηi jklulHk + 2u[iE j],
+(S4)
+where here ηijkl = √−gδijkl is the Levi-Civita tensor (with δi jkl the Levi-Civita alternating symbol in four dimensions) and T[abc... ]
+denotes the antisymmetrization of the tensor with respect to the indices in square brackets. The Maxwell equations can then be
+projected in the ui direction or orthogonal to it using the projection operator into the rest frame of ui, hi j = gi j + uiuj. The end
+result is, in the case the observer field is chosen as ui = δi
+0/ √−g00
+δαβγ∂βHγ − ∂0Dα = 0; ∂lDl = 0
+(S5)
+δαβγ∂βEγ + ∂0Bα = 0; ∂lBl = 0,
+(S6)
+where the first two equations come from Maxwell equations (S2) (with δαβγ the Levi-Civita alternating symbol in three dimen-
+sions) while the second two from eq. (S1). Here, Hα = √−g00Hα, Eα = √−g00Eα, Greeks indices run from 1 to 3, and
+Dα = − √−ggαβ
+g00
+Eβ − δαβγ g0γ
+g00
+Hβ
+(S7)
+Bα = − √−ggαβ
+g00
+Hβ + δαβγ g0γ
+g00
+Eβ.
+(S8)
+From here one can see that these equations are actually equivalent to Maxwell equations in flat spacetime in the presence of an
+optical medium whose constitutive relations are characterized by a dielectric (εαβ
+sp ) and magnetic permeability (µαβ
+sp ) given by
+µαβ
+sp = εαβ
+sp = − √−ggαβ
+g00
+.
+(S9)
+As shown in the main text, when a physical optical medium whose rest frame is characterized by ui is added, we can follow
+the same derivation starting from Maxwell’s equations in curved spacetime and with a material medium
+∇kF∗ i j = 0
+(S10)
+∇kGik = 0,
+(S11)
+
+2
+where now G and F are related by the material constitutive equations. Then we have
+Ba = −1
+2ηabcdubFcd; Ei = Fi juj
+(S12)
+Ha = −1
+2ηabcdubGcd; Di = Gi ju j
+(S13)
+Fab = −ηcd
+abudBc + 2u[aEb]
+(S14)
+Gab = −ηcd
+abudHc + 2u[aDb],
+(S15)
+where we have introduced the electric and magnetic excitation, Da and Ha respectively, on top of the electric and magnetic
+strength Ea and Ba.
+Note that the definitions of E, B, F are equivalent to the vacuum case, since the homogeneous Maxwell equations are the
+same. The inhomogeneous equations have also the same form as in the vacuum case, but with the substitution of E, B, F with
+D, H,G, where the definition of G with respect to H, D is the same as F with respect to E, B. From this simple observation we
+can immediately deduce that the projection of Maxwell equations in 3-dimensional form will, in the case the observer field is
+chosen as ui = δi
+0/ √−g00, lead to
+δαβγ∂βHγ − ∂0Dα = 0; ∂lDl = 0
+(S16)
+δαβγ∂βEγ + ∂0Bα = 0; ∂lBl = 0,
+(S17)
+where Eα = √−g00Eα, Hα = √−g00Hα, and
+Dα = − √−ggαβ
+g00
+Dβ − δαβγ g0γ
+g00
+Hβ
+(S18)
+Bα = − √−ggαβ
+g00
+Bβ + δαβγ g0γ
+g00
+Eβ,
+(S19)
+with Bα = √−g00Bα, and Dα = √−g00Dα. Once again, these equations are equivalent to Maxwell’s equations in flat spacetime
+in the presence of an effective optical medium.
+Consider the case of a linear, dispersionless medium. We can then write Gi j =
+1
+2χi j klFkl, with the material’s constitutive
+tensor χij kl, containing all material properties, which is symmetric under the exchange of the first and second pair of indices
+and antisymmetric with respect to the swap within an index pair. In particular, we can also write Da = εb
+aEb, and Ba = µb
+aHb,
+which are the constitutive relations in the reference frame of the observer in which the medium is at rest, neglecting magneto-
+electric effects. For an isotropic medium, we also have that the dielectric and permeability tensor assume the simplified form
+εb
+a = ε(δb
+a + UbUa) and µb
+a = µ(δb
+a + UbUa) for some scalar, positive functions ε and µ. The effective optical medium is such that
+its constitutive relations are then characterized by a dielectric and inverse magnetic permeability given by
+˜εαβ = − √−ggαγ
+g00
+ε β
+γ ,
+(S20)
+˜µαβ = − √−ggαγ
+g00
+µ β
+γ ,
+(S21)
+while the antisymmetric parts of the constitutive tensor are completely characterized by the vacuum spacetime properties5.
+Non-linear media, with a Kerr-type non-linearity, can be treated analogously by promoting the dielectric and permeability
+tensors to explicitly depend on the field strengths. If also dispersion needs to be included in the game, we need to consider,
+as usual, the dispersion relation in frequency space in order to write it in a local form. Note that we can always write D =
+ε0E + P (and analogously for the magnetic field and excitation), moving all non-linearity and dispersion in the polarization
+(magnetization) vector. Thus, from eq. (9) we can conclude that the effective medium will give rise to an effective electric
+excitation
+De f f = εsp(ε0E + P),
+(S22)
+which can then be written, for the dispersive case of interest, locally in frequency space for the effective medium “living” in flat,
+Minkowski spacetime.
+5 More in general, one could also include in this description materials for
+which the magnetoelectric entries of the constitutive tensor are not negli-
+gible. In such a case, Da = εb
+aEb + γb
+aHb, and Ba = µb
+aHb − γb
+aEb with
+γab the antisymmetric part of the constitutive tensor. In this case, the same
+derivation still stands, with the only difference that the antisymmetric parts
+of the constitutive tensor for the effective medium are given by
+˜γαβ = −δαβγ g0γ
+g00
+− √−ggαδ
+g00
+γβ
+δ.
+
+3
+DERIVATION OF THE NLSE: TECHNICAL DETAILS
+Let us consider now Maxwell’s equations for the effective medium, thus in flat spacetime, written in the usual notation
+∇ · B = 0, ∇ · D = 0
+(S23)
+∇ × E = −∂tB, ∇ × H = ∂tD,
+(S24)
+with D = ˜εE and H = B/˜µ. Here, we consider the case of a spherically symmetric spacetime in isotropic coordinates. The
+metric can then be written, in full generality, as
+ds2 = −
+� B(t, r)
+A(t, r)
+�2
+dt2 + a2(t)A4(t, r)δαβdxαdxβ,
+(S25)
+with r =
+�
+δαβxαxβ, A(t, r), B(t, r) real functions, and a(t) a scale factor analogous to the one appearing in FRLW spacetime.
+Note that this metric can be rewritten as
+ds2 = Ω−1
+�
+−
+B2(t, r)
+a2(t)A6(t, r)dt2 + δαβdxαdxβ
+�
+,
+(S26)
+where the “conformal factor” Ω = a(t)−2A(t, r)−4.
+In particular, we specialize to the case for which a(t) = 1, A = A(r), B = B(r) and such that,
+in frequency space,
+˜ε(E, r, ω) = ε0A(r)3B(r)−1 �
+1 + χ(1)(ω) + 3χ(3)|E|2/Ω
+�
+and ˜µ = ˜µ(r) = µ0µsp = µ0A(r)3B(r)−1, i.e., we are considering a non-
+magnetic material, where all the magnetic properties are induced by the curved background, with a Kerr non-linearity. As we
+previously discussed, ω is the physical frequency defined with respect to the stationary observer uµ that we assume to be the rest
+frame of the physical medium. The conformal factor Ω appearing in the non-linear term in ˜ε arises due to the fact that EaEa in
+curved spacetime corresponds to |E|2/Ω, with |E|2 = EaEbδab the flat spacetime norm squared of the electric strength, in the flat
+spacetime of the effective medium, as can be easily seen directly from eq. (S26).
+For the sake of notation clarity, let us emphasized that, in the following, tilded quantities refer to quantities pertaining to
+the effective medium in flat spacetime while the untilded ones represent the optical properties of the physical medium that is
+stationary in (physical) curved spacetime.
+In the following, we focus on Schwarzschild’s spacetime, for which
+A(r) = 1 + rS
+4r
+(S27)
+B(r) = 1 − rS
+4r,
+(S28)
+where rS is the Schwarzschild’s radius.
+From Maxwell’s equations, taking the curl of the third one, we obtain
+∇2E − ∇ (∇ · E) = ∂t
+�
+˜µ∂tD − B × ∇˜µ
+˜µ
+�
+.
+(S29)
+and thus
+∇2E − ∇(∇ · E) − ˜µ∂2
+t D = −(∇ log(µsp)) × (∇ × E).
+(S30)
+Note that this last expression is valid for ∂t ˜µ = 0, which includes the case of Schwarzschild spacetime. For a generic spherically
+symmetric metric, as in eq. (S26), additional terms would be present due to the explicit time dependence of ˜µ. Moving now to
+frequency space, where we indicate with ν the conjugate variable to the coordinate time t in the flat spacetime of the effective
+medium, and writing D = ˜εℓE + PNL, where ˜εℓ is the linear part of the dielectric permeability and PNL contains the nonlinear
+components of the polarization, we obtain
+∇2E − ∇(∇ · E) + ˜µ˜εℓν2E = −˜µν2PNL − (∇ log(µsp)) × (∇ × E).
+(S31)
+This is our starting point for the derivation of the scalar NLSE. Note that, apart from the last term, the equation resembles the
+textbook wave equation modulo the inhomogeneity of the medium encoded in the coordinate dependence of ˜ε, ˜µ [12].
+Before starting the derivation of the NLSE, an observation is in order. In curved spacetime, the linear dispersion relation of
+the medium assumes the simple form, in the rest frame of the medium,
+n(ω) = c
+�
+µ0ε0εr(ω) = c κ
+ω,
+(S32)
+
+4
+with κ the modulus of the spatial projection of the wave 4-vector , εr = 1 + χ(1)(ω), and ω the physical frequency6, i.e., the
+frequency measured by an observer in curved spacetime. Thus, we write the dispersion relation for our effective medium as
+˜n = c ˜κ
+ν,
+(S33)
+where ˜n = nspn with n2
+sp = (Ω|g00|)−1. Eq. (S33) is the expression that we will use in deriving the NLSE. Note once again that
+here ν = ω √−g00 where ν is the conjugate Fourier variable to the coordinate time t in flat spacetime. Since for consistency we
+want the two dispersion relations to be equivalent, we see that ˜κ = κnsp
+√−g00. Once again, in the dispersive case, we will need
+to consider ˜n = ˜n(ω) since otherwise the two dispersion relations would not remain equivalent.
+Derivation of the standard NLSE
+Before delving into the derivation of the NLSE for our effective, inhomogeneous medium, we summarize here the derivation
+of the NLSE in the standard case, following [12].
+In the standard case of a homogeneous, non-magnetic material in flat spacetime, writing the displacement electric field D as
+the sum of a linear part and the non-linear polarization, we have the wave equation in frequency space
+∇2E − ∇(∇ · E) + µ0εℓν2E = −µ0ν2PNL.
+(S34)
+Note that in this section we always work with untilded quantities that refer to the optical properties of the physical medium
+that is considered in flat spacetime. Indeed, in this case the effective medium coincides with the physical one since the optical
+properties of flat spacetime are trivial. Note however that, as previously specified, from the next section we will go back to
+consider the case of curved spacetime. Thus, we will need to distinguish once again between physical and effective medium,
+with the latter represented by tilded quantities in flat spacetime.
+We recall that µ0 = 1/(ε0c2). We then neglect the vectorial operator −∇(∇ · E) due to the fact that the homogeneous Maxwell
+equation for D implies this term to be in general negligible – and get
+∇2E(ν) + εr(ν)ν2
+c2 E(ν) = − ν2
+ε0c2 PNL(ν),
+(S35)
+with εr(ν) = εℓ/ε0 the linear, relative polarizability.
+For a linearly polarized field, this equation becomes a scalar one. We can then write the electric field as a slowly varying,
+complex amplitude E(r, t) times a plane wave propagating in the z direction with central frequency ν0
+E(r, t) = E(r, t)ei(κ0z−ν0t) + cc., where κ0 = n(ν0)ν0
+c
+.
+(S36)
+Using the Fourier transform w.r.t. t for E and the one for the amplitude7 E, eq. (S36) can be rewritten in frequency space as a
+sum of terms dependent on ν ± ν0. We can then discard the fast rotating, high frequency (ν + ν0) components. Indeed, the slowly
+varying in time envelope E(r, t) in which we are interested does not possess high-frequency Fourier components [12]. We thus
+obtain
+E(r, ν) ≈ E(r, ν − ν0)eiκ0z.
+(S37)
+The scalar wave equation for the amplitude then becomes
+∇2
+⊥E + ∂2
+zE + 2iκ0∂zE + [κ2(ν) − κ2
+0]E = − ν2
+ε0c2 PNLe−iκ0z,
+(S38)
+6 This is connected to the frequency in flat spacetime via ω = ( √−g00)−1ν.
+7 We follow [12] in defining,
+E(r, t) =
+� ∞
+−∞
+dν
+2π E(r, ν)e−iνt = E(r, t)ei(κ0z−ν0t) + E∗(r, t)e−i(κ0z−ν0t)
+=
+� ∞
+−∞
+dν
+2π E(r, ν)e−i(ν+ν0)teiκ0z +
+� ∞
+−∞
+dν
+2π E∗(r, ν)e−i(ν−ν0)te−iκ0z
+=
+� ∞
+−∞
+dν
+2π E(r, ν − ν0)e−iνteiκ0z +
+� ∞
+−∞
+dν
+2π E∗(r, ν + ν0)e−iνte−iκ0z.
+From these expressions we then obtain
+E(r, ν) = E(r, ν − ν0)eiκ0z + E∗(r, ν + ν0)e−iκ0z
+
+5
+with κ(ν) = n(ν)ν/c. At this point, we approximate κ(ν) as a power series in ν − ν0
+κ(ν) = κ0 + κ1(ν − ν0) + D,
+(S39)
+with D = κ2(ν − ν0)2/2 + O
+�
+(ν − ν0)3�
+, such that
+κ(ν)2 = κ2
+0 + 2κ0κ1(ν − ν0) + 2κ0D + 2κ1D(ν − ν0) + κ2
+1(ν − ν0)2 + D2.
+(S40)
+Here κ1 is the inverse of the group velocity �g. We will neglect D2 terms and convert back to the time domain8 to obtain
+�
+∇2
+⊥ + ∂2
+z + 2iκ0(∂z + κ1∂t) + 2iκ1 ¯D∂t + 2κ0 ¯D − κ2
+1∂2
+t
+�
+E(r, t) =
+1
+ε0c2 ∂2
+t (PNL(r, t)) e−i(κ0z−ν0t).
+(S41)
+Note that now ¯D is a differential operator with ¯D = −(κ2/2)∂2
+t + .... Finally, by writing also the polarization PNL(r, t) =
+p(r, t)eiκ0z−ν0t + c.c., i.e., as a slowly-varying amplitude p(r, t) times a plane wave eiκ0z−ν0t propagating in the z direction, one can
+see that the right-hand side (RHS) becomes9
+1
+ε0c2 ∂2
+t PNL(r, t)e−i(κ0z−ν0t) = − ν2
+0
+ε0c2
+�
+1 + i
+ν0
+∂t
+�2
+p(r, t)+c.c..
+(S42)
+This is the starting point for implementing the slowly varying envelope approximation (SVEA). It usually involves moving
+to the frame moving with the pulse group velocity κ−1
+1 , and then neglecting terms with second derivatives in the propagation
+direction. Let us sketch the procedure here:
+• The retarded frame is defined as z′ = z and τ = t − z/�g = t − κ1z.
+• Thus, ∂z = ∂z′ − κ1∂τ, and ∂t = ∂τ =⇒ ∂2
+z = ∂2
+z′ − 2κ1∂z′∂τ + κ2
+1∂2
+τ.
+• The wave equation thus becomes
+�
+∇2
+⊥ + ∂2
+z′E − 2κ1∂z′∂τ + 2iκ0∂z′ + 2iκ1 ¯D∂τ + 2κ0 ¯D
+�
+E = − ν2
+0
+ε0c2
+�
+1 + i
+ν0
+∂τ
+�2
+p.
+(S43)
+• Now the SVEA in space is valid when the pulse is longer than just a few wavelengths so that ∂2
+z′E ≪ κ0∂z′E. With this
+approximation
+�
+∇2
+⊥ − 2κ1∂z′∂τ + 2iκ0∂z′ + 2iκ1 ¯D∂τ + 2κ0 ¯D
+�
+E = − ν2
+0
+ε0c2
+�
+1 + i
+ν0
+∂τ
+�2
+p.
+(S44)
+• Moreover, one can also implement a SVEA in time since10 κ1/κ0 = (�ph/�g)ν−1
+0
+≈ ν−1
+0 where �ph and �g are the phase and
+group velocities respectively. When the pulse length Tpulse is long enough to contain more than just a few optical cycles,
+with Toptical = 2π/ν0, within the envelope, then (κ1/κ0)∂τ ≈ Toptical/Tpulse ≪ 1 so that
+�
+∇2
+⊥ + 2iκ0∂z′ + 2κ0 ¯D
+�
+E(r, t) = − ν2
+0
+ε0c2 p(r, t),
+(S45)
+where the time derivative of the slowly varying polarization envelope has been ignored, compared to the constant term, on
+the same basis that Toptical/Tpulse ≪ 1. This approximation of the polarization term on the right hand side of eq. (S44) is
+equivalent to neglecting the self-steepening effect [11].
+8 This is achieved by multiplying the equation by e−i(ν−ν0)t and integrating
+over all values of ν − ν0. Recall that PNL(r, t) =
+�
+PNL(r, ν)e−iνtdν/2π.
+9 Here we can write
+∂2
+t P(r, t)e−i(κ0z−ν0t) = ∂2
+t
+�
+p(r, t)ei(κ0z−ν0t)�
+e−i(κ0z−ν0t)
+=
+�
+−ν2
+0 − 2iν0∂t + ∂2
+t
+�
+p(r, t)
+10 This is not true, for example, in slow light materials.
+
+6
+Setting up some important relations
+In the case of the inhomogeneous effective medium, we need to investigate some relation between the effective medium
+quantities and the one of the physical material before delving into the derivation of the NLSE. We have seen that the dielectric
+permeability and the magnetic one can be written in frequency space as
+˜ε(E, r, ω) = ε0εsp(r)
+�
+1 + χ(1)(ω) + 3χ(3)|E|2/Ω
+�
+(S46)
+˜µ(r, ω) = µsp(r)µ0
+(S47)
+where χ(1)(ω) is the material linear dielectric permeability, including the effect of dispersion. Note also that εsp(r) = µsp(r) in
+isotropic coordinates (that we are working with), so that nsp(r) ≡ √εspµsp = εsp(r). Thus, we have
+˜n(ω, r) = nsp(r)n(ω) = cεsp(r)
+�
+µ0ε0(1 + χ(1)(ω)).
+(S48)
+In the wave equation eq. (S31), we have the term −˜µ˜εℓν2E with ˜εℓ = ε0εsp(1 + χ1(ω)) . In light of the previous considerations,
+this term can be written as
+− ˜µ˜εℓν2E = −(˜n2/c2)ν2E.
+(S49)
+When we move to the frequency space for the effective medium, we use the conjugate variable (ν) to Minkowski time. As we
+already noticed, this is related to the frequency measured by an observer at rest with respect to the medium in curved spacetime
+by ν = ω √−g00. The effective dispersion relation is thus
+˜n2ν2 = c2˜κ2,
+(S50)
+as previously discussed (see eq. (S33)). In expanding in power series ˜κ around ν0 we will then have
+˜κ = ˜κ0 + ˜κ1(ν − ν0) + ˜D.
+(S51)
+By comparing the dispersion relation in curved spacetime and the one of the effective medium it is easy to see that
+˜κ0 = √−g00nspκ0
+(S52)
+˜κ1 = nspκ1
+(S53)
+˜κ2 = (nsp/ √−g00)κ2
+˜D = 1
+2 ˜κ2(ν − ν0)2 + ... = nsp
+√−g00D,
+(S54)
+where the κi(ω0) appearing in these expressions are the analogues of their tilded versions, i.e.,
+κ0 = κ|ω0
+(S55)
+κ1 = ∂ωκ|ω0
+(S56)
+κ2 = ∂2
+ωκ|ω0,
+(S57)
+and refer to the tabulated optical properties of the physical medium we are considering.
+The expression in eq. (S53) implies that the group velocity in the effective medium is related to the physical one in curved
+spacetime by
+˜�g = �g/nsp.
+(S58)
+Note that this is consistent with the way the phase-velocity in the effective medium is related to the one in curved spacetime via
+˜�ph ≡ 1/˜n = 1/(nspn) = �ph/nsp. More in general, this is consistent with the relation between the coordinate velocity ˜� = dx/dt,
+characterizing the propagation in the effective medium in flat spacetime, and the velocity with respect to an observer comoving
+with the dielectric � = dχ/dτ where τ = √−g00t is the proper time with respect to the stationary observer and χi = xi/
+√
+Ω
+represents the proper length. Indeed, we see immediately that � = dχ/dτ = ˜�nsp.
+
+7
+Derivation of the NLSE for the effective medium
+First let us notice that, in order for the effective medium description to be equivalent to the physical one in curved spacetime,
+we need to require that:
+1. the dielectric permeability and magnetization are dependent on the radial coordinate with the expressions given in the
+previous section
+2. dispersion enters via the physical frequency ω = ν/ √−g00 which corresponds to a position dependent correction to the
+Fourier variable ν.
+Note that the rest of the relations in the previous section are not necessary in the derivation of the NLSE, but they are nonetheless
+important for connecting the effective medium properties with the ones of the physical medium in curved spacetime.
+In order to derive the NLSE in this case, we go back to the wave equation in eq. (S31) that we report here for convenience
+∇2E − ∇(∇ · E) + ˜µ˜εℓν2E = −˜µν2PNL − (∇ log(µsp)) × (∇ × E).
+(S59)
+To proceed further, as discussed in the main text, we can make use of the homogeneous Maxwell equation for D in order to
+write
+∇ · D = 0 =⇒ ∇ · E = −(∇ log ˜εl) · E − 1
+˜εl
+∇ · PNL
+−∇(∇ · E) = (E · ∇)∇ log ˜εl + �(∇ log ˜εl) · ∇� E + (∇ log ˜εl) × (∇ × E) + ∇
+� 1
+˜εl
+(∇ · PNL)
+�
+,
+where we have used that E × �∇ × ∇ log ˜εl
+� = 0 since the curl of the gradient vanishes. We obtain
+∇2E + ˜µ˜εℓν2E = −˜µν2PNL − ∇
+� 1
+˜εl
+(∇ · PNL)
+�
+− (E · ∇)∇ log εsp −
+�
+(∇ log εsp) · ∇
+�
+E −
+�
+∇ log µsp + ∇ log εsp
+�
+× (∇ × E). (S60)
+As discussed in the main text, eq. (S60) does not allow, in general, to write down a scalar propagation equation since even by
+starting from a linearly polarized electric field we end up having coupled equations between all the components of the electric
+field. This is in general also true whenever one does not ignore the vectorial term ∇(∇ · E).
+In order to bypass these problems, we resort to considering two cases of interest, which are the ones analyzed in the main text.
+See also Fig. S1. Before doing so, let us emphasize that we will be interested in the specific case of a Kerr non-linear medium.
+Thus, we write the (slow envelope of the) non-linear polarization of the effective medium as
+p(r, t) = 3ε0nsp(r)χ(3)|E|2E/Ω,
+(S61)
+which includes the non-linearity of the material and the contribution coming from the curved spacetime. Using the expression
+for ˜κ0 in eq. (S52), the term containing the polarization can be written as
+− nsp(r)ν2
+0
+2˜κ0ε0c2 p(r, t) = −n2ν0nsp(r)ε0|E|2E/Ω,
+(S62)
+where n2 = 3χ(3)/2n(ω0)cε0 is the nonlinear index of the Kerr material. As before, we are also going to neglect the self-
+steepening effect [11]. Furthermore, in our simulations we use the parameters of a single-mode, fused silica optical fiber
+employed in [4] that we summarize here in Tab. I.
+Horizontal propagation
+As we have seen in the main text, considering linearly polarized light propagating – in a medium stationary on Earth – for
+distances much smaller than Earth’s radius, the horizontal motion can be considered as happening at constant radius r ≥ r⊕. We
+can then follow the derivation in [4] where the pulse propagation in a single-mode optical fiber was considered.
+In a nutshell, whenever the coefficients in eq.(S59) are constant, so that the very last term vanishes since ∇ log(µsp) = 0, we
+find an equation
+∇2E − ∇(∇ · E) + ˜µ˜εℓν2E = −˜µν2PNL,
+(S63)
+
+8
+Symbol
+Name
+Value
+Soliton pulse properties from [4]:
+T0
+Duration (this corresponds to 70 fs total pulse length)
+40 fs
+Es
+Generating pulse energy (not used, only for reference)
+5 pJ
+λ0 = 2πc/ν0
+central soliton wavelength
+803 nm
+Fiber properties:
+κ0(ν0)
+n(ν0)ν0/c, assuming n(ν0) = 1.5
+1.17 · 107 /m
+κ1(ν0)
+1/�g(ν0), assuming �g(ν0) = 0.65c
+5.13176 · 10−9 s/m
+κ2(ν0)
+Group velocity dispersion from [4]
+−9.5 · 10−27 s2/m
+n2
+Kerr non-linearity of silica from [29]
+2.19 · 10−20 m2/W
+Aeff
+Effective transverse mode area
+π (1.6 µm/2)2
+Properties of fused silica:
+P11 22
+Component for transverse stress of the photoelastic tensor from [30, 31] 0.271
+cs
+Speed of sound tabulated in [32]
+5720 m/s
+Miscellaneous:
+r⊕
+Earth equatorial radius
+6378137 m
+rS (Earth)
+Schwarzschild radius of Earth
+9 · 10−3 m
+TABLE I: Specifics of all the parameters entering the numerical simulations of the NLSE(s). The material parameters are
+extrapolated from Philbin et al. [4], κ0/ν0 = 1.5/c and κ−1
+1 ≈ 0.65c. Consistently, we use the properties of the Crystal Fibre
+NL-PM 750 from NKT photonics [4].
+which is equivalent to eq. (S1) of [4] in frequency space.
+Following [4], and considering the propagation of light pulses in an optical fiber, this equation can be solved by separation
+of variables between an amplitude that depends on the propagation direction and a vectorial part depending on the transverse
+directions, i.e., E(ν, r) = E(ν, z)U(ν, x, y) in frequency space and with U(ν, x, y) a 3-dimensional vector. By solving the eigenvalue
+problem for the transverse part, we then remain with a one-dimensional problem given by
+∂2
+z E(z) + ˜n2
+c2 ν2E(z) = −˜µν2PNL(z)
+(S64)
+where the refractive index is set by the eigenvalue of the transverse fiber mode and accounts for the property of the fiber’s core
+and of the transverse profile. In our case, we can then assume to start directly from this equation, where the property of the
+effective medium accounts also for the non-trivial spacetime background via nsp.
+At this point, the derivation of the NLSE proceeds as in the standard case discussed above. We introduce the field scalar
+amplitude via E(z, ν) ≈ E(z, ν − ν0)ei˜κ0z in our equation to obtain
+∂2
+zE + 2i˜κ0∂zE + [˜κ2(ν) − ˜κ2
+0]E = −ν2 ˜µPNLe−i˜κ0z.
+(S65)
+We then proceed as before by expanding
+˜κ = ˜κ0 + ˜κ1(ν − ν0) + ˜D
+(S66)
+to get, neglecting ˜D2 terms and converting back to the time domain,
+�
+∂2
+z + 2i˜κ0(∂z + ˜κ1∂t) + 2i˜κ1 ¯˜D∂t + 2˜κ0 ¯˜D − ˜κ2
+1∂2
+t
+�
+E(z, t) = ˜µ∂2
+t (PNL(z, t)) e−i(˜κ0z−ν0t),
+(S67)
+where ¯˜D = −(˜κ2/2)∂2
+t + ... in complete analogy with the standard derivation outlined above.
+At this point, by neglecting the second derivatives in z as well as terms (˜κ1/˜κ0)∂t and using eq. (S62) we arrive at eq. (20) of
+the main text, i.e.,
+i(∂z + ˜κ1∂t)E − ˜κ2
+2 ∂2
+t E = −n2ν0nsp(r⊕)ε0
+|E|2
+Ω E.
+(S68)
+
+9
+y
+x
+z
+(i)
+g
+z
+x
+y
+(ii)
+FIG. S1: Geometry of the problem. The two cases considered are labelled by (i) and (ii). In (i), the light pulse propagates in a
+horizontal fiber positioned at r⊕ = r ∼ constant. In (ii), the light pulse propagates in a vertically positioned fiber.
+Radial motion
+As we already discussed, in the case of vertical motion, in which we identify the radial direction with the propagation direction
+along z with r = r⊕ + z, the effective medium becomes a gradient-index medium with the refractive index changing along the
+propagation direction. We assume that all the quantities entering the wave equation depend solely on z. Upon considering
+linearly polarized light along a direction orthogonal to z, we end up with the system of three decoupled equations in eq. (22) of
+the main text that we report here for completeness
+∂2
+z Ex(y) + ˜µ˜εℓν2Ex(y) = −˜µν2PNL,x(y) − (∂z ln ˜εℓ)∂zEx(y) + (∂z(ln ˜εℓ + ln ˜µ)) ∂zEx(y)
+(S69)
+∂2
+z Ez + ˜µ˜εℓν2Ez = −˜µν2PNL,z − ∂z
+� 1
+˜εℓ
+∂zPNL,z
+�
+− 2(∂z ln ˜εℓ)∂zEz − Ez∂2
+z ln ˜εℓ
+(S70)
+We can then: (1.) use the ansatz Ex(z, t) ∝ E(z, t)ei(˜κ0(z)z−ν0t) + cc.; (2.) proceed as before in expanding the dispersion relation
+around the central frequency, i.e., expanding ˜κ(z, ν) around ν0; (3.) neglect ˜D2 terms, to arrive at
+1
+2˜κ0
+∂2
+zE + i(∂z + ˜κ1∂t)E − ˜κ2
+2 ∂2
+t E − 2i ˜κ1˜κ2
+4˜κ0
+∂3
+t E − ˜κ2
+1
+2˜κ0
+∂2
+t E + 2i∂z˜κ0
+2˜κ0
+E + 2iz∂z˜κ0
+2˜κ0
+∂zE + iz∂2
+z ˜κ0
+2˜κ0
+E − z∂z˜κ0E − z2 (∂z˜κ0)2
+2˜κ0
+E
+(S71)
+= −n2ν0nsp(r)ε0|E|2E/Ω + ∂z ln nsp
+2˜κ0
+(i˜κ0E + ∂zE + iz(∂z˜κ0) E).
+Upon using the SVEA approximation(s), that entail that ∂2
+zE ≪ κ0∂zE and (˜κ1/˜κ0)∂t ≪ 1, we then obtain the NLSE given by
+eq. (24) in the main text. It should also be noted that, in the weak field approximation, the terms −z2 �
+(∂z˜κ0)2/(2˜κ0)
+�
+E and
+(∂z ln nsp)/(2˜κ0)iz(∂z˜κ0)E are negligible since at least quadratic in rS /r⊕.
+SOLUTION OF THE 1D EQUATIONS
+As discussed in the main text, in the case of horizontal propagation and considering a material with anomalous dispersion, we
+can solve eq. (20) analytically. Borrowing the solution from eq.(S74) of the supplementary material in [4] the analytical solution
+is given by (see also Fig. S2)
+E(t, z) =
+�
+Ω|˜κ2|
+ν0n2nspε0T 2
+0
+cosh
+�t − ˜κ1z
+T0
+�−1
+exp
+������
+iz|˜κ2|
+2T 2
+0
+������ ,
+(S72)
+where T0 is the pulse length, and 1/˜κ1 is its speed of propagation. This solution reduces exactly to eq.(S4) of [4] in the limit of
+rS → 0. Note that the propagation speed of the soliton is ˜�g(ν0) = �g(ω0)/nsp. This is exactly the proper velocity with respect to
+the observer’s proper time and proper length in curved spacetime, as found above in eq. (S58).
+
+10
+t(s)
+z(m)
+|A(z, t)|
+FIG. S2: Propagation of the 1D analytic soliton, eq. (20), for rS = 9 × 10−3 m and r⊕ = 6 × 106 m.
+In the case of the vertical propagation, we solve eq. (24) by way of the split-step Fourier method as showcased in [11]. In
+particular, we have adapted the Matlab code reported in [11] to our needs. In solving numerically the NLSE, we assign as initial
+temporal profile the soliton solution in flat spacetime of Philbin et al. [4], which coincides with the solution in the horizontal 1D
+propagation at z = 0 and for rS → 0.
+Schematically, the split-step Fourier method consists in rewriting the NLSE as
+∂zE =
+� ˆD + ˆN
+�
+E,
+(S73)
+where the non-linear operator ˆN = ˆN(z, |E|2) accounts for the non-linearity and the diffusive dynamics is enclosed in the operator
+ˆD = ˆD(z, ∂t). We then need to separate the action of the non-linear term and the dissipative one by dividing the propagation
+distance in small steps such that
+E(z + h, t) ≈ e ˆNh/2e ˆDhe ˆNh/2E(z, t).
+(S74)
+This can be easily accomplished by alternating the use of the fast-Fourier/inverse Fourier transform algorithm in order to apply
+ˆD in frequency space as a multiplicative operator and going back to the time domain at each step. Furthermore, since our
+operator ˆD = ˆD(z, ∂t) depends on the z coordinate, a more precise implementation of the method would see to apply at each step
+exp
+�� z+h
+z
+ˆD
+�
+, which however is well approximated by e ˆDh in our simulations.
+PHOTOELASTICITY – INCLUDING THE EFFECT OF MATERIAL DEFORMATION ON THE REFRACTIVE INDEX
+As we have discussed so far, the optical medium in curved spacetime turns out to be equivalent to an effective one in flat
+spacetime, where the optical properties have a contribution coming from the curved spacetime background. However, whenever
+our physical medium is stationary in a curved spacetime, i.e., it follows the trajectories of the timelike Killing vector, it will also
+be subject to forces that can deform it. As discussed in the main text, deformations due to gravity of our physical medium lead
+to a change in the refractive index via the photoelastic effect [17].
+Given our previous considerations, we will be interested in the effect of photoelasticity only for the vertical propagation
+equation. In order to include this effect and separate the contributions coming from the curvature of spacetime and the inertial
+acceleration of the fiber, we follow the discussion in [18] on the description of a deformable resonator. We choose to ignore the
+potential effects of photoelasticity on the nonlinear properties of the material, i.e., the nonlinear susceptibility χ(3), as they would
+be mediated through different mechanisms compared to the effect on the linear refractive index.
+Consider then the situation depicted in Fig. S3. A fiber of length L and constant mass density ρm is hanging from a support
+located at r = r0 ≡ r⊕ + L. In Schwarzschild spacetime, for an observer given by the stationary Killing vector ∂t/∥∂t∥, the
+
+1×108
+5×107
+1. × 10-12
+5.×10-13
+-0.0002
+-0.0001
+0.0000
+0.0001
+0.000211
+z
+x
+y
+L
+r = r⊕ + L
+A⊘
+g
+FIG. S3: Geometry of a fiber hanging in a weak gravitational field.
+proper acceleration of the observer fixed at the support, i.e., an observer at constant radius in isotropic coordinates, and the local
+curvature projected into the proper detector frame of this observer are given by [33]
+aJ =
+����������0, 0,
+rS
+2r2
+0 c2
+�
+1 − rS
+4r0
+� �
+1 + rS
+4r0
+�3
+����������
+(S75)
+R0J0J =
+rS
+r3
+0
+�
+1 + rS
+4r0
+�6
+�1
+2, 1
+2, −1
+�
+,
+(S76)
+where we chose for the z direction to be aligned radially. Furthermore, consistently with the notation we have used so far, we
+want to consider the origin of our coordinate at r = r⊕. This entails shifting z → z − L to translate the origin from the support
+at r0 to r = r⊕. Note that the proper detector frame is determined by an orthonormal tetrad Fermi-Walker transported along the
+timelike trajectory of the support of the fiber which, in our set-up, corresponds to a stationary observer [34].
+We can now compute the acceleration of test particles in the proper detector frame by following the derivation in [18]. At
+linear order in (z − L)/(r⊕ + L), the acceleration is given by az
+p = −
+�
+az + c2R0z0z(z − L)
+�
+. It should be noted that this expression
+is derived by neglecting acceleration squared terms in the proper detector frame metric as well as working at first order in the
+perturbations around flat spacetime (see discussion in [18]). This calls for care when wanting to extrapolate these expressions as
+generally valid. Each segment of the fiber is then stressed by the force Fz(z) of the parts of the fiber hanging “below” it
+Fz(z) =
+� z
+0
+dz′ρmA⊘az
+p(z′) = −ρmA⊘c2 rS
+2r2
+0
+����������
+z
+�
+1 − rS
+4r0
+� �
+1 + rS
+4r0
+�3 −
+z2 − 2Lz
+r0
+�
+1 + rS
+4r0
+�6
+���������� ,
+(S77)
+where A⊘ is the cross-section of the fiber.
+More generally, the fiber is subject to a stress σkl = Fk/Al, where Fk is the force in direction ˆek and Al is the differential area
+normal to ˆel upon which the force acts, caused by the inertial and tidal forces within the fiber. As long as we are considering
+strains within the elastic limit of the material, which is the case of interest here, we can employ Hooke’s law and find that the
+strain in the fiber is Skl = σkl/Y, where Y is the Young modulus of the material. The relation to the electric permeability tensor
+εr is then given by ∆(εr)−1
+kl = Pkl mnSmn, where P is the photoelastic tensor [17]. The fact that the change in the inverse of εr is
+linear in the strain holds for small or moderate strain. Limiting ourselves to isotropic materials, and a diagonal stress tensor, the
+equations reduce in complexity to
+∆(εr)−1
+kk = Pkk llSll = Pkk ll
+Y
+σll
+(S78)
+In our set-up, the stress and then the strain on the fiber are given explicitly by
+σzz(z) = F(z)
+A⊘
+= ρmc2 rS
+2r2
+0
+����������
+z
+�
+1 − rS
+4r0
+� �
+1 + rS
+4r0
+�3 −
+z2 − 2Lz
+r0
+�
+1 + rS
+4r0
+�6
+����������
+(S79)
+Szz(z) = c2
+c2s
+rS
+2r2
+0
+����������
+z
+�
+1 − rS
+4r0
+� �
+1 + rS
+4r0
+�3 −
+z2 − 2Lz
+r0
+�
+1 + rS
+4r0
+�6
+���������� ,
+(S80)
+
+12
+z (m)
+rS (m)
+z (m)
+z (m)
+0.08
+0.06
+0.04
+0.02
+0
+0.3
+0.2
+0.2
+0
+FIG. S4: Comparison between the full expression for ∆εr and the approximate one that are appearing in eq. (S81). Left panel:
+Here we have used the parameters tabulated in Tab. I and chosen a quite large rS = 3 km. The solid, blue curve represents the
+approximate expression for ∆εr, the dashed red curve the exact value of ∆εr, while the dot-dashed black curve is the value of εr
+in the absence of photoelasticity. We see that (i) the full and approximate expressions start to deviate from propagation
+distances O(1m) onward and (ii) for relatively small propagation distances ∆εr is not anymore a small correction to the relative
+permeability εr but becomes equal or greater than εr. Central panel: Fractional difference between the full and approximated
+expressions for the photoelastic correction ∆εr i.e.,
+�
+|∆εr|approx − |∆εr|full
+�
+/
+�
+|∆εr|full + |∆εr|approx
+�
+. Here rS goes from zero to 104
+times the Schwarzschild radius of Earth and the propagation distance reaches 100 m. We see that the difference between the
+two expressions remains below 10%. Right panel: |∆εr|full. The value of ∆εr, for rS from zero to 104 times the Schwarzschild
+radius of Earth and propagation distance up to 100 m, is always well below the value of the relative permeability εr ≈ 2.25.
+where we used that the speed of sound in the fiber is cs =
+�
+Y/ρm. Note that the strain and stress have a positive sign due to the
+force being directed in the negative z direction or, in other words, since we are considering an elongation of the fiber. Due to the
+axial symmetry of the problem, and the irrelevance of two directions orthogonal to the z-axis for the 1D case, the photoelastic
+tensor is a scalar.
+The perturbation to the electric permeability, promoting εr → εr + ∆εr, is then also a scalar, and is given by
+∆εr = −
+ε2
+r∆(ε−1
+r )
+1 + ε0r∆(ε−1
+r ) ≈ −(ε0
+r)2∆(ε−1
+r ),
+(S81)
+where εr indicates the electric permeability in the absence of photoelasticity and the last expression holds whenever the photoe-
+lastic effect is a small correction to the material properties giving11
+∆εr ≈ −(ε0
+r)2P11 22Szz(z) = −(ε0
+r)2P11 22
+c2
+c2s
+rS
+2r2
+0
+����������
+z
+�
+1 − rS
+4r0
+� �
+1 + rS
+4r0
+�3 −
+z2 − 2Lz
+r0
+�
+1 + rS
+4r0
+�6
+���������� .
+(S82)
+Photoelasticity represents an additional correction to the electric permeability on top of the other effects accounting for the
+effective medium as described in the previous sections. For fused silica, the tabulated values in [30, 31] give P11 22 = 0.271 and
+cs = 5720 m/s [32]. Then, from eq. (S81), for a 10 cm long fiber in the gravitational field of Earth, the contribution of the inertial
+acceleration (first term in eq. (S82)) at the end of the fiber to ∆εr is on the order 10−8 while the tidal acceleration (second term
+in eq. (S82)) contributes a term of order 10−16. Note that, while the tidal contribution is clearly negligible, the correction to the
+relative permeability induced by the inertial acceleration is between one and two orders of magnitude greater than the correction
+due to the vacuum curved spacetime optical properties in our effective picture as quantified by 1 − εnp ∼ 10−9.
+It is easy to check that, considering a P11 22 ≈ 0.271, the approximate expression in eq. (S82) will start to fail around a
+propagation length of 2 m if we consider to be at one Earth’s radius distance from an object whose mass corresponds to a
+11 Note that here we have ∆(ε−1
+r ) = P11 22Szz. The indices are determined by
+the fact that we are considering an electric field linearly polarized in the
+x direction, we identify {x, y, z} ↔ {1, 2, 3}, and we consider an isotropic
+material. Thus, (1) the only component of the perturbation tensor of in-
+terest is ∆(ε−1
+r )11, (2) the only component of the strain is S 33, and (3) we
+have P11 33 = P11 22. See Appendix D of [17] where the notation and the
+example of isotropic materials are discussed in detailed.
+
+2.5
+2.0
+1.5
+Approx
+Full
+1.0
+0.5
+0.0
+0
+5
+10
+15
+20100
+0.08
+80
+0.06
+60
+40
+0.04
+20
+0.02
+0
+0
+20
+40
+60
+80
+100100
+80
+0.3
+60
+0.2
+40
+0.1
+20
+0
+0
+20
+40
+60
+80
+10013
+Schwarzschild radius of 3 km. Indeed, Fig. S4 shows this failure as well as the fact that for such extreme values of rS , ∆εr starts
+to be comparable or greater than εr at propagation distances less than 10 m. At the same time, the same figure shows that, for rS
+up to 104 times the one of Earth, both the conditions for the validity of the approximate expression for ∆εr and the fact that the
+correction to εr is small are well satisfied.
+CONDITIONS FOR VALIDITY OF THE UNIDIRECTIONAL APPROACH
+As discussed in the main text, when considering the propagation of a light pulse in a gradient-index medium, we should
+account for the fact that the position-dependent refractive index will cause some light to be backscattered – this effect has also
+technological application in distributed acoustic sensing for seismology, see [35] and references therein. However, when solving
+the NLSE using the SSF method, the boundary condition completely ignores this fact – it would require already knowing the
+solution to include the backscattered light in the boundary condition. That this is a drawback of using the NLSE – which is a
+unidirectional equation for the validity of which, by definition, back-propagating fields must be negligible – in conjunction with
+the SSF method is well known [24, 25].
+However, back-propagating fields cannot always be simply ignored. A formalism fully accounting for this issue would require
+to solve a system of coupled bidirectional equations, or just solve the full Maxwell equations. However, as argued in [25], we
+can define conditions that guarantee us that the backward reflected light is negligible. In our case, this sets a restriction on the
+parameter space that we can explore, where the description given by our solution to the NLSE can be trusted. Essentially, this
+regime corresponds to the one of weak-field and not large propagation distances. Indeed, physically, for vertical propagation,
+longer propagation distances and stronger gravitational accelerations would imply greater changes to the refractive index giving
+potentially rise to non-negligible back-propagating fields. To make this observation more quantitative, we follow here [25] where
+a more detailed discussion can be found.
+We start from eq. (22) that we report here for convenience
+∂2
+z Ex + ˜µ˜εℓν2Ex = −˜µν2PNL,x + (∂z ln ˜µ)∂zEx.
+(S83)
+Following [25], we can rewrite this equation as
+(∂2
+z + β2)Ex(z) = −Q(z, Ex),
+(S84)
+where β is a reference momentum that can contain the dispersive character of the physical medium but no z dependence and
+that forms our underlying dynamics on top of which we have some perturbation encoded in Q, the residual terms. In our case, a
+sensible choice for β is
+β2 = n2
+0ν2
+c2 ,
+(S85)
+where n0 = √εr( ¯ω0) is the material refractive index without any additional effect from the spacetime (and ignoring the effect of
+redshift combined with the dispersion of the material) and not accounting for the photoelasticity. With this choice we have12
+− Q(z, Ex) = −˜µν2PNL,x(z) + (∂z log ˜µ)∂zEx(z) + β2[1 − n2
+sp(1 + ∆εr/εr)]Ex(z).
+(S86)
+Now, we can decompose the field in forward and backward directed (in time) fields Ex = E+ + E− and find the equivalent
+system of two equations [25]
+∂zE± = ±iβE± ± i
+2βQ.
+(S87)
+The question is then when, starting with E− = 0, E− remains negligible. Indeed, if E− remains negligible then we are left with a
+unidirectional equation and, more importantly, we know that the reflected light can be safely neglected even in comparison with
+the unperturbed propagation in flat spacetime in a linear medium.
+12 In a nutshell, from eq. (S83) we have
+∂2
+z Ex + µ0ϵ0(εr + ∆εr)ν2µspεspEx = −µ0µspν2PNL,x + (∂z ln ˜µ)∂zEx
+Writing then µ0ϵ0(εr + ∆εr)ν2µspεsp
+=
+µ0ϵ0ν2εr(1 − (1 − n2
+sp)) +
+µ0ϵ0ν2n2
+sp∆εr we arrive at
+∂2
+z Ex + β2Ex = β2
+�
+1 − n2
+sp
+�
+1 + ∆εr
+εr
+��
+Ex − µ0µspν2PNL,x + (∂z ln ˜µ)∂zEx
+
+14
+rS (m)
+z (m)
+z (m)
+Slow condition
+(a)
+(b)
+0
+0.02
+0.04
+0.06
+0.08
+0
+0.00005
+0.00010
+0.00015
+FIG. S5: Panel (a) shows the right-hand side of the slow-evolution condition of eq. (S88) in the case in which photoelasticity is
+not considered; Panel (b) shows the same when also photoelasticity in included. We see that without photoelasticity the
+condition is very well satisfied for a large set of parameters. When including photoelasticity, we see violations of the condition
+for values of rS or the propagation length for which, from Fig. S4 and the corresponding discussion, we know that the ∆εr starts
+to be not anymore a small correction to the relative permeability.
+Photoelasticity is the main culprit for the possible significance of the reflected light since, as we have argued before, it is the
+dominant effect giving rise to an effective gradient-index medium. It enters only in the term linear in the electric field. Thus,
+we focus solely on this term in the following. As discussed in [25], the first condition for the backward propagating light to be
+negligible is that the residual terms contained in the term Q/2β in eq. (S87) are negligible with respect to βEx. This translates to
+the condition
+1 − n2
+sp(1 + ∆εr/εr) ≪ 1.
+(S88)
+The second condition arises from considering the backward-evolving part of E−(cf. the discussion in the appendix of [25])
+∂zE−, backwards ≈
+∂zχ
+(k + β)2 ,
+(S89)
+where Q = χEx and k2(z) ≡ β2 + Q(z)
+Ex(z). For small Q and ignoring non-linearities, requiring that the change in the medium
+parameters does not cause significant back-propagation on the order of a wavelength leads to
+∂z
+�
+n2
+sp(1 + ∆εr/εr)
+�
+(3/2 + n2sp(1 + ∆εr/εr))2 ≪ β.
+(S90)
+This no-accumulation condition requires that the derivative of the backpropagating fields is negligible and encodes the fact that
+there is no-accumulation of the reflected light giving in the end a non-negligible contribution.
+From Fig. S5 and Fig. S6 we can see that, in the absence of photoelasticity, i.e., considering a rigid dielectric, these conditions
+are very well satisfied for the parameters in our simulations also when considering relatively large values of rS and propagation
+lengths. When turning on photoelasticity, the situation changes, and we can arrive to regimes of large rS and large propagation
+distances where the conditions are not satisfied anymore. In particular, from Fig. S5 and Fig. S6 we see that the main limiting
+factor is the slow-evolution condition. However, it should be noted that the slow-evolution condition starts to be violated in
+the same range of parameters in which ∆εr cannot anymore be considered a small correction and when it is arguable if the
+treatment of the photoelasticity as linear in the stresses is valid. To corroborate these observations, in the regime in which the
+slow-evolution condition is clearly violated we observe a non-negligible energy loss in the numerical solutions of the vertical
+propagation equation (see Fig. S7).
+
+1000
+800
+600
+400
+200
+0
+0
+20
+40
+60
+80
+100100
+80
+60
+40
+20
+0
+0
+20
+40
+60
+80
+10015
+rS (m)
+z (m)
+z (m)
+accu condition
+(b)
+(a)
+0
+5.0 × 10-10
+1.0 × 10-9
+1.5 × 10-9
+0
+0.1
+0.2
+0.3
+0.4
+× 10−17
+FIG. S6: Panel (a) shows the no-accumulation condition by depicting the ratio between the left and the right-hand sides of
+eq. (S90) in the case in which photoelasticity is not considered; Panel (b) shows the same when also photoelasticity in
+included. We see that this condition is actually well satisfied in both cases, while it remains that without photoelasticity the
+condition is much better satisfied. This analysis shows that the slow-evolution condition is the relevant one for the problem that
+we are considering.
+COEFFICIENTS FOR THE NUMERICAL SIMULATIONS:
+Finally, we report here the explicit expressions for the different coefficients entering the vertical propagation equation that we
+use in our simulations.
+First, let us recall that, when including photoelasticity, we have
+n(ω) =
+�
+εr(ω) + ∆εr(ω) =
+�
+1 + χ1(ω) + ∆εr(ω) =
+�
+n0(ω)2 + ∆εr(ω),
+(S91)
+where n0(ω) is the refractive index in the absence of photoelasticity. We can then proceed to compute all the κi coefficients of
+interest
+κ0 = κ|ω0
+(S92)
+κ1 = ∂ωκ|ω0
+(S93)
+κ2 = ∂2
+ωκ|ω0,
+(S94)
+where κ = nω/c. We start from κ0, where we have
+κ0(ω0) = n(ω0)ω0
+c
+.
+(S95)
+We now consider the case in which the pulse propagates from the bottom of the vertically oriented fiber, which is the case we
+simulate numerically. We thus refer the various quantities of interest to the initial physical frequency ¯ω0, i.e., the frequency
+measured by the stationary observer at the bottom of the fiber. We can then write
+κ0(ω0) =
+n
+�
+¯ω0
+√
+−g00(r⊕)
+√
+−g00(r⊕+z)
+�
+¯ω0
+√
+−g00(r⊕)
+√
+−g00(r⊕+z)
+c
+.
+(S96)
+Note that, even for extreme values of rS and z, like rS = 10−2r⊕ and z = 100 m we have that 1 −
+�
+−g00(r⊕)/
+�
+−g00(r⊕ + z) is
+negligible when considering the dispersive properties of realistic materials at the optical frequencies of interest, i.e., the changes
+would be on scales way too fine-grained with respect to the tabulated values of the refractive index at the µm scale [36]. To
+account for this fact, calling ζ = 1 −
+�
+−g00(r⊕)/
+�
+−g00(r⊕ + z) we perform an expansion of eq. (S96) at the first order in ζ.
+In the following we report the expressions for all the coefficients necessary to simulate the vertical propagation of the pulse
+at first order in ζ. Note however that, for what concerns the simulations reported in the work, we can always safely neglect
+
+1000
+800
+600
+400
+200
+0
+0
+20
+40
+60
+80
+1001000
+800
+600
+400
+200
+0
+0
+20
+40
+60
+80
+10016
+L = 0.1 m
+L = 1 m
+rS/r⊕
+I(L)/I(0)
+0.0000
+0.0002
+0.0004
+0.0006
+0.0008
+0.0010
+0.7
+0.8
+0.9
+1.0
+L = 0.1 m, P1122 = 0
+FIG. S7: Energy loss due to photoelasticity. We show the ratio between the final and initial energy
+I(L)/I(0) ∼ εr
+�
+dt|E|2���z=L / εr
+�
+dt|E|2���z=0, in the proper detector frame. The orange, square points correspond to the case of a
+propagation length of 1 m while the blue, round points to a propagation length of 0.1 m. The lines represent the linear fit of the
+corresponding data. We obtain slopes of −360.3rs/r⊕ and −43.2rs/r⊕ respectively. The green, rhomboidal points correspond to
+the case without photoelasticity and are compatible with energy conservation up to a negligible energy loss accounted for by
+purely gravitational redshift.
+also the corrections to the zeroth order terms for all the ˜κi. The same holds true also for the terms ∂z˜κ0 and ∂2
+z ˜κ0 as far as
+photoelasticity is considered since the z−dependence is dominated by the photoelasticity. However, when considering the case
+with no photoelasticity, neglecting the z−dependence coming from the redshift factors in κ0 amounts to a relative error of one
+part in 103. While still small, we have performed the simulations in which photoelasticity is not included considering also the ζ
+corrections in full to account for this small discrepancy.
+The full expression including all the corrections at order ζ are reported in the following. Starting with κ0 we have
+κ0(ω0) ≈ κ0( ¯ω0) − ζ ¯ω0κ1( ¯ω0)
+(S97)
+= κb
+0
+�
+1 + ∆εr
+εr
+− ζ
+������������
+κb
+0
+�
+1 + ∆εr
+εr
++ ¯ω2
+0
+c
+ε′
+r + ∆ε′
+r
+2n0
+�
+1 + ∆εr
+εr
+������������
+,
+where κ1( ¯ω0) = (n( ¯ω0) + ¯ω0∂ωn| ¯ω0)/c, while n0 and κb
+0 are the tabulated refractive index and corresponding κ0 of the material,
+without photoelasticity, i.e. n0 = √εr( ¯ω0) and κb
+0 = ¯ω0n0/c, and a prime indicates the derivative with respect to the frequency.
+Note that ε′
+r = χ′
+1. We derive the expression for the latter, in terms of tabulated values, below. Before doing so, however, let
+us compute the derivatives of ˜κ0 at first order in ζ. Using the fact that ˜κ0 =
+�
+−g00(r⊕ + z)nsp(z)κ0(ω0), we find
+∂z˜κ0 ≈κb
+0
+�
+−g00(r⊕)
+������������
+∂znsp
+�
+∆εr
+εr
++ 1 +
+nsp∂z∆εr
+2εr
+�
+∆εr
+εr + 1
+������������
++
+¯ω2
+0
+4c(∆εr + εr( ¯ω0))3/2
+�
+−2ζ(∂znsp)
+�
+−g00 (r⊕ + z)(∆εr + εr( ¯ω0)) �∆ε′
+r + ε′
+r
+�
+(S98)
+−2ζnsp(∂z∆ε′
+r)
+�
+−g00 (r⊕ + z)(∆εr + εr) + ζnsp(∂z∆εr)
+�
+−g00 (r⊕ + z) �∆ε′
+r + ε′
+r
+�
+−2nsp(∂zζ)
+�
+−g00 (r⊕ + z)(∆εr + εr) �∆ε′
+r + ε′
+r
+� − 2ζnsp∂z
+�
+−g00(r⊕ + z)(∆εr + εr) �∆ε′
+r + ε′
+r
+��
+,
+
+1.0
+0.9
+0.8
+0.7
+0.0000
+0.0002
+0.0004
+0.0006
+0.0008
+0.001017
+∂2
+z ˜κ0 ≈κb
+0
+�
+−g00(r⊕)
+������������
+∂2
+znsp
+�
+∆εr
+εr
++ 1 + ∂znsp(∂z∆εr)
+εr
+�
+∆εr
+εr + 1
++ nsp
+������������
+∂2
+z∆εr
+2εr
+�
+∆εr
+εr + 1
+−
+(∂z∆εr)2
+4ε2
+r
+� ∆εr
+εr + 1
+�
+3/2
+������������
+������������
+(S99)
+− ¯ω2
+0
+�
+−g00(r⊕ + z)ζ(∂2
+znsp) �∆ε′
+r + ε′
+r
+�
+2c √∆εr + εr
+−
+¯ω2
+0(∂znsp)
+�
+2ζ(∂z
+�
+−g00(r⊕ + z))(∆εr + εr) �∆ε′
+r + ε′
+r
+� −
+�
+−g00(r⊕ + z)ζ∂z∆εr
+�∆ε′
+r + ε′
+r
+��
+2c(∆εr + εr)3/2
+−
+¯ω2
+0(∂znsp)
+�
+2
+�
+−g00(r⊕ + z)ζ(∂z∆ε′
+r)(∆εr + εr) + 2
+�
+−g00(r⊕ + z)(∂zζ)(∆εr + εr) �∆ε′
+r + ε′
+r
+��
+2c(∆εr + εr)3/2
+−
+¯ω2
+0nsp
+�
+4(∆εr + εr)
+�
+2(∂z
+�
+−g00(r⊕ + z))(∆εr + εr) −
+�
+−g00(r⊕ + z)∂z∆εr
+� �ζ(∂z∆ε′
+r) + (∂zζ) �∆ε′
+r + ε′
+r
+���
+8c(∆εr + εr)5/2
+−
+¯ω2
+0nsp
+�
+−ζ �∆ε′
+r + ε′
+r
+� �
+−4(∂2
+z
+�
+−g00(r⊕ + z))(∆εr + εr)2 + 4(∂z
+�
+−g00(r⊕ + z))∂z∆εr(∆εr + εr) + 2
+�
+−g00(r⊕ + z)∂2
+z∆εr(∆εr + εr)
+��
+8c(∆εr + εr)5/2
+−
+¯ω2
+0nsp
+�
+−ζ �∆ε′
+r + ε′
+r
+� �
+−3
+�
+−g00(r⊕ + z)(∂z∆εr)2��
+8c(∆εr + εr)5/2
+−
+¯ω2
+0nsp
+�
+4
+�
+−g00(r⊕ + z)(∆εr + εr)2 �
+ζ(∂2
+z∆ε′
+r) + 2(∂z∆ε′
+r)(∂zζ) + (∂2
+zζ) �∆ε′
+r + ε′
+r
+���
+8c(∆εr + εr)5/2
+Considering that
+∆εr = −
+ε2
+r∆(ε−1
+r )
+1 + εr∆(ε−1
+r ),
+(S100)
+with
+∆(ε−1
+r ) =
+c2P1122rS
+�
+z
+�
+1−
+rS
+4(L+r⊕)
+��
+rS
+4(L+r⊕) +1
+�3 −
+(z−L)2−L2
+(L+r⊕)
+�
+rS
+4(L+r⊕) +1
+�6
+�
+2c2s(L + r⊕)2
+(S101)
+we also have
+∂z∆εr = −
+ε2
+r∂z∆(ε−1
+r )
+(εr∆(ε−1
+r ) + 1)2
+(S102)
+∂2
+z∆εr =
+ε2
+r
+�
+2εr(∂z∆(ε−1
+r ))2 − (εr∆(ε−1
+r ) + 1)∂2
+z∆(ε−1
+r )
+�
+(εr∆(ε−1
+r ) + 1)3
+.
+(S103)
+Moving on, for κ1 we need
+κ1(ω0) = (n(ω0) + ω0∂νn(ν)|ω0)/c
+(S104)
+We proceed with the same approximation at first order in ζ as done above. We get
+κ1(ω0) ≈c−1
+�������
+�
+εr + ∆εr + ¯ω0
+ϵ′
+r + ∆ε′
+r
+2 √εr + ∆εr
+�
+−g00(r⊕)
+�
+−g00(r⊕ + z)
+�������
+(S105)
++ c−1
+�
+ζ
+�1
+2 ¯ω0
+�− ¯ω0∆ε′′
+r − ¯ω0ε′′
+r
+√∆εr + εr
++
+� ¯ω0∆ε′
+r + ¯ω0ε′
+r
+2(∆εr + εr)3/2 −
+1
+√∆εr + εr
+� �∆ε′
+r + ε′
+r
+����
+,
+where all quantities on the right-hand side are evaluated at ¯ω0.
+Following the same notation as before, we indicate with κb
+1 the tabulated optical parameter for the material without photoelas-
+ticity. This tabulated quantity enters the previous expression through ε′
+r = χ′
+1( ¯ω0). Indeed, from κb
+1 = (n0 + ¯ω0∂νn0(ν)| ¯ω0)/c, we
+have
+κb
+1 = c−1(n0 + ¯ω0∂νn0(ν)| ¯ω0) = c−1
+�
+n0 + ¯ω0
+χ′
+1
+2n0
+�
+,
+(S106)
+from which we can read χ′
+1 = 2c2 �
+−(κb
+0)2 + κb
+0κb
+1 ¯ω0
+�
+/ ¯ω3
+0.
+
+18
+We thus remain with identifying ∆ε′
+r. Let us consider the full form of ∆εr in eq. (S100) and notice that ∆(ε−1
+r ) in there, as
+given in eq. (S101), does not depend on the frequency but only on the stresses and strains. Thus we get,
+∆ε′
+r = −
+ε′
+rεr∆(ε−1
+r )
+�
+2 + εr∆(ε−1
+r )
+�
+(1 + εr∆(ε−1
+r ))2
+,
+(S107)
+with
+ε′
+r = χ′
+1,
+(S108)
+Finally, under the same assumption as before, we have
+κ2(ω0) = c−1(2n(ν)′|ω0 + ω0n(ν)′′|ω0).
+(S109)
+Thus we end up with
+κ2(ω0) ≈c−1
+�������
+∆ε′
+r + ε′
+r
+√∆εr + εr
++ ¯ω0
+2 (∆εr + εr) �∆ε′′
+r + ε′′
+r
+� − �∆ε′
+r + ε′
+r
+� 2
+4 (∆εr + εr) 3/2
+�
+−g00(r⊕)
+�
+−g00(r⊕ + z)
+�������
+(S110)
++
+ζ ¯ω0
+�
+− �−3 ¯ω0
+�∆ε′
+r + ε′
+r
+� + 4∆εr + 4εr
+� �
+2(∆εr + εr) �∆ε′′
+r + ε′′
+r
+� − �∆ε′
+r + ε′
+r
+�2��
+8c(∆εr + εr)5/2
+− ζ ¯ω2
+0
+�∆ε′′′
+r + ε′′′
+r
+�
+2c(∆εr + εr)1/2 ,
+where all the quantities on the right-hand side are evaluated at ¯ω0.
+As before, indicating the tabulated optical property of the material with κb
+2 and κb
+3 without photoelasticity, it is immediate to
+derive an expression for χ′′
+1 ( ¯ω0)
+χ′′
+1 = 2c2
+¯ω4
+0
+�
+3(κb
+0)2 − 4κb
+0κb
+1 ¯ω0 +
+�
+κb
+1 ¯ω0
+�2 + κb
+0κb
+2 ¯ω2
+0
+�
+,
+(S111)
+and
+χ′′′
+1 =
+2c2 �
+−12(κb
+0)2 − 6 ¯ω2
+0
+�
+κb
+0κb
+2 + (κb
+1)2�
++ ¯ω3
+0(κb
+0κb
+3 + 3κb
+1κb
+2) + 18κb
+0κb
+1 ¯ω0
+�
+¯ω5
+0
+(S112)
+while
+∆ε′′
+r =
+∆(ε−1
+r )
+�
+−εr
+�
+∆(ε−1
+r )εr + 1
+� �
+∆(ε−1
+r )εr + 2
+�
+ε′′
+r − 2ε′
+r
+2�
+�∆(ε−1
+r )εr + 1� 3
+,
+(S113)
+and
+∆ε′′′
+r = −
+∆(ε−1
+r )
+�
+εrε′′′
+r (∆(ε−1
+r )εr + 1)2(∆(ε−1
+r )εr + 2) − 6∆(ε−1
+r )ε′3
+r + 6(∆(ε−1
+r )εr + 1)ε′
+rε′′
+r
+�
+(∆(ε−1
+r )εr + 1)4
+,
+(S114)
+with
+ε′′
+r = χ′′
+1 and ε′′′
+r = χ′′′
+1 .
+(S115)
+In Eq.(S112), we also neglect κb
+3 since this term is negligible.
+WIDTH OF THE PULSE
+While until now we have considered only the effect of a gravitational field on the propagation velocity of the optical pulse,
+we can also look at the width of the pulse while it propagates. In the horizontal case, the width remains constant, as can be seen
+from the analytical solution of eq. (20). In the vertical propagation case, however, this is no longer true. From Fig. S8, we see
+that spacetime effects, in conjunction with photoelasticity, reduce the width of the pulse. This is clearly negligible for realistic
+values of rS , and it becomes relevant only at extreme values but shows, nonetheless, that gravity has a focusing effect on the
+propagating pulse.
+
+19
+Photoelasticity
+No-Photoelasticity
+rS/r⊕
+0.0000
+0.0002
+0.0004
+0.0006
+0.0008
+0.0010
+0.94
+0.96
+0.98
+1.00
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+8.4×10-12
+8.5×10-12
+8.6×10-12
+8.7×10-12
+8.8×10-12
+8.9×10-12
+9. ×10-12
+9.1×10-12
+9.2×10-12
+rS = 10−3r⊕
+rS = 10−2r⊕
+z (m)
+rS = 10−2r⊕, P1122 = 0
+FWHM (s)
+FWHM[z = L]/FWHM[z = 0]
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+8.4×10-12
+8.5×10-12
+8.6×10-12
+8.7×10-12
+8.8×10-12
+8.9×10-12
+9. ×10-12
+9.1×10-12
+9.2×10-12
+FIG. S8: Left: FWHM (full width at half maximum) of the pulse at z = L normalized by the FWHM of the initial pulse at z = 0
+as a function of rS /r⊕ for an initial pulse with T0 = 40 · 10−13 s propagating for 0.1 m. The blue solid line shows the case
+including the effect of photoelasticity, the red dotted line represents the case without photoelasticity. Right: pulse FWHM (in
+seconds) as a function of the propagation distance z for two different values of rS , again using a pulse with T0 = 40 · 10−13 s,
+longer than the one previously considered, for better numerical precision. The green dot-dashed line represents the case without
+photoelasticity.
+
diff --git a/e9E4T4oBgHgl3EQfRAxR/content/tmp_files/load_file.txt b/e9E4T4oBgHgl3EQfRAxR/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..765c05170ca17bfdfc7f2c2c20f04745fbd311b0
--- /dev/null
+++ b/e9E4T4oBgHgl3EQfRAxR/content/tmp_files/load_file.txt
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+page_content=' Eberhard-Karls-Universit¨at T¨ubingen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' 72076 T¨ubingen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
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+page_content=' Am Fallturm 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
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+page_content=' Institut f¨ur Physik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
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+page_content=' Germany  (Dated: January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' 2023)  Light propagation in curved spacetime is at the basis of some of the most stringent tests of Einstein’s general  relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' At the same time, light propagation in media is at the basis of several communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Given  the ubiquity of the gravitational field, and the exquisite level of sensitivity of optical measurements, the time is  ripe for investigations combining these two aspects and studying light propagation in media located in curved  spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In this work, we focus on the effect of a weak gravitational field on the propagation of optical  solitons in non-linear optical media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We derive a non-linear Schr¨odinger equation describing the propagation of  an optical pulse in an effective, gradient-index medium in flat spacetime, encoding both the material properties  and curved spacetime effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In analyzing the special case of propagation in a 1D optical fiber, we also include  the effect of mechanical deformations and show it to be the dominant effect for a fiber oriented in the radial  direction in Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  INTRODUCTION  The properties of light propagating in optical media is a  subject as old as optics itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In recent years, the possibility  to engineer novel metamaterials has opened the door to the  so-called transformation optics [1], a field promising to en-  hance existing devices and create novel ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' At the basis of  this revolution is the fact that, in the geometric optics limit  – and neglecting dispersion –, light rays propagate in media  following the geodesics of an effective Lorentzian metric, the  so-called optical metric [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' This has also led to the investi-  gation of light in optical media as an analogue gravity model,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', a model in which field perturbations propagate as if in a  curved spacetime background, particularly useful in the inves-  tigation of kinematic effects of quantum field theory in curved  spacetime, like the Hawking radiation and cosmological par-  ticle production [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' When also the effect of dispersion is  considered, the metric description can be cast aside for a more  powerful Hamiltonian formalism, giving rise to the so-called  ray-optical structures [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  This analogy between optical media and curved space-  times can be pushed even further by showing that Maxwell  equations in vacuum, curved spacetime are equivalent to  flat-spacetime Maxwell equations in the presence of a bi-  anisotropic moving medium whose dielectric permittivity and  magnetic permeability are determined entirely by the space-  time metric [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Spacetime itself can then be described as an  optical medium at the level of full electromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' It is then  natural to wonder what would happen if light were to propa-  gate in an optical medium placed in a curved spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Far  from being a far-fetched situation, this is exactly the case for  light propagating in media on Earth due to the non-vanishing,  albeit weak, gravitational field of our planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In this work, we  are interested in exactly this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In particular, while at  the geometric optics level the formalism of ray-optical struc-  tures can be used, we aim here at a description, analogous to  the one in [8], at the level of full Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Indeed,  such a description allows for the modelling of the propagation  of intense pulses in situations of physical interest, like soliton  propagation in optical fibers, taking into account the effect of  a weak gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  We show that light propagation in a medium in curved  spacetime is equivalent to propagation in an effective medium  in flat spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We then use this formalism to investigate the  propagation of intense light pulses in non-linear media, giving  rise to optical solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Solitons, and more in general propa-  gating pulses, in optical fibers are at the basis of several com-  munication protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Given that fibers on Earth are de facto  in a curved spacetime due to our planet’s gravitational field,  it is relevant to analyze how gravity influences light-pulses  propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Our result allows us to set up a framework for  the analysis of the effect of acceleration and curvature on the  propagation of pulses in optical fibers in curved spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  We numerically investigate some of these effects for the sim-  ple case of 1D propagation in the weak-field limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  AN EFFECTIVE “SPACETIME MEDIUM”  While light in media can propagate as in a curved space-  time, curved spacetime can also be seen as an effective  medium with non-trivial permeability and permittivity [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  It is not difficult to generalize the derivations in [8, 9] to the  case in which light propagates in an optical medium placed  in curved spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Also in this case it can be shown that  Maxwell’s equations are equivalent to Maxwell’s equations  in flat spacetime for an effective medium whose properties  encode both the ones of the physical medium and of curved  spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Indeed, consider a dielectric and permeable medium in  curved spacetime characterized by a Lorentzian metric gµν  with mostly plus signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We follow here the notation of [7],  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='04986v1  [gr-qc]  12 Jan 2023  2  also reported in the Supplemental Material [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Maxwell’s  equations in the absence of free charges and currents are given  by  ∇kF∗ ik = 0  (1)  ∇kGik = 0,  (2)  where F∗ is the Hodge dual of the electromagnetic tensor F,  and G and F are related by the constitutive equations of the  material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Choosing an observer field ui, the electric and mag-  netic field strengths can be defined with respect to it as  Ba = −1  2ηabcdubFcd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Ei = Fijuj  (3)  Ha = −1  2ηabcdubGcd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Di = Giju j  (4)  Fab = −ηcd  abudBc + 2u[aEb]  (5)  Gab = −ηcd  abudHc + 2u[aDb],  (6)  in the reference frame of the observer in which the medium  is assumed to be at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Here ηijkl = √−gδijkl is the Levi-  Civita tensor and T[abc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' ] denotes the antisymmetrization of  the tensor with respect to the indices in square brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  As discussed in [10], choosing ui = δi  0/ √−g00, the projec-  tion of Maxwell’s equations in 3-dimensional form leads to  δαβγ∂βHγ − ∂0Dα = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' ∂lDl = 0  (7)  δαβγ∂βEγ + ∂0Bα = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' ∂lBl = 0,  (8)  where Eα = √−g00Eα, Hα = √−g00Hα, and  Dα = − √−ggαβ  g00  Dβ − δαβγ g0γ  g00  Hβ  (9)  Bα = − √−ggαβ  g00  Bβ + δαβγ g0γ  g00  Eβ,  (10)  with Bα = √−g00Bα, and Dα = √−g00Dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' These expres-  sions are equivalent to Maxwell’s equations in flat space-  time in the presence of an optical medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  In particular,  for a non-dispersive medium characterized by constitutive re-  lations Da = εb  aEb, and Ba = µb  aHb, the effective medium  will be characterized by a dielectric and magnetic permeabil-  ity given by the product of the material ones and the ones  characterizing the curved spacetime [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Indeed, expressing  Dα = ˜εαβEβ+ ˜γβ  αHβ and correspondingly Bα = ˜µαβHβ− ˜γβ  αEβ,  where ˜γβ  α encode magnetoelectric effects, we see that  ˜µαβ = − √−ggαγ  g00  µ β  γ  (11)  ˜εαβ = − √−ggαγ  g00  ε β  γ ,  (12)  and ˜γαβ = −δαβγg0γ/g001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' As a direct consequence, whenever  the refractive index of the effective medium can be defined, it  1 Note that, in the case the material itself possesses magnetoelectric terms in  the constitutive equations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', Da = εb  aEb + γb  aHb, and Ba = µb  aHb − γb  aEb  then ˜γαβ = −δαβγ g0γ  g00 − √−g gαδ  g00 γβ  δ  will also be the product of the material refractive index times  the vacuum spacetime effective one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The same result can be  easily obtained at the level of geometric optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Finally, we make two observations relevant for the study of  the propagation of light pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Firstly, a non-magnetic mate-  rial in curved spacetime corresponds to a magnetic effective  medium in Minkowski due to the “magnetic permeability”  of the background spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Secondly, when considering  a non-linear material, we see that the non-linearity will also  be affected by the curvature of spacetime as well as the linear  polarizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  PULSE PROPAGATION: NON-LINEAR SCHR ¨ODINGER  EQUATION  We next consider the propagation of light pulses in a Kerr  non-linear, non-magnetic material in curved spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  In  particular, we focus on the case in which the material is in a  stationary orbit of Schwarzschild spacetime and use isotropic  coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' This situation well-captures the cases of interest  for optical communication and laboratory experiments like,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', optical fibers hanging still above Earth’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  In flat spacetime, the non-linear Schr¨odinger equation  (NLSE) is often used when considering the propagation of  light pulses whose amplitude is well-described by a scalar  envelope slowly varying with respect to the light period and  wavelength [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In the case of a medium stationary in  Schwarzschild’ spacetime, by employing the correspondence  with an effective medium in flat spacetime as described in the  previous section, the usual derivation of the NLSE can be car-  ried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' However, the effective medium will be inhomoge-  neous due to the curved spacetime contribution to the polar-  izability and permeability of the material medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' This gives  rise to extra terms in the NLSE which are of purely gravita-  tional origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Furthermore, another source of inhomogeneity  in the medium can be included when considering the effect of  tidal forces on the material that, through photoelasticity, ren-  der the refractive index position-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Neglecting for the moment photoelasticity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', consider-  ing a rigid dielectric, we can write Maxwell’s equation in flat  spacetime for the effective medium in the familiar notation,  using the fields and field strengths that we indicate with plain  capital letters from now on,  ∇ · B = 0, ∇ · D = 0  (13)  ∇ × E = −∂tB, ∇ × H = ∂tD,  (14)  where D  =  ˜εE and H  =  B/˜µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Here ˜µ  =  ˜µ(r) and  ˜ε = ˜ε(E, r, ω) in frequency space, allowing us to account  for the effect of material dispersion, are the permeability  and permittivity of the effective medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Expressing the  Schwarzschild’ spacetime metric in isotropic coordinates as  ds2 = − (B(t, r)/A(t, r))2 dt2 + A4(t, r)δαβdxαdxβ, with A(r) =  1 + rS /4r and B(r) = 1 − rS /4r, with rS the Schwarzschild  3  radius, we have  ˜ε(E, r, ω) = ε0εspε = ε0  A(r)3  B(r)  �  1 + χ(1)(ω) + 3χ(3) |E|2  Ω  �  ,  (15)  ˜µ = ˜µ(r) = µ0µsp = µ0A(r)3B(r)−1,  (16)  with Ω = A(r)−4 the conformal factor relating the spacial part  of the metric with the flat, Euclidean one2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The explicit ra-  dial dependence in the linear part of these effective quantities  comes from the curved spacetime optical properties encoded  in the diagonal terms √−ggαα/g00 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (11)) that we define  as εsp = µsp = A(r)3B(r)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The field dependency of ˜ε takes  into account the non-linearity of the physical medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Note  also that dispersion implies that the dielectric permeability is  a function of the physical frequency ω defined with respect to  our stationary observer uµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (13), and writing D = ˜εℓE + PNL, where ˜εℓ =  ε0εsp(1 + χ(1)(ω)) is the linear part of the dielectric permeabil-  ity in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (15) and PNL is the non-linear polarization, we can  then obtain the wave equation, in frequency space,  ∇2E − ∇(∇ · E) + ˜µ˜εℓν2E = −˜µν2PNL − (∇ log(µsp)) × (∇ × E) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (17)  Here we indicate with ν the conjugate variable to the coordi-  nate time t in the flat spacetime of the effective medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Note  that the homogeneous Maxwell equations imply that  ∇ · E = −(∇ log ˜εℓ) · E − 1  ˜εℓ  ∇ · PNL,  (18)  and thus  −∇(∇ · E) = (E · ∇)∇ log ˜εℓ + �(∇ log ˜εℓ) · ∇� E  (19)  + (∇ log ˜εℓ) × (∇ × E) + ∇  � 1  ˜εℓ  (∇ · PNL)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (18) makes evident that ∇ · E is of the same order as the  non-linearities and inhomogeneities in the electric permittiv-  ity, which is also why it is usually safely neglected in deriva-  tions of the NLSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  The wave equation in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (17) is equivalent to Maxwell  equations and, as such, presents the same level of complexity  if analytical or numerical solutions are attempted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The NLSE  is a scalar propagation equation for the electric field’s slowly  varying amplitude that allows one to numerically simulate the  pulse propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We thus want to write the electric field as  the product of a slowly varying amplitude times a phase prop-  agating along the propagation direction, that we will identify  with the z direction in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In this context, notice  that the dispersion relation of the physical medium, in its rest  2 This conformal factor arises due to the fact that EaEa in curved space-  time corresponds to |E|2/Ω with |E|2 = EaEbδab the flat spacetime norm  squared of the electric strength field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  frame, is given simply by n(ω) = cκ/ω, with κ the modulus of  the spatial projection of the wave 4-vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' For the effective  medium, this relation reads ˜n = c˜κ/ν, where ˜n = √εspµspn  is the product of the material refractive index and the “space-  time refractive index” nsp = √εspµsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Moreover, since ν is  the frequency defined with respect to Minkowski coordinate  time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', the conjugate Fourier variable to t, it is related to  the physical frequency, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', the one measured by a physical  observer in curved spacetime, by the gravitational redshift  ν = ω √−g00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' From the equivalence of the dispersion rela-  tions, we see that ˜κ(r) = κnsp(r)  �  −g00(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We will thus write  E(r, t) ∝ E(r)ei(˜κ0z−ν0t) + cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', with ˜κ0 = ˜κ(r, ν0) evaluated at a  central frequency ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  In order to proceed with the derivation of the NLSE, and to  further simplify our equations, we consider two separate sit-  uations of physical interest: (i) pulse propagation at approxi-  mately constant radius;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (ii) pulse propagating radially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Horizontal motion at (almost) constant radius  We assume the propagation direction of the light pulse to  be the z axis taken to be perpendicular to the radial direction  for horizontal motion, and consider linearly polarized light  propagating in a medium stationary on Earth for concreteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Then, for propagation distances much smaller than Earth’s ra-  dius (r⊕), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', z ≪ r⊕, the horizontal motion can be considered  as happening at constant radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' With these approximations,  the spacetime permeability and permittivity are constant func-  tions of r⊕, µsp = εsp = A(r⊕)3B(r⊕)−1 and also the physical  frequency is not changing with z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Thus, we see that in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (17)  the last term on the right-hand side vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  We follow the derivation in [4] where the pulse propaga-  tion in a single-mode optical fiber was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Indeed,  for µsp εsp constant, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (17) is formally equivalent to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S1)  of [4] in frequency space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We thus end up with an effective  one dimensional problem for the slowly varying envelope, and  the derivation of the NLSE is the textbook one [10, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In  particular, recall that the slowly varying envelope approxima-  tion(s) (SVEA) consists in neglecting terms ∂2  zE ≪ ˜κ0∂zE and  (˜κ1/˜κ0)∂t ≪ 1 on the basis that the envelope will contain many  wavelengths and optical cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' If we apply now the SVEA  we end up with, in the time domain,  i(∂z + ˜κ1∂t)E − ˜κ2  2 ∂2  t E = −n2ν0nsp(r⊕)ε0  |E|2  Ω E,  (20)  where ˜κi(ν0) are the coefficients of the power series expansion  ˜κ(ν) = �  n ˜κn(ν0)/n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (ν − ν0)n in ν − ν0 and we are considering  Kerr non-linear media for which the nonlinear index is n2 =  3χ(3)/(2n(ω0)cε0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Considering  an  anomalous  dispersive  material,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=',  κ2(ν0) < 0, an analytical solution of the NLSE can be found  (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', [4]) and reads  E(t, z) =  �  Ω|˜κ2|  ν0n2nspε0T 2  0  cosh  �t − ˜κ1z  T0  �−1  exp  ������  iz|˜κ2|  2T 2  0  ������ , (21)  4  z (m)  rS/r⊕ = 10−3  rS/r⊕ = 10−2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='9487 ×108  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='9488 ×108  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='9489 ×108  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='9490 ×108  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='9491 ×108  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='9492 ×108  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='9493 ×108  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='950× 108  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='952× 108  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='954× 108  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='956× 108  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='958× 108  (m/s)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' 1: Velocity of the soliton along the fiber, with respect to  an observer comoving with the segment of the dielectric  material where the (peak of the) soliton is located, for  L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='1 m, rs = 10−3r⊕, and including photoelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The  red, dashed and blue, solid curves represent the analytical  expression in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (25) including or in the absence,  respectively, of photoelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The red points and blue  squares are obtained by numerical simulations and agrees  perfectly with the analytical formula of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The inset  shows the case with photoelasticity in which rs = 10−2r⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  This shows a deviation from a purely linear relation between  the velocity and the propagation distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  where T0 is the pulse length, and 1/˜κ1 is its speed of prop-  agation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' This reduces to the result from Philbin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' [4] –  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S74) of the supplementary material in [4] – in the limit  of rS → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' From this expression, combined with the fact  that ˜κ1(ν0) = nspκ1(ω0), we can conclude that the velocity of  the horizontally propagating soliton in curved spacetime with  respect to an observer comoving with the segment of the di-  electric material3 is given simply by κ1(ω0)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Radial motion  Let us now consider the case in which the light pulse propa-  gates radially along the z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Care is in order here, since  now all the quantities appearing in the wave equation will  change along the propagation direction, including the physical  frequency that will be subject to gravitational redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Moti-  vated by the symmetry of the problem, and in order to obtain a  scalar, one-dimensional equation whose solution can be sim-  ulated, we assume that all the quantities entering the wave  equation depend solely on z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' This is tantamount to identifying  the radial direction with the z-axis and work close to x = y = 0  so that r = r⊕ + z, which is a reasonable assumption since we  are considering the vertical propagation of a well localized  pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' With this approximation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' the wave equations (17) re-  duce to a system of three decoupled equations [13]  ∂2  z Ex(y) + ˜µ˜εℓν2Ex(y) = − ˜µν2PNL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='x(y) + (∂z(ln ˜µ)) ∂zEx(y)  (22)  ∂2  z Ez + ˜µ˜εℓν2Ez = − ˜µν2PNL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='z − ∂z  � 1  ˜εℓ  ∂zPNL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='z  �  (23)  − 2(∂z ln ˜εℓ)∂zEz − Ez∂2  z ln ˜εℓ  It is immediate to realize that Ez = 0 is a solution of the cor-  responding equation so that we can consider the propagation  of linearly polarized light (in a direction orthogonal to z) and  we end up with a single equation of the form of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Proceeding as before with substituting the ansatz E(z, t) ∝  E(z, t)ei(˜κ0(z)z−ν0t) + cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', expanding ˜κ(z, ν) around ν0, and using  the SVEA approximation(s) we obtain the NLSE given by  i(∂z + ˜κ1∂t)E − ˜κ2  2 ∂2  t E + 2i∂z˜κ0  2˜κ0  E + 2iz∂z˜κ0  2˜κ0  ∂zE + iz∂2  z ˜κ0  2˜κ0  E − z∂z˜κ0E − z2 (∂z˜κ0)2  2˜κ0  E = −n2ν0nsp(r)ε0|E|2E/Ω + ∂z ln nsp  2˜κ0  (i˜κ0E + ∂zE + iz(∂z˜κ0)E) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (24)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (24) contains several additional terms with respect to the  equation for the horizontal propagation due to the fact that  now the wavevector ˜κ0 depends explicitly on the coordinate  along the propagation direction and so does the refractive  index, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', we are propagating in a gradient-index medium  (GRIN)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' All geometrical quantities appearing in the equation  3 Indeed note that proper length and proper time for an observer comoving  with the segment of the dielectric material and in connection with coor-  dinate quantities are given by ℓ = A2 z and τ = t B/A so that � ≡ ℓ/τ =  A3B−1z/t = nsp˜�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  4 See also [14–16] for early studies of soliton propagation in inhomogeneous  are evaluated at r⊕ + z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Finally, consistently with the horizon-  tal propagation case, upon setting ˜κ0 constant, we return to  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  INCLUDING PHOTOELASTICITY  Up until now, we have considered rigid dielectrics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', di-  electric media in which the speed of sound is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' For  media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  U5  realistic materials, this is of course never the case and the di-  electric gets deformed by the action of forces, including the  tidal ones in our set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Let us consider an optical fiber as a  paradigmatic example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In this case, the deformation due to the  action of gravity will be relevant only for the case of vertical  propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Deformations of a dielectric lead to a change in the relative  permeability of the material, and thus of the refractive index,  a phenomenon known as photoelasticity [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The contribu-  tions to this effect coming from the curvature of spacetime  and the inertial acceleration of the fiber can be separately ac-  counted for following the discussion in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Consider a fiber  of length L hanging from at support located at r⊕ + L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' As  far as the strain is within the elastic limit of the material, we  can relate it with the stresses through a linear relation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=',  Hooke’s law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Thus, we write the strain tensor as Skl = 1  Y σkl,  where Y is the Young’s modulus of the material and σkl = Fk  Al  is the stress tensor given by the ratio between the force Fk in  direction ˆek and the cross-sectional area Al normal to ˆel upon  which the force acts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The photoelastic (or acousto-optic) ef-  fect consists in the change of the relative electric permeability  by ∆(εr)−1  kl = Pkl mnSmn, where P is the photoelastic tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  In the following, we limit ourselves to the case of isotropic  materials and a diagonal stress tensor (see [10] for the de-  tails of the computation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' It should be noted that photoelastic-  ity is far from negligible in the case under investigation and  becomes the dominant effect in the vertical propagation sce-  nario, overwhelming the effect related to the optical properties  of the background spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  While photoelasticity introduces a further radial depen-  dence in the optical properties of the effective medium, this  does not affect the form of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (24), which remains valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The  only difference is in the expressions for the quantities ˜κi and  their derivatives, due to the fact that now the refractive index  of the medium is given by n(ω) =  �  1 + χ1(ω) + ∆εr(ω) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  NUMERICAL RESULTS  While the wave equation in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (17) gives us the full  Maxwell equations, including possibly interesting effects re-  lated to the vectorial nature of the electric field, and thus to the  interplay between gravity and the light polarization, its numer-  ical investigation is beyond the scope of the current work, and  it is left for future investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Here, we focus on the prop-  agation of light pulses as described by the simplified eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (24),  motivated by light propagation in optical fibers [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Note that  in the case of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (20) an analytical solution was presented in  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Equation (24) for the vertical propagation is solved numeri-  cally – being a non-linear PDE with coordinate dependent co-  efficients – using the split-step Fourier (SSF) method [11] and  taking into account also the effect of the fiber deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  For this purpose, we utilize the same fiber parameters as in [4]  (see also table I in [10]) and initialize the temporal profile at  0  20  40  60  80  100  0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='×10-14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='×10-13  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='5×10-13  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='×10-13  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='5×10-13  δτ (s)  × 10−13  z (m)  0  20  40  60  80  100  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='×10-21  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='×10-21  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='×10-21  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='×10-21  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='×10-21  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='×10-21  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' 2: Time of arrival of the soliton for the case of  propagation in the gravitational field of Earth for which we  assume rS = 9 × 10−3 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The main figure shows the  difference in time of arrival, with respect to an observer  comoving with the segment of the dielectric material where  the soliton is located, between vertically and horizontally  propagating solitons over the propagation coordinate length  z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The inset shows the same in the case photoelasticity is  neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  z = 0 as the one of the input pulse in the same reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  The intuition based on the SSF method– where the propa-  gation equation (24) is rewritten in the form ∂zE =  � ˆD + ˆN  �  E  with the diffusive dynamics enclosed in the operator ˆD =  ˆD(z, ∂t) [10] – allows us to formulate the educated guess that  the propagation speed of the soliton, in the effective flat space-  time, is given by  ˜� = 1 + z ˜κ′  0(z)/˜κ0(z)  ˜κ1(z)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (25)  Indeed, this appears as (the real part of) the inverse of the  coefficient of the time derivative in ˆD(z, ∂t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Then, in order to  translate this result into the speed measured by an observer co-  moving with the segment of the dielectric material where the  soliton peak is located, we need to just multiply eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (25) by the  spacetime refractive index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' That this intuition is indeed cor-  rect is verified by the numerical simulations reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  We see that the z-dependence of the propagation velocity is  strongly enhanced by the effects of mechanical deformation  of the fiber with respect to the case in which photoelasticity is  ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The z-dependence of the vertical propagation veloc-  ity without photoelasticity is weak, and the velocity is close to  the one of the horizontal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' To quantify the latter statement,  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' 2 we show the difference in the (proper) time of arrival  of the soliton for the case of propagation in the gravitational  field of Earth, corresponding to a Schwarzschild radius that  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='5 × 10-13  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' × 10-13  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='5× 10-13  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' × 10-13  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' × 10-14  0  0  20  40  60  80  1006  we take as rS = 9 × 10−3 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The main figure shows  δτ = |z(  �  −g00(r⊕ + z)˜�−1  ↑ ) −  �  −g00(r⊕)˜�−1  → )|,  (26)  with ˜�↑ and ˜�→ the propagation velocities, in the effective flat  spacetime, for vertical and horizontal propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The inset  shows instead the case in which for the vertical propagation  the photoelasticity is neglected, showing a much weaker de-  pendence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Finally, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' 3 we show the deviation of the average ve-  locity along the vertical direction �a�(rS ) with respect to the  constant velocity at rS = 0 as a function of the dimension-  less ratio rS /r⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The average velocity is obtained numeri-  cally from the simulations as the ratio of the total length L  and the propagation time of the soliton and transformed into  the frame of the observer comoving with the fiber at its upper  end-point – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', multiplied by nsp(r⊕ + L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Analytically, we  use �a� = (  � L  0 � dz)/L with � = nsp˜� and ˜� given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' 3 shows once again the agreement between the simulated  data and our analytical ansatz and it also shows that the pho-  toelasticity is the main effect that allows one to have a sizable  difference between the flat and curved spacetime propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Another quantity characterizing the propagating pulse is its  temporal width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In the horizontal propagation case, the du-  ration of the pulse is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The same is not, in general,  true when considering the vertical propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In the Supple-  mental Material [10], we report the evolution of the temporal  width along the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In particular, our simulations show a  focusing of the pulse which is however sizable only in the  presence of photoelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  CONCLUSIONS  We have considered the propagation of light pulses in non-  linear, non-magnetic media stationary in curved spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Taking some intuition from the seminal work of Plebanski [8],  we showed that light propagation in such media can be equiv-  alently described as the propagation in an effective medium  in flat spacetime whose electric and magnetic properties ac-  quire a multiplicative factor encoding the spacetime structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Having done that, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (17) describes the propagation of light  in the effective medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' It is interesting to note, even though  we did not investigate it in this work, that the vectorial na-  ture of this equation encodes the interplay between the light  polarization and the gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Such interplay should  be expected on the basis of the fact that the effective medium  is an inhomogeneous, gradient-index medium for which it is  well known that the propagation of light is influenced by its  own polarization [19–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Furthermore, the effect of polariza-  tion on the propagation of light in curved, vacuum spacetime  has been extensively considered in the literature and shown to  take place also for static spacetimes [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Neglecting the aforementioned effects, which would be un-  doubtedly small, by virtue of approximations we have been  able to derive a scalar NLSE describing the propagation of  rS/r⊕  L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='1 m  L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='1 m, P1122 = 0  L = 1 m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0020  (rS)/  (rS = 0) − 1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' 3: Change in average velocity (�a�) of the soliton in the  fiber – with respect to the observer comoving with the  dielectric – compared to the case with rS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Orange, square  points corresponds to the case of a L = 1 m propagation with  photoelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Blue, round points correspond to the case of  a L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='1 m propagation with photoelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Green,  diamonds correspond to the case of a L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='1 m propagation  without photoelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The lines correspond to the  analytical result that fits perfectly the different sets of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  a light pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' It is important to notice that, when solving the  NLSE employing the SSF method, we are implicitly consider-  ing a unidirectional equation and ignoring any possible back-  propagating field in the boundary conditions imposed, for all  times, at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' This means that backscattered light from the  pulse is assumed negligible relative to the pulse itself, a condi-  tion common to all unidirectional envelope propagation equa-  tions [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' While this is not a problem for the horizontal prop-  agation, in which case only the weak non-linearity could give  rise to back-reflection, in the case of the vertical propagation  light is effectively propagating in a gradient-index medium  with the refracting index slowly varying in the propagation  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  This by itself can give rise to back-propagating  fields, and effectively limits the validity of our treatment to  regimes in which the photoelasticity allows to employ a uni-  directional equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Luckily, the regime of validity of the  equation – which depends on the parameter chosen for the  physical medium – can be readily estimated by following the  discussion in [25] as we detail in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Given these caveats, the NLSE that we have derived shows  that an optical pulse propagating radially in a Kerr non-liner  medium stationary in Schwarzschild spacetime experiences a  change in its propagation velocity captured by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' This  effect is mostly due to photoelasticity which overwhelms the  purely spatiotemporal effects encoded in nsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The difference  in propagation velocity between the vertically and horizon-  UVav7  tally propagating pulses results, in turn, in a difference of the  time of arrival of two pulses of the order of hundreds of fem-  toseconds in Earth gravitational field, a fact that puts this dif-  ference in the reach of current technologies (see [26–28] and  references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  The authors thank Francesco Marino for interesting dis-  cussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Belenchia and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Braun acknowledge support  from the Deutsche Forschungsgemeinschaft (DFG, German  Research Foundation) project number BR 5221/4-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' R¨atzel  acknowledges funding by the Federal Ministry of Education  and Research of Germany in the project “Open6GHub” (grant  number:  16KISK016) and support through the Deutsche  Forschungsgemeinschaft (DFG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
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+page_content='  School of Mathematics and Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Queen’s University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Belfast BT7 1NN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' United Kingdom  3Humboldt Universit¨at zu Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Institut f¨ur Physik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Newtonstraße 15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' 12489 Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Germany  In this supplemental material,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' we collect the detailed derivations of the results in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  VACUUM SPACETIME AS AN OPTICAL MEDIUM & THE EFFECTIVE MEDIUM DESCRIPTION  Thanks to the seminal work of Plebanski in the ’60s [8], it is well known that electromagnetism in curved spacetime is  equivalent to propagation in an optical medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Following the derivation presented in [9], Maxwell vacuum equations in curved  spacetime are written as  ∇kF∗ ik = 0  (S1)  ∇kFik = 0,  (S2)  where F∗ is the Hodge dual of the e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' tensor, Latin indices run from 0 to 3, and the metric gi j has mostly plus signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' As in  the main text, we consider the case with no currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Choosing an observer field ui, the electric and magnetic field strength can be defined with respect to it as  Hi = F∗ i juj, Ei = Fi juj  (S3)  Fi j = ηi jklulHk + 2u[iE j],  (S4)  where here ηijkl = √−gδijkl is the Levi-Civita tensor (with δi jkl the Levi-Civita alternating symbol in four dimensions) and T[abc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' ]  denotes the antisymmetrization of the tensor with respect to the indices in square brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The Maxwell equations can then be  projected in the ui direction or orthogonal to it using the projection operator into the rest frame of ui, hi j = gi j + uiuj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The end  result is, in the case the observer field is chosen as ui = δi  0/ √−g00  δαβγ∂βHγ − ∂0Dα = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' ∂lDl = 0  (S5)  δαβγ∂βEγ + ∂0Bα = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' ∂lBl = 0,  (S6)  where the first two equations come from Maxwell equations (S2) (with δαβγ the Levi-Civita alternating symbol in three dimen-  sions) while the second two from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Here, Hα = √−g00Hα, Eα = √−g00Eα, Greeks indices run from 1 to 3, and  Dα = − √−ggαβ  g00  Eβ − δαβγ g0γ  g00  Hβ  (S7)  Bα = − √−ggαβ  g00  Hβ + δαβγ g0γ  g00  Eβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S8)  From here one can see that these equations are actually equivalent to Maxwell equations in flat spacetime in the presence of an  optical medium whose constitutive relations are characterized by a dielectric (εαβ  sp ) and magnetic permeability (µαβ  sp ) given by  µαβ  sp = εαβ  sp = − √−ggαβ  g00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S9)  As shown in the main text, when a physical optical medium whose rest frame is characterized by ui is added, we can follow  the same derivation starting from Maxwell’s equations in curved spacetime and with a material medium  ∇kF∗ i j = 0  (S10)  ∇kGik = 0,  (S11)  2  where now G and F are related by the material constitutive equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Then we have  Ba = −1  2ηabcdubFcd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Ei = Fi juj  (S12)  Ha = −1  2ηabcdubGcd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Di = Gi ju j  (S13)  Fab = −ηcd  abudBc + 2u[aEb]  (S14)  Gab = −ηcd  abudHc + 2u[aDb],  (S15)  where we have introduced the electric and magnetic excitation, Da and Ha respectively, on top of the electric and magnetic  strength Ea and Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Note that the definitions of E, B, F are equivalent to the vacuum case, since the homogeneous Maxwell equations are the  same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The inhomogeneous equations have also the same form as in the vacuum case, but with the substitution of E, B, F with  D, H,G, where the definition of G with respect to H, D is the same as F with respect to E, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' From this simple observation we  can immediately deduce that the projection of Maxwell equations in 3-dimensional form will, in the case the observer field is  chosen as ui = δi  0/ √−g00, lead to  δαβγ∂βHγ − ∂0Dα = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' ∂lDl = 0  (S16)  δαβγ∂βEγ + ∂0Bα = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' ∂lBl = 0,  (S17)  where Eα = √−g00Eα, Hα = √−g00Hα, and  Dα = − √−ggαβ  g00  Dβ − δαβγ g0γ  g00  Hβ  (S18)  Bα = − √−ggαβ  g00  Bβ + δαβγ g0γ  g00  Eβ,  (S19)  with Bα = √−g00Bα, and Dα = √−g00Dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Once again, these equations are equivalent to Maxwell’s equations in flat spacetime  in the presence of an effective optical medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Consider the case of a linear, dispersionless medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We can then write Gi j =  1  2χi j klFkl, with the material’s constitutive  tensor χij kl, containing all material properties, which is symmetric under the exchange of the first and second pair of indices  and antisymmetric with respect to the swap within an index pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In particular, we can also write Da = εb  aEb, and Ba = µb  aHb,  which are the constitutive relations in the reference frame of the observer in which the medium is at rest, neglecting magneto-  electric effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' For an isotropic medium, we also have that the dielectric and permeability tensor assume the simplified form  εb  a = ε(δb  a + UbUa) and µb  a = µ(δb  a + UbUa) for some scalar, positive functions ε and µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The effective optical medium is such that  its constitutive relations are then characterized by a dielectric and inverse magnetic permeability given by  ˜εαβ = − √−ggαγ  g00  ε β  γ ,  (S20)  ˜µαβ = − √−ggαγ  g00  µ β  γ ,  (S21)  while the antisymmetric parts of the constitutive tensor are completely characterized by the vacuum spacetime properties5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Non-linear media, with a Kerr-type non-linearity, can be treated analogously by promoting the dielectric and permeability  tensors to explicitly depend on the field strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' If also dispersion needs to be included in the game, we need to consider,  as usual, the dispersion relation in frequency space in order to write it in a local form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Note that we can always write D =  ε0E + P (and analogously for the magnetic field and excitation), moving all non-linearity and dispersion in the polarization  (magnetization) vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Thus, from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (9) we can conclude that the effective medium will give rise to an effective electric  excitation  De f f = εsp(ε0E + P),  (S22)  which can then be written, for the dispersive case of interest, locally in frequency space for the effective medium “living” in flat,  Minkowski spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  5 More in general, one could also include in this description materials for  which the magnetoelectric entries of the constitutive tensor are not negli-  gible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In such a case, Da = εb  aEb + γb  aHb, and Ba = µb  aHb − γb  aEb with  γab the antisymmetric part of the constitutive tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In this case, the same  derivation still stands, with the only difference that the antisymmetric parts  of the constitutive tensor for the effective medium are given by  ˜γαβ = −δαβγ g0γ  g00  − √−ggαδ  g00  γβ  δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  3  DERIVATION OF THE NLSE: TECHNICAL DETAILS  Let us consider now Maxwell’s equations for the effective medium, thus in flat spacetime, written in the usual notation  ∇ · B = 0, ∇ · D = 0  (S23)  ∇ × E = −∂tB, ∇ × H = ∂tD,  (S24)  with D = ˜εE and H = B/˜µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Here, we consider the case of a spherically symmetric spacetime in isotropic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The  metric can then be written, in full generality, as  ds2 = −  � B(t, r)  A(t, r)  �2  dt2 + a2(t)A4(t, r)δαβdxαdxβ,  (S25)  with r =  �  δαβxαxβ, A(t, r), B(t, r) real functions, and a(t) a scale factor analogous to the one appearing in FRLW spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Note that this metric can be rewritten as  ds2 = Ω−1  �  −  B2(t, r)  a2(t)A6(t, r)dt2 + δαβdxαdxβ  �  ,  (S26)  where the “conformal factor” Ω = a(t)−2A(t, r)−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  In particular, we specialize to the case for which a(t) = 1, A = A(r), B = B(r) and such that,  in frequency space,  ˜ε(E, r, ω) = ε0A(r)3B(r)−1 �  1 + χ(1)(ω) + 3χ(3)|E|2/Ω  �  and ˜µ = ˜µ(r) = µ0µsp = µ0A(r)3B(r)−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', we are considering a non-  magnetic material, where all the magnetic properties are induced by the curved background, with a Kerr non-linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' As we  previously discussed, ω is the physical frequency defined with respect to the stationary observer uµ that we assume to be the rest  frame of the physical medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The conformal factor Ω appearing in the non-linear term in ˜ε arises due to the fact that EaEa in  curved spacetime corresponds to |E|2/Ω, with |E|2 = EaEbδab the flat spacetime norm squared of the electric strength, in the flat  spacetime of the effective medium, as can be easily seen directly from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  For the sake of notation clarity, let us emphasized that, in the following, tilded quantities refer to quantities pertaining to  the effective medium in flat spacetime while the untilded ones represent the optical properties of the physical medium that is  stationary in (physical) curved spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  In the following, we focus on Schwarzschild’s spacetime, for which  A(r) = 1 + rS  4r  (S27)  B(r) = 1 − rS  4r,  (S28)  where rS is the Schwarzschild’s radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  From Maxwell’s equations, taking the curl of the third one, we obtain  ∇2E − ∇ (∇ · E) = ∂t  �  ˜µ∂tD − B × ∇˜µ  ˜µ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S29)  and thus  ∇2E − ∇(∇ · E) − ˜µ∂2  t D = −(∇ log(µsp)) × (∇ × E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S30)  Note that this last expression is valid for ∂t ˜µ = 0, which includes the case of Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' For a generic spherically  symmetric metric, as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S26), additional terms would be present due to the explicit time dependence of ˜µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Moving now to  frequency space, where we indicate with ν the conjugate variable to the coordinate time t in the flat spacetime of the effective  medium, and writing D = ˜εℓE + PNL, where ˜εℓ is the linear part of the dielectric permeability and PNL contains the nonlinear  components of the polarization, we obtain  ∇2E − ∇(∇ · E) + ˜µ˜εℓν2E = −˜µν2PNL − (∇ log(µsp)) × (∇ × E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S31)  This is our starting point for the derivation of the scalar NLSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Note that, apart from the last term, the equation resembles the  textbook wave equation modulo the inhomogeneity of the medium encoded in the coordinate dependence of ˜ε, ˜µ [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Before starting the derivation of the NLSE, an observation is in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In curved spacetime, the linear dispersion relation of  the medium assumes the simple form, in the rest frame of the medium,  n(ω) = c  �  µ0ε0εr(ω) = c κ  ω,  (S32)  4  with κ the modulus of the spatial projection of the wave 4-vector , εr = 1 + χ(1)(ω), and ω the physical frequency6, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', the  frequency measured by an observer in curved spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Thus, we write the dispersion relation for our effective medium as  ˜n = c ˜κ  ν,  (S33)  where ˜n = nspn with n2  sp = (Ω|g00|)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S33) is the expression that we will use in deriving the NLSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Note once again that  here ν = ω √−g00 where ν is the conjugate Fourier variable to the coordinate time t in flat spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Since for consistency we  want the two dispersion relations to be equivalent, we see that ˜κ = κnsp  √−g00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Once again, in the dispersive case, we will need  to consider ˜n = ˜n(ω) since otherwise the two dispersion relations would not remain equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Derivation of the standard NLSE  Before delving into the derivation of the NLSE for our effective, inhomogeneous medium, we summarize here the derivation  of the NLSE in the standard case, following [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  In the standard case of a homogeneous, non-magnetic material in flat spacetime, writing the displacement electric field D as  the sum of a linear part and the non-linear polarization, we have the wave equation in frequency space  ∇2E − ∇(∇ · E) + µ0εℓν2E = −µ0ν2PNL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S34)  Note that in this section we always work with untilded quantities that refer to the optical properties of the physical medium  that is considered in flat spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Indeed, in this case the effective medium coincides with the physical one since the optical  properties of flat spacetime are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Note however that, as previously specified, from the next section we will go back to  consider the case of curved spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Thus, we will need to distinguish once again between physical and effective medium,  with the latter represented by tilded quantities in flat spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  We recall that µ0 = 1/(ε0c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We then neglect the vectorial operator −∇(∇ · E) due to the fact that the homogeneous Maxwell  equation for D implies this term to be in general negligible – and get  ∇2E(ν) + εr(ν)ν2  c2 E(ν) = − ν2  ε0c2 PNL(ν),  (S35)  with εr(ν) = εℓ/ε0 the linear, relative polarizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  For a linearly polarized field, this equation becomes a scalar one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We can then write the electric field as a slowly varying,  complex amplitude E(r, t) times a plane wave propagating in the z direction with central frequency ν0  E(r, t) = E(r, t)ei(κ0z−ν0t) + cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', where κ0 = n(ν0)ν0  c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S36)  Using the Fourier transform w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' t for E and the one for the amplitude7 E, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S36) can be rewritten in frequency space as a  sum of terms dependent on ν ± ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We can then discard the fast rotating, high frequency (ν + ν0) components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Indeed, the slowly  varying in time envelope E(r, t) in which we are interested does not possess high-frequency Fourier components [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We thus  obtain  E(r, ν) ≈ E(r, ν − ν0)eiκ0z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S37)  The scalar wave equation for the amplitude then becomes  ∇2  ⊥E + ∂2  zE + 2iκ0∂zE + [κ2(ν) − κ2  0]E = − ν2  ε0c2 PNLe−iκ0z,  (S38)  6 This is connected to the frequency in flat spacetime via ω = ( √−g00)−1ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  7 We follow [12] in defining,  E(r, t) =  � ∞  −∞  dν  2π E(r, ν)e−iνt = E(r, t)ei(κ0z−ν0t) + E∗(r, t)e−i(κ0z−ν0t)  =  � ∞  −∞  dν  2π E(r, ν)e−i(ν+ν0)teiκ0z +  � ∞  −∞  dν  2π E∗(r, ν)e−i(ν−ν0)te−iκ0z  =  � ∞  −∞  dν  2π E(r, ν − ν0)e−iνteiκ0z +  � ∞  −∞  dν  2π E∗(r, ν + ν0)e−iνte−iκ0z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  From these expressions we then obtain  E(r, ν) = E(r, ν − ν0)eiκ0z + E∗(r, ν + ν0)e−iκ0z  5  with κ(ν) = n(ν)ν/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' At this point, we approximate κ(ν) as a power series in ν − ν0  κ(ν) = κ0 + κ1(ν − ν0) + D,  (S39)  with D = κ2(ν − ν0)2/2 + O  �  (ν − ν0)3�  , such that  κ(ν)2 = κ2  0 + 2κ0κ1(ν − ν0) + 2κ0D + 2κ1D(ν − ν0) + κ2  1(ν − ν0)2 + D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S40)  Here κ1 is the inverse of the group velocity �g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We will neglect D2 terms and convert back to the time domain8 to obtain  �  ∇2  ⊥ + ∂2  z + 2iκ0(∂z + κ1∂t) + 2iκ1 ¯D∂t + 2κ0 ¯D − κ2  1∂2  t  �  E(r, t) =  1  ε0c2 ∂2  t (PNL(r, t)) e−i(κ0z−ν0t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S41)  Note that now ¯D is a differential operator with ¯D = −(κ2/2)∂2  t + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='. Finally, by writing also the polarization PNL(r, t) =  p(r, t)eiκ0z−ν0t + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', as a slowly-varying amplitude p(r, t) times a plane wave eiκ0z−ν0t propagating in the z direction, one can  see that the right-hand side (RHS) becomes9  1  ε0c2 ∂2  t PNL(r, t)e−i(κ0z−ν0t) = − ν2  0  ε0c2  �  1 + i  ν0  ∂t  �2  p(r, t)+c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='.  (S42)  This is the starting point for implementing the slowly varying envelope approximation (SVEA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' It usually involves moving  to the frame moving with the pulse group velocity κ−1  1 , and then neglecting terms with second derivatives in the propagation  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Let us sketch the procedure here:  The retarded frame is defined as z′ = z and τ = t − z/�g = t − κ1z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Thus, ∂z = ∂z′ − κ1∂τ, and ∂t = ∂τ =⇒ ∂2  z = ∂2  z′ − 2κ1∂z′∂τ + κ2  1∂2  τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  The wave equation thus becomes  �  ∇2  ⊥ + ∂2  z′E − 2κ1∂z′∂τ + 2iκ0∂z′ + 2iκ1 ¯D∂τ + 2κ0 ¯D  �  E = − ν2  0  ε0c2  �  1 + i  ν0  ∂τ  �2  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S43)  Now the SVEA in space is valid when the pulse is longer than just a few wavelengths so that ∂2  z′E ≪ κ0∂z′E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' With this  approximation  �  ∇2  ⊥ − 2κ1∂z′∂τ + 2iκ0∂z′ + 2iκ1 ¯D∂τ + 2κ0 ¯D  �  E = − ν2  0  ε0c2  �  1 + i  ν0  ∂τ  �2  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S44)  Moreover, one can also implement a SVEA in time since10 κ1/κ0 = (�ph/�g)ν−1  0  ≈ ν−1  0 where �ph and �g are the phase and  group velocities respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' When the pulse length Tpulse is long enough to contain more than just a few optical cycles,  with Toptical = 2π/ν0, within the envelope, then (κ1/κ0)∂τ ≈ Toptical/Tpulse ≪ 1 so that  �  ∇2  ⊥ + 2iκ0∂z′ + 2κ0 ¯D  �  E(r, t) = − ν2  0  ε0c2 p(r, t),  (S45)  where the time derivative of the slowly varying polarization envelope has been ignored, compared to the constant term, on  the same basis that Toptical/Tpulse ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' This approximation of the polarization term on the right hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S44) is  equivalent to neglecting the self-steepening effect [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  8 This is achieved by multiplying the equation by e−i(ν−ν0)t and integrating  over all values of ν − ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Recall that PNL(r, t) =  �  PNL(r, ν)e−iνtdν/2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  9 Here we can write  ∂2  t P(r, t)e−i(κ0z−ν0t) = ∂2  t  �  p(r, t)ei(κ0z−ν0t)�  e−i(κ0z−ν0t)  =  �  −ν2  0 − 2iν0∂t + ∂2  t  �  p(r, t)  10 This is not true, for example, in slow light materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  6  Setting up some important relations  In the case of the inhomogeneous effective medium, we need to investigate some relation between the effective medium  quantities and the one of the physical material before delving into the derivation of the NLSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We have seen that the dielectric  permeability and the magnetic one can be written in frequency space as  ˜ε(E, r, ω) = ε0εsp(r)  �  1 + χ(1)(ω) + 3χ(3)|E|2/Ω  �  (S46)  ˜µ(r, ω) = µsp(r)µ0  (S47)  where χ(1)(ω) is the material linear dielectric permeability, including the effect of dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Note also that εsp(r) = µsp(r) in  isotropic coordinates (that we are working with), so that nsp(r) ≡ √εspµsp = εsp(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Thus, we have  ˜n(ω, r) = nsp(r)n(ω) = cεsp(r)  �  µ0ε0(1 + χ(1)(ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S48)  In the wave equation eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S31), we have the term −˜µ˜εℓν2E with ˜εℓ = ε0εsp(1 + χ1(ω)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In light of the previous considerations,  this term can be written as  − ˜µ˜εℓν2E = −(˜n2/c2)ν2E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S49)  When we move to the frequency space for the effective medium, we use the conjugate variable (ν) to Minkowski time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' As we  already noticed, this is related to the frequency measured by an observer at rest with respect to the medium in curved spacetime  by ν = ω √−g00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The effective dispersion relation is thus  ˜n2ν2 = c2˜κ2,  (S50)  as previously discussed (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S33)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In expanding in power series ˜κ around ν0 we will then have  ˜κ = ˜κ0 + ˜κ1(ν − ν0) + ˜D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S51)  By comparing the dispersion relation in curved spacetime and the one of the effective medium it is easy to see that  ˜κ0 = √−g00nspκ0  (S52)  ˜κ1 = nspκ1  (S53)  ˜κ2 = (nsp/ √−g00)κ2  ˜D = 1  2 ˜κ2(ν − ν0)2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' = nsp  √−g00D,  (S54)  where the κi(ω0) appearing in these expressions are the analogues of their tilded versions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=',  κ0 = κ|ω0  (S55)  κ1 = ∂ωκ|ω0  (S56)  κ2 = ∂2  ωκ|ω0,  (S57)  and refer to the tabulated optical properties of the physical medium we are considering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  The expression in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S53) implies that the group velocity in the effective medium is related to the physical one in curved  spacetime by  ˜�g = �g/nsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S58)  Note that this is consistent with the way the phase-velocity in the effective medium is related to the one in curved spacetime via  ˜�ph ≡ 1/˜n = 1/(nspn) = �ph/nsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' More in general, this is consistent with the relation between the coordinate velocity ˜� = dx/dt,  characterizing the propagation in the effective medium in flat spacetime, and the velocity with respect to an observer comoving  with the dielectric � = dχ/dτ where τ = √−g00t is the proper time with respect to the stationary observer and χi = xi/  √  Ω  represents the proper length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Indeed, we see immediately that � = dχ/dτ = ˜�nsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  7  Derivation of the NLSE for the effective medium  First let us notice that, in order for the effective medium description to be equivalent to the physical one in curved spacetime,  we need to require that:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' the dielectric permeability and magnetization are dependent on the radial coordinate with the expressions given in the  previous section  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' dispersion enters via the physical frequency ω = ν/ √−g00 which corresponds to a position dependent correction to the  Fourier variable ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Note that the rest of the relations in the previous section are not necessary in the derivation of the NLSE, but they are nonetheless  important for connecting the effective medium properties with the ones of the physical medium in curved spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  In order to derive the NLSE in this case, we go back to the wave equation in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S31) that we report here for convenience  ∇2E − ∇(∇ · E) + ˜µ˜εℓν2E = −˜µν2PNL − (∇ log(µsp)) × (∇ × E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S59)  To proceed further, as discussed in the main text, we can make use of the homogeneous Maxwell equation for D in order to  write  ∇ · D = 0 =⇒ ∇ · E = −(∇ log ˜εl) · E − 1  ˜εl  ∇ · PNL  −∇(∇ · E) = (E · ∇)∇ log ˜εl + �(∇ log ˜εl) · ∇� E + (∇ log ˜εl) × (∇ × E) + ∇  � 1  ˜εl  (∇ · PNL)  �  ,  where we have used that E × �∇ × ∇ log ˜εl  � = 0 since the curl of the gradient vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We obtain  ∇2E + ˜µ˜εℓν2E = −˜µν2PNL − ∇  � 1  ˜εl  (∇ · PNL)  �  − (E · ∇)∇ log εsp −  �  (∇ log εsp) · ∇  �  E −  �  ∇ log µsp + ∇ log εsp  �  × (∇ × E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S60)  As discussed in the main text, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S60) does not allow, in general, to write down a scalar propagation equation since even by  starting from a linearly polarized electric field we end up having coupled equations between all the components of the electric  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' This is in general also true whenever one does not ignore the vectorial term ∇(∇ · E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  In order to bypass these problems, we resort to considering two cases of interest, which are the ones analyzed in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  See also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Before doing so, let us emphasize that we will be interested in the specific case of a Kerr non-linear medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Thus, we write the (slow envelope of the) non-linear polarization of the effective medium as  p(r, t) = 3ε0nsp(r)χ(3)|E|2E/Ω,  (S61)  which includes the non-linearity of the material and the contribution coming from the curved spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Using the expression  for ˜κ0 in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S52), the term containing the polarization can be written as  − nsp(r)ν2  0  2˜κ0ε0c2 p(r, t) = −n2ν0nsp(r)ε0|E|2E/Ω,  (S62)  where n2 = 3χ(3)/2n(ω0)cε0 is the nonlinear index of the Kerr material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' As before, we are also going to neglect the self-  steepening effect [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Furthermore, in our simulations we use the parameters of a single-mode, fused silica optical fiber  employed in [4] that we summarize here in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Horizontal propagation  As we have seen in the main text, considering linearly polarized light propagating – in a medium stationary on Earth – for  distances much smaller than Earth’s radius, the horizontal motion can be considered as happening at constant radius r ≥ r⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We  can then follow the derivation in [4] where the pulse propagation in a single-mode optical fiber was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  In a nutshell, whenever the coefficients in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S59) are constant, so that the very last term vanishes since ∇ log(µsp) = 0, we  find an equation  ∇2E − ∇(∇ · E) + ˜µ˜εℓν2E = −˜µν2PNL,  (S63)  8  Symbol  Name  Value  Soliton pulse properties from [4]:  T0  Duration (this corresponds to 70 fs total pulse length)  40 fs  Es  Generating pulse energy (not used, only for reference)  5 pJ  λ0 = 2πc/ν0  central soliton wavelength  803 nm  Fiber properties:  κ0(ν0)  n(ν0)ν0/c, assuming n(ν0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='17 · 107 /m  κ1(ν0)  1/�g(ν0), assuming �g(ν0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='65c  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='13176 · 10−9 s/m  κ2(ν0)  Group velocity dispersion from [4]  −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='5 · 10−27 s2/m  n2  Kerr non-linearity of silica from [29]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='19 · 10−20 m2/W  Aeff  Effective transverse mode area  π (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='6 µm/2)2  Properties of fused silica:  P11 22  Component for transverse stress of the photoelastic tensor from [30, 31] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='271  cs  Speed of sound tabulated in [32]  5720 m/s  Miscellaneous:  r⊕  Earth equatorial radius  6378137 m  rS (Earth)  Schwarzschild radius of Earth  9 · 10−3 m  TABLE I: Specifics of all the parameters entering the numerical simulations of the NLSE(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The material parameters are  extrapolated from Philbin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' [4], κ0/ν0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='5/c and κ−1  1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='65c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Consistently, we use the properties of the Crystal Fibre  NL-PM 750 from NKT photonics [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  which is equivalent to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S1) of [4] in frequency space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Following [4], and considering the propagation of light pulses in an optical fiber, this equation can be solved by separation  of variables between an amplitude that depends on the propagation direction and a vectorial part depending on the transverse  directions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', E(ν, r) = E(ν, z)U(ν, x, y) in frequency space and with U(ν, x, y) a 3-dimensional vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' By solving the eigenvalue  problem for the transverse part, we then remain with a one-dimensional problem given by  ∂2  z E(z) + ˜n2  c2 ν2E(z) = −˜µν2PNL(z)  (S64)  where the refractive index is set by the eigenvalue of the transverse fiber mode and accounts for the property of the fiber’s core  and of the transverse profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In our case, we can then assume to start directly from this equation, where the property of the  effective medium accounts also for the non-trivial spacetime background via nsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  At this point, the derivation of the NLSE proceeds as in the standard case discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We introduce the field scalar  amplitude via E(z, ν) ≈ E(z, ν − ν0)ei˜κ0z in our equation to obtain  ∂2  zE + 2i˜κ0∂zE + [˜κ2(ν) − ˜κ2  0]E = −ν2 ˜µPNLe−i˜κ0z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S65)  We then proceed as before by expanding  ˜κ = ˜κ0 + ˜κ1(ν − ν0) + ˜D  (S66)  to get, neglecting ˜D2 terms and converting back to the time domain,  �  ∂2  z + 2i˜κ0(∂z + ˜κ1∂t) + 2i˜κ1 ¯˜D∂t + 2˜κ0 ¯˜D − ˜κ2  1∂2  t  �  E(z, t) = ˜µ∂2  t (PNL(z, t)) e−i(˜κ0z−ν0t),  (S67)  where ¯˜D = −(˜κ2/2)∂2  t + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' in complete analogy with the standard derivation outlined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  At this point, by neglecting the second derivatives in z as well as terms (˜κ1/˜κ0)∂t and using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S62) we arrive at eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (20) of  the main text, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=',  i(∂z + ˜κ1∂t)E − ˜κ2  2 ∂2  t E = −n2ν0nsp(r⊕)ε0  |E|2  Ω E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S68)  9  y  x  z  (i)  g  z  x  y  (ii)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' S1: Geometry of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The two cases considered are labelled by (i) and (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In (i), the light pulse propagates in a  horizontal fiber positioned at r⊕ = r ∼ constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In (ii), the light pulse propagates in a vertically positioned fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Radial motion  As we already discussed, in the case of vertical motion, in which we identify the radial direction with the propagation direction  along z with r = r⊕ + z, the effective medium becomes a gradient-index medium with the refractive index changing along the  propagation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We assume that all the quantities entering the wave equation depend solely on z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Upon considering  linearly polarized light along a direction orthogonal to z, we end up with the system of three decoupled equations in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (22) of  the main text that we report here for completeness  ∂2  z Ex(y) + ˜µ˜εℓν2Ex(y) = −˜µν2PNL,x(y) − (∂z ln ˜εℓ)∂zEx(y) + (∂z(ln ˜εℓ + ln ˜µ)) ∂zEx(y)  (S69)  ∂2  z Ez + ˜µ˜εℓν2Ez = −˜µν2PNL,z − ∂z  � 1  ˜εℓ  ∂zPNL,z  �  − 2(∂z ln ˜εℓ)∂zEz − Ez∂2  z ln ˜εℓ  (S70)  We can then: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=') use the ansatz Ex(z, t) ∝ E(z, t)ei(˜κ0(z)z−ν0t) + cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=') proceed as before in expanding the dispersion relation  around the central frequency, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', expanding ˜κ(z, ν) around ν0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=') neglect ˜D2 terms, to arrive at  1  2˜κ0  ∂2  zE + i(∂z + ˜κ1∂t)E − ˜κ2  2 ∂2  t E − 2i ˜κ1˜κ2  4˜κ0  ∂3  t E − ˜κ2  1  2˜κ0  ∂2  t E + 2i∂z˜κ0  2˜κ0  E + 2iz∂z˜κ0  2˜κ0  ∂zE + iz∂2  z ˜κ0  2˜κ0  E − z∂z˜κ0E − z2 (∂z˜κ0)2  2˜κ0  E  (S71)  = −n2ν0nsp(r)ε0|E|2E/Ω + ∂z ln nsp  2˜κ0  (i˜κ0E + ∂zE + iz(∂z˜κ0) E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Upon using the SVEA approximation(s), that entail that ∂2  zE ≪ κ0∂zE and (˜κ1/˜κ0)∂t ≪ 1, we then obtain the NLSE given by  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (24) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' It should also be noted that, in the weak field approximation, the terms −z2 �  (∂z˜κ0)2/(2˜κ0)  �  E and  (∂z ln nsp)/(2˜κ0)iz(∂z˜κ0)E are negligible since at least quadratic in rS /r⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  SOLUTION OF THE 1D EQUATIONS  As discussed in the main text, in the case of horizontal propagation and considering a material with anomalous dispersion, we  can solve eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (20) analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Borrowing the solution from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S74) of the supplementary material in [4] the analytical solution  is given by (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' S2)  E(t, z) =  �  Ω|˜κ2|  ν0n2nspε0T 2  0  cosh  �t − ˜κ1z  T0  �−1  exp  ������  iz|˜κ2|  2T 2  0  ������ ,  (S72)  where T0 is the pulse length, and 1/˜κ1 is its speed of propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' This solution reduces exactly to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S4) of [4] in the limit of  rS → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Note that the propagation speed of the soliton is ˜�g(ν0) = �g(ω0)/nsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' This is exactly the proper velocity with respect to  the observer’s proper time and proper length in curved spacetime, as found above in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  10  t(s)  z(m)  |A(z, t)|  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' S2: Propagation of the 1D analytic soliton, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (20), for rS = 9 × 10−3 m and r⊕ = 6 × 106 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  In the case of the vertical propagation, we solve eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (24) by way of the split-step Fourier method as showcased in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In  particular, we have adapted the Matlab code reported in [11] to our needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In solving numerically the NLSE, we assign as initial  temporal profile the soliton solution in flat spacetime of Philbin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' [4], which coincides with the solution in the horizontal 1D  propagation at z = 0 and for rS → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Schematically, the split-step Fourier method consists in rewriting the NLSE as  ∂zE =  � ˆD + ˆN  �  E,  (S73)  where the non-linear operator ˆN = ˆN(z, |E|2) accounts for the non-linearity and the diffusive dynamics is enclosed in the operator  ˆD = ˆD(z, ∂t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We then need to separate the action of the non-linear term and the dissipative one by dividing the propagation  distance in small steps such that  E(z + h, t) ≈ e ˆNh/2e ˆDhe ˆNh/2E(z, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S74)  This can be easily accomplished by alternating the use of the fast-Fourier/inverse Fourier transform algorithm in order to apply  ˆD in frequency space as a multiplicative operator and going back to the time domain at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Furthermore, since our  operator ˆD = ˆD(z, ∂t) depends on the z coordinate, a more precise implementation of the method would see to apply at each step  exp  �� z+h  z  ˆD  �  , which however is well approximated by e ˆDh in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  PHOTOELASTICITY – INCLUDING THE EFFECT OF MATERIAL DEFORMATION ON THE REFRACTIVE INDEX  As we have discussed so far, the optical medium in curved spacetime turns out to be equivalent to an effective one in flat  spacetime, where the optical properties have a contribution coming from the curved spacetime background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' However, whenever  our physical medium is stationary in a curved spacetime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', it follows the trajectories of the timelike Killing vector, it will also  be subject to forces that can deform it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' As discussed in the main text, deformations due to gravity of our physical medium lead  to a change in the refractive index via the photoelastic effect [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Given our previous considerations, we will be interested in the effect of photoelasticity only for the vertical propagation  equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In order to include this effect and separate the contributions coming from the curvature of spacetime and the inertial  acceleration of the fiber, we follow the discussion in [18] on the description of a deformable resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We choose to ignore the  potential effects of photoelasticity on the nonlinear properties of the material, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', the nonlinear susceptibility χ(3), as they would  be mediated through different mechanisms compared to the effect on the linear refractive index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Consider then the situation depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' A fiber of length L and constant mass density ρm is hanging from a support  located at r = r0 ≡ r⊕ + L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In Schwarzschild spacetime, for an observer given by the stationary Killing vector ∂t/∥∂t∥, the  1×108  5×107  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' × 10-12  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='×10-13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='000211  z  x  y  L  r = r⊕ + L  A⊘  g  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' S3: Geometry of a fiber hanging in a weak gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  proper acceleration of the observer fixed at the support, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', an observer at constant radius in isotropic coordinates, and the local  curvature projected into the proper detector frame of this observer are given by [33]  aJ =  ����������0, 0,  rS  2r2  0 c2  �  1 − rS  4r0  � �  1 + rS  4r0  �3  ����������  (S75)  R0J0J =  rS  r3  0  �  1 + rS  4r0  �6  �1  2, 1  2, −1  �  ,  (S76)  where we chose for the z direction to be aligned radially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Furthermore, consistently with the notation we have used so far, we  want to consider the origin of our coordinate at r = r⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' This entails shifting z → z − L to translate the origin from the support  at r0 to r = r⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Note that the proper detector frame is determined by an orthonormal tetrad Fermi-Walker transported along the  timelike trajectory of the support of the fiber which, in our set-up, corresponds to a stationary observer [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  We can now compute the acceleration of test particles in the proper detector frame by following the derivation in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' At  linear order in (z − L)/(r⊕ + L), the acceleration is given by az  p = −  �  az + c2R0z0z(z − L)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' It should be noted that this expression  is derived by neglecting acceleration squared terms in the proper detector frame metric as well as working at first order in the  perturbations around flat spacetime (see discussion in [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' This calls for care when wanting to extrapolate these expressions as  generally valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Each segment of the fiber is then stressed by the force Fz(z) of the parts of the fiber hanging “below” it  Fz(z) =  � z  0  dz′ρmA⊘az  p(z′) = −ρmA⊘c2 rS  2r2  0  ����������  z  �  1 − rS  4r0  � �  1 + rS  4r0  �3 −  z2 − 2Lz  r0  �  1 + rS  4r0  �6  ���������� ,  (S77)  where A⊘ is the cross-section of the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  More generally, the fiber is subject to a stress σkl = Fk/Al, where Fk is the force in direction ˆek and Al is the differential area  normal to ˆel upon which the force acts, caused by the inertial and tidal forces within the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' As long as we are considering  strains within the elastic limit of the material, which is the case of interest here, we can employ Hooke’s law and find that the  strain in the fiber is Skl = σkl/Y, where Y is the Young modulus of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The relation to the electric permeability tensor  εr is then given by ∆(εr)−1  kl = Pkl mnSmn, where P is the photoelastic tensor [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The fact that the change in the inverse of εr is  linear in the strain holds for small or moderate strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Limiting ourselves to isotropic materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' and a diagonal stress tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' the  equations reduce in complexity to  ∆(εr)−1  kk = Pkk llSll = Pkk ll  Y  σll  (S78)  In our set-up,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' the stress and then the strain on the fiber are given explicitly by  σzz(z) = F(z)  A⊘  = ρmc2 rS  2r2  0  ����������  z  �  1 − rS  4r0  � �  1 + rS  4r0  �3 −  z2 − 2Lz  r0  �  1 + rS  4r0  �6  ����������  (S79)  Szz(z) = c2  c2s  rS  2r2  0  ����������  z  �  1 − rS  4r0  � �  1 + rS  4r0  �3 −  z2 − 2Lz  r0  �  1 + rS  4r0  �6  ���������� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S80)  12  z (m)  rS (m)  z (m)  z (m)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='02  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='2  0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' S4: Comparison between the full expression for ∆εr and the approximate one that are appearing in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Left panel:  Here we have used the parameters tabulated in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' I and chosen a quite large rS = 3 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The solid, blue curve represents the  approximate expression for ∆εr, the dashed red curve the exact value of ∆εr, while the dot-dashed black curve is the value of εr  in the absence of photoelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We see that (i) the full and approximate expressions start to deviate from propagation  distances O(1m) onward and (ii) for relatively small propagation distances ∆εr is not anymore a small correction to the relative  permeability εr but becomes equal or greater than εr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Central panel: Fractional difference between the full and approximated  expressions for the photoelastic correction ∆εr i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=',  �  |∆εr|approx − |∆εr|full  �  /  �  |∆εr|full + |∆εr|approx  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Here rS goes from zero to 104  times the Schwarzschild radius of Earth and the propagation distance reaches 100 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We see that the difference between the  two expressions remains below 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Right panel: |∆εr|full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The value of ∆εr, for rS from zero to 104 times the Schwarzschild  radius of Earth and propagation distance up to 100 m, is always well below the value of the relative permeability εr ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  where we used that the speed of sound in the fiber is cs =  �  Y/ρm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Note that the strain and stress have a positive sign due to the  force being directed in the negative z direction or, in other words, since we are considering an elongation of the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Due to the  axial symmetry of the problem, and the irrelevance of two directions orthogonal to the z-axis for the 1D case, the photoelastic  tensor is a scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  The perturbation to the electric permeability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' promoting εr → εr + ∆εr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' is then also a scalar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' and is given by  ∆εr = −  ε2  r∆(ε−1  r )  1 + ε0r∆(ε−1  r ) ≈ −(ε0  r)2∆(ε−1  r ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S81)  where εr indicates the electric permeability in the absence of photoelasticity and the last expression holds whenever the photoe-  lastic effect is a small correction to the material properties giving11  ∆εr ≈ −(ε0  r)2P11 22Szz(z) = −(ε0  r)2P11 22  c2  c2s  rS  2r2  0  ����������  z  �  1 − rS  4r0  � �  1 + rS  4r0  �3 −  z2 − 2Lz  r0  �  1 + rS  4r0  �6  ���������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S82)  Photoelasticity represents an additional correction to the electric permeability on top of the other effects accounting for the  effective medium as described in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' For fused silica, the tabulated values in [30, 31] give P11 22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='271 and  cs = 5720 m/s [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Then, from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S81), for a 10 cm long fiber in the gravitational field of Earth, the contribution of the inertial  acceleration (first term in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S82)) at the end of the fiber to ∆εr is on the order 10−8 while the tidal acceleration (second term  in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S82)) contributes a term of order 10−16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Note that, while the tidal contribution is clearly negligible, the correction to the  relative permeability induced by the inertial acceleration is between one and two orders of magnitude greater than the correction  due to the vacuum curved spacetime optical properties in our effective picture as quantified by 1 − εnp ∼ 10−9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  It is easy to check that, considering a P11 22 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='271, the approximate expression in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S82) will start to fail around a  propagation length of 2 m if we consider to be at one Earth’s radius distance from an object whose mass corresponds to a  11 Note that here we have ∆(ε−1  r ) = P11 22Szz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The indices are determined by  the fact that we are considering an electric field linearly polarized in the  x direction, we identify {x, y, z} ↔ {1, 2, 3}, and we consider an isotropic  material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Thus, (1) the only component of the perturbation tensor of in-  terest is ∆(ε−1  r )11, (2) the only component of the strain is S 33, and (3) we  have P11 33 = P11 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' See Appendix D of [17] where the notation and the  example of isotropic materials are discussed in detailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='5  Approx  Full  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0  0  5  10  15  20100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='08  80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='06  60  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='04  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='02  0  0  20  40  60  80  100100  80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='3  60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='2  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='1  20  0  0  20  40  60  80  10013  Schwarzschild radius of 3 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Indeed, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' S4 shows this failure as well as the fact that for such extreme values of rS , ∆εr starts  to be comparable or greater than εr at propagation distances less than 10 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' At the same time, the same figure shows that, for rS  up to 104 times the one of Earth, both the conditions for the validity of the approximate expression for ∆εr and the fact that the  correction to εr is small are well satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  CONDITIONS FOR VALIDITY OF THE UNIDIRECTIONAL APPROACH  As discussed in the main text, when considering the propagation of a light pulse in a gradient-index medium, we should  account for the fact that the position-dependent refractive index will cause some light to be backscattered – this effect has also  technological application in distributed acoustic sensing for seismology, see [35] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' However, when solving  the NLSE using the SSF method, the boundary condition completely ignores this fact – it would require already knowing the  solution to include the backscattered light in the boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' That this is a drawback of using the NLSE – which is a  unidirectional equation for the validity of which, by definition, back-propagating fields must be negligible – in conjunction with  the SSF method is well known [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  However, back-propagating fields cannot always be simply ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' A formalism fully accounting for this issue would require  to solve a system of coupled bidirectional equations, or just solve the full Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' However, as argued in [25], we  can define conditions that guarantee us that the backward reflected light is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In our case, this sets a restriction on the  parameter space that we can explore, where the description given by our solution to the NLSE can be trusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Essentially, this  regime corresponds to the one of weak-field and not large propagation distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Indeed, physically, for vertical propagation,  longer propagation distances and stronger gravitational accelerations would imply greater changes to the refractive index giving  potentially rise to non-negligible back-propagating fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' To make this observation more quantitative, we follow here [25] where  a more detailed discussion can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  We start from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (22) that we report here for convenience  ∂2  z Ex + ˜µ˜εℓν2Ex = −˜µν2PNL,x + (∂z ln ˜µ)∂zEx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S83)  Following [25], we can rewrite this equation as  (∂2  z + β2)Ex(z) = −Q(z, Ex),  (S84)  where β is a reference momentum that can contain the dispersive character of the physical medium but no z dependence and  that forms our underlying dynamics on top of which we have some perturbation encoded in Q, the residual terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In our case, a  sensible choice for β is  β2 = n2  0ν2  c2 ,  (S85)  where n0 = √εr( ¯ω0) is the material refractive index without any additional effect from the spacetime (and ignoring the effect of  redshift combined with the dispersion of the material) and not accounting for the photoelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' With this choice we have12  − Q(z, Ex) = −˜µν2PNL,x(z) + (∂z log ˜µ)∂zEx(z) + β2[1 − n2  sp(1 + ∆εr/εr)]Ex(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S86)  Now, we can decompose the field in forward and backward directed (in time) fields Ex = E+ + E− and find the equivalent  system of two equations [25]  ∂zE± = ±iβE± ± i  2βQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S87)  The question is then when, starting with E− = 0, E− remains negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Indeed, if E− remains negligible then we are left with a  unidirectional equation and, more importantly, we know that the reflected light can be safely neglected even in comparison with  the unperturbed propagation in flat spacetime in a linear medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  12 In a nutshell, from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S83) we have  ∂2  z Ex + µ0ϵ0(εr + ∆εr)ν2µspεspEx = −µ0µspν2PNL,x + (∂z ln ˜µ)∂zEx  Writing then µ0ϵ0(εr + ∆εr)ν2µspεsp  =  µ0ϵ0ν2εr(1 − (1 − n2  sp)) +  µ0ϵ0ν2n2  sp∆εr we arrive at  ∂2  z Ex + β2Ex = β2  �  1 − n2  sp  �  1 + ∆εr  εr  ��  Ex − µ0µspν2PNL,x + (∂z ln ˜µ)∂zEx  14  rS (m)  z (m)  z (m)  Slow condition  (a)  (b)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='08  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='00005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='00010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='00015  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' S5: Panel (a) shows the right-hand side of the slow-evolution condition of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S88) in the case in which photoelasticity is  not considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Panel (b) shows the same when also photoelasticity in included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We see that without photoelasticity the  condition is very well satisfied for a large set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' When including photoelasticity, we see violations of the condition  for values of rS or the propagation length for which, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' S4 and the corresponding discussion, we know that the ∆εr starts  to be not anymore a small correction to the relative permeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Photoelasticity is the main culprit for the possible significance of the reflected light since, as we have argued before, it is the  dominant effect giving rise to an effective gradient-index medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' It enters only in the term linear in the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Thus,  we focus solely on this term in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' As discussed in [25], the first condition for the backward propagating light to be  negligible is that the residual terms contained in the term Q/2β in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S87) are negligible with respect to βEx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' This translates to  the condition  1 − n2  sp(1 + ∆εr/εr) ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S88)  The second condition arises from considering the backward-evolving part of E−(cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' the discussion in the appendix of [25])  ∂zE−, backwards ≈  ∂zχ  (k + β)2 ,  (S89)  where Q = χEx and k2(z) ≡ β2 + Q(z)  Ex(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' For small Q and ignoring non-linearities, requiring that the change in the medium  parameters does not cause significant back-propagation on the order of a wavelength leads to  ∂z  �  n2  sp(1 + ∆εr/εr)  �  (3/2 + n2sp(1 + ∆εr/εr))2 ≪ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S90)  This no-accumulation condition requires that the derivative of the backpropagating fields is negligible and encodes the fact that  there is no-accumulation of the reflected light giving in the end a non-negligible contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' S5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' S6 we can see that, in the absence of photoelasticity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', considering a rigid dielectric, these conditions  are very well satisfied for the parameters in our simulations also when considering relatively large values of rS and propagation  lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' When turning on photoelasticity, the situation changes, and we can arrive to regimes of large rS and large propagation  distances where the conditions are not satisfied anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In particular, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' S5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' S6 we see that the main limiting  factor is the slow-evolution condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' However, it should be noted that the slow-evolution condition starts to be violated in  the same range of parameters in which ∆εr cannot anymore be considered a small correction and when it is arguable if the  treatment of the photoelasticity as linear in the stresses is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' To corroborate these observations, in the regime in which the  slow-evolution condition is clearly violated we observe a non-negligible energy loss in the numerical solutions of the vertical  propagation equation (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  1000  800  600  400  200  0  0  20  40  60  80  100100  80  60  40  20  0  0  20  40  60  80  10015  rS (m)  z (m)  z (m)  accu condition  (b)  (a)  0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0 × 10-10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0 × 10-9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='5 × 10-9  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='4  × 10−17  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' S6: Panel (a) shows the no-accumulation condition by depicting the ratio between the left and the right-hand sides of  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S90) in the case in which photoelasticity is not considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Panel (b) shows the same when also photoelasticity in  included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We see that this condition is actually well satisfied in both cases, while it remains that without photoelasticity the  condition is much better satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' This analysis shows that the slow-evolution condition is the relevant one for the problem that  we are considering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  COEFFICIENTS FOR THE NUMERICAL SIMULATIONS:  Finally, we report here the explicit expressions for the different coefficients entering the vertical propagation equation that we  use in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  First, let us recall that, when including photoelasticity, we have  n(ω) =  �  εr(ω) + ∆εr(ω) =  �  1 + χ1(ω) + ∆εr(ω) =  �  n0(ω)2 + ∆εr(ω),  (S91)  where n0(ω) is the refractive index in the absence of photoelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We can then proceed to compute all the κi coefficients of  interest  κ0 = κ|ω0  (S92)  κ1 = ∂ωκ|ω0  (S93)  κ2 = ∂2  ωκ|ω0,  (S94)  where κ = nω/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We start from κ0, where we have  κ0(ω0) = n(ω0)ω0  c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S95)  We now consider the case in which the pulse propagates from the bottom of the vertically oriented fiber, which is the case we  simulate numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We thus refer the various quantities of interest to the initial physical frequency ¯ω0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', the frequency  measured by the stationary observer at the bottom of the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We can then write  κ0(ω0) =  n  �  ¯ω0  √  −g00(r⊕)  √  −g00(r⊕+z)  �  ¯ω0  √  −g00(r⊕)  √  −g00(r⊕+z)  c  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S96)  Note that, even for extreme values of rS and z, like rS = 10−2r⊕ and z = 100 m we have that 1 −  �  −g00(r⊕)/  �  −g00(r⊕ + z) is  negligible when considering the dispersive properties of realistic materials at the optical frequencies of interest, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=', the changes  would be on scales way too fine-grained with respect to the tabulated values of the refractive index at the µm scale [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' To  account for this fact, calling ζ = 1 −  �  −g00(r⊕)/  �  −g00(r⊕ + z) we perform an expansion of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S96) at the first order in ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  In the following we report the expressions for all the coefficients necessary to simulate the vertical propagation of the pulse  at first order in ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Note however that, for what concerns the simulations reported in the work, we can always safely neglect  1000  800  600  400  200  0  0  20  40  60  80  1001000  800  600  400  200  0  0  20  40  60  80  10016  L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='1 m  L = 1 m  rS/r⊕  I(L)/I(0)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0  L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='1 m, P1122 = 0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' S7: Energy loss due to photoelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We show the ratio between the final and initial energy  I(L)/I(0) ∼ εr  �  dt|E|2���z=L / εr  �  dt|E|2���z=0, in the proper detector frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The orange, square points correspond to the case of a  propagation length of 1 m while the blue, round points to a propagation length of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The lines represent the linear fit of the  corresponding data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We obtain slopes of −360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='3rs/r⊕ and −43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='2rs/r⊕ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The green, rhomboidal points correspond to  the case without photoelasticity and are compatible with energy conservation up to a negligible energy loss accounted for by  purely gravitational redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  also the corrections to the zeroth order terms for all the ˜κi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The same holds true also for the terms ∂z˜κ0 and ∂2  z ˜κ0 as far as  photoelasticity is considered since the z−dependence is dominated by the photoelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' However, when considering the case  with no photoelasticity, neglecting the z−dependence coming from the redshift factors in κ0 amounts to a relative error of one  part in 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' While still small, we have performed the simulations in which photoelasticity is not included considering also the ζ  corrections in full to account for this small discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  The full expression including all the corrections at order ζ are reported in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Starting with κ0 we have  κ0(ω0) ≈ κ0( ¯ω0) − ζ ¯ω0κ1( ¯ω0)  (S97)  = κb  0  �  1 + ∆εr  εr  − ζ  ������������  κb  0  �  1 + ∆εr  εr  + ¯ω2  0  c  ε′  r + ∆ε′  r  2n0  �  1 + ∆εr  εr  ������������  ,  where κ1( ¯ω0) = (n( ¯ω0) + ¯ω0∂ωn| ¯ω0)/c, while n0 and κb  0 are the tabulated refractive index and corresponding κ0 of the material,  without photoelasticity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' n0 = √εr( ¯ω0) and κb  0 = ¯ω0n0/c, and a prime indicates the derivative with respect to the frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Note that ε′  r = χ′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We derive the expression for the latter, in terms of tabulated values, below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Before doing so, however, let  us compute the derivatives of ˜κ0 at first order in ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Using the fact that ˜κ0 =  �  −g00(r⊕ + z)nsp(z)κ0(ω0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' we find  ∂z˜κ0 ≈κb  0  �  −g00(r⊕)  ������������  ∂znsp  �  ∆εr  εr  + 1 +  nsp∂z∆εr  2εr  �  ∆εr  εr + 1  ������������  +  ¯ω2  0  4c(∆εr + εr( ¯ω0))3/2  �  −2ζ(∂znsp)  �  −g00 (r⊕ + z)(∆εr + εr( ¯ω0)) �∆ε′  r + ε′  r  �  (S98)  −2ζnsp(∂z∆ε′  r)  �  −g00 (r⊕ + z)(∆εr + εr) + ζnsp(∂z∆εr)  �  −g00 (r⊕ + z) �∆ε′  r + ε′  r  �  −2nsp(∂zζ)  �  −g00 (r⊕ + z)(∆εr + εr) �∆ε′  r + ε′  r  � − 2ζnsp∂z  �  −g00(r⊕ + z)(∆εr + εr) �∆ε′  r + ε′  r  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='001017  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='z ˜κ0 ≈κb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−g00(r⊕)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='znsp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='∆εr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='εr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='+ 1 + ∂znsp(∂z∆εr)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='εr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='∆εr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='εr + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='+ nsp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='z∆εr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='2εr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='∆εr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='εr + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='(∂z∆εr)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='4ε2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='� ∆εr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='εr + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='3/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='(S99)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='− ¯ω2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−g00(r⊕ + z)ζ(∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='znsp) �∆ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r + ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='2c √∆εr + εr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='¯ω2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0(∂znsp)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='2ζ(∂z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−g00(r⊕ + z))(∆εr + εr) �∆ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r + ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='� −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−g00(r⊕ + z)ζ∂z∆εr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�∆ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r + ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='2c(∆εr + εr)3/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='¯ω2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0(∂znsp)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−g00(r⊕ + z)ζ(∂z∆ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r)(∆εr + εr) + 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−g00(r⊕ + z)(∂zζ)(∆εr + εr) �∆ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r + ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='2c(∆εr + εr)3/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='¯ω2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0nsp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='4(∆εr + εr)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='2(∂z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−g00(r⊕ + z))(∆εr + εr) −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−g00(r⊕ + z)∂z∆εr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='� �ζ(∂z∆ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r) + (∂zζ) �∆ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r + ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='8c(∆εr + εr)5/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='¯ω2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0nsp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−ζ �∆ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r + ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='� �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−4(∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−g00(r⊕ + z))(∆εr + εr)2 + 4(∂z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−g00(r⊕ + z))∂z∆εr(∆εr + εr) + 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−g00(r⊕ + z)∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='z∆εr(∆εr + εr)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='8c(∆εr + εr)5/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='¯ω2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0nsp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−ζ �∆ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r + ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='� �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−g00(r⊕ + z)(∂z∆εr)2��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='8c(∆εr + εr)5/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='¯ω2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0nsp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='−g00(r⊕ + z)(∆εr + εr)2 �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='ζ(∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='z∆ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r) + 2(∂z∆ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r)(∂zζ) + (∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='zζ) �∆ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r + ε′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='8c(∆εr + εr)5/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='Considering that  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='∆εr = −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='ε2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r∆(ε−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='1 + εr∆(ε−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='r ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S100)  with  ∆(ε−1  r ) =  c2P1122rS  �  z  �  1−  rS  4(L+r⊕)  ��  rS  4(L+r⊕) +1  �3 −  (z−L)2−L2  (L+r⊕)  �  rS  4(L+r⊕) +1  �6  �  2c2s(L + r⊕)2  (S101)  we also have  ∂z∆εr = −  ε2  r∂z∆(ε−1  r )  (εr∆(ε−1  r ) + 1)2  (S102)  ∂2  z∆εr =  ε2  r  �  2εr(∂z∆(ε−1  r ))2 − (εr∆(ε−1  r ) + 1)∂2  z∆(ε−1  r )  �  (εr∆(ε−1  r ) + 1)3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S103)  Moving on, for κ1 we need  κ1(ω0) = (n(ω0) + ω0∂νn(ν)|ω0)/c  (S104)  We proceed with the same approximation at first order in ζ as done above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' We get  κ1(ω0) ≈c−1  �������  �  εr + ∆εr + ¯ω0  ϵ′  r + ∆ε′  r  2 √εr + ∆εr  �  −g00(r⊕)  �  −g00(r⊕ + z)  �������  (S105)  + c−1  �  ζ  �1  2 ¯ω0  �− ¯ω0∆ε′′  r − ¯ω0ε′′  r  √∆εr + εr  +  � ¯ω0∆ε′  r + ¯ω0ε′  r  2(∆εr + εr)3/2 −  1  √∆εr + εr  � �∆ε′  r + ε′  r  ����  ,  where all quantities on the right-hand side are evaluated at ¯ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  Following the same notation as before, we indicate with κb  1 the tabulated optical parameter for the material without photoelas-  ticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' This tabulated quantity enters the previous expression through ε′  r = χ′  1( ¯ω0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Indeed, from κb  1 = (n0 + ¯ω0∂νn0(ν)| ¯ω0)/c, we  have  κb  1 = c−1(n0 + ¯ω0∂νn0(ν)| ¯ω0) = c−1  �  n0 + ¯ω0  χ′  1  2n0  �  ,  (S106)  from which we can read χ′  1 = 2c2 �  −(κb  0)2 + κb  0κb  1 ¯ω0  �  / ¯ω3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  18  We thus remain with identifying ∆ε′  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Let us consider the full form of ∆εr in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S100) and notice that ∆(ε−1  r ) in there, as  given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S101), does not depend on the frequency but only on the stresses and strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Thus we get,  ∆ε′  r = −  ε′  rεr∆(ε−1  r )  �  2 + εr∆(ε−1  r )  �  (1 + εr∆(ε−1  r ))2  ,  (S107)  with  ε′  r = χ′  1,  (S108)  Finally, under the same assumption as before, we have  κ2(ω0) = c−1(2n(ν)′|ω0 + ω0n(ν)′′|ω0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S109)  Thus we end up with  κ2(ω0) ≈c−1  �������  ∆ε′  r + ε′  r  √∆εr + εr  + ¯ω0  2 (∆εr + εr) �∆ε′′  r + ε′′  r  � − �∆ε′  r + ε′  r  � 2  4 (∆εr + εr) 3/2  �  −g00(r⊕)  �  −g00(r⊕ + z)  �������  (S110)  +  ζ ¯ω0  �  − �−3 ¯ω0  �∆ε′  r + ε′  r  � + 4∆εr + 4εr  � �  2(∆εr + εr) �∆ε′′  r + ε′′  r  � − �∆ε′  r + ε′  r  �2��  8c(∆εr + εr)5/2  − ζ ¯ω2  0  �∆ε′′′  r + ε′′′  r  �  2c(∆εr + εr)1/2 ,  where all the quantities on the right-hand side are evaluated at ¯ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  As before,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' indicating the tabulated optical property of the material with κb  2 and κb  3 without photoelasticity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' it is immediate to  derive an expression for χ′′  1 ( ¯ω0)  χ′′  1 = 2c2  ¯ω4  0  �  3(κb  0)2 − 4κb  0κb  1 ¯ω0 +  �  κb  1 ¯ω0  �2 + κb  0κb  2 ¯ω2  0  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S111)  and  χ′′′  1 =  2c2 �  −12(κb  0)2 − 6 ¯ω2  0  �  κb  0κb  2 + (κb  1)2�  + ¯ω3  0(κb  0κb  3 + 3κb  1κb  2) + 18κb  0κb  1 ¯ω0  �  ¯ω5  0  (S112)  while  ∆ε′′  r =  ∆(ε−1  r )  �  −εr  �  ∆(ε−1  r )εr + 1  � �  ∆(ε−1  r )εr + 2  �  ε′′  r − 2ε′  r  2�  �∆(ε−1  r )εr + 1� 3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S113)  and  ∆ε′′′  r = −  ∆(ε−1  r )  �  εrε′′′  r (∆(ε−1  r )εr + 1)2(∆(ε−1  r )εr + 2) − 6∆(ε−1  r )ε′3  r + 6(∆(ε−1  r )εr + 1)ε′  rε′′  r  �  (∆(ε−1  r )εr + 1)4  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S114)  with  ε′′  r = χ′′  1 and ε′′′  r = χ′′′  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  (S115)  In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (S112), we also neglect κb  3 since this term is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  WIDTH OF THE PULSE  While until now we have considered only the effect of a gravitational field on the propagation velocity of the optical pulse,  we can also look at the width of the pulse while it propagates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In the horizontal case, the width remains constant, as can be seen  from the analytical solution of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' In the vertical propagation case, however, this is no longer true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' S8, we see  that spacetime effects, in conjunction with photoelasticity, reduce the width of the pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' This is clearly negligible for realistic  values of rS , and it becomes relevant only at extreme values but shows, nonetheless, that gravity has a focusing effect on the  propagating pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='  19  Photoelasticity  No-Photoelasticity  rS/r⊕  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='0010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='98  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='10  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='4×10-12  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='5×10-12  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='6×10-12  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='7×10-12  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='8×10-12  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='9×10-12  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' ×10-12  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='1×10-12  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='2×10-12  rS = 10−3r⊕  rS = 10−2r⊕  z (m)  rS = 10−2r⊕, P1122 = 0  FWHM (s)  FWHM[z = L]/FWHM[z = 0]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='10  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='4×10-12  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='5×10-12  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='6×10-12  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='7×10-12  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='8×10-12  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='9×10-12  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' ×10-12  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='1×10-12  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='2×10-12  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' S8: Left: FWHM (full width at half maximum) of the pulse at z = L normalized by the FWHM of the initial pulse at z = 0  as a function of rS /r⊕ for an initial pulse with T0 = 40 · 10−13 s propagating for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content='1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The blue solid line shows the case  including the effect of photoelasticity, the red dotted line represents the case without photoelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' Right: pulse FWHM (in  seconds) as a function of the propagation distance z for two different values of rS , again using a pulse with T0 = 40 · 10−13 s,  longer than the one previously considered, for better numerical precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
+page_content=' The green dot-dashed line represents the case without  photoelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfRAxR/content/2301.04986v1.pdf'}
diff --git a/eNFST4oBgHgl3EQfFjgg/content/tmp_files/2301.13718v1.pdf.txt b/eNFST4oBgHgl3EQfFjgg/content/tmp_files/2301.13718v1.pdf.txt
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@@ -0,0 +1,853 @@
+arXiv:2301.13718v1  [math.CO]  31 Jan 2023
+Naturally emerging maps for derangements and
+nonderangements
+Melanie Ferreri
+February 1, 2023
+Abstract
+A derangement is a permutation with no fixed point. We denote by Dn the set of
+derangements of [n] contained in the symmetric group Sn. Similarly, we denote the set
+of nonderangements, or permutations with at least one fixed point, by Dn. We denote
+by En the set of permutations with exactly one fixed point, and we use En to denote
+the complement of En in Sn, i.e. the set of permutations that do not have exactly
+one fixed point. We describe a recursive map that gives a bijection between Dn and
+En excluding one element. We then show the combinatorial description of this map
+and how it compares with other known bijections, and present an involution on Sn
+which exchanges elements of Dn and En excluding one element. This map can be used
+to obtain a map from Dn to En excluding one element, and can be combined with a
+bijection from En to [n] × Dn−1 to give a bijective proof of the one-term identity for
+nonderangements.
+1
+Introduction
+A derangement is a permutation σ ∈ Sn such that for all i ∈ [n],
+σ(i) ̸= i,
+i.e. a permutation which does not fix any element. We denote by Dn the set of derangements
+on n elements, and let dn = |Dn|. Let En be the set of permutations of n elements with
+exactly one fixed point, and let en = |En|. Two well-known recurrence relations for counting
+derangements are
+dn = (n − 1)dn−1 + (n − 1)dn−2
+(1)
+and
+dn = ndn−1 + (−1)n
+(2)
+with d0 = 1 and d1 = 0. The equation (1) has a classical bijective proof, and (2) can be
+proven from (1) by induction.
+In [6], Stanley states that a bijective proof for the one-term recurrence takes considerably
+more work than that of the two-term recurrence. There are several bijective proofs in the
+1
+
+literature. In [5], Remmel proves the one-term identity, along with a q-analog of the equation.
+D´esarm´enien and Wilf also give bijective proofs in [1] and [7].
+Our objective is to show that a bijective proof of (2) emerges naturally from the two-term
+map and the inductive proof of the one-term identity. In order to do this, we show that a
+bijection proving the recurrence dn = (n−1)dn−1 +en−1 can be applied recursively to obtain
+a bijection demonstrating of the identity dn = en + (−1)n, which has a direct combinatorial
+interpretation. From there, composing with a map from En to [n] × Dn−1 yields the desired
+bijection proving (2).
+Furthermore, the map we obtain can be modified to take in elements of both Dn and
+En, yielding an involution which exchanges derangements and permutations with exactly
+one fixed point, excluding one element. Extending this to all elements of Sn, we obtain an
+involution on the entire symmetric group.
+The recurrence relations for derangements can be used to obtain recurrence relations for
+nonderangements as well, which are permutations with at least one fixed point. We denote
+by Dn the set of nonderangements in Sn. Similarly, we let En be the set of permutation in
+Sn which fix either zero elements or at least two elements. In Section 5, we discuss these
+sets in more detail, and show how the previously obtained maps can also be used to define a
+bijection from Dn to En, and another bijection from En to [n] × Dn−1. This yields bijective
+proofs for recurrence relations of nonderangements.
+2
+Recurrence relations for dn and en
+The recurrence relation
+dn = (n − 1)dn−1 + (n − 1)dn−2,
+(1)
+with d0 = 1 and d1 = 0, has a straightforward bijective proof; see [6].
+The number of
+permutations of [n] with exactly one fixed point satisfies the identity
+en = ndn−1,
+(3)
+since a permutation on [n] with exactly one fixed point can be obtained by choosing one of
+the n elements to be fixed, and then deranging the remaining n − 1 elements. The identity
+(3) can also be shown with a bijection
+fn : [n] × Dn−1 → En
+(m, σ) �→ (mn)σ(mn),
+which constructs a permutation with exactly one fixed point by replacing m with n in the
+disjoint cycle decomposition of σ and fixing m if m < n, and otherwise just appending the
+one-cycle (n). When we have a sequence of cycles that are not disjoint, we will use the
+convention of composing cycles from left to right.
+To define the inverse map, we make use of some notation provided in [3]: Given a
+permutation σ ∈ Sn and a ∈ [n], we denote by σ \ a the permutation given by removing a
+from the disjoint cycle decomposition of σ. Then the map f −1
+n
+is given by
+f −1
+n
+: En → [n] × Dn−1
+τ �→ (a, (an)τ(an) \ n),
+2
+
+where a is the unique fixed point of τ. This takes the fixed point for the first coordinate, and
+for the second coordinate, it essentially swaps the fixed point of τ with n and then removes
+(n) to get a permutation in Dn−1. So, equation (1) may be rewritten as
+dn = (n − 1)dn−1 + en−1,
+(4)
+with d0 = 1, d1 = 0, and en = ndn−1. For n > 1, equation (4) can be proven by exhibiting a
+bijection between Dn and [n − 1] × Dn−1 ∪ En−1. Let
+ϕn : Dn → [n − 1] × Dn−1 ∪ En−1
+σ �→
+�
+(σ(n), σ \ n)
+if σ \ n ∈ Dn−1
+σ \ n
+otherwise.
+Removing n from the cycle factorization of a derangement yields a permutation in En−1
+exactly when n was in a transposition in σ, so the first case occurs exactly when n is not in
+a transposition in σ. The inverse map ϕ−1
+n
+is as follows:
+ϕ−1
+n : [n − 1] × Dn−1 ∪ En−1 → Dn
+(m, σ) �→ σ ◦ (nm)
+τ �→ (an)τ
+where τ ∈ En−1 and a is the unique fixed point of τ. In the first case, we essentially wedge
+n in the cycle of σ containing m just before m, which undoes the operation of ϕn to get a
+derangement once again. In the second case, we are just taking the fixed point a and putting
+it in a transposition with n, which undoes the second case of ϕn.
+The recurrence (2) can be proven by induction. Letting d0 = 1, we obtain that d1 =
+1 + (−1) = 0, which is indeed the number of derangements on [1]. Let n > 1, and suppose
+for induction that the result holds for dn−1. Then, using the first relation, we have
+dn = (n − 1)dn−1 + (n − 1)dn−2
+= (n − 1)dn−1 + (dn−1 − (−1)n−1)
+(by the inductive hypothesis)
+= ndn−1 − (−1)n−1
+= ndn−1 + (−1)n.
+Then, substituting (3) into (2), we obtain
+dn = en + (−1)n.
+(5)
+From here, we will use the map ϕn to obtain a bijection demonstrating the relation (5).
+3
+Defining maps
+Given functions f : A → C and g : B → D, we define
+f ⊕ g : A ∪ B → C ∪ D
+x �→
+�
+f(x)
+if x ∈ A
+g(x)
+if x ∈ B.
+3
+
+Define the element πn ∈ Sn by
+πn =
+�
+(1 2)(3 4) · · ·(n − 1 n)
+if n is even,
+(1 2)(3 4) · · ·(n − 2 n − 1)(n)
+if n is odd.
+In other words, πn is a product of disjoint simple transpositions, with n in a cycle by itself
+if n is odd. Let Πn be the singleton set containing the permutation πn. Define the map
+ℓn : Πn → Πn−1
+πn �→ πn \ n.
+We define the inverse ℓ−1
+n : Πn−1 → Πn to send πn−1 to πn−1 ◦ (n − 1 n) if n is even, and if n
+is odd, πn−1 �→ πn−1 ◦ (n).
+Also define
+gn : [n] × Dn−1 → [n − 1] × Dn−1 ∪ Dn−1
+(m, σ) �→
+�
+(m, σ)
+if m ̸= n
+σ
+if m = n.
+This map just removes the first coordinate if it is n, and otherwise does nothing. Its inverse
+is
+g−1
+n
+: [n − 1] × Dn−1 ∪ Dn−1 → [n] × Dn−1
+(m, σ) �→ (m, σ)
+σ �→ (n, σ).
+To obtain a bijection proving the identity dn = en + (−1)n, we will construct a map αn,
+which will have an extra element in either its domain or codomain, depending on the parity
+of n. Denote by () the empty permutation from ∅ to ∅. We give the following base cases: If
+n = 0, define
+α0 : D0 → E0 ∪ Π0
+() �→ π0.
+If n = 1, define
+α1 : D1 ∪ Π1 → E1
+π1 �→ (1).
+For n > 1, we define the maps αn and α−1
+n recursively. If n is even, define αn : Dn → En ∪Πn
+as follows:
+αn : Dn
+ϕn
+−→ ([n − 1] × Dn−1) ∪ En−1
+id[n−1]×Dn−1⊕α−1
+n−1
+−−−−−−−−−−−−→ ([n − 1] × Dn−1) ∪ Dn−1 ∪ Πn−1
+g−1
+n ⊕ℓ−1
+n
+−−−−−→ [n] × Dn−1 ∪ Πn
+fn⊕idΠn
+−−−−−→ En ∪ Πn
+4
+
+If n is odd, we define αn : Dn ∪ Πn → En as follows:
+αn : Dn ∪ Πn
+ϕn⊕ℓn
+−−−−→ ([n − 1] × Dn−1) ∪ En−1 ∪ Πn−1
+id[n−1]×Dn−1⊕α−1
+n−1
+−−−−−−−−−−−−→ [n − 1] × Dn−1 ∪ Dn−1
+g−1
+n
+−−→ [n] × Dn−1
+fn
+−→ En.
+We also define the inverse maps.
+We let α−1
+0
+: E0 ∪ Π0 → D0 send π0 �→ (), and let
+α−1
+1
+: E1 → D1 ∪ Π1 send (1) �→ π1.
+For n even, define α−1
+n
+: En ∪ Πn → Dn as follows:
+α−1
+n : En ∪ Πn
+f−1
+n ⊕idΠn
+−−−−−−→ ([n] × Dn−1) ∪ Πn
+gn⊕ℓn
+−−−→ ([n − 1] × Dn−1) ∪ Dn−1 ∪ Πn−1
+id[n−1]×Dn−1⊕αn−1
+−−−−−−−−−−−−→ ([n − 1] × Dn−1) ∪ En−1
+ϕ−1
+n
+−−→ Dn.
+For n odd, define α−1
+n
+: En → Dn ∪ Πn as follows:
+α−1
+n
+: En
+f−1
+n
+−−→ [n] × Dn−1
+gn
+−→ ([n − 1] × Dn−1) ∪ Dn−1
+id[n−1]×Dn−1⊕αn−1
+−−−−−−−−−−−−→ ([n − 1] × Dn−1) ∪ En−1 ∪ Πn−1
+ϕ−1
+n ⊕ℓ−1
+n
+−−−−−→ Dn ∪ Πn.
+Note that by construction of αn, for n even, the composition of αn with the map f −1
+n ⊕idΠn
+has image [n]×Dn−1 ∪Πn, and for n odd, the composition of αn with the map f −1
+n
+has image
+[n] × Dn−1. Let
+An =
+�
+(f −1
+n
+⊕ idΠn) ◦ αn
+if n is even,
+f −1
+n
+◦ αn
+if n is odd.
+Then the composition An is a bijection showing the identity (2).
+3.1
+Combinatorial description of αn
+After defining αn, we can trace through the recursion to obtain a direct description of the
+image of a derangement σ. First, αn sends πn to πn always. Let σ ∈ Dn, σ ̸= πn. To find
+αn(σ) ∈ En, we look at the disjoint cycle decomposition of σ, and find the smallest j such
+that
+σ = (· · · )(j j + 1) · · ·(n − 1 n)
+where the initial (· · · ) is any combination of cycles. If there is no trailing pattern of simple
+transpositions like this, we let j = n + 1.
+Case 1. If j − 1 is in a 2-cycle, we have
+σ = (· · · )(· · ·j − 2 b · · ·)(j − 1 a)(j j + 1) · · ·(n − 1 n)
+which is sent to
+αn(σ) = (· · ·)(· · · j − 2 a b · · · )(j − 1 j)(j + 1 j + 2) · · ·(n − 2 n − 1)(n)
+where any values above n are excluded from the disjoint cycle notation. In particular, if
+j − 1 = n, then we have n fixed.
+5
+
+Case 2. If j − 1 is not in a 2-cycle, we have
+σ = (· · · )(· · ·j − 1 a b · · · )(j j + 1) · · · (n − 1 n)
+which is sent to
+αn(σ) = (· · · )(· · · j − 1 b · · · )(j a)(j + 1 j + 2) · · · (n − 2 n − 1)(n),
+where again any values above n are excluded from the cycles. In particular, if j − 1 = n,
+then in this case we have a fixed.
+The inverse map operates similarly to αn. We describe it directly in Section 4.
+Example 1 (n = 7).
+Case 1:
+(12)(346)(57) — (12)(3465)(7)
+Case 2:
+(12)(34)(567) — (12)(34)(5)(67)
+(124)(35)(67) — (1243)(56)(7)
+(12345)(67) — (16)(2345)(7)
+(13)(27)(456) — (13)(2456)(7)
+(12)(34675) — (12)(3467)(5)
+We can check that this description indeed matches the effect of applying αn by tracing
+through the recursive definitions of the maps. Proofs can be found in Appendix A.
+Lemma 1. Let πn ∈ Sn be as defined earlier. For all n ≥ 1, αn and α−1
+n
+fix πn.
+Theorem 1. The combinatorial description of αn matches the recursive definition of αn.
+Note that the special case for sending πn to itself does not overlap with the other cases.
+Suppose σ ∈ Dn, σ ̸= πn. In Case 1 of the combinatorial description, the resulting permu-
+tation always has a cycle of length at least 3. In Case 2, the resulting permutation has a
+transposition (j a), where a = σ(j − 1) < j − 1, so (j a) is not a simple transposition. Thus
+neither of the two cases can yield a permutation which has only simple transpositions in its
+disjoint cycle decomposition.
+3.2
+Comparing with similar maps
+The bijection presented in [3] sends derangements in Dn to permutations in En, the set of
+permutations with exactly one fixed point, via the following map ψ:
+Let σ ∈ Dn, and let k be the largest non-negative integer such that the disjoint cycle
+notation of σ starts with (1, 2)(3, 4) . . .(2k − 1, 2k). Then,
+(i) If the cycle containing 2k + 1 has at least 3 elements, then σ and ψ(σ) are as follows:
+σ = (1, 2)(3, 4)...(2k − 1, 2k)(2k + 1, a1, a2, ..., aj)...
+ψ(σ) = (1)(2, 3)(4, 5)...(2k, a1)(2k + 1, a2, ..., aj)...
+Where if k = 0, then {1, 2, ..., 2k} = ∅ and the fixed point in ψ(σ) is a1.
+6
+
+(ii) Otherwise, σ and ψ(σ) are as follows:
+σ = (1, 2)(3, 4)...(2k − 1, 2k)(2k + 1, a1)(2k + 2, a2, ..., aj)...
+ψ(σ) = (1)(2, 3)(4, 5)...(2k, 2k + 1)(2k + 2, a1, a2, ..., aj)...
+The map ψ is conjugate to the map αn by an involution on Sn. To see this, consider
+a derangement σ. If we “element-reverse” the derangement (that is, swap k for n − k + 1
+for all k in the disjoint cycle decomposition of σ), then apply ψ, and then element-reverse
+again, we obtain αn(σ). So we see that the combinatorial proof in [3] can be derived from
+the combinatorial proof for the identity (4).
+The map described in [5] also operates similarly on σ in the case that n appears in a cycle
+of length at least 3. In this case, σ is mapped to the pair (i, σ \ n) where i is the position
+of n in the word W(σ), defined in [5]. This is similar to f −1
+n
+◦ αn in this case: For αn, if n
+appears in a cycle of length at least 3, then σ(n) is removed from the cycle it appears in and
+becomes fixed. After applying f −1
+n , the permutation is sent to (σ(n), σ \ n). So for this case,
+both maps essentially remove n from the cycle decomposition and record where it appeared.
+4
+An involution on Sn
+Using the map αn, we can extend to permutations with one fixed point as follows: First
+define γn : En → Dn+1 to be the map which sends to σ to σ ◦ (n + 1 m), where m is the
+unique fixed point of σ. Then let
+λn : Dn ∪ En → Dn ∪ En
+σ �→
+�
+αn(σ)
+if σ ∈ Dn
+αn+1(γn(σ)) \ (n + 1)
+if σ ∈ En.
+The restriction of λn to Dn is the map αn, and applying λn to a permutation σ ∈ En
+amounts to changing σ into a derangement by adding n + 1 as a placeholder, applying αn,
+and then removing n + 1 from the resulting permutation. If σ = πn, then λn sends σ to
+itself. Rephrasing the combinatorial description of αn gives a combinatorial description of
+λn(σ) for a general σ ∈ Dn ∪ En. We can check that this defines an involution on Dn ∪ En
+which fixes πn and exchanges the other elements Dn and En.
+Theorem 2. Let σ ∈ Dn ∪ En, with σ ̸= πn. If σ ∈ Dn, then λ(σ) ∈ En. If σ ∈ En, then
+λ(σ) ∈ Dn. Also, λn(λn(σ)) = σ.
+Example 2 (n = 5).
+λ5((12)(345)) :
+The permutation (12)(345) is in D5, so we apply α5. This goes to
+(12)(3)(45).
+λ5((12)(3)(45)):
+First we change the permutation to (12)(36)(45), adding the 6 as a
+placeholder. Then we apply α6 and remove the 6. The permutation
+gets sent to (12)(345).
+7
+
+λ5((1234)(5)):
+First we change the permutation to (1234)(56) and then apply α6
+and remove the 6. This goes to (15)(234).
+λ5((15)(234)):
+The permutation (15)(234) is in D5, so we apply α5. This goes to
+(1234)(5).
+Having defined λn, we can now directly describe α−1
+n . Let τ ∈ En, τ ̸= πn. Since αn is
+equivalent to λn |Dn±Πn, and λn is an involution, it follows that α−1
+n
+= λn |En∓Πn. So we
+have that α−1
+n (τ) = λn(τ) for τ ∈ En.
+Fixing all other permutations, we can also define
+Λn : Sn → Sn
+σ �→
+�
+σ
+if σ /∈ Dn ∪ En
+λn(σ)
+otherwise,
+which is an involution on Sn.
+5
+Maps of nonderangements
+Let Dn denote Sn \ Dn, the set of nonderangements of [n]. Similarly let En denote Sn \ En,
+the set of permutations of [n] which do not have exactly one fixed point. Having found the
+bijection αn : Dn → En(±Πn), we can also define
+αn : Dn → En (∓Πn)
+σ �→
+�
+σ
+if σ ∈ En
+α−1
+n (σ)
+if σ ∈ En.
+We have πn ∈ En ⊆ Dn only when n is odd. In this case, πn is fixed and we have En ∪ Πn
+as the image of αn. If n is even, then πn ∈ Dn ⊆ En. However, there is nothing in Dn that
+maps to πn, so in this case the image of αn is En \ Πn.
+Let dn = |Dn| = n! − dn. Subtracting the equation (2) from the equation n! = n(n − 1)!,
+we have
+n! − dn = n(n − 1)! − ndn−1 − (−1)n
+= n((n − 1)! − dn−1) − (−1)n
+dn = ndn−1 − (−1)n.
+(6)
+Using the method of subtraction described in [2], we can also obtain a bijection showing
+the 1-term identity (6).
+We have just found αn : Dn → En (∓Πn). It remains to find a map ζ : En → [n]×Dn−1.
+Then composing the maps yields a bijection from Dn to [n] × Dn−1 (∓Πn). The map ζn can
+be obtained by subtraction. First define
+Hn : Sn → [n] × Sn−1
+σ �→ (σ(n), σ \ n).
+8
+
+This map records where n is in the disjoint cycle decomposition of σ, and then removes n
+from the cycle in which it appears. This is a bijection since if we are given a pair (m, σ′), we
+can recover the preimage by inserting n into the cycle of σ′ containing m, right before m. In
+Section 2 we also defined
+f −1
+n
+: En → [n] × Dn−1
+τ �→ (a, (an)σ(an) \ n),
+where a is the unique fixed point of τ. Then subtracting f −1
+n
+from Hn gives the desired map
+ζ : En → [n] × Dn−1.
+Let σ ∈ En. Applying Hn sends σ to (σ(n), σ\n). If (σ(n), σ\n) ∈ [n]×Dn−1, then either
+σ was not a derangement, or σ was a derangement with n in a cycle of length 2. Otherwise
+σ \ n ∈ Dn−1. This implies either σ was a derangement or σ had only n as a fixed point.
+However, the latter case cannot have happened since σ ∈ En. Then σ was a derangement,
+where n must have appeared in a cycle of length at least 3. In this case we send the pair
+(σ(n), σ\n) through the map fn. This yields the permutation given by replacing σ(n) with n
+in the disjoint cycle decomposition and fixing σ(n). So if σ = (· · · )(· · · n σ(n) σ(σ(n)) · · ·),
+we now have fn(Hn(σ)) = (· · ·)(· · · n σ(σ(n)) · · · )(σ(n)). This permutation is then sent
+through Hn again, to obtain the pair (σ(σ(n)), σ\{n, σ(n)}◦(σ(n))) ∈ [n]×Dn−1. Collecting
+these results, we have
+ζn : En → [n] × Dn−1
+σ �→
+�
+(σ(n), σ \ n)
+if σ ∈ Dn or σ ∈ Dn with n in a 2-cycle
+(σ(σ(n)), σ \ {n, σ(n)} ◦ (σ(n)))
+if σ ∈ Dn with n in a cycle of length ≥ 3.
+Finally we define Zn : Dn → [n] × Dn−1 ∓ Πn to be ζn ◦ αn. Tracing through possible
+cases, we can directly write down Zn as follows:
+Zn : Dn → [n] × Dn−1 ∓ Πn
+σ �→
+�
+(σ(n), σ \ n)
+if σ has a fixed point ̸= n
+(α−1
+n (σ)(n), α−1
+n (σ) \ n)
+if n is the unique fixed point of σ.
+This description of Zn is shown in the proof of Theorem 3, found in Appendix A.
+The recurrence (6) is also proven bijectively in [4], via a map θ. This map takes elements
+of Dn to Mn, which they define as the set of permutations that have a marked fixed point
+and at least one unmarked fixed point. It is not immediately clear if there is a direct way to
+translate between the map θ and Zn.
+Lastly, we note that the map Zn is a composition of maps found via subtraction, and we
+could have instead directly subtracted the map An defined in Section 3 from Hn. This yields
+a map Yn = Hn \ An equivalent to the composition Zn.
+Theorem 3. The map Yn obtained by subtracting An from Hn is equivalent to Zn. That is,
+Yn = Hn \ An = Hn \ (f −1
+n
+◦ αn) = (Hn \ f −1
+n ) ◦ αn = (Hn \ f −1
+n ) ◦ (id \ αn) = ζn ◦ αn = Zn.
+9
+
+Acknowledgement
+I would like to thank my advisor Peter Doyle for many helpful insights on this work.
+Appendix A
+Proofs
+Lemma 1. Let πn ∈ Sn be as defined earlier. For all n ≥ 1, αn and α−1
+n
+fix πn.
+Proof of Lemma 1. We will prove the result by induction. For the base cases n = 0, 1, we
+see that each of α0, α1, and their inverses map πn to itself by definition. Now suppose for
+induction that αn−1(πn−1) = πn−1 and α−1
+n−1(πn−1) = πn−1.
+If n is even, αn first sends πn though ϕn, which sends πn to πn \ n. This is the same
+permutation as πn−1, and since n is even, πn−1 ∈ En. Then this is sent through α−1
+n−1. By
+the inductive hypothesis, α−1
+n−1 fixes πn−1. So we get back πn−1, which is then sent through
+ℓ−1
+n , which takes πn−1 to πn. Finally, πn is then sent through the identity, so we obtain
+αn(πn) = πn for n even.
+Now suppose n is odd. In this case, αn first sends πn through ℓn, to obtain πn−1. This is
+then sent through α−1
+n−1, which fixes πn−1 by the inductive hypothesis. Then πn−1 is mapped
+via g−1
+n
+to the pair (n, πn−1), which is then sent through fn, which just adds back the 1-cycle
+(n) to πn−1, resulting in the permutation πn. So αn(πn) = πn when n is odd.
+Since αn(πn) = πn for all n ≥ 1, it follows that α−1
+n (πn) = πn as well. Thus the result
+holds by induction.
+Theorem 1. The combinatorial description of αn matches the recursive definition of αn.
+Proof of Theorem 1. Let n ≥ 1. First consider πn, which is the product of simple transpo-
+sitions if n is even, or the product of simple transpositions with the 1-cycle (n) if n is odd.
+The bijection maps πn to itself always, by Lemma 1.
+Now, let σ ∈ Dn, σ ̸= π. We will show the result by induction.
+The smallest base case where Dn or Dn ∪ Πn contains an element other than πn is when
+n = 3. We have D3 ∪ Π3 = {(123), (132), (12)(3)}. Applying α3 to each of these elements,
+we have
+(123)
+ϕn
+−→ (1, (12)) → (1, (12)) → (1, (12)) → (1, (12))
+fn
+−→ (1)(23) = α3((123)),
+(132)
+ϕn
+−→ (2, (12)) → (2, (12)) → (2, (12)) → (2, (12))
+fn
+−→ (13)(2) = α3((132)).
+And by the lemma, α3((12)(3)) = α3(π3) = (12)(3), which matches the combinatorial de-
+scription on πn.
+To see that the mapping for the other elements of D3 ∪ Π3 matches the combinatorial
+description, for (123), there is no pattern of simple transpositions at the end, so j = 4.
+Then j − 1 = 3, which appears in a cycle of length greater than 2, so we are in case two
+of the description, with a = 1. Then αn((123)) = (1)(23). Similarly, we can check that the
+combinatorial description maps (132) to (13)(2), so the base case holds.
+Now suppose for induction that the combinatorial description matches for αi, for all
+0 ≤ i ≤ n − 1. Then, when sending σ ∈ Dn through the map αn, we first apply ϕn. If σ
+10
+
+had n in a cycle of length greater than 2, then σ is sent to (σ(n), σ \ n), which remains fixed
+until we apply the map fn. This sends (σ(n), σ \ n) to the permutation obtained from σ \ n
+by replacing σ(n) with n, and then fixing σ(n). This is equivalent to pulling σ(n) out of the
+disjoint cycle decomposition of σ and fixing it.
+To see that this matches the combinatorial description, if n is in a cycle of length greater
+than 2, then there is no pattern of simple transpositions at the end of σ, so we let j = n + 1.
+Then j − 1 = n, which is in a cycle of length greater than 2, so we apply case two of the
+description. Then σ = (· · · n σ(n)) is sent to (· · · n · · ·)(σ(n)), which is the same as the
+permutation obtained from σ by removing σ(n) from the disjoint cycle decomposition of σ
+and fixing it.
+Otherwise if σ had n in a 2-cycle, then upon applying ϕn, σ is sent to σ \ n ∈ En, which
+is then sent through α−1
+n−1.
+We now check where α−1
+n−1 sends this element.
+First, this sends σ \ n through f −1
+n−1,
+which gives an ordered pair whose first coordinate is the fixed point σ(n), and for the second
+coordinate, gives the permutation obtained from σ \ n by swapping the fixed point, σ(n),
+with n − 1, and then removing (n − 1).
+Case 1. If σ(n) was n − 1, then σ was of the form (· · · )(n − 1 n). Applying f −1
+n−1 to σ \ n
+yields the pair (n − 1, σ \ {n, n − 1}) ∈ [n − 1] × Dn−2. The next step in the definition α−1
+n−1
+is to apply gn−1 to this, which in this case sends the pair to σ \ {n, n−1} ∈ Dn−2. Following
+the next arrow in the definition of α−1
+n−1, we send this through αn−2.
+Case 1a. If σ \ {n, n − 1} is of the form
+(· · · )(· · ·j − 2 b · · · )(j − 1 a)(j j + 1) · · ·(n − 3 n − 2),
+then by the inductive hypothesis,
+αn−2(σ \ {n, n − 1}) = (· · · )(· · · j − 2 a b · · · )(j − 1 j)(j + 1 j + 2) · · · (n − 4 n − 3)(n − 2),
+with any elements above n − 2 excluded. Then α−1
+n−1 finally sends this through ϕ−1
+n−1, which
+just adds n − 1 to the cycle containing the fixed point. So we have
+α−1
+n−1(σ \ n) = (· · · )(· · · j − 2 a b · · · )(j − 1 j)(j + 1 j + 2) · · ·(n − 2 n − 1) ∈ Dn−1
+in this case.
+Case 1b. If instead σ \ {n, n − 1} was of the form
+(· · · )(· · ·j − 1 a b · · · )(j j + 1) · · ·(n − 3 n − 2),
+then by the inductive hypothesis,
+αn−2(σ \ {n, n − 1}) = (· · · )(· · · j − 1 b · · · )(j a)(j + 1 j + 2) · · · (n − 4 n − 3)(n − 2),
+with any elements above n − 2 excluded. Then α−1
+n−1 finally sends this through ϕ−1
+n−1, which
+again just adds n − 1 to the cycle containing the fixed point. So we have
+α−1
+n−1(σ \ n) = (· · · )(· · · j − 1 b · · · )(j a)(j + 1 j + 2) · · ·(n − 2 n − 1) ∈ Dn−1
+in this case.
+11
+
+Next in the definition of αn, we apply g−1
+n , which sends α−1
+n−1(σ \ n) to the ordered pair
+with n as the first coordinate and the same permutation α−1
+n−1(σ\n) as the second coordinate,
+and then finally this is sent through fn, which just adds the 1-cycle (n) to the permutation
+α−1
+n−1(σ \ n).
+So for a permutation σ of the form (· · · )(n − 1 n), we have that if
+σ = (· · · )(· · · j − 2 b · · · )(j − 1 a)(j j + 1) · · ·(n − 1 n),
+then
+αn(σ) = (· · · )(· · · j − 2 a b · · · )(j − 1 j)(j + 1 j + 2) · · ·(n − 2 n − 1)(n),
+and if
+σ = (· · · )(· · · j − 1 a b · · · )(j j + 1) · · ·(n − 1 n),
+then
+αn(σ) = (· · · )(· · · j − 1 b · · · )(j a)(j + 1 j + 2) · · ·(n − 2 n − 1)(n).
+Case 2. If σ(n) = a ̸= n, then f −1
+n−1 sends σ \ n = (· · · )(n − 1 b · · · )(a) to the pair
+(a, (· · · )(a b · · · )) ∈ [n − 1] × Dn−2. This is then fixed by gn, fixed by the following arrow in
+the definition of α−1
+n−1, and then sent through ϕ−1
+n−1. This adds n−1 into the cycle containing
+a right before a, to yield a derangement (· · · )(n − 1 a b · · · ) ∈ Dn−1.
+Next in the definition of αn, we apply g−1
+n , which sends this to the ordered pair with n as
+the first coordinate and the same permutation (· · · )(n − 1 a b · · · ) as the second coordinate,
+and then finally this is sent through fn, which just adds the 1-cycle (n) to the permutation
+(· · · )(n−1 a b · · · ). So in this case, αn sends a derangement of the form (· · ·)(n−1 b · · · )(a n)
+with a ̸= n−1 to (· · · )(n−1 a b · · · )(n). This matches Case 1 of the combinatorial description
+in Section 3.1.
+So in all cases, the recursive definition of αn is equivalent to the combinatorial description;
+thus the result holds by induction.
+Theorem 2. Let σ ∈ Dn ∪ En, with σ ̸= πn. If σ ∈ Dn, then λ(σ) ∈ En. If σ ∈ En, then
+λ(σ) ∈ Dn. Also, λn(λn(σ)) = σ.
+Proof of Theorem 2. We will use the following alternative interpretation of the map λn: Let
+σ ∈ Dn ∪ En. If σ has any fixed point m, we add the 2-cycle (n + 1 m). The n + 1 is a
+placeholder and will be discarded after applying the map to σ. Let N = n if n + 1 was not
+added as a placeholder, and otherwise let N = n + 1.
+Define πN,j to be a permutation on [N] \ [j − 1] given by
+πN,j =
+�
+(j j + 1) · · ·(N − 1 N)
+if N − j + 1 is even
+(j j + 1) · · ·(N − 2 N − 1)(N)
+if N − j + 1 is odd.
+Then look at the disjoint cycle decomposition of σ, and find the smallest j such that
+σ = δ ◦ πN,j
+for δ ∈ DN−1, having no copy of πj−1,i at the end. If there is no pattern of simple transpo-
+sitions like this, we let j = N + 1. Then we send σ through the following map.
+12
+
+Case 1. If δ has j − 1 in a 2-cycle, we have
+σ = (· · ·)(· · · j − 2 · · ·)(j − 1 a) ◦ πN,j
+and the we change it to
+τ = (· · · )(· · · j − 2 a · · · )(j − 1 j) ◦ πN,j+1
+where any values above n are excluded from the cycles.
+Case 2. If δ does not have j − 1 in a 2-cycle, we have
+σ = (· · · )(· · ·j − 1 a · · · ) ◦ πN,j
+and then we change it to
+τ = (· · · )(· · · j − 1 · · · )(j a) ◦ πN,j+1
+Again, any values above n are excluded from the cycles. We show that this equivalent
+map has the desired property. Let σ ∈ Dn ∪ En, σ ̸= πn.
+Case 1. Suppose σ ∈ Dn. Since σ has no fixed point, we will not add n+1 to the disjoint
+cycle decomposition, and N = n.
+Now look at the disjoint cycle decomposition of σ.
+Case 1a. If σ is of the form (· · · )(· · · j − 2 · · ·)(j − 1 a) ◦ πn,j for some minimal j, then
+λn(σ) = (· · · )(· · · j − 2 a · · · )(j − 1 j) ◦ πn,j+1
+where any values above n are excluded from the cycles.
+Note that if j was equal to n + 1 when we first applied λn, then j − 1 = n is now fixed
+by λn(σ) since we are in the first case of the definition of λn. Otherwise if j was less than
+n+1, then σ ended with a nontrivial copy of πn,j, and since σ ∈ Dn, it follows that n−j +1
+is even. Then n − (j + 1) + 1 is odd, so πn,j+1 contains the 1-cycle (n). Also since j was less
+than n + 1, every element up to j appears in a cycle of length 2 or more in λn(σ), so the
+only fixed point of λn(σ) occurs in πn,j+1. Either way, in the first case of the definition of
+λn, n becomes fixed by λn(σ), so λn(σ) ∈ En.
+Then, upon applying λn again, we add n + 1 to the cycle (n), so we apply λn to the
+permutation (· · · )(· · · j − 2 a · · · )(j − 1 j) ◦ πn+1,j+1. Then we have
+λn(λn(σ)) = λn((· · · )(· · ·j − 2 a · · · )(j − 1 j) ◦ πn+1,j+1)
+= (· · · )(· · ·j − 2 · · ·)(j − 1 a) ◦ πn+1,j,
+which is equal to σ after removing any elements greater than n.
+Case 1b. Otherwise, σ is of the form (· · · )(· · · j −1 a · · · )◦πn,j for some minimal j. Then
+λn(σ) = (· · · )(· · · j − 1 · · ·)(j a) ◦ πn,j+1.
+where any values above n are excluded from the cycles.
+13
+
+If j was equal to n + 1 when we first applied λn, then σ had no copy of πn,j at the end,
+so σ = (· · · )(· · ·j − 1 a · · ·). In this case, a = σ(j − 1) is now fixed by λn(σ), so λn(σ) ∈ En.
+Then when applying λn again, we are applying λn to the permutation
+(· · · )(· · · j − 1 · · ·)(n + 1 a),
+which gets mapped under the first case of λn to (· · · )(· · · j − 1 a · · · ) = σ.
+If j was less than n + 1, then, as argued previously, πn,j+1 contains n as a fixed point.
+Also, every element up to j appears in a cycle of length 2 or more in λn(σ), so the only fixed
+point of λn(σ) occurs in πn,j+1; thus λn(σ) ∈ En.
+Also in this case, when applying λn again, we add n + 1 to the cycle (n), so we apply λn
+to the permutation (· · · )(· · · j − 1 · · ·)(j a) ◦ πn+1,j+1. Then we have
+λn(λn(σ)) = λn((· · ·)(· · · j − 1 · · ·)(j a) ◦ πn+1,j+1)
+= (· · · )(· · ·j − 1 a · · · )(j j + 1) ◦ πn,j+2
+= (· · · )(· · ·j − 1 a · · · ) ◦ πn+1,j,
+which is equal to σ after removing any elements greater than n.
+Case 2. Suppose σ ∈ En. In this case σ has some fixed point m, so we add n + 1 to the
+cycle containing m. After making this modification, look at the disjoint cycle decomposition
+of σ.
+Case 2a. If σ is of the form (· · · )(· · · j − 2 · · ·)(j − 1 a) ◦ πn+1,j for some minimal j, then
+λn(σ) = (· · · )(· · ·j −2 a · · · )(j −1 j)◦πn+1,j+1, where any values above n are excluded from
+the cycles, so this is equivalent to
+λn(σ) = (· · ·)(· · · j − 2 a · · ·)(j − 1 j) ◦ πn,j+1,
+where any values above n are excluded from the cycles. Note that if j was equal to n+2 when
+we first applied λn, then j −1 = n+ 1 was in a 2-cycle with a, which was originally the fixed
+point of σ. Then in λn(σ), a is put into a cycle after j − 2 = n, so it is no longer fixed. Also
+if j was equal to n+ 2, then λn(σ) has no nontrivial copy of πn,j−1 at the end after removing
+the elements larger than n, so λn(σ) = (· · · )(· · · j − 2 a · · ·) = (· · · )(· · · n a · · · ) ∈ Dn.
+Also, applying λn again, we have that
+λn(λn(σ)) = λn((· · · )(· · · n a · · · ))
+= (· · · )(· · · n · · · )(n + 1 a).
+which is equal to the original σ after removing n + 1 since it is larger than n.
+If j was less than n + 2, then σ ended with a nontrivial copy of πn+1,j, and since σ was
+originally in En and we added n + 1 so that σ would have no fixed point, it follows that
+n + 1 − j + 1 is even. Then n + 1 − (j + 1) + 1 is odd, so πn+1,j+1 contains 1 fixed point,
+which can only be n + 1. Also since j was less than n + 1, every element up to j appears in
+a cycle of length 2 or more in λn(σ), so the only fixed point in λn(σ) can be n + 1, but this
+will be removed in the end since n + 1 > n, leaving λn(σ) with no remaining fixed points.
+Thus λn(σ) ∈ Dn. Also, applying λn again, we have
+λn(λn(σ)) = λn((· · · )(· · ·j − 2 a · · · )(j − 1 j) ◦ πn,j+1)
+= λn((· · · )(· · ·j − 2 a · · · ) ◦ πn,j−1)
+= (· · · )(· · · j − 2 · · ·)(j − 1 a) ◦ πn,j,
+14
+
+which is equal to the original σ after removing elements larger than n.
+Case 2b. Otherwise, σ is of the form (· · · )(· · ·j − 1 a · · · ) ◦ πn+1,j for some minimal j.
+Then λn(σ) = (· · · )(· · · j −1 · · · )(j a)◦πn+1,j+1, where any values above n are excluded from
+the cycles, so this is equivalent to
+λn(σ) = (· · · )(· · · j − 1 · · ·)(j a) ◦ πn,j+1,
+where any values above n are excluded from the cycles. In this case, j cannot have been
+equal to n+2, because if it were, then σ had no copy of πn+1,j at the end, so σ = (· · · )(· · · j−
+1 a · · · ) = (· · · )(· · ·n + 1 a · · ·). However, we could only add n + 1 in a transposition with
+the original fixed point of σ, so n + 1 would not appear in a cycle of length greater than
+2. Also, j cannot equal n + 1 since if it did, then σ would have ended with the 1-cycle
+πn+1,n+1 = (n + 1), and this cannot happen because again, n + 1 can only appear in a
+transposition.
+Thus j must have been less than n + 1, so a nontrivial copy of πn+1,j appeared in σ. As
+argued previously, it follows that πn+1,j+1 contains 1 fixed point, which can only be n + 1.
+Again, since j was less than n + 1, every element up to j appears in a cycle of length at
+least 2, so the only fixed point in λn(σ) can be n+ 1, which will be removed at the end since
+n + 1 > n. Thus λn(σ) has no fixed point, so λn(σ) ∈ Dn.
+Also, when applying λn again, we have
+λn(λn(σ)) = λn((· · ·)(· · · j − 1 · · ·)(j a) ◦ πn,j+1)
+= (· · · )(· · ·j − 1 a · · · )(j j + 1) ◦ πn,j+2
+= (· · · )(· · ·j − 1 a · · · ) ◦ πn,j,
+which is equal to the original σ after removing any elements greater than n.
+So for σ ̸= πn, if σ ∈ Dn, then λn(σ) ∈ En, and if σ ∈ En, then λn(σ) ∈ Dn. Also, in all
+cases, λn(λn(σ)) = σ; thus λn is an involution on Dn ∪ En which swaps the elements of Dn
+and En excluding πn.
+Theorem 3. The map Yn obtained by subtracting An from Hn is equivalent to Zn. That is,
+Yn = Hn \ An = Hn \ (f −1
+n
+◦ αn) = (Hn \ f −1
+n ) ◦ αn = (Hn \ f −1
+n ) ◦ (id \ αn) = ζn ◦ αn = Zn.
+Proof of Theorem 3. The map Yn : Dn → [n] × Dn−1 ∓ Πn obtained by subtracting An
+from Hn can be found as follows: Let σ ∈ Dn. Then apply Hn to obtain (σ(n), σ \ n).
+If σ \ n ∈ Dn−1, then we are done. This occurs when σ had a fixed point other than n.
+Otherwise, if n was the unique fixed point of σ, then σ\n ∈ Dn−1 so we apply A−1
+n = α−1
+n ◦fn
+to (σ(n), σ \ n). Since n was the unique fixed point, (σ(n), σ \ n) = (n, σ \ n), and fn sends
+this to σ \ n ◦ (n) = σ. Then we apply α−1
+n
+and reapply Hn. Thus if n was the unique fixed
+point, Yn sends σ to (α−1
+n (σ)(n), α−1
+n (σ) \ n).
+This is our final description of Yn:
+Yn : Dn → [n] × Dn−1 ∓ Πn
+σ �→
+�
+(σ(n), σ \ n)
+if σ has a fixed point ̸= n
+(α−1
+n (σ)(n), α−1
+n (σ) \ n)
+if n is the unique fixed point of σ.
+15
+
+Now, Zn = ζn ◦ αn maps as follows: Let σ ∈ Dn. Then apply αn. If σ ∈ En, this does
+nothing. Then we apply ζn to σ. Since σ ∈ Dn, it gets sent to (σ(n), σ \ n).
+Suppose otherwise that σ ∈ En. In this case, αn(σ) = α−1
+n (σ). Let τ = α−1
+n (σ) ∈ Dn.
+Then this gets sent through ζn. If τ has n in a 2-cycle, then ζn(τ) = (τ(n), τ \ n). Otherwise
+if n appears in τ in a cycle of length 3 or more, ζn(τ) = (τ(τ(n)), τ \ {n, τ(n)} ◦ (τ(n)).
+We claim that n appears in a transposition in τ = α−1
+n (σ) if and only if n was a fixed
+point of σ. Suppose that n appears in a transposition in τ. To recover σ, we apply αn to
+τ. If τ(n) = n − 1, then in the combinatorial description of αn, we have j < n. Then in
+σ = αn(τ), n is fixed. Otherwise if τ(n) = a ̸= n−1, then we have j = n+1 and so n = j −1
+is in a transposition, so this is in Case 1 of the combinatorial description of αn, which results
+in n being fixed in αn(τ).
+On the other hand, suppose n was fixed in σ. We can check where n appears in α−1
+n (σ) =
+λn(σ). Since n was fixed, we have n + 1 in a transposition in γn(σ). Then applying αn+1 to
+this, we have j < n + 1, so n + 1 becomes fixed and n appears in a transposition in α−1
+n (σ).
+So we have that τ has n in a 2-cycle if and only if n is fixed in σ, and σ ∈ En, so n is the
+unique fixed point of σ. Thus ζn(α−1
+n (σ)) = (τ(n), τ \ n) when σ had n in a 2-cycle.
+It follows from the claim that if n appears in a cycle of length 3 or more in τ, then n was
+not fixed in σ. In this case, σ has one of the following cycle decompositions:
+Case 1. σ = (· · · )(m)(n a), for some m, a ̸= n. In this case, τ = α−1
+n (σ) = (· · · )(n m a).
+Case 2.
+σ = (· · · )(m)(n a b · · · ), for some m, a, b ̸= n.
+In this case, τ = α−1
+n (σ) =
+(· · · )(n m a b · · · ).
+In both cases, the original fixed point m is put into the cycle containing n just after n
+and just before σ(n). So m = τ(n), and τ(τ(n)) = σ(n).
+Also, since τ = α−1(σ) is obtained by adding m to the cycle containing n after n, it
+follows that τ \ {n, τ(n)} ◦ (τ(n)) = σ \ n, since removing n and m and then fixing m in τ
+is the same as simply removing n from σ.
+So if n was not the unique fixed point of σ ∈ En, we have that ζn(α−1
+n (σ)) = (σ(n), σ \n).
+Collecting these results, we have the following description of Zn:
+Zn : Dn → [n] × Dn−1 ∓ Πn
+σ �→
+�
+(σ(n), σ \ n)
+if σ ∈ En or if σ has a unique fixed point m ̸= n
+(τ(n), τ \ n)
+if n is the unique fixed point of σ.
+This is the same as
+Zn : Dn → [n] × Dn−1 ∓ Πn
+σ �→
+�
+(σ(n), σ \ n)
+if σ has a fixed point ̸= n
+(α−1
+n (σ)(n), α−1
+n (σ) \ n)
+if n is the unique fixed point of σ.
+which is exactly the description of Yn.
+References
+[1]
+Jacques D´esarm´enien. “Une autre interpr´etation du nombre des d´erangements”. In:
+S´eminaire Lotharingien de Combinatoire B08b (1984).
+16
+
+[2]
+Peter G. Doyle. Let’s reinvent subtraction. 2022. doi: 10.48550/ARXIV.2203.13836.
+url: https://arxiv.org/abs/2203.13836.
+[3]
+Sergi Elizalde. A simple bijective proof of a familiar derangement recurrence. 2020. doi:
+10.48550/ARXIV.2005.11312. url: https://arxiv.org/abs/2005.11312.
+[4]
+Sergi Elizalde. Bijections for restricted inversion sequences and permutations with fixed
+points. 2020. doi: 10.48550/ARXIV.2006.13842. url: https://arxiv.org/abs/2006
+.13842.
+[5]
+J.B. Remmel. “A Note on a Recursion for the Number of Derangements”. In: European
+Journal of Combinatorics 4.4 (1983), pp. 371–374. issn: 0195-6698. doi: https://doi
+.org/10.1016/S0195-6698(83)80034-1. url: https://www.sciencedirect.com/sc
+ience/article/pii/S0195669883800341.
+[6]
+R.P. Stanley. Enumerative Combinatorics: Volume 1. Cambridge Studies in Advanced
+Mathematics. Cambridge University Press, 2012. isbn: 9781107015425. url: https://b
+ooks.google.com/books?id=0wmJntp8IBQC.
+[7]
+Herbert S. Wilf. “A Bijection in the Theory of Derangements”. In: Mathematics Mag-
+azine 57.1 (1984), pp. 37–40. issn: 0025570X, 19300980. url: http://www.jstor.org
+/stable/2690295.
+17
+
diff --git a/eNFST4oBgHgl3EQfFjgg/content/tmp_files/load_file.txt b/eNFST4oBgHgl3EQfFjgg/content/tmp_files/load_file.txt
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@@ -0,0 +1,486 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf,len=485
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='13718v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='CO]  31 Jan 2023  Naturally emerging maps for derangements and  nonderangements  Melanie Ferreri  February 1, 2023  Abstract  A derangement is a permutation with no fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' We denote by Dn the set of  derangements of [n] contained in the symmetric group Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Similarly, we denote the set  of nonderangements, or permutations with at least one fixed point, by Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' We denote  by En the set of permutations with exactly one fixed point, and we use En to denote  the complement of En in Sn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' the set of permutations that do not have exactly  one fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' We describe a recursive map that gives a bijection between Dn and  En excluding one element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' We then show the combinatorial description of this map  and how it compares with other known bijections, and present an involution on Sn  which exchanges elements of Dn and En excluding one element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' This map can be used  to obtain a map from Dn to En excluding one element, and can be combined with a  bijection from En to [n] × Dn−1 to give a bijective proof of the one-term identity for  nonderangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  1  Introduction  A derangement is a permutation σ ∈ Sn such that for all i ∈ [n],  σ(i) ̸= i,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' a permutation which does not fix any element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' We denote by Dn the set of derangements  on n elements, and let dn = |Dn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Let En be the set of permutations of n elements with  exactly one fixed point, and let en = |En|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Two well-known recurrence relations for counting  derangements are  dn = (n − 1)dn−1 + (n − 1)dn−2  (1)  and  dn = ndn−1 + (−1)n  (2)  with d0 = 1 and d1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' The equation (1) has a classical bijective proof, and (2) can be  proven from (1) by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  In [6], Stanley states that a bijective proof for the one-term recurrence takes considerably  more work than that of the two-term recurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' There are several bijective proofs in the  1  literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In [5], Remmel proves the one-term identity, along with a q-analog of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  D´esarm´enien and Wilf also give bijective proofs in [1] and [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Our objective is to show that a bijective proof of (2) emerges naturally from the two-term  map and the inductive proof of the one-term identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In order to do this, we show that a  bijection proving the recurrence dn = (n−1)dn−1 +en−1 can be applied recursively to obtain  a bijection demonstrating of the identity dn = en + (−1)n, which has a direct combinatorial  interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' From there, composing with a map from En to [n] × Dn−1 yields the desired  bijection proving (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Furthermore, the map we obtain can be modified to take in elements of both Dn and  En, yielding an involution which exchanges derangements and permutations with exactly  one fixed point, excluding one element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Extending this to all elements of Sn, we obtain an  involution on the entire symmetric group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  The recurrence relations for derangements can be used to obtain recurrence relations for  nonderangements as well, which are permutations with at least one fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' We denote  by Dn the set of nonderangements in Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Similarly, we let En be the set of permutation in  Sn which fix either zero elements or at least two elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In Section 5, we discuss these  sets in more detail, and show how the previously obtained maps can also be used to define a  bijection from Dn to En, and another bijection from En to [n] × Dn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' This yields bijective  proofs for recurrence relations of nonderangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  2  Recurrence relations for dn and en  The recurrence relation  dn = (n − 1)dn−1 + (n − 1)dn−2,  (1)  with d0 = 1 and d1 = 0, has a straightforward bijective proof;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' see [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  The number of  permutations of [n] with exactly one fixed point satisfies the identity  en = ndn−1,  (3)  since a permutation on [n] with exactly one fixed point can be obtained by choosing one of  the n elements to be fixed, and then deranging the remaining n − 1 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' The identity  (3) can also be shown with a bijection  fn : [n] × Dn−1 → En  (m, σ) �→ (mn)σ(mn),  which constructs a permutation with exactly one fixed point by replacing m with n in the  disjoint cycle decomposition of σ and fixing m if m < n, and otherwise just appending the  one-cycle (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' When we have a sequence of cycles that are not disjoint, we will use the  convention of composing cycles from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  To define the inverse map, we make use of some notation provided in [3]: Given a  permutation σ ∈ Sn and a ∈ [n], we denote by σ \\ a the permutation given by removing a  from the disjoint cycle decomposition of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then the map f −1  n  is given by  f −1  n  : En → [n] × Dn−1  τ �→ (a, (an)τ(an) \\ n),  2  where a is the unique fixed point of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' This takes the fixed point for the first coordinate, and  for the second coordinate, it essentially swaps the fixed point of τ with n and then removes  (n) to get a permutation in Dn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' So, equation (1) may be rewritten as  dn = (n − 1)dn−1 + en−1,  (4)  with d0 = 1, d1 = 0, and en = ndn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' For n > 1, equation (4) can be proven by exhibiting a  bijection between Dn and [n − 1] × Dn−1 ∪ En−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Let  ϕn : Dn → [n − 1] × Dn−1 ∪ En−1  σ �→  �  (σ(n), σ \\ n)  if σ \\ n ∈ Dn−1  σ \\ n  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Removing n from the cycle factorization of a derangement yields a permutation in En−1  exactly when n was in a transposition in σ, so the first case occurs exactly when n is not in  a transposition in σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' The inverse map ϕ−1  n  is as follows:  ϕ−1  n : [n − 1] × Dn−1 ∪ En−1 → Dn  (m, σ) �→ σ ◦ (nm)  τ �→ (an)τ  where τ ∈ En−1 and a is the unique fixed point of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In the first case, we essentially wedge  n in the cycle of σ containing m just before m, which undoes the operation of ϕn to get a  derangement once again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In the second case, we are just taking the fixed point a and putting  it in a transposition with n, which undoes the second case of ϕn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  The recurrence (2) can be proven by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Letting d0 = 1, we obtain that d1 =  1 + (−1) = 0, which is indeed the number of derangements on [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Let n > 1, and suppose  for induction that the result holds for dn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then, using the first relation, we have  dn = (n − 1)dn−1 + (n − 1)dn−2  = (n − 1)dn−1 + (dn−1 − (−1)n−1)  (by the inductive hypothesis)  = ndn−1 − (−1)n−1  = ndn−1 + (−1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Then, substituting (3) into (2), we obtain  dn = en + (−1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  (5)  From here, we will use the map ϕn to obtain a bijection demonstrating the relation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  3  Defining maps  Given functions f : A → C and g : B → D, we define  f ⊕ g : A ∪ B → C ∪ D  x �→  �  f(x)  if x ∈ A  g(x)  if x ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  3  Define the element πn ∈ Sn by  πn =  �  (1 2)(3 4) · · ·(n − 1 n)  if n is even,  (1 2)(3 4) · · ·(n − 2 n − 1)(n)  if n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  In other words, πn is a product of disjoint simple transpositions, with n in a cycle by itself  if n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Let Πn be the singleton set containing the permutation πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Define the map  ℓn : Πn → Πn−1  πn �→ πn \\ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  We define the inverse ℓ−1  n : Πn−1 → Πn to send πn−1 to πn−1 ◦ (n − 1 n) if n is even, and if n  is odd, πn−1 �→ πn−1 ◦ (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Also define  gn : [n] × Dn−1 → [n − 1] × Dn−1 ∪ Dn−1  (m, σ) �→  �  (m, σ)  if m ̸= n  σ  if m = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  This map just removes the first coordinate if it is n, and otherwise does nothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Its inverse  is  g−1  n  : [n − 1] × Dn−1 ∪ Dn−1 → [n] × Dn−1  (m, σ) �→ (m, σ)  σ �→ (n, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  To obtain a bijection proving the identity dn = en + (−1)n, we will construct a map αn,  which will have an extra element in either its domain or codomain, depending on the parity  of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Denote by () the empty permutation from ∅ to ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' We give the following base cases: If  n = 0, define  α0 : D0 → E0 ∪ Π0  () �→ π0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  If n = 1, define  α1 : D1 ∪ Π1 → E1  π1 �→ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  For n > 1, we define the maps αn and α−1  n recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If n is even, define αn : Dn → En ∪Πn  as follows:  αn : Dn  ϕn  −→ ([n − 1] × Dn−1) ∪ En−1  id[n−1]×Dn−1⊕α−1  n−1  −−−−−−−−−−−−→ ([n − 1] × Dn−1) ∪ Dn−1 ∪ Πn−1  g−1  n ⊕ℓ−1  n  −−−−−→ [n] × Dn−1 ∪ Πn  fn⊕idΠn  −−−−−→ En ∪ Πn  4  If n is odd, we define αn : Dn ∪ Πn → En as follows:  αn : Dn ∪ Πn  ϕn⊕ℓn  −−−−→ ([n − 1] × Dn−1) ∪ En−1 ∪ Πn−1  id[n−1]×Dn−1⊕α−1  n−1  −−−−−−−−−−−−→ [n − 1] × Dn−1 ∪ Dn−1  g−1  n  −−→ [n] × Dn−1  fn  −→ En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  We also define the inverse maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  We let α−1  0  : E0 ∪ Π0 → D0 send π0 �→ (), and let  α−1  1  : E1 → D1 ∪ Π1 send (1) �→ π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  For n even, define α−1  n  : En ∪ Πn → Dn as follows:  α−1  n : En ∪ Πn  f−1  n ⊕idΠn  −−−−−−→ ([n] × Dn−1) ∪ Πn  gn⊕ℓn  −−−→ ([n − 1] × Dn−1) ∪ Dn−1 ∪ Πn−1  id[n−1]×Dn−1⊕αn−1  −−−−−−−−−−−−→ ([n − 1] × Dn−1) ∪ En−1  ϕ−1  n  −−→ Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  For n odd, define α−1  n  : En → Dn ∪ Πn as follows:  α−1  n  : En  f−1  n  −−→ [n] × Dn−1  gn  −→ ([n − 1] × Dn−1) ∪ Dn−1  id[n−1]×Dn−1⊕αn−1  −−−−−−−−−−−−→ ([n − 1] × Dn−1) ∪ En−1 ∪ Πn−1  ϕ−1  n ⊕ℓ−1  n  −−−−−→ Dn ∪ Πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Note that by construction of αn, for n even, the composition of αn with the map f −1  n ⊕idΠn  has image [n]×Dn−1 ∪Πn, and for n odd, the composition of αn with the map f −1  n  has image  [n] × Dn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Let  An =  �  (f −1  n  ⊕ idΠn) ◦ αn  if n is even,  f −1  n  αn  if n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Then the composition An is a bijection showing the identity (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='1  Combinatorial description of αn  After defining αn, we can trace through the recursion to obtain a direct description of the  image of a derangement σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' First, αn sends πn to πn always.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Let σ ∈ Dn, σ ̸= πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' To find  αn(σ) ∈ En, we look at the disjoint cycle decomposition of σ, and find the smallest j such  that  σ = (· · · )(j j + 1) · · ·(n − 1 n)  where the initial (· · · ) is any combination of cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If there is no trailing pattern of simple  transpositions like this, we let j = n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If j − 1 is in a 2-cycle, we have  σ = (· · · )(· · ·j − 2 b · · ·)(j − 1 a)(j j + 1) · · ·(n − 1 n)  which is sent to  αn(σ) = (· · ·)(· · · j − 2 a b · · · )(j − 1 j)(j + 1 j + 2) · · ·(n − 2 n − 1)(n)  where any values above n are excluded from the disjoint cycle notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In particular, if  j − 1 = n, then we have n fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  5  Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If j − 1 is not in a 2-cycle, we have  σ = (· · · )(· · ·j − 1 a b · · · )(j j + 1) · · · (n − 1 n)  which is sent to  αn(σ) = (· · · )(· · · j − 1 b · · · )(j a)(j + 1 j + 2) · · · (n − 2 n − 1)(n),  where again any values above n are excluded from the cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In particular, if j − 1 = n,  then in this case we have a fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  The inverse map operates similarly to αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' We describe it directly in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Example 1 (n = 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Case 1:  (12)(346)(57) — (12)(3465)(7)  Case 2:  (12)(34)(567) — (12)(34)(5)(67)  (124)(35)(67) — (1243)(56)(7)  (12345)(67) — (16)(2345)(7)  (13)(27)(456) — (13)(2456)(7)  (12)(34675) — (12)(3467)(5)  We can check that this description indeed matches the effect of applying αn by tracing  through the recursive definitions of the maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Proofs can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Let πn ∈ Sn be as defined earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' For all n ≥ 1, αn and α−1  n  fix πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' The combinatorial description of αn matches the recursive definition of αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Note that the special case for sending πn to itself does not overlap with the other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Suppose σ ∈ Dn, σ ̸= πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In Case 1 of the combinatorial description, the resulting permu-  tation always has a cycle of length at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In Case 2, the resulting permutation has a  transposition (j a), where a = σ(j − 1) < j − 1, so (j a) is not a simple transposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Thus  neither of the two cases can yield a permutation which has only simple transpositions in its  disjoint cycle decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='2  Comparing with similar maps  The bijection presented in [3] sends derangements in Dn to permutations in En, the set of  permutations with exactly one fixed point, via the following map ψ:  Let σ ∈ Dn, and let k be the largest non-negative integer such that the disjoint cycle  notation of σ starts with (1, 2)(3, 4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' (2k − 1, 2k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then,  (i) If the cycle containing 2k + 1 has at least 3 elements, then σ and ψ(σ) are as follows:  σ = (1, 2)(3, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='(2k − 1, 2k)(2k + 1, a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=', aj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  ψ(σ) = (1)(2, 3)(4, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='(2k, a1)(2k + 1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=', aj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Where if k = 0, then {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=', 2k} = ∅ and the fixed point in ψ(σ) is a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  6  (ii) Otherwise, σ and ψ(σ) are as follows:  σ = (1, 2)(3, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='(2k − 1, 2k)(2k + 1, a1)(2k + 2, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=', aj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  ψ(σ) = (1)(2, 3)(4, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='(2k, 2k + 1)(2k + 2, a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=', aj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  The map ψ is conjugate to the map αn by an involution on Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' To see this, consider  a derangement σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If we “element-reverse” the derangement (that is, swap k for n − k + 1  for all k in the disjoint cycle decomposition of σ), then apply ψ, and then element-reverse  again, we obtain αn(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' So we see that the combinatorial proof in [3] can be derived from  the combinatorial proof for the identity (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  The map described in [5] also operates similarly on σ in the case that n appears in a cycle  of length at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In this case, σ is mapped to the pair (i, σ \\ n) where i is the position  of n in the word W(σ), defined in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' This is similar to f −1  n  αn in this case: For αn, if n  appears in a cycle of length at least 3, then σ(n) is removed from the cycle it appears in and  becomes fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' After applying f −1  n , the permutation is sent to (σ(n), σ \\ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' So for this case,  both maps essentially remove n from the cycle decomposition and record where it appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  4  An involution on Sn  Using the map αn, we can extend to permutations with one fixed point as follows: First  define γn : En → Dn+1 to be the map which sends to σ to σ ◦ (n + 1 m), where m is the  unique fixed point of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then let  λn : Dn ∪ En → Dn ∪ En  σ �→  �  αn(σ)  if σ ∈ Dn  αn+1(γn(σ)) \\ (n + 1)  if σ ∈ En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  The restriction of λn to Dn is the map αn, and applying λn to a permutation σ ∈ En  amounts to changing σ into a derangement by adding n + 1 as a placeholder, applying αn,  and then removing n + 1 from the resulting permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If σ = πn, then λn sends σ to  itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Rephrasing the combinatorial description of αn gives a combinatorial description of  λn(σ) for a general σ ∈ Dn ∪ En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' We can check that this defines an involution on Dn ∪ En  which fixes πn and exchanges the other elements Dn and En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Let σ ∈ Dn ∪ En, with σ ̸= πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If σ ∈ Dn, then λ(σ) ∈ En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If σ ∈ En, then  λ(σ) ∈ Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Also, λn(λn(σ)) = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Example 2 (n = 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  λ5((12)(345)) :  The permutation (12)(345) is in D5, so we apply α5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' This goes to  (12)(3)(45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  λ5((12)(3)(45)):  First we change the permutation to (12)(36)(45), adding the 6 as a  placeholder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then we apply α6 and remove the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' The permutation  gets sent to (12)(345).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  7  λ5((1234)(5)):  First we change the permutation to (1234)(56) and then apply α6  and remove the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' This goes to (15)(234).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  λ5((15)(234)):  The permutation (15)(234) is in D5, so we apply α5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' This goes to  (1234)(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Having defined λn, we can now directly describe α−1  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Let τ ∈ En, τ ̸= πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Since αn is  equivalent to λn |Dn±Πn, and λn is an involution, it follows that α−1  n  = λn |En∓Πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' So we  have that α−1  n (τ) = λn(τ) for τ ∈ En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Fixing all other permutations, we can also define  Λn : Sn → Sn  σ �→  �  σ  if σ /∈ Dn ∪ En  λn(σ)  otherwise,  which is an involution on Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  5  Maps of nonderangements  Let Dn denote Sn \\ Dn, the set of nonderangements of [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Similarly let En denote Sn \\ En,  the set of permutations of [n] which do not have exactly one fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Having found the  bijection αn : Dn → En(±Πn), we can also define  αn : Dn → En (∓Πn)  σ �→  �  σ  if σ ∈ En  α−1  n (σ)  if σ ∈ En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  We have πn ∈ En ⊆ Dn only when n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In this case, πn is fixed and we have En ∪ Πn  as the image of αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If n is even, then πn ∈ Dn ⊆ En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' However, there is nothing in Dn that  maps to πn, so in this case the image of αn is En \\ Πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Let dn = |Dn| = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' − dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Subtracting the equation (2) from the equation n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' = n(n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=',  we have  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' − dn = n(n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' − ndn−1 − (−1)n  = n((n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' − dn−1) − (−1)n  dn = ndn−1 − (−1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  (6)  Using the method of subtraction described in [2], we can also obtain a bijection showing  the 1-term identity (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  We have just found αn : Dn → En (∓Πn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' It remains to find a map ζ : En → [n]×Dn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Then composing the maps yields a bijection from Dn to [n] × Dn−1 (∓Πn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' The map ζn can  be obtained by subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' First define  Hn : Sn → [n] × Sn−1  σ �→ (σ(n), σ \\ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  8  This map records where n is in the disjoint cycle decomposition of σ, and then removes n  from the cycle in which it appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' This is a bijection since if we are given a pair (m, σ′), we  can recover the preimage by inserting n into the cycle of σ′ containing m, right before m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In  Section 2 we also defined  f −1  n  : En → [n] × Dn−1  τ �→ (a, (an)σ(an) \\ n),  where a is the unique fixed point of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then subtracting f −1  n  from Hn gives the desired map  ζ : En → [n] × Dn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Let σ ∈ En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Applying Hn sends σ to (σ(n), σ\\n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If (σ(n), σ\\n) ∈ [n]×Dn−1, then either  σ was not a derangement, or σ was a derangement with n in a cycle of length 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Otherwise  σ \\ n ∈ Dn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' This implies either σ was a derangement or σ had only n as a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  However, the latter case cannot have happened since σ ∈ En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then σ was a derangement,  where n must have appeared in a cycle of length at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In this case we send the pair  (σ(n), σ\\n) through the map fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' This yields the permutation given by replacing σ(n) with n  in the disjoint cycle decomposition and fixing σ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' So if σ = (· · · )(· · · n σ(n) σ(σ(n)) · · ·),  we now have fn(Hn(σ)) = (· · ·)(· · · n σ(σ(n)) · · · )(σ(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' This permutation is then sent  through Hn again, to obtain the pair (σ(σ(n)), σ\\{n, σ(n)}◦(σ(n))) ∈ [n]×Dn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Collecting  these results, we have  ζn : En → [n] × Dn−1  σ �→  �  (σ(n), σ \\ n)  if σ ∈ Dn or σ ∈ Dn with n in a 2-cycle  (σ(σ(n)), σ \\ {n, σ(n)} ◦ (σ(n)))  if σ ∈ Dn with n in a cycle of length ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Finally we define Zn : Dn → [n] × Dn−1 ∓ Πn to be ζn ◦ αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Tracing through possible  cases, we can directly write down Zn as follows:  Zn : Dn → [n] × Dn−1 ∓ Πn  σ �→  �  (σ(n), σ \\ n)  if σ has a fixed point ̸= n  (α−1  n (σ)(n), α−1  n (σ) \\ n)  if n is the unique fixed point of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  This description of Zn is shown in the proof of Theorem 3, found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  The recurrence (6) is also proven bijectively in [4], via a map θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' This map takes elements  of Dn to Mn, which they define as the set of permutations that have a marked fixed point  and at least one unmarked fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' It is not immediately clear if there is a direct way to  translate between the map θ and Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Lastly, we note that the map Zn is a composition of maps found via subtraction, and we  could have instead directly subtracted the map An defined in Section 3 from Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' This yields  a map Yn = Hn \\ An equivalent to the composition Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' The map Yn obtained by subtracting An from Hn is equivalent to Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' That is,  Yn = Hn \\ An = Hn \\ (f −1  n  αn) = (Hn \\ f −1  n ) ◦ αn = (Hn \\ f −1  n ) ◦ (id \\ αn) = ζn ◦ αn = Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  9  Acknowledgement  I would like to thank my advisor Peter Doyle for many helpful insights on this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Appendix A  Proofs  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Let πn ∈ Sn be as defined earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' For all n ≥ 1, αn and α−1  n  fix πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' We will prove the result by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' For the base cases n = 0, 1, we  see that each of α0, α1, and their inverses map πn to itself by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Now suppose for  induction that αn−1(πn−1) = πn−1 and α−1  n−1(πn−1) = πn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  If n is even, αn first sends πn though ϕn, which sends πn to πn \\ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' This is the same  permutation as πn−1, and since n is even, πn−1 ∈ En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then this is sent through α−1  n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' By  the inductive hypothesis, α−1  n−1 fixes πn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' So we get back πn−1, which is then sent through  ℓ−1  n , which takes πn−1 to πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Finally, πn is then sent through the identity, so we obtain  αn(πn) = πn for n even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Now suppose n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In this case, αn first sends πn through ℓn, to obtain πn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' This is  then sent through α−1  n−1, which fixes πn−1 by the inductive hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then πn−1 is mapped  via g−1  n  to the pair (n, πn−1), which is then sent through fn, which just adds back the 1-cycle  (n) to πn−1, resulting in the permutation πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' So αn(πn) = πn when n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Since αn(πn) = πn for all n ≥ 1, it follows that α−1  n (πn) = πn as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Thus the result  holds by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' The combinatorial description of αn matches the recursive definition of αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Let n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' First consider πn, which is the product of simple transpo-  sitions if n is even, or the product of simple transpositions with the 1-cycle (n) if n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  The bijection maps πn to itself always, by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Now, let σ ∈ Dn, σ ̸= π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' We will show the result by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  The smallest base case where Dn or Dn ∪ Πn contains an element other than πn is when  n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' We have D3 ∪ Π3 = {(123), (132), (12)(3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Applying α3 to each of these elements,  we have  (123)  ϕn  −→ (1, (12)) → (1, (12)) → (1, (12)) → (1, (12))  fn  −→ (1)(23) = α3((123)),  (132)  ϕn  −→ (2, (12)) → (2, (12)) → (2, (12)) → (2, (12))  fn  −→ (13)(2) = α3((132)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  And by the lemma, α3((12)(3)) = α3(π3) = (12)(3), which matches the combinatorial de-  scription on πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  To see that the mapping for the other elements of D3 ∪ Π3 matches the combinatorial  description, for (123), there is no pattern of simple transpositions at the end, so j = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Then j − 1 = 3, which appears in a cycle of length greater than 2, so we are in case two  of the description, with a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then αn((123)) = (1)(23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Similarly, we can check that the  combinatorial description maps (132) to (13)(2), so the base case holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Now suppose for induction that the combinatorial description matches for αi, for all  0 ≤ i ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then, when sending σ ∈ Dn through the map αn, we first apply ϕn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If σ  10  had n in a cycle of length greater than 2, then σ is sent to (σ(n), σ \\ n), which remains fixed  until we apply the map fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' This sends (σ(n), σ \\ n) to the permutation obtained from σ \\ n  by replacing σ(n) with n, and then fixing σ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' This is equivalent to pulling σ(n) out of the  disjoint cycle decomposition of σ and fixing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  To see that this matches the combinatorial description, if n is in a cycle of length greater  than 2, then there is no pattern of simple transpositions at the end of σ, so we let j = n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Then j − 1 = n, which is in a cycle of length greater than 2, so we apply case two of the  description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then σ = (· · · n σ(n)) is sent to (· · · n · · ·)(σ(n)), which is the same as the  permutation obtained from σ by removing σ(n) from the disjoint cycle decomposition of σ  and fixing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Otherwise if σ had n in a 2-cycle, then upon applying ϕn, σ is sent to σ \\ n ∈ En, which  is then sent through α−1  n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  We now check where α−1  n−1 sends this element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  First, this sends σ \\ n through f −1  n−1,  which gives an ordered pair whose first coordinate is the fixed point σ(n), and for the second  coordinate, gives the permutation obtained from σ \\ n by swapping the fixed point, σ(n),  with n − 1, and then removing (n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If σ(n) was n − 1, then σ was of the form (· · · )(n − 1 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Applying f −1  n−1 to σ \\ n  yields the pair (n − 1, σ \\ {n, n − 1}) ∈ [n − 1] × Dn−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' The next step in the definition α−1  n−1  is to apply gn−1 to this, which in this case sends the pair to σ \\ {n, n−1} ∈ Dn−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Following  the next arrow in the definition of α−1  n−1, we send this through αn−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Case 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If σ \\ {n, n − 1} is of the form  (· · · )(· · ·j − 2 b · · · )(j − 1 a)(j j + 1) · · ·(n − 3 n − 2),  then by the inductive hypothesis,  αn−2(σ \\ {n, n − 1}) = (· · · )(· · · j − 2 a b · · · )(j − 1 j)(j + 1 j + 2) · · · (n − 4 n − 3)(n − 2),  with any elements above n − 2 excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then α−1  n−1 finally sends this through ϕ−1  n−1, which  just adds n − 1 to the cycle containing the fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' So we have  α−1  n−1(σ \\ n) = (· · · )(· · · j − 2 a b · · · )(j − 1 j)(j + 1 j + 2) · · ·(n − 2 n − 1) ∈ Dn−1  in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Case 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If instead σ \\ {n, n − 1} was of the form  (· · · )(· · ·j − 1 a b · · · )(j j + 1) · · ·(n − 3 n − 2),  then by the inductive hypothesis,  αn−2(σ \\ {n, n − 1}) = (· · · )(· · · j − 1 b · · · )(j a)(j + 1 j + 2) · · · (n − 4 n − 3)(n − 2),  with any elements above n − 2 excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then α−1  n−1 finally sends this through ϕ−1  n−1, which  again just adds n − 1 to the cycle containing the fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' So we have  α−1  n−1(σ \\ n) = (· · · )(· · · j − 1 b · · · )(j a)(j + 1 j + 2) · · ·(n − 2 n − 1) ∈ Dn−1  in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  11  Next in the definition of αn, we apply g−1  n , which sends α−1  n−1(σ \\ n) to the ordered pair  with n as the first coordinate and the same permutation α−1  n−1(σ\\n) as the second coordinate,  and then finally this is sent through fn, which just adds the 1-cycle (n) to the permutation  α−1  n−1(σ \\ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  So for a permutation σ of the form (· · · )(n − 1 n), we have that if  σ = (· · · )(· · · j − 2 b · · · )(j − 1 a)(j j + 1) · · ·(n − 1 n),  then  αn(σ) = (· · · )(· · · j − 2 a b · · · )(j − 1 j)(j + 1 j + 2) · · ·(n − 2 n − 1)(n),  and if  σ = (· · · )(· · · j − 1 a b · · · )(j j + 1) · · ·(n − 1 n),  then  αn(σ) = (· · · )(· · · j − 1 b · · · )(j a)(j + 1 j + 2) · · ·(n − 2 n − 1)(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If σ(n) = a ̸= n, then f −1  n−1 sends σ \\ n = (· · · )(n − 1 b · · · )(a) to the pair  (a, (· · · )(a b · · · )) ∈ [n − 1] × Dn−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' This is then fixed by gn, fixed by the following arrow in  the definition of α−1  n−1, and then sent through ϕ−1  n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' This adds n−1 into the cycle containing  a right before a, to yield a derangement (· · · )(n − 1 a b · · · ) ∈ Dn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Next in the definition of αn, we apply g−1  n , which sends this to the ordered pair with n as  the first coordinate and the same permutation (· · · )(n − 1 a b · · · ) as the second coordinate,  and then finally this is sent through fn, which just adds the 1-cycle (n) to the permutation  (· · · )(n−1 a b · · · ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' So in this case, αn sends a derangement of the form (· · ·)(n−1 b · · · )(a n)  with a ̸= n−1 to (· · · )(n−1 a b · · · )(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' This matches Case 1 of the combinatorial description  in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  So in all cases, the recursive definition of αn is equivalent to the combinatorial description;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  thus the result holds by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Let σ ∈ Dn ∪ En, with σ ̸= πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If σ ∈ Dn, then λ(σ) ∈ En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If σ ∈ En, then  λ(σ) ∈ Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Also, λn(λn(σ)) = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' We will use the following alternative interpretation of the map λn: Let  σ ∈ Dn ∪ En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If σ has any fixed point m, we add the 2-cycle (n + 1 m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' The n + 1 is a  placeholder and will be discarded after applying the map to σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Let N = n if n + 1 was not  added as a placeholder, and otherwise let N = n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Define πN,j to be a permutation on [N] \\ [j − 1] given by  πN,j =  �  (j j + 1) · · ·(N − 1 N)  if N − j + 1 is even  (j j + 1) · · ·(N − 2 N − 1)(N)  if N − j + 1 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Then look at the disjoint cycle decomposition of σ, and find the smallest j such that  σ = δ ◦ πN,j  for δ ∈ DN−1, having no copy of πj−1,i at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If there is no pattern of simple transpo-  sitions like this, we let j = N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then we send σ through the following map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  12  Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If δ has j − 1 in a 2-cycle, we have  σ = (· · ·)(· · · j − 2 · · ·)(j − 1 a) ◦ πN,j  and the we change it to  τ = (· · · )(· · · j − 2 a · · · )(j − 1 j) ◦ πN,j+1  where any values above n are excluded from the cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If δ does not have j − 1 in a 2-cycle, we have  σ = (· · · )(· · ·j − 1 a · · · ) ◦ πN,j  and then we change it to  τ = (· · · )(· · · j − 1 · · · )(j a) ◦ πN,j+1  Again, any values above n are excluded from the cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' We show that this equivalent  map has the desired property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Let σ ∈ Dn ∪ En, σ ̸= πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Suppose σ ∈ Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Since σ has no fixed point, we will not add n+1 to the disjoint  cycle decomposition, and N = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Now look at the disjoint cycle decomposition of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Case 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If σ is of the form (· · · )(· · · j − 2 · · ·)(j − 1 a) ◦ πn,j for some minimal j, then  λn(σ) = (· · · )(· · · j − 2 a · · · )(j − 1 j) ◦ πn,j+1  where any values above n are excluded from the cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Note that if j was equal to n + 1 when we first applied λn, then j − 1 = n is now fixed  by λn(σ) since we are in the first case of the definition of λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Otherwise if j was less than  n+1, then σ ended with a nontrivial copy of πn,j, and since σ ∈ Dn, it follows that n−j +1  is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then n − (j + 1) + 1 is odd, so πn,j+1 contains the 1-cycle (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Also since j was less  than n + 1, every element up to j appears in a cycle of length 2 or more in λn(σ), so the  only fixed point of λn(σ) occurs in πn,j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Either way, in the first case of the definition of  λn, n becomes fixed by λn(σ), so λn(σ) ∈ En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Then, upon applying λn again, we add n + 1 to the cycle (n), so we apply λn to the  permutation (· · · )(· · · j − 2 a · · · )(j − 1 j) ◦ πn+1,j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then we have  λn(λn(σ)) = λn((· · · )(· · ·j − 2 a · · · )(j − 1 j) ◦ πn+1,j+1)  = (· · · )(· · ·j − 2 · · ·)(j − 1 a) ◦ πn+1,j,  which is equal to σ after removing any elements greater than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Case 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Otherwise, σ is of the form (· · · )(· · · j −1 a · · · )◦πn,j for some minimal j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then  λn(σ) = (· · · )(· · · j − 1 · · ·)(j a) ◦ πn,j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  where any values above n are excluded from the cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  13  If j was equal to n + 1 when we first applied λn, then σ had no copy of πn,j at the end,  so σ = (· · · )(· · ·j − 1 a · · ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In this case, a = σ(j − 1) is now fixed by λn(σ), so λn(σ) ∈ En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Then when applying λn again, we are applying λn to the permutation  (· · · )(· · · j − 1 · · ·)(n + 1 a),  which gets mapped under the first case of λn to (· · · )(· · · j − 1 a · · · ) = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  If j was less than n + 1, then, as argued previously, πn,j+1 contains n as a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Also, every element up to j appears in a cycle of length 2 or more in λn(σ), so the only fixed  point of λn(σ) occurs in πn,j+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' thus λn(σ) ∈ En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Also in this case, when applying λn again, we add n + 1 to the cycle (n), so we apply λn  to the permutation (· · · )(· · · j − 1 · · ·)(j a) ◦ πn+1,j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then we have  λn(λn(σ)) = λn((· · ·)(· · · j − 1 · · ·)(j a) ◦ πn+1,j+1)  = (· · · )(· · ·j − 1 a · · · )(j j + 1) ◦ πn,j+2  = (· · · )(· · ·j − 1 a · · · ) ◦ πn+1,j,  which is equal to σ after removing any elements greater than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Suppose σ ∈ En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In this case σ has some fixed point m, so we add n + 1 to the  cycle containing m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' After making this modification, look at the disjoint cycle decomposition  of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Case 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If σ is of the form (· · · )(· · · j − 2 · · ·)(j − 1 a) ◦ πn+1,j for some minimal j, then  λn(σ) = (· · · )(· · ·j −2 a · · · )(j −1 j)◦πn+1,j+1, where any values above n are excluded from  the cycles, so this is equivalent to  λn(σ) = (· · ·)(· · · j − 2 a · · ·)(j − 1 j) ◦ πn,j+1,  where any values above n are excluded from the cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Note that if j was equal to n+2 when  we first applied λn, then j −1 = n+ 1 was in a 2-cycle with a, which was originally the fixed  point of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then in λn(σ), a is put into a cycle after j − 2 = n, so it is no longer fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Also  if j was equal to n+ 2, then λn(σ) has no nontrivial copy of πn,j−1 at the end after removing  the elements larger than n, so λn(σ) = (· · · )(· · · j − 2 a · · ·) = (· · · )(· · · n a · · · ) ∈ Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Also, applying λn again, we have that  λn(λn(σ)) = λn((· · · )(· · · n a · · · ))  = (· · · )(· · · n · · · )(n + 1 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  which is equal to the original σ after removing n + 1 since it is larger than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  If j was less than n + 2, then σ ended with a nontrivial copy of πn+1,j, and since σ was  originally in En and we added n + 1 so that σ would have no fixed point, it follows that  n + 1 − j + 1 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then n + 1 − (j + 1) + 1 is odd, so πn+1,j+1 contains 1 fixed point,  which can only be n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Also since j was less than n + 1, every element up to j appears in  a cycle of length 2 or more in λn(σ), so the only fixed point in λn(σ) can be n + 1, but this  will be removed in the end since n + 1 > n, leaving λn(σ) with no remaining fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Thus λn(σ) ∈ Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Also, applying λn again, we have  λn(λn(σ)) = λn((· · · )(· · ·j − 2 a · · · )(j − 1 j) ◦ πn,j+1)  = λn((· · · )(· · ·j − 2 a · · · ) ◦ πn,j−1)  = (· · · )(· · · j − 2 · · ·)(j − 1 a) ◦ πn,j,  14  which is equal to the original σ after removing elements larger than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Case 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Otherwise, σ is of the form (· · · )(· · ·j − 1 a · · · ) ◦ πn+1,j for some minimal j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Then λn(σ) = (· · · )(· · · j −1 · · · )(j a)◦πn+1,j+1, where any values above n are excluded from  the cycles, so this is equivalent to  λn(σ) = (· · · )(· · · j − 1 · · ·)(j a) ◦ πn,j+1,  where any values above n are excluded from the cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In this case, j cannot have been  equal to n+2, because if it were, then σ had no copy of πn+1,j at the end, so σ = (· · · )(· · · j−  1 a · · · ) = (· · · )(· · ·n + 1 a · · ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' However, we could only add n + 1 in a transposition with  the original fixed point of σ, so n + 1 would not appear in a cycle of length greater than  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Also, j cannot equal n + 1 since if it did, then σ would have ended with the 1-cycle  πn+1,n+1 = (n + 1), and this cannot happen because again, n + 1 can only appear in a  transposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Thus j must have been less than n + 1, so a nontrivial copy of πn+1,j appeared in σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' As  argued previously, it follows that πn+1,j+1 contains 1 fixed point, which can only be n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Again, since j was less than n + 1, every element up to j appears in a cycle of length at  least 2, so the only fixed point in λn(σ) can be n+ 1, which will be removed at the end since  n + 1 > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Thus λn(σ) has no fixed point, so λn(σ) ∈ Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Also, when applying λn again, we have  λn(λn(σ)) = λn((· · ·)(· · · j − 1 · · ·)(j a) ◦ πn,j+1)  = (· · · )(· · ·j − 1 a · · · )(j j + 1) ◦ πn,j+2  = (· · · )(· · ·j − 1 a · · · ) ◦ πn,j,  which is equal to the original σ after removing any elements greater than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  So for σ ̸= πn, if σ ∈ Dn, then λn(σ) ∈ En, and if σ ∈ En, then λn(σ) ∈ Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Also, in all  cases, λn(λn(σ)) = σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' thus λn is an involution on Dn ∪ En which swaps the elements of Dn  and En excluding πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' The map Yn obtained by subtracting An from Hn is equivalent to Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' That is,  Yn = Hn \\ An = Hn \\ (f −1  n  αn) = (Hn \\ f −1  n ) ◦ αn = (Hn \\ f −1  n ) ◦ (id \\ αn) = ζn ◦ αn = Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' The map Yn : Dn → [n] × Dn−1 ∓ Πn obtained by subtracting An  from Hn can be found as follows: Let σ ∈ Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then apply Hn to obtain (σ(n), σ \\ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  If σ \\ n ∈ Dn−1, then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' This occurs when σ had a fixed point other than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Otherwise, if n was the unique fixed point of σ, then σ\\n ∈ Dn−1 so we apply A−1  n = α−1  n ◦fn  to (σ(n), σ \\ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Since n was the unique fixed point, (σ(n), σ \\ n) = (n, σ \\ n), and fn sends  this to σ \\ n ◦ (n) = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then we apply α−1  n  and reapply Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Thus if n was the unique fixed  point, Yn sends σ to (α−1  n (σ)(n), α−1  n (σ) \\ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  This is our final description of Yn:  Yn : Dn → [n] × Dn−1 ∓ Πn  σ �→  �  (σ(n), σ \\ n)  if σ has a fixed point ̸= n  (α−1  n (σ)(n), α−1  n (σ) \\ n)  if n is the unique fixed point of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  15  Now, Zn = ζn ◦ αn maps as follows: Let σ ∈ Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then apply αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If σ ∈ En, this does  nothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then we apply ζn to σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Since σ ∈ Dn, it gets sent to (σ(n), σ \\ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Suppose otherwise that σ ∈ En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In this case, αn(σ) = α−1  n (σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Let τ = α−1  n (σ) ∈ Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Then this gets sent through ζn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If τ has n in a 2-cycle, then ζn(τ) = (τ(n), τ \\ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Otherwise  if n appears in τ in a cycle of length 3 or more, ζn(τ) = (τ(τ(n)), τ \\ {n, τ(n)} ◦ (τ(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  We claim that n appears in a transposition in τ = α−1  n (σ) if and only if n was a fixed  point of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Suppose that n appears in a transposition in τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' To recover σ, we apply αn to  τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' If τ(n) = n − 1, then in the combinatorial description of αn, we have j < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then in  σ = αn(τ), n is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Otherwise if τ(n) = a ̸= n−1, then we have j = n+1 and so n = j −1  is in a transposition, so this is in Case 1 of the combinatorial description of αn, which results  in n being fixed in αn(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  On the other hand, suppose n was fixed in σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' We can check where n appears in α−1  n (σ) =  λn(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Since n was fixed, we have n + 1 in a transposition in γn(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Then applying αn+1 to  this, we have j < n + 1, so n + 1 becomes fixed and n appears in a transposition in α−1  n (σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  So we have that τ has n in a 2-cycle if and only if n is fixed in σ, and σ ∈ En, so n is the  unique fixed point of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Thus ζn(α−1  n (σ)) = (τ(n), τ \\ n) when σ had n in a 2-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  It follows from the claim that if n appears in a cycle of length 3 or more in τ, then n was  not fixed in σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In this case, σ has one of the following cycle decompositions:  Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' σ = (· · · )(m)(n a), for some m, a ̸= n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In this case, τ = α−1  n (σ) = (· · · )(n m a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  σ = (· · · )(m)(n a b · · · ), for some m, a, b ̸= n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  In this case, τ = α−1  n (σ) =  (· · · )(n m a b · · · ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  In both cases, the original fixed point m is put into the cycle containing n just after n  and just before σ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' So m = τ(n), and τ(τ(n)) = σ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Also, since τ = α−1(σ) is obtained by adding m to the cycle containing n after n, it  follows that τ \\ {n, τ(n)} ◦ (τ(n)) = σ \\ n, since removing n and m and then fixing m in τ  is the same as simply removing n from σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  So if n was not the unique fixed point of σ ∈ En, we have that ζn(α−1  n (σ)) = (σ(n), σ \\n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  Collecting these results, we have the following description of Zn:  Zn : Dn → [n] × Dn−1 ∓ Πn  σ �→  �  (σ(n), σ \\ n)  if σ ∈ En or if σ has a unique fixed point m ̸= n  (τ(n), τ \\ n)  if n is the unique fixed point of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  This is the same as  Zn : Dn → [n] × Dn−1 ∓ Πn  σ �→  �  (σ(n), σ \\ n)  if σ has a fixed point ̸= n  (α−1  n (σ)(n), α−1  n (σ) \\ n)  if n is the unique fixed point of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  which is exactly the description of Yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  References  [1]  Jacques D´esarm´enien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' “Une autre interpr´etation du nombre des d´erangements”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In:  S´eminaire Lotharingien de Combinatoire B08b (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  16  [2]  Peter G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Doyle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Let’s reinvent subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='13836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  url: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='org/abs/2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='13836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  [3]  Sergi Elizalde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' A simple bijective proof of a familiar derangement recurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='11312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' url: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='org/abs/2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='11312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  [4]  Sergi Elizalde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Bijections for restricted inversion sequences and permutations with fixed  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='13842.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' url: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='org/abs/2006  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='13842.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  [5]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Remmel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' “A Note on a Recursion for the Number of Derangements”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In: European  Journal of Combinatorics 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='4 (1983), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' 371–374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' issn: 0195-6698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' doi: https://doi  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='1016/S0195-6698(83)80034-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' url: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='com/sc  ience/article/pii/S0195669883800341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  [6]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Stanley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Enumerative Combinatorics: Volume 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Cambridge Studies in Advanced  Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Cambridge University Press, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' isbn: 9781107015425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' url: https://b  ooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='com/books?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='id=0wmJntp8IBQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  [7]  Herbert S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' Wilf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' “A Bijection in the Theory of Derangements”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' In: Mathematics Mag-  azine 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='1 (1984), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' 37–40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' issn: 0025570X, 19300980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content=' url: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='jstor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='org  /stable/2690295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
+page_content='  17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFST4oBgHgl3EQfFjgg/content/2301.13718v1.pdf'}
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+1
+This work has been submitted to the IEEE for possible publication. Copyright may be trans-
+ferred without notice, after which this version may no longer be accessible.
+RIC-O: Efficient placement of a disaggregated
+and distributed RAN Intelligent Controller with
+dynamic clustering of radio nodes
+Gabriel M. Almeida, Gustavo Z. Bruno, Alexandre Huff,
+Matti Hiltunen, Elias P. Duarte Jr., Cristiano B. Both, Kleber V. Cardoso
+Abstract
+The Radio Access Network (RAN) is the segment of cellular networks that provides wireless
+connectivity to end-users. O-RAN Alliance has been transforming the RAN industry by proposing
+open RAN specifications and the programmable Non-Real-Time and Near-Real-Time RAN Intelligent
+Controllers (Non-RT RIC and Near-RT RIC). Both RICs provide platforms for running applications
+called rApps and xApps, respectively, to optimize the behavior of the RAN. We investigate a disaggre-
+gation strategy of the Near-RT RIC so that its components meet stringent latency requirements while
+presenting a cost-effective solution. We propose the novel RIC Orchestrator (RIC-O) that optimizes
+the deployment of the Near-RT RIC components across the cloud-edge continuum. Edge computing
+nodes often present limited resources and are expensive compared to cloud computing. For example, in
+the O-RAN Signalling Storm Protection, Near-RT RIC is expected to support end-to-end control loop
+latencies as low as 10ms. Therefore, performance-critical components of Near-RT RIC and certain xApps
+Gabriel M. Almeida and Kleber V. Cardoso are with the Universidade Federal de Goi´as (UFG), Brazil. E-mail:
+{gabrielmatheus, kleber}@inf.ufg.br
+Gustavo Z. Bruno and Cristiano B. Both are with the University of Vale do Rio dos Sinos (UNISINOS), Brazil. E-mail:
+zanattabruno@edu.unisinos.br, cbboth@unisinos.br
+Alexandre Huff is with Universidade Tecnol´ogica Federal do Paran´a, Brazil. E-mail:alexandrehuff@utfpr.edu.br
+Matti Hiltunen is with AT&T Labs Research, USA. E-mail: hiltunen@research.att.com
+Elias P. Duarte Jr. is with Universidade Federal do Paran´a, Brazil. E-mail: elias@inf.ufpr.br
+arXiv:2301.02760v1  [cs.NI]  7 Jan 2023
+
+2
+should run at the edge while other components can run on the cloud. Furthermore, RIC-O employs an
+efficient strategy to react to sudden changes and re-deploy components dynamically. We evaluate our
+proposal through analytical modeling and real-world experiments in an extended Kubernetes deployment
+implementing RIC-O and disaggregated Near-RT RIC.
+Index Terms
+RAN Intelligent Controller, O-RAN, Near-RT RIC, placement, disaggregation.
+I. INTRODUCTION
+The Radio Access Network (RAN) is considered the most critical segment of a cellular
+network [1]. A RAN comprises network nodes that provide direct wireless connection to end-
+users. It requires continuous innovations to improve the overall mobile network performance
+and user experience. Novel RAN technologies have included innovations related to wireless
+communication technologies (e.g., mmWave, massive MIMO, intelligent reflective surfaces,
+and THz communications). Other innovations are related to current networking technologies,
+particularly network softwarization, virtualization, programmability, and management (e.g., NFV,
+SDN, Network Slicing, and Zero touch network & Service Management). Traditionally, the
+RAN accounts for most of the CAPEX and OPEX costs of a wireless mobile network. Network
+operators also have to face several hurdles, such as interoperability issues, resulting from the
+diversity of hardware from different vendors and the burden of continuously supporting new
+services and applications. In this context, the O-RAN Alliance [2] has been transforming the
+RAN industry by proposing open, virtualized, fully interoperable, and intelligent mobile networks
+[3].
+The O-RAN Alliance defines a series of specifications describing open interfaces to ensure
+multi-vendor interoperability among the main components that form the RAN. That “openness”
+has been carefully introduced to allow interoperability without hurting the intellectual property of
+the participating companies. Given the significant investments necessary to promote innovation
+of RAN technologies, this approach is essential to keep the interest of those companies in
+contributing to this vital area. Yet another advantage of an open RAN is that it decreases the
+barriers for newcomers that can contribute to solving specific problems and move the technology
+further, particularly in academia. On a side note, it is important to highlight that O-RAN
+
+3
+specifications [4] are aligned with 3GPP standards [5]. Therefore, it is reasonable to assume that
+as the 3GPP introduces enhancements, they are promptly incorporated by the O-RAN Alliance.
+The new RAN architecture [4] stands out as a key contribution among the O-RAN specifica-
+tions. The architecture combines concepts from SDN and NFV and also takes into account cloud-
+native technologies while largely adopting Artificial Intelligence (AI) and Machine Learning
+(ML) technologies [6]. It adopts SDN concepts, such as control and data planes separation and
+the possibility of having a remote RAN controller. The design of a RAN controller presents
+formidable challenges, as it must support the execution of tens to hundreds of RAN functions
+while featuring a large protocol stack. The O-RAN architecture splits the controller into two main
+building blocks: the Near-Real-Time RAN Intelligent Controller (Near-RT RIC) for time-sensitive
+operations and the Non-Real-Time RAN Intelligent Controller (Non-RT RIC) for operations that
+present fewer time restrictions. Moreover, the O-RAN specifications standardize open interfaces
+among the architecture components, which run as virtual (network) functions (or services) on
+cloud-native infrastructures.
+Those RAN controllers run AI/ML-based applications that establish control loops with the
+RAN nodes under their management. The Non-RT RIC runs applications called rApps that
+demand control loops to time intervals above 1s. The Near-RT RIC runs applications called
+xApps that establish control loops constrained to time intervals between 10ms and 1s. The time
+constraint of a given control loop depends on the RAN function under the management of the
+corresponding xApp. For example, an xApp related to medium access management may need
+to complete the control loop under the 10ms threshold, while an xApp related to user session
+management may tolerate longer delays of up to 1s. In a large RAN, the Near-RT RIC (or some
+of its components) and latency-sensitive xApps must be replicated and assigned to manage a
+limited set of RAN nodes, i.e., a cluster of RAN nodes. Determining the minimum number of
+Near-RT RIC instances and where they must run is a non-trivial resource allocation problem.
+The problem becomes even more challenging considering the dynamics of a mobile wireless
+network. While this problem has been previously identified [7], some works rely on multiple
+instances of the Near-RT RIC [8]. Also, to the best of our knowledge, there is no comprehensive
+solution to the problem as we describe in the present work.
+In this work, we propose a disaggregation strategy of the Near-RT RIC so that individual
+components can be distributed and placed across the cloud-edge continuum. We also propose a
+RIC Orchestrator (RIC-O) to deploy (or place) and monitor the Near-RT RIC components so that
+
+4
+they can meet the stringent latency requirements. The RIC-O employs optimization to deploy
+the Near-RT RIC components across the cloud-edge continuum while keeping the overall cost
+as low as possible. Edge computing nodes often present limited resources and are expensive
+compared to cloud computing nodes. While performance-critical components of the Near-RT
+RIC platform and certain xApps should run at the edge, other components can run on cloud
+nodes. Furthermore, RIC-O employs a fast and efficient strategy to react dynamically to sudden
+changes and redeploy components. We also explore the flexibility of the O-RAN architecture to
+introduce a proposal that replicates some specific components of the Near-RT RIC. We evaluate
+the proposal through analytical modeling and real-world experiments in an extended Kubernetes
+deployment that runs the RIC-O and the disaggregated Near-RT RIC.
+The main contributions of this work can be summarized as follows:
+• Proposes a disaggregation strategy to place the Near-RT RIC components across the cloud-
+edge continuum;
+• Formalizes the problem of minimizing the overall cost of the placement of Near-RT RIC
+components while ensuring the latency-sensitive control loop;
+• Proposes a hybrid approach that combines heuristic and optimal strategies to quickly provide
+cost-efficient solutions for placing Near-RT RIC components;
+• Provides performance evaluation results to illustrate the advantage of the proposed approach;
+• All implementations are publicly available, including the source code*, thus, allowing the
+work reproducibility.
+Section II presents an overview of the O-RAN architecture, particularly of the Near-RT
+RIC. Section III describes the related work. Section IV provides the system model, problem
+formulation, and the optimal and heuristic strategies for the placement of the Near-RT RIC
+components. Section V introduces the RIC orchestrator, i.e., the proposed architecture that allows
+efficient placement of Near-RT RIC components and clustering of RAN nodes across the cloud-
+edge continuum. The implementation of the RIC orchestrator is described in Section VI. Real-
+world experiments and an analytical performance evaluation are presented in Section VII. Finally,
+Section VIII concludes the article and discusses future work.
+*https://github.com/LABORA-INF-UFG/paper-GGAMECK-2023
+
+5
+II. BACKGROUND
+The O-RAN architecture defines two RICs responsible for controlling and managing RAN
+nodes on different time scales. While the Non-RT RIC is in charge of tasks that can present a
+latency above 1s, the Near-RT RIC is responsible for tasks that demand a latency between 10ms
+and 1s. Figure 1 illustrates the O-RAN architecture highlighting the Near-RT RIC internal compo-
+nents. Several of those components are not latency-sensitive, such as the Management Services,
+the xApp Manager, the Subscription Manager, and A1 and O1 terminations. However, E2
+Nodes, E2 termination (E2T), xApps, and Shared Layers can be sensitive to latency and require
+correct placement. The xApp Manager provides a flexible way for deploying and managing the
+near real-time applications (xApps). The xApps are applications that run services responsible
+for improving the performance of the RAN. Each xApp runs a well-defined function to access,
+control, and monitor E2 Nodes. While some xApps may implement latency-sensitive functions,
+other xApps can also tolerate latencies up to 1s depending on the tasks they execute. Moreover,
+xApps can provide services to other xApps. The Subscription Manager is responsible for
+providing E2 subscriptions from xApps to E2 nodes. The Management Services includes several
+services provided by the Near-RT RIC to the xApps and E2 nodes, such as routing management,
+alarm notification service, logging, and E2 node management.
+A1
+O1
+E2T
+      Messaging Infrastructure
+Termination
+xAppA xAppB
+xAppZ
+…
+Subscription
+Manager
+xApp
+Manager
+Management
+Services
+…
+R-NIB, UE-NIB, …
+Service Management and Orchestration (SMO)
+Near-RT RIC Internal Architecture
+Non-RT RIC
+E2 Nodes (RAN)
+E2AP (Insert)
+E2AP (Control)
+O1
+Shared Layers
+Fig. 1. O-RAN architecture.
+The A1 termination also shown in Fig. 1 provides an interface between Non-RT RIC and
+Near-RT RIC to exchange information about policies and machine learning models that are exe-
+
+6
+cuted by xApps. The O1 termination represents the interface between the Service Management
+and Orchestration (SMO) framework with network functions and E2 nodes. This termination
+enables the SMO to execute Fault, Configuration, Accounting, Policy, and Security (FCAPS)
+operations on those network functions and E2 nodes. All O-RAN components are expected to
+support the O1 termination when exchanging messages with SMO. The SMO is responsible for
+managing and orchestrating the entire RAN, relying on the Non-RT RIC for RAN optimization,
+and the O1 termination for RAN instrumentation. Moreover, SMO is in charge of managing and or-
+chestrating any cloud infrastructure in use [4]. The E2T is responsible for connecting the Near-RT
+RIC with one or more E2 Nodes using E2 Application Protocol (E2AP) and E2 Service Model
+(E2SM) which are described further. The Messaging Infrastructure element corresponds to
+the underlying messaging subsystem employed by all the Near-RT RIC components to exchange
+information [9]. The Shared Layers correspond to the Shared Data Layer (SDL) and Shared
+Time-Series Layer (STSL), which are high-speed interfaces for accessing shared data storage
+by several stateless Near-RT RIC components and xApps. R-NIB and UE-NIB are examples of
+databases that store network information for RAN and User Equipment (UE), respectively.
+According to the O-RAN specifications [4], RAN nodes (i.e., DUs, CUs, or O-RAN-compliant
+LTE eNBs) are E2 Nodes controlled through the E2 interface. This interface allows the Near-
+RT RIC to control procedures and functionalities of those nodes and is logically organized
+into two parts: E2AP and E2SM. E2AP enables the communication between the Near-RT RIC
+and E2 nodes, which provide four services to the Near-RT RIC: Report, Insert, Control, and
+Policy [9]. The Report service allows xApps to subscribe to E2 nodes to receive information
+about specific RAN events. The Insert service also provides RAN information to xApps, but
+also allows xApps to configure E2 nodes through control messages. The Control service allows
+the Near-RT RIC and xApps to send control messages to E2 nodes, which can initiate new
+procedures or resume a previously suspended procedure associated with that E2 node. Such a
+procedure should specify exactly and completely the functional behavior of a given E2 node.
+Finally, the Near-RT RIC uses the Policy service to establish policy-driven monitoring and control
+the behavior of the corresponding E2 node [10]. These services can be combined in different
+ways to implement E2SMs. An E2SM [9] can be described as a contract between xApps and
+the RAN functions on E2 nodes. Each RAN function exposed by a given E2SM allows the
+Near-RT RIC to monitor, suspend, stop, override or even control the behavior of the RAN on
+that E2 node. Therefore, xApps and their corresponding E2 nodes must implement the same
+
+7
+E2SM definitions to communicate and control the RAN.
+The E2AP Insert service is the most latency-sensitive loop of the Near-RT RIC. This loop
+starts when a given E2 node sends a message to the E2T using the E2AP (Insert) service. The
+E2T, in its turn, delivers that message to the corresponding xApp. This xApp then processes the
+message and sends back the corresponding reply using the E2AP Control service. Moreover,
+the control loop can involve access to Shared Layers (i.e., SDL or STSL), it may also have to
+access one or more databases (e.g., R-NIB and UE-NIB) and even other xApps, as illustrated in
+Fig. 1. We note that in some use cases, such as the O-RAN Signaling Storm Protection [11], it
+is essential to deliver the RIC control message back to the E2 node within the 10ms threshold.
+In this example use case, the E2 node suspends the current procedure execution and waits for
+the corresponding reply from an E2AP (Control) service to resume its operation. Although an
+E2 node can apply a default action when the 10ms threshold is exceeded, this usually implies
+applying further optimization procedures and performance penalties for the RAN. In this work,
+we propose to disaggregate and distribute the Near-RT RIC components to guarantee the latency
+restrictions of the time-sensitive xApps, which execute under a control loop limit of 10ms.
+The O-RAN Software Community (OSC) provides an open-source reference implementa-
+tion [12] of the Near-RT RIC. This implementation also functions as a baseline that allows the
+community to explore alternative deployments, including, for instance, replication, among other
+innovations. The SMO framework must be particularly aware of any alternative configuration
+so that it can dynamically (1) detect any latency violation (e.g., latency above the limit in the
+critical loop); (2) redeploy components to solve the violations; (3) re-orchestrate the Near-RT
+RIC components; and, (4) reconfigure all involved elements. As illustrated in Fig. 1, the SMO
+communicates with all RAN elements, including the E2 Nodes, through the O1 interface that
+offers the necessary means to reconfigure and reorganize the cluster of E2 nodes that each E2T
+and xApps are responsible for. Current SMOs do not fully implement all tasks in a coordinated
+manner. We claim one of the contributions of the current work is to present an SMO extension that
+can provide all necessary capabilities to accomplish those tasks coherently. Another contribution
+is related to task (2), which comprises the resource allocation problem, for which we formulate
+and propose optimal and heuristic strategies.
+
+8
+III. RELATED WORK
+This section presents an overview of related work that also has a focus on the RIC architecture.
+In a small RAN with a few dozen or less E2 nodes, it may be possible to deploy a single Near-RT
+RIC that satisfies the latency-sensitive control loop of the corresponding xApps. However, larger
+RANs with hundreds or even thousands of E2 nodes demand a different approach. Dryjanski and
+Kliks [7] present two options for implementing the Near-RT RIC: centralized and distributed.
+In the centralized option, every E2 node (i.e., the whole gNB or eNB) is handled by the same
+and only Near-RT RIC, which can take unified decisions for an individual E2 node and globally
+optimize operations. In the distributed option, each E2 node type (i.e., O-CU, O-DU, or O-eNB)
+is handled by a specialized instance of the Near-RT RIC that allows optimizing these individual
+types of E2 nodes. The authors discuss the impact of these two implementation options in the
+design of the E2 interface. However, they do not tackle the distribution and replication of the
+Near-RT RIC components.
+Singh and Nguyen [13] propose a framework called O-RANFed to deploy and optimize a set
+of Federated Learning (FL) tasks that provide 5G slicing services in the context of the O-RAN
+specifications. To be precise: the authors introduce a theoretical model of the RIC architecture
+with support for FL. Moreover, the authors present an optimization model for the problem of
+local learning selection and resource allocation. The performance of FL improves with modeling
+and training done in every iteration. However, some details of the proposed FL-supporting RIC
+architecture are missing.
+Huff, Hiltunen, and Duarte [14] discuss and evaluate techniques to make the RIC fault-tolerant
+while preserving high scalability. The fundamental assumption of this work is that traditional
+replication techniques cannot sustain high throughput and low latency as required by RAN
+elements. The authors propose techniques that use state partitioning, partial replication, and fast
+re-route with role awareness to decrease the overhead. Moreover, the authors implemented the
+fault tolerance techniques as a library called RFT (RIC Fault Tolerance) considering a distributed
+RIC, but do not deal with the problem of the placement of RIC components in a disaggregated
+virtual infrastructure.
+D’Oro et al. [8] introduced a novel orchestration framework called OrchestRAN for the Non-
+RT RIC. OrchestRAN allows network operators to specify high-level control and inference
+objectives. The orchestrator defines the optimal set of data-driven algorithms and their execution
+
+9
+locations to achieve intents specified by the network operators. The work assumes that a complete
+instance of the Near-RT RIC is deployed to control each cluster of E2 nodes. Moreover, the
+authors show that the intelligence orchestration problem in O-RAN is NP-hard and design low-
+complexity approaches to support real-world applications. A prototype of OrchestRAN was
+implemented and tested at scale on Colosseum, i.e., the world’s largest wireless network emulator
+with hardware in the loop.
+Schimidt, Irazabal, and Nikaein [15] presented FlexRIC, a software development kit to build
+specialized service-oriented controllers. FlexRIC has a centralized modular architecture with
+a minimal footprint and is designed for extensibility. The authors validate FlexRIC building
+implementations for two use cases: (i) a recursive RAN controller to allow multiple tenants
+to concurrently control and operate their services in a shared infrastructure over heterogeneous
+5G networks, and (ii) an SD-RAN controller supplying programmability for RAN slicing with
+multi-radio technology, and flow-based traffic control aiming at low-latency communications.
+Balasubramanian et al. [16] disaggregate the traditional monolithic control plane RAN ar-
+chitecture. The authors introduce a Near-RT RIC platform that decouples the control and data
+planes of the RAN. The motivation of the project is to work towards intelligent and continuously
+evolving radio networks by fostering network openness and empowering network intelligence
+with AI-enabled applications. The authors provide functional and software architecture of the
+Near-RT RIC and discuss related design challenges. Moreover, they elaborate on how the Near-
+RT RIC can enable network optimization in 5G for the dual-connectivity use case using machine
+learning control loops. In this context, the Near-RT RIC architecture design is generic, providing
+several options for its implementation and deployment.
+Other works focus on developing xApps and rApps. For example, Cao et al. [17] [18] propose
+an intelligent user access control scheme with Deep Reinforcement Learning (DRL). A federated
+DRL-based method is proposed with a global model server running on RIC that updates the
+distributed deep Q-networks (DQNs) parameters to optimize the performance of DQNs trained
+by UE. Johnson, Maas, and Van Der Merwe [19] introduce NexRAN, a use case of the Powder
+mobile and wireless research platform, allowing closed-loop control of an approach for O-RAN
+slicing. O-RAN slicing, in this case, is implemented in the srsRAN open-source mobility stack
+and is exposed through a custom service model to xApps, executing on a Near-RT RIC.
+IV. SYSTEM MODEL, PROBLEM FORMULATION,
+
+10
+AND THE PROPOSED STRATEGY
+In the following, we describe the system model, formulate the problem, and describe the
+optimal and heuristic strategies proposed to solve the problem.
+A. System Model
+We assume a Radio Access Network that follows the O-RAN specifications and is composed
+of a set N = {n1, n2, ..., n|N|} of E2 nodes and a set C = {c1, c2, ..., c|C|} of edge computing
+nodes (CNs), where each CN is a host consisting of a general purpose hardware characterized
+by its processing capability (cProc
+m
+), memory (cMem
+m
+), and storage (cSto
+m ) resources. Moreover, we
+consider a node c0 representing a cloud computing infrastructure (e.g., a large data center) with
+unlimited processing, memory, and storage resources but positioned far from the E2 nodes. We
+propose disaggregating the Near-RT RIC so that different components can run on various nodes.
+The node c0 and each CN cm ∈ C can run any disaggregated RIC function as long as it does
+not exceed its resources. To represent the overlay communication network among all nodes, we
+define a graph G = (V, E) with V = VC ∪N representing the nodes (where VC = {c0}∪C), and
+E = {eij; vi, vj ∈ V} represent the set of links, where each link is characterized by its latency
+eLat
+i,j . If vi = vj, then eLat
+i,j = 0, which is relevant for different components running on the same
+computing node. As illustrated in Fig. 2, c0 and each CN cm ∈ C can communicate with each
+other, while each E2 node ni ∈ N can communicate with c0 or any CN cm ∈ C. Therefore,
+the paths starting at E2 nodes and ending at computing nodes (that run RIC components) are
+defined over this graph G.
+Cloud node
+Edge computing node
+E2 node
+n2
+n1
+n|N|
+c1
+c2
+c|C|
+c0
+Fig. 2. Overlay communication network among E2 nodes and CNs.
+We assume that the Near-RT RIC can be disaggregated and distributed into five main groups
+of components: Near-RT RIC Management (RIC Man), E2T, SDL/STSL, NIBs, and xApps, as
+
+11
+illustrated in Fig. 3. Each E2 node ni ∈ N is connected to an E2T component, which servers a set
+of xApps A = {xApp1, xApp2, ..., xApp|A|}. While the control loop of a latency-sensitive xApp
+must be monitored per E2 node (ni ∈ N), we are also interested in creating clusters of E2 nodes
+to minimize the replication of Near-RT RIC components, as illustrated in Fig. 3. The red lines
+with arrows in this figure represent the control loop that each latency-sensitive xApp establishes
+with each E2 node. Given the computing nodes where the Near-RT RIC components are running
+and the graph from Fig. 2, it is possible to compute the round-trip latency starting from an E2
+node ni ∈ N, going through an E2T until an xApp (and possibly other RIC components), and
+back by the same path.
+For each E2 node ni ∈ N, there are four atomic (or indivisible) groups: RIC Man, E2T,
+SDL/STSL, and NIBs, which the placement is represented by a tuple p = (r, t, s, d). We also
+employ the following auxiliary variables: Rcm = {0, 1} that indicates if CN cm ∈ VC runs
+at least an instance of RIC Man, Tcm = {0, 1} that indicates if CN cm ∈ VC runs at least
+an instance of E2T, Scm = {0, 1} that indicates if CN cm ∈ VC runs at least an instance of
+SDL/STSL, and Dcm = {0, 1} that indicates if CN cm ∈ VC hosts at least one NIB instance. For
+example, the tuple p = (c0, c1, c1, c1) represents RIC Man running on c0 and the other atomic
+groups running on c1, i.e., Rc0 = 1, Tc1 = 1, Sc1 = 1, and Dc1 = 1. Finally, we define a set
+of configurations P = {p1, p2, ..., p|P|} that lists all combinations of atomic groups running in
+the available computing nodes. However, the group of xApps can be further disaggregated and
+distributed into multiple computing nodes and, therefore, is not part of this configuration.
+E2TA
+xAppA
+xAppZ
+…
+Near-RT
+RIC Management
+SDL/STSL
+NIBs
+Near-RT RIC 
+Summarized
+Internal Architecture
+E2 Nodes cluster A
+E2TZ
+xAppA'
+xAppZ'
+…
+SDL/STSL
+NIBs
+…
+E2 Nodes cluster Z
+…
+Fig. 3. Summarized version of the Near-RT RIC architecture.
+
+12
+B. Problem Formulation
+The objective function minimizes the total cost to run a disaggregated and distributed Near-
+RT RIC. The cost has two parts, one fixed and another variable. The fixed cost corresponds to
+activating or leasing a CN, which is paid independently of the number of consumed resources.
+While the variable cost depends on the number of Near-RT RIC components running on a CN.
+Each computing node may be assigned a fixed cost, a variable cost, or both.
+Let the decision variable xp
+i ∈ {0, 1} represent the choice of configuration p ∈ P to serve E2
+node ni ∈ N. Let U p
+cm = {0, 1} indicate if CN cm ∈ VC is used in configuration p ∈ P. The
+activation (or leasing) of CN cm ∈ VC is given by:
+Ucm =
+�
+�
+�
+�
+�
+�
+�
+1,
+if �
+ni∈N
+�
+p∈P
+(xp
+i U p
+cm) > 0
+0,
+otherwise.
+(1)
+The fixed cost is defined as:
+ΦFix =
+�
+cm∈VC
+(UcmF(cm)),
+(2)
+where F(cm) ∈ R represents the cost to activate CN cm ∈ VC. While the fixed cost may be less
+common in cloud-native infrastructures, it is included in the model to make it flexible enough
+to represent configurations such as bare-metal services.
+Let the decision variable ya
+i,m ∈ {0, 1} represent the choice of xApp a ∈ A, running on
+cm ∈ VC, to control the E2 node ni ∈ N. The utilization of CN cm ∈ VC to run xApp a ∈ A is
+given by:
+Aa
+cm =
+�
+�
+�
+�
+�
+�
+�
+1,
+if �
+ni∈N
+ya
+i,m > 0
+0,
+otherwise.
+(3)
+The variable cost is defined as:
+ΦV ar =
+�
+cm∈VC
+�
+RcmCR(cm) + TcmCE(cm) + ScmCS(cm)
++ DcmCD(cm)
+�
++
+�
+a∈A
+Aa
+cmCAa(cm),
+(4)
+where CR(cm), CE(cm), CS(cm), CD(cm), and CAa(cm) represent the cost of running RIC Man,
+E2T, SDL/STSL, NIBs, and xApp a ∈ A, respectively, on a given CN cm ∈ VC.
+
+13
+The objective function of minimizing the total cost is finally defined as:
+minimize
+ΦFix + ΦV ar.
+(5)
+Next, we present the constraints of the problem. For each E2 node ni ∈ N, exactly a single
+configuration p ∈ P must be selected:
+�
+p∈P
+xp
+i = 1,
+∀ni ∈ N.
+(6)
+The latency-sensitive control loop has two mandatory segments. The latency of the first one
+is measured between E2 node ni ∈ N and CN cm ∈ VC, which runs the corresponding E2T:
+LcT
+i
+=
+�
+cm∈VC
+(δ(xp
+i )TcmeLat
+ni,cm),
+where δ(xp
+i ) = {0, 1} indicates if CN cm ∈ VC runs the E2T that controls the E2 node ni ∈ N.
+Concerning the second segment of the latency-sensitive control loop, its latency is measured
+between CN cm ∈ VC that runs E2T (associated with the E2 node ni ∈ N) and CN cm′ ∈ VC
+that runs xApp a ∈ A (responsible for controlling the same E2 node ni ∈ N):
+LcT ,cA
+i
+=
+�
+cm,cm′∈VC
+(δ(xp
+i , ya
+i,m′)Aa
+cm′eLat
+cm,cm′),
+where δ(xp
+i , ya
+i,m′) = {0, 1} indicates if CNs cm, cm′ ∈ VC run, respectively, E2T and xApp
+a ∈ A responsible for controlling the E2 node ni ∈ N.
+In a latency-sensitive control loop, however, an xApp a ∈ A (running on CN cm ∈ VC)
+may need to access an SDL/STSL (running on CN cm′ ∈ VC) to reach a NIB (running in CN
+cm′′ ∈ VC). This communication has the following latency:
+LcA,cS,cD
+i
+= ψ(a)
+�
+cm,cm′,cm′′∈VC
+�
+δ(ya
+i,m, xp
+i )Scm′eLat
+cm,cm′
++δ′(ya
+i,m, xp
+i )Dcm′′eLat
+cm′,cm′′
+�
+,
+where ψ(a) = {0, 1} indicates if xApp a ∈ A needs to access SDL/STSL and NIB; δ(ya
+i,m, xp
+i ) =
+{0, 1} indicates if CNs cm, cm′ ∈ VC run, respectively, xApp a ∈ A and SDL/STSL; and
+δ′(ya
+i,m, xp
+i ) = {0, 1} indicates if CNs cm, cm′′ ∈ VC run, respectively, xApp a ∈ A and NIB.
+These components are related to E2 node ni ∈ N.
+
+14
+Additionally, in a latency-sensitive control loop, an xApp a ∈ A (running on CN cm ∈ VC)
+may need to interact with other xApp a′ ∈ A (running on CN cm′ ∈ VC), which generates the
+following latency:
+L
+cA,cA′
+i
+= ψ′(a)
+�
+cm,cm′∈VC
+(δ(ya
+i,m, ya′
+i,m′)Aa′
+cm′eLat
+cm,cm′),
+where ψ′(a) = {0, 1} indicates if xApp a ∈ A needs to communicate with another xApp, and
+δ(ya
+i,m, ya
+i,m′) = {0, 1} indicates if CNs cm, cm′ ∈ VC run, respectively, xApps a, a′ ∈ A, which
+control the E2 node ni ∈ N.
+Considering the potential chain of xApps Ach ⊆ A involved in the control loop of a certain
+xApp a ∈ A, its total latency is the sum of the four previous equations and cannot exceed the
+threshold ρa:
+LcT
+i
++ LcT ,cA
+i
++ LcA,cS,cD
+i
++
+(7)
+�
+a,a′∈Ach
+�
+L
+cA,cA′
+i
++ L
+cA′,cS,cD
+i
+�
+≤ ρa, ∀ni ∈ N, ∀a ∈ A.
+The Near-RT RIC components running on a CN cm ∈ VC must not exceed its processing
+capability (cProc
+m
+), memory (cMem
+m
+), and storage (cSto
+m ) resources:
+RcmRProc + TcmT Proc + ScmSProc + DcmDProc+
+�
+a∈Acm⊆A
+(Aa
+cmAa,Proc) ≤ cProc
+m
+,
+∀cm ∈ VC,
+(8)
+RcmRMem + TcmT Mem + ScmSMem + DcmDMem+
+�
+a∈Acm⊆A
+(Aa
+cmAa,Mem) ≤ cMem
+m
+,
+∀cm ∈ VC,
+(9)
+RcmRSto + TcmT Sto + ScmSSto + DcmDSto+
+�
+a∈Acm⊆A
+(Aa
+cmAa,Sto) ≤ cSto
+m ,
+∀cm ∈ VC,
+(10)
+where R•, T •, S•, D•, and Aa,• represent processing, memory, and storage demands for RIC Man,
+E2T, SDL/STSL, NIBs, and xApp a ∈ Acm ⊆ A, respectively.
+As discussed in [20], the problem represented by Equation (5) is NP-Complete because it
+corresponds to a mixed-integer quadratic programming (MIQP) decision. The proof shows that
+there is a polynomial-length certificate for yes-instances of the MIQP decision so that the MIQP
+decision is in NP. Moreover, the proof that the MIQP decision is NP-complete consists of
+
+15
+providing a polynomial-time reduction of the problem for determining whether there is a cut of
+cardinality at least k in a graph G (which is NP-complete) to an instance of the MIQP decision.
+C. Heuristic Strategy
+MIQP problems such as the one represent by Equation (5) can be optimally solved using
+traditional solvers, e.g., IBM ILOG CPLEX. However, this problem has a large number of
+decision variables that can be estimated as (|N|·|VC|)comp+(|N|·|VC|)|A|, where comp represents
+the number of Near-RT RIC components. For example, a configuration with |N| = 100 E2 nodes,
+comp = 4, |VC| = 5 computing nodes, and |A| = 5 xApps involves more than 3·1013 optimization
+variables. Therefore, due to the computational time and the number of resources required, which
+are extremely high, solving the problem optimally is impractical, even for small instances of the
+problem, as we present in Section VII.
+To deal with large instances of the problem and quickly provide solutions for RIC-O re-
+orchestration requests, the efficient heuristic strategy described as Algorithm 1 is proposed. The
+heuristic starts with a feasible solution and then seeks to improve the cost efficiency through a
+greedy strategy. The heuristic takes as input the same parameters used in the optimal strategy and
+produces as output the placement of the Near-RT RIC components across CNs, including xApps.
+The output is represented by Places[ni, ‘Near-RT RIC component’], where ni ∈ N identifies the
+E2 node that is associated with a ‘Near-RT RIC component’, which can be RIC Man (Ri), E2T
+(Ti), SDL/STSL (Si), NIBs (Di), or xApp ai ∈ Ai. Moreover, let (Zi, ⪯) = Ti ∪ Ai ∪ Si ∪ Di
+be the ordered set of components involved in the latency-sensitive control loop, as described in
+Section IV-A and illustrated in Fig. 3. We present additional details in the following.
+Initially, for each ni ∈ N, the heuristic strategy seeks to place every Near-RT RIC component
+ζi ∈ (Zi, ⪯) at the closest CN cm ∈ VC with enough available resources, as shown in Algorithm 1
+(lines 1–3). This corresponds to the best solution regarding latency, but it ignores the cost of CNs
+and causes unnecessary replication of Near-RT RIC components, e.g., E2T and xApps, which
+also increases cost. Next (lines 4, 5), in CNcost[ˆζi], the algorithm sorts in decreasing order of
+cost the list of CNs that are available for running each Near-RT RIC component ˆζi ∈ Ri∪(Zi, ⪯).
+RIC Man is included because it is a Near-RT RIC component, despite of not being involved in
+the latency-sensitive control loop. The last part of the algorithm (lines 6–9) focuses on improving
+cost efficiency. First (line 7), the RIC Man component is placed on the cheapest CN ni ∈ N
+that has enough available resources. RIC Man tends to be placed on c0 since this tends to be
+
+16
+Algorithm 1: Heuristic strategy
+Input : The same input of the optimal strategy
+Output: Placement of the Near-RT RIC components
+1 forall ni ∈ N do
+2
+forall ζi ∈ (Zi, ⪯) do
+3
+Places[ni, ζi] ← closestCN(ni, ζi)
+4 forall ˆζi ∈ Ri ∪ (Zi, ⪯) do
+5
+CNcost[ˆζi] ← sortByDecreasingCost(ˆζi,VC)
+6 forall ni ∈ N do
+7
+Places[ni, Ri] ← rePlace(Ri, CNcost[Ri])
+8
+forall ζi ∈ (Zi, ⪯) do
+9
+Places[ni, ζi] ← rePlace(ζi, CNcost[ζi])
+10 return Places[•, •]
+the cheapest CN, and, generally, function rePlace determines that node has enough available
+resources. Moreover, a single RIC Man is probably able to serve all E2 nodes. Last (lines 8,
+9), the algorithm seeks to find the cheapest CN ni ∈ N for each Near-RT RIC component
+ζi ∈ (Zi, ⪯). Therefore, in addition to checking CN resource availability, function rePlace also
+checks compliance with the latency-sensitive control loop. These final steps of the heuristic also
+tend to reduce the number of instances of Near-RT RIC components because a single component
+may be able to ensure the latency is within the required threshold for multiple E2 nodes.
+The proposed heuristic strategy solves the problem in polynomial time and produces sat-
+isfactory results regarding the placement of the Near-RT RIC components, as we present in
+Section VII. To be precise, the asymptotic complexity of the Algorithm 1 is O((comp + |A|)2 ·
+|N| · |VC|), which represents the placement of all Near-RT RIC components that serve all E2
+nodes, considering, in the worst case, all available CNs.
+V. RIC ORCHESTRATOR
+This section describes the architecture of the RIC-O which is responsible for the dynamic
+and efficient placement of the disaggregated Near-RT RIC components. The architecture of the
+RIC-O is shown in Fig. 4, along with some of the RIC components required to describe the
+
+17
+orchestrator. RIC-O was designed considering components that run in both Non-RT RIC and
+Near-RT RIC. The RIC-O Optimizer and RIC-O Deployer run on the Non-RT RIC, while
+the RIC-O Triggers execute on the Metrics Server of the Near-RT RIC. The Monitoring
+System is in charge of monitoring resource usage of the Near-RT RIC components, E2 Nodes,
+and O-Cloud infrastructure, in addition to throwing alerts whenever it detects threshold violations
+in the latency-sensitive control loop.
+Near-RT RIC
+Near-RT RIC 
+Management
+Service Management and Orchestration (SMO)
+Non-RT RIC
+E2 Nodes (RAN)
+SDL/STSL
+RIC-O Optimizer
+RIC-O Deployer
+O2 
+O-Cloud
+NIBs
+NIBs
+xApps
+xApps
+xApp
+E2 
+O1
+A1
+Monitoring System
+Alert Manager
+Metrics Server
+RIC-O Triggers
+TSDB
+xApps
+xApps
+E2T
+Fig. 4. RIC-O architecture.
+The Metrics Server is a monitoring module that collects metrics information from all
+xApp and E2T instances running on the Near-RT RIC platform. Example metrics include CPU
+utilization, memory usage, and the communication latency to monitored instances. The latency
+to access SDL/STSL shared layers and databases are also collected by the Metrics Server.
+All metrics are stored in a time-series database (TSDB) for further processing and reporting.
+The RIC-O Triggers component consists of predefined alert rules (e.g., the latency is above a
+given threshold) upon the occurrence of which an alert is generated. As the Metrics Server
+detects some metric violation that is associated with some trigger, it notifies the Alert Manager
+that a metric has been violated. The Alert Manager, in its turn, sends a message to the RIC-O
+
+18
+Optimizer requesting a new round of optimization to reconfigure the system so that it can satisfy
+the metric that has been violated.
+When a request is delivered to the RIC-O Optimizer, it fetches the current metric measure-
+ments from the Metrics Server, executing the optimization strategies described in Section IV.
+The heuristic and optimal optimization strategies run in parallel. The heuristic strategy should
+find a new solution quickly. However, the optimal strategy is executed even if the heuristic
+completes earlier. The solution generated by the optimal strategy is only applied if is superior to
+the heuristic and if it becomes available before a new notification of metric violation. The RIC-O
+Optimizer sends the configuration of the computed solution to the RIC-O Deployer, which
+redeploys the Near-RT RIC components accordingly. The redeployment is only executed if the
+RIC-O Optimizer finds a solution different from the previous one. Lastly, the RIC-O Deployer
+uses the standardized O2 interface defined by O-RAN to exchange messages with the O-Cloud
+element to effectively apply the new placement.
+VI. IMPLEMENTATION
+We implemented a proof-of-concept prototype of RIC-O to validate and evaluate our proposal.
+The prototype is described in this section, along with the testbed used for its evaluation. RIC-O
+Optimizer and RIC-O Deployer were implemented using the Python language. The Monitoring
+System component comprises the monitoring subsystem of the Near-RT RIC platform [21] that
+runs on Kubernetes (K8S). The Near-RT RIC uses Prometheus to collect metrics, and Metrics
+Server corresponds to the main Prometheus server. Metrics are collected at the pod and cluster
+levels and stored as time series data in the internal Prometheus database (TSDB). Prometheus
+was a natural choice to collect metrics in our proposal since it is the native solution of K8S for
+data collection and is part of the Near-RT RIC platform from the O-RAN SC.
+We also employed the Alert Manager component from Prometheus to handle alerts issued by
+different client applications (e.g., Metrics Server). Alert handling may include deduplicating,
+grouping, silencing, inhibiting, and routing alerts to other endpoints. The Alert Manager notifies
+RIC-O Optimizer of the need for starting a new optimization round when a given RIC-O trigger
+detects a latency threshold has been violated. The RIC-O Triggers element was implemented
+using the Prometheus Query Language (PromQL). In this context, our prototype monitors CPU,
+memory, and network latency from K8S system pods, Near-RT RIC pods, and K8S nodes.
+One of the monitored metrics is the latency-sensitive control loop of the Near-RT RIC, which
+
+19
+cannot surpass a given threshold (i.e., above 10ms for certain control loops). Upon reaching this
+threshold, a trigger launches the RIC-O Optimizer to compute a new placement solution for
+the Near-RT RIC components. After the optimization is computed, a notification is delivered to
+RIC-O Deployer, which redeploys the Near-RT RIC components based on the outcome of the
+RIC-O Optimizer.
+Figure 5 shows the message exchange between the RIC-O components, Monitoring System,
+Near-RT RIC components, and the O-Cloud element while orchestrating the optimization and
+deployment tasks. Initially, the deployment is executed as a response to the original request
+coming from the network operator. Subsequently, a re-deploy is executed only after the Alert
+Manager sends a notification to the RIC-O Optimizer due to a latency violation in the Near-RT
+RIC control loop or due to the unavailability of required computing resources in the O-Cloud
+element.
+Notify RIC (Re)deployment
+Send metrics data
+Request metrics data
+SMO
+RIC-O
+Deployer
+RIC-O
+Optimizer
+Near-RT RIC
+Components
+O-Cloud
+Alert 
+Manager
+Deploy (Re)allocation Plan
+Instantiate RIC Components
+Notify Instantiation
+Deallocate RIC Components
+Notify Deallocation
+Run Optimization
+5
+8
+9
+10
+11
+12
+13
+14
+15
+Monitoring
+System
+Send metrics data
+Send metrics data from RIC and O-Cloud
+1
+2
+3
+4
+6
+7
+Only on
+Redeploying
+For
+Each
+Demand Reallocation Plan
+Request metrics data
+Request metrics data from RIC and O-Cloud
+Send (Re)allocation Plan
+Monitoring
+Loop
+Fig. 5. Messages exchanged for Near-RT RIC optimization.
+The Monitoring System shown in Fig. 5 collects metrics regularly from the Near-RT RIC
+Components and the O-Cloud element. These metrics include CPU and memory usage of
+
+20
+Near-RT RIC pods, K8S nodes, and their communication latency (messages 1, 2, 3, and 4).
+The Monitoring Loop is executed regularly at every second. Once Metrics Server notifies
+Alert Manager that a given RIC-O trigger has been violated, message 5 is delivered to RIC-O
+Optimizer.
+Afterward, the RIC-O Optimizer starts a reallocation plan to compute the placement of the
+Near-RT RIC Components that run on the O-Cloud element. RIC-O Optimizer then requests
+metrics information that includes the available computing resources of K8S pods, CNs, and the
+communication latency between them (messages 6 and 7), which is the information required
+as input for the optimization strategies. Next, the heuristic and optimal strategies are initiated
+and run in parallel (message 8). Once an outcome from the optimization strategies becomes
+available, it is sent to the RIC-O Deployer in message 9. Eventually, if there is enough time
+for computing the optimal solution, two outcomes are generated. The RIC-O Deployer, in turn,
+starts deploying the new (re)allocation plan in O-Cloud through the O2 interface defined by
+O-RAN (message 10). Near-RT RIC components are instantiated by O-Cloud, which is notified
+when all the components are deployed (messages 11 and 12). In the particular case when the
+Near-RT RIC components were deployed previously, O-Cloud releases the resources of the Near-
+RT RIC components that are no longer required (messages 13 and 14). Finally, O-Cloud notifies
+the RIC-O Deployer that the deployment has finished (message 15).
+We also implemented an extended xApp and an E2 simulator (E2Sim), both coded in C++
+language. Our implementation is based on the open-source versions of the Bouncer xApp* and
+the E2Sim† from the O-RAN SC. Moreover, we included new features and updated the service
+models of both the Bouncer xApp and E2Sim. Originally, the Bouncer and E2Sim employed the
+Key Performance Measurement service model (E2SM-KPM) to expose performance measure-
+ments of logical functions running on E2 Nodes [22]. However, E2SM-KPM does not support
+RAN control through the E2AP Control service. Therefore, we implemented the E2SM-RC
+(RAN Control) [23] service model, which provides RAN control. Figure 6 shows the message
+exchange between the Bouncer xApp, E2Sim, and Near-RT RIC components. The goal is to
+monitor the latency-sensitive control loop between the E2 nodes and the corresponding xApp.
+We assume that all the Near-RT RIC components and E2Sim are already set up and running
+*https://gerrit.o-ran-sc.org/r/admin/repos/ric-app/bouncer
+†https://gerrit.o-ran-sc.org/r/admin/repos/sim/e2-interface
+
+21
+when Bouncer xApp is deployed.
+Control 
+Loop
+Near-RT RIC Management
+loop
+Registration Request
+Registration Response
+REST Subscription Request
+RIC Subscription Response
+RIC Subscription Request
+Subscription
+Manager
+Routing
+Manager
+xApp
+Manager
+E2T
+E2Sim
+REST Subscription Response
+Route Create
+E2 Subscription Request
+E2 Subscription Response
+REST Subscription Notification
+3.1
+3.2
+3.3
+4
+6
+5
+7
+8
+Bouncer
+xApp List Request
+RIC Insert Request
+RIC Insert Request
+1.1
+1.2
+9
+10
+RIC Control
+RIC Control
+11
+12
+3.4
+xApp List Response
+Route Create
+a) Update Routes
+b) Update Routes
+a) Update Routes
+b) Update Routes
+2.1
+2.2
+2.3
+2.4
+2.4
+3.4
+par
+Fig. 6. Messages exchanged for xApp deployment and execution.
+When Bouncer xApp initializes, it registers itself with the Near-RT RIC platform by sending
+a registration message to xApp Manager, which sends back a reply indicating the outcome
+(messages 1.1 and 1.2). In the meantime, the Routing Manager periodically retrieves the list
+of xApps registered in xApp Manager to set up the corresponding routing rules in the Near-RT
+RIC platform (messages 2.1 and 2.2). Moreover, the Routing Manager builds a routing table
+for all registered xApps (message 2.3) and then distributes it to Bouncer and E2T (messages
+2.4a and 2.4b). These routes are required to allow transparent and dynamic message delivery to
+the endpoints concerned in the communication. During the registration, the Bouncer xApp also
+
+22
+subscribes to Subscription Manager to receive messages from the E2AP Insert service from
+the RAN (messages 3.1 and 3.2). Upon receiving the subscription request, the Subscription
+Manager notifies Routing Manager to create the corresponding routing rules to deliver those
+messages to Bouncer xApp (message 3.3). Next, the Routing Manager distributes the generated
+routing rules to the respective endpoints (messages 3.4a and 3.4b).
+Additionally, the Subscription Manager issues a new subscription request (message 4) to
+the E2T, which implements the E2 interface to communicate with E2 nodes. The E2T then
+deals with the subscription communication with the E2Sim (messages 5 and 6) according to the
+subscription that the Bouncer xApp has generated in message 3.1. The communication between
+E2T and E2Sim is established through Stream Control Transmission Protocol (SCTP). Messages
+7 and 8 notify Subscription Manager and Bouncer xApp about the subscription outcome. At
+this point, Bouncer xApp is fully deployed in the Near-RT RIC platform, and communication
+with E2Sim can take place through the E2T component.
+The Control Loop block in Fig. 6 illustrates the latency-sensitive control loop of 10ms. When
+a given E2 node issues a RIC Insert Request message to the Near-RT RIC, the E2 node suspends
+its execution and waits for the corresponding reply message (i.e., RIC Control) to resume its
+operation. Messages 9-12 illustrate this control loop that needs to complete within the 10ms
+threshold. If no RIC Control message is delivered within this threshold time, a default action is
+executed by the E2 node, which usually implies penalties in the optimization and performance of
+RAN. Moreover, we implemented the capability of computing the control loop latency between
+E2Sim and Bouncer xApp. In this case, E2Sim collects the timestamps on sending message 9 and
+receiving message 12. The latency-sensitive control loop is exported to the Prometheus metrics
+collector.
+VII. EVALUATION
+This section presents a performance evaluation of RIC-O both from a theoretical perspective,
+using analytical modeling (in Subsection VII-A), and from a practical perspective, employing
+real-world experiments (in Subsection VII-B). The first part of the evaluation focuses on scala-
+bility, while the second mainly deals with timing issues.
+
+23
+A. Analytical modeling
+In this part of the evaluation, we assume as RAN topology a next-generation hierarchical
+network with 512 E2 nodes organized in three main tiers [24]. The top one (Tier-1) is closest
+to the core and so to the cloud node (i.e., c0). The one-way latency (in ms) between the tiers
+is a uniform random choice from the set {1, 2, 2, 3, 3}, and the one-way latency from Tier-1
+to c0 is 4ms. We vary the number of edge computing nodes (i.e., cm ∈ C) along the network
+tiers and, for each number of CNs, extract an overlay topology equivalent to the one shown
+in Fig. 2. There is always a cloud node, and all others are edge computing nodes distributed
+from bottom to top in the hierarchical RAN network as the number of CNs increases. The cloud
+node has no fixed cost and presents the following variable cost for Near-RT RIC components:
+two (RIC Man, E2T), one (SDL/STSL, NIBs), and one (xApp). There are two xApps, and both
+of them access the database, i.e., SDL/STSL and NIBs. Additional information about underlay
+RAN topology, and Near-RT RIC is presented in Table I.
+Parameters
+Tier-1
+Tier-2
+Tier-3
+Number of E2 nodes
+5
+20
+487
+CN fixed cost
+10
+20
+30
+CN variable cost: (RIC Man, E2T),
+(SDL/STSL, NIBs), (xApps)
+4, 2, 1
+8, 4, 2
+16, 8, 4
+cP roc
+m
+(cores)
+32
+16
+8
+cMem
+m
+(GB)
+64
+32
+16
+cSto
+m
+(GB)
+256
+256
+256
+Near-RT RIC requirements
+Proc
+Mem
+Sto
+RIC Man
+4
+8
+4
+E2T
+2
+4
+2
+SDL/STSL
+2
+4
+1
+NIBs
+1
+2
+50
+xApps
+1
+2
+1
+TABLE I
+PARAMETERS OF THE UNDERLAY RAN TOPOLOGY AND REQUIREMENTS OF THE NEAR-RT RIC COMPONENTS.
+We first investigate to which extent the complexity of the exact optimization model (Sub-
+section IV-B) limits its scalability. Although our heuristic strategy (Subsection IV-C) is very
+efficient, it is necessary to investigate the quality of its solutions, i.e., how close to optimal those
+solutions tend to be. Therefore, we evaluate and compare the optimal and heuristic strategies
+
+24
+regarding computing time and objective function as presented in Fig. 7. The computing time is
+related to the scalability of each strategy, and the objective function, i.e., the total cost, represents
+the quality of the solution. Moreover, the metrics vary as a function of the number of CNs.
+Fig. 7. Comparison between the optimal and heuristic strategies regarding computing time and total cost (objective function).
+We can observe in Fig. 7 that the optimal strategy scales poorly, being unable to find a solution
+for a RAN with 12 CNs even after running for an entire day. Therefore, the optimal strategy
+is rarely helpful for large real-world instances. Nevertheless, the heuristic strategy scales very
+well and finds a solution for 512 CNs in less than 10 seconds. Concerning the quality of the
+solutions, as expected, the heuristic strategy cannot always find the optimal solution. However,
+a decreasing trend can be discerned in the objective function that suggests the heuristic strategy
+is in the right direction, including in scenarios with a large number of CNs, e.g., 512 CNs. In
+those cases, the cheapest edge computing nodes (i.e., those in Tier-1) are readily available, and
+the solution improves noticeably.
+We further investigate the optimization strategies by evaluating the number of instances of the
+main Near-RT RIC components involved in the latency-sensitive control loop, i.e., xApps and
+E2T. As presented in Fig. 8, our heuristic strategy finds the best (i.e., the smallest) number of E2T
+instances, matching the result obtained by the optimal strategy when this is available and keeping
+it at the minimum (i.e., one instance) as the number of CNs increases. Moreover, the heuristic
+strategy finds some non-optimal solutions for the number of xApps instances, e.g., with four
+
+108
+Heuristic
+106
+-Optimal
+104
+102
+100
+2
+4
+6
+8
+10
+12
+16
+32
+64
+128
+256
+512
+CNs (#)Objective Function (#)
+170
+130
+90
+50
+10
+2
+4
+6
+8
+10
+12
+16
+32
+64
+128
+256
+512
+CNs (#)25
+2
+4
+6
+8
+10
+12
+16
+32
+64
+128
+256
+512
+CNs (#)
+0
+1
+2
+3
+4
+5
+Instances (#)
+Heuristic
+Optimal
+xApps
+xApps
+E2T
+E2T
+Fig. 8. Comparison between optimal and heuristic strategies regarding the number of xApps and E2T instances.
+and six CNs. This behavior is expected from our heuristic strategy because it prioritizes Near-
+RT RIC components involved in the latency-sensitive control loop in the following order: E2T,
+xApps, SDL/STSL, and NIBs. Since latency optimization and cost optimization are performed
+in this order and without revisiting previous decisions for improvement, the heuristic strategy
+potentially finds non-optimal solutions involving xApps, SDL/STSL, or NIBs.
+B. Real-world experiments
+In this part of the evaluation, we run the experiments in a scenario with five CNs, which are
+virtual machines (VMs) with the following configuration: 4 vCPUs, 8 GB RAM, and 50 GB
+of the virtual disk. One CN represents the cloud node (i.e., c0), and the others represent the
+edge computing nodes (i.e., cm ∈ C). These CNs are worker nodes in a K8S cluster managed
+by a master node running a sixth VM with the following configuration: eight vCPUs, 16 GB
+RAM, and 100 GB of the virtual disk. All VMs are hosted on a DELL PowerEdge M610 server
+with four Intel Xeon X5660 processors and 192 GB RAM, which runs VMware ESXi 6.7 as the
+hypervisor. Additional details about the software tools employed in the experiments are available
+in the public repository of this article. We also employed an E2 simulator to represent four E2
+nodes that must be served by the Near-RT RIC. Overlay and underlay topologies are the same
+in this part of the evaluation and correspond to the one presented in Fig. 2.
+To illustrate the orchestration capabilities of RIC-O, we designed two scenarios in which the
+latency-sensitive control loop is disrupted and show how our proposal acts to bring the Near-RT
+RIC back to normal operation. In the first scenario, RIC-O must deal with a sudden and high
+increase in the latency of the path used to serve a certain E2 node. The second scenario is
+more challenging because RIC-O needs to deal with a CN that becomes unavailable, i.e., any
+
+26
+latency-sensitive control loop involving this CN disappears since the Near-RT RIC components
+running in it suddenly become inaccessible.
+In a real-world RAN, the latency between a pair of nodes may change due to a (re)route
+decision in the underlying network, for example. Since our underlay network matches the overlay
+one, we emulate the sudden increase in the latency between the E2 node and its corresponding
+E2T by reconfiguring the latency in the virtual link connecting these nodes. Figure 9 illustrates the
+main events occurring along the time in this scenario. This figure shows the status of the control
+loop between each E2 node and its corresponding xApp. In addition, the figure presents the CPU
+utilization of some essential software components (i.e., RIC-O Deployer, RIC-O Optimizer,
+xApps, and E2T), which helps keep track of the actions performed by RIC-O.
+Control loop 
+failure
+Control loop
+satisfied
+Start 
+redeploy
+Optimization
+trigger
+Fig. 9. Reaction to a sudden violation of requirements of the latency-sensitive control loop.
+As illustrated in Fig. 9, the first scenario is initially in a fully operational state, and the
+latency-sensitive control loop of each E2 node is satisfied by the Near-RT RIC thanks to the
+initial orchestration defined by RIC-O. Then, at time instant 150s, the latency of the control loop
+from E2 node 4 increases suddenly and remains persistently above 10ms, as indicated by event
+Control loop failure in the figure. After 10 seconds, the Monitoring System considers that the
+event is a consistent control loop violation and notifies the RIC-O Optimizer to compute a
+new solution, at time instant 160s, as indicated by the event Optimization trigger. The heuristic
+strategy of the RIC-O Optimizer quickly finds a solution and requests RIC-O Deployer to apply
+
+Control loop (ms)
+E2Node 1
+102
+E2Node 2
+E2Node 3
+E2Node 4
+101
+M
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+Time (s)101
+CPU (milicores)
+Deployer
+Optimizer
+xApps
+E2T
+100
+10-1
+10-2
+10
+=
+1027
+this new placement nearly 5 seconds later, as indicated by the event Start redeploy. The RIC-O
+Optimizer keeps running the optimal strategy thread. Finally, the redeploy of the Near-RT RIC
+components and reconfiguration of E2 nodes completes at time instant 200s, as indicated by
+event Control loop satisfied, when the latency-sensitive control loop is again limited to 10ms.
+Figure 9, and also Fig. 10 show a few measurements of the control loop that go above 10ms.
+This behavior is related to the underlying operating system and virtualization platform (i.e.,
+hypervisor). A tight threshold of 10ms is on such a sensitive scale that even a traditional process
+scheduler may sometimes cause a small variation. Since fine-tuning those systems are out of the
+scope of this work, we configured the Monitoring System to report only persistent violations
+of the 10ms threshold in latency-sensitive control loops.
+The second scenario, in which a CN suddenly crashes, may either represent a software or
+hardware failure, or network outage. We emulate this behavior by abruptly forcing a shutdown
+of the VM running the CN. Figure 10 illustrates the main events occurring along the time in
+this scenario. Similar to Fig. 9, Fig. 10 also shows the status of the control loop between each
+E2 node and its corresponding xApp. However, we have not identified relevant information that
+justified presenting measurements related to Near-RT RIC nor RIC-O components.
+CN down
+Finish 
+redeploy
+Start 
+redeploy
+Optimization
+trigger
+Fig. 10. Reaction to sudden unavailability of CN under use.
+As illustrated in Fig. 10, the second scenario also starts from a fully operational state, where
+the latency-sensitive control loops of all E2 nodes are satisfied by the Near-RT RIC thanks to the
+initial orchestration defined by RIC-O. The CN running Near-RT RIC components responsible
+for serving the E2 node 4 suddenly becomes unavailable, as indicated by the event CN down
+in the figure. In this case, the latency-sensitive control loop of this E2 node is totally disrupted,
+i.e., there are no more measurements of the control loop. After 50 seconds, the Monitoring
+System detects the problem and notifies the RIC-O Optimizer to compute a new solution, as
+
+(sw) dooj
+E2Node 1
+102
+E2Node 2
+E2Node 3
+E2Node 4
+101
+Control 
+AMMA
+100
+20
+40
+60
+80
+100
+120
+140
+160
+180
+Time (s)28
+indicated by the event Optimization trigger. This time interval for reporting the problem may
+seem long, but it is the default K8S policy for detecting worker node unavailability. Again,
+after nearly 5 seconds, the heuristic strategy of RIC-O Optimizer finds a solution and requests
+RIC-O Deployer to apply the new placement, as indicated by the event Start redeploy. Finally,
+the redeployment of the Near-RT RIC components finishes, and the latency-sensitive control loop
+of E2 node 4 is reestablished at time instant 130s, as indicated by the event Finish redeploy.
+VIII. CONCLUSION
+In this work, we claim that the most efficient approach to ensure that the Near-RT RIC is able to
+deal with a large real-world RAN is by disaggregating and distributing its components. However,
+it is necessary to orchestrate these components properly across the edge-cloud continuum to
+ensure the latency-sensitive control loop that involves xApps running over the Near-RT RIC.
+Therefore, we propose the RIC orchestrator (RIC-O) that employs a hybrid strategy combining
+optimal and heuristic strategies for efficient placement of the Near-RT RIC components. We
+evaluate RIC-O through analytical modeling and real-world experiments to illustrate its properties
+and benefits. Future work includes the investigation of machine learning-based time series
+forecasting for improving the orchestration of Near-RT RIC components. Moreover, the adoption
+of the O-RAN architecture by large-scale production networks will provide additional data on
+its usage, thus allowing extensive investigation of novel optimization techniques. Finally, we
+envision the potential of the RIC-O being adopted as part of the Non-RT RIC, running as an
+rApp, which will demand further experiments and software integration.
+ACKNOWLEDGMENT
+This work was conducted with partial financial support from the National Council for Scientific
+and Technological Development (CNPq) under Grant Nos. 405111/2021-5 and 130555/2019-3
+and from the Coordination for the Improvement of Higher Education Personnel (CAPES) -
+Finance Code 001, Brazil. Moreover, RNP partially supported the work, with resources from
+MCTIC, Grant No. 01245.010604/2020-14, under the 6G Mobile Communications Systems
+project and by MCTIC/CGI.br/S˜ao Paulo Research Foundation (FAPESP) through the Project
+SAMURAI - Smart 5G Core And MUltiRAn Integration under Grant 2020/05127-2 and Pro-
+grammability, ORchestration and VIRtualization of 5G Networks (PORVIR-5G) under Grant No.
+2020/05182-3.
+
+29
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+
diff --git a/gNE0T4oBgHgl3EQf6QIN/content/tmp_files/load_file.txt b/gNE0T4oBgHgl3EQf6QIN/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf,len=848
+page_content='1  This work has been submitted to the IEEE for possible publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Copyright may be trans-  ferred without notice, after which this version may no longer be accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  RIC-O: Efficient placement of a disaggregated  and distributed RAN Intelligent Controller with  dynamic clustering of radio nodes  Gabriel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Almeida, Gustavo Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Bruno, Alexandre Huff,  Matti Hiltunen, Elias P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Duarte Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', Cristiano B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Both, Kleber V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Cardoso  Abstract  The Radio Access Network (RAN) is the segment of cellular networks that provides wireless  connectivity to end-users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' O-RAN Alliance has been transforming the RAN industry by proposing  open RAN specifications and the programmable Non-Real-Time and Near-Real-Time RAN Intelligent  Controllers (Non-RT RIC and Near-RT RIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Both RICs provide platforms for running applications  called rApps and xApps, respectively, to optimize the behavior of the RAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' We investigate a disaggre-  gation strategy of the Near-RT RIC so that its components meet stringent latency requirements while  presenting a cost-effective solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' We propose the novel RIC Orchestrator (RIC-O) that optimizes  the deployment of the Near-RT RIC components across the cloud-edge continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Edge computing  nodes often present limited resources and are expensive compared to cloud computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' For example, in  the O-RAN Signalling Storm Protection, Near-RT RIC is expected to support end-to-end control loop  latencies as low as 10ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Therefore, performance-critical components of Near-RT RIC and certain xApps  Gabriel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Almeida and Kleber V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Cardoso are with the Universidade Federal de Goi´as (UFG), Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' E-mail:  {gabrielmatheus, kleber}@inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='ufg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='br  Gustavo Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Bruno and Cristiano B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Both are with the University of Vale do Rio dos Sinos (UNISINOS), Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' E-mail:  zanattabruno@edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='unisinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='br, cbboth@unisinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='br  Alexandre Huff is with Universidade Tecnol´ogica Federal do Paran´a, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' E-mail:alexandrehuff@utfpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='br  Matti Hiltunen is with AT&T Labs Research, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' E-mail: hiltunen@research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='att.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='com  Elias P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Duarte Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' is with Universidade Federal do Paran´a, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' E-mail: elias@inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='ufpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='br  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='02760v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='NI]  7 Jan 2023  2  should run at the edge while other components can run on the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Furthermore, RIC-O employs an  efficient strategy to react to sudden changes and re-deploy components dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' We evaluate our  proposal through analytical modeling and real-world experiments in an extended Kubernetes deployment  implementing RIC-O and disaggregated Near-RT RIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Index Terms  RAN Intelligent Controller, O-RAN, Near-RT RIC, placement, disaggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' INTRODUCTION  The Radio Access Network (RAN) is considered the most critical segment of a cellular  network [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' A RAN comprises network nodes that provide direct wireless connection to end-  users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' It requires continuous innovations to improve the overall mobile network performance  and user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Novel RAN technologies have included innovations related to wireless  communication technologies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', mmWave, massive MIMO, intelligent reflective surfaces,  and THz communications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Other innovations are related to current networking technologies,  particularly network softwarization, virtualization, programmability, and management (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', NFV,  SDN, Network Slicing, and Zero touch network & Service Management).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Traditionally, the  RAN accounts for most of the CAPEX and OPEX costs of a wireless mobile network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Network  operators also have to face several hurdles, such as interoperability issues, resulting from the  diversity of hardware from different vendors and the burden of continuously supporting new  services and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' In this context, the O-RAN Alliance [2] has been transforming the  RAN industry by proposing open, virtualized, fully interoperable, and intelligent mobile networks  [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  The O-RAN Alliance defines a series of specifications describing open interfaces to ensure  multi-vendor interoperability among the main components that form the RAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' That “openness”  has been carefully introduced to allow interoperability without hurting the intellectual property of  the participating companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Given the significant investments necessary to promote innovation  of RAN technologies, this approach is essential to keep the interest of those companies in  contributing to this vital area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Yet another advantage of an open RAN is that it decreases the  barriers for newcomers that can contribute to solving specific problems and move the technology  further, particularly in academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' On a side note, it is important to highlight that O-RAN  3  specifications [4] are aligned with 3GPP standards [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Therefore, it is reasonable to assume that  as the 3GPP introduces enhancements, they are promptly incorporated by the O-RAN Alliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  The new RAN architecture [4] stands out as a key contribution among the O-RAN specifica-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The architecture combines concepts from SDN and NFV and also takes into account cloud-  native technologies while largely adopting Artificial Intelligence (AI) and Machine Learning  (ML) technologies [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' It adopts SDN concepts, such as control and data planes separation and  the possibility of having a remote RAN controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The design of a RAN controller presents  formidable challenges, as it must support the execution of tens to hundreds of RAN functions  while featuring a large protocol stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The O-RAN architecture splits the controller into two main  building blocks: the Near-Real-Time RAN Intelligent Controller (Near-RT RIC) for time-sensitive  operations and the Non-Real-Time RAN Intelligent Controller (Non-RT RIC) for operations that  present fewer time restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Moreover, the O-RAN specifications standardize open interfaces  among the architecture components, which run as virtual (network) functions (or services) on  cloud-native infrastructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Those RAN controllers run AI/ML-based applications that establish control loops with the  RAN nodes under their management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The Non-RT RIC runs applications called rApps that  demand control loops to time intervals above 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The Near-RT RIC runs applications called  xApps that establish control loops constrained to time intervals between 10ms and 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The time  constraint of a given control loop depends on the RAN function under the management of the  corresponding xApp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' For example, an xApp related to medium access management may need  to complete the control loop under the 10ms threshold, while an xApp related to user session  management may tolerate longer delays of up to 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' In a large RAN, the Near-RT RIC (or some  of its components) and latency-sensitive xApps must be replicated and assigned to manage a  limited set of RAN nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', a cluster of RAN nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Determining the minimum number of  Near-RT RIC instances and where they must run is a non-trivial resource allocation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  The problem becomes even more challenging considering the dynamics of a mobile wireless  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' While this problem has been previously identified [7], some works rely on multiple  instances of the Near-RT RIC [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Also, to the best of our knowledge, there is no comprehensive  solution to the problem as we describe in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  In this work, we propose a disaggregation strategy of the Near-RT RIC so that individual  components can be distributed and placed across the cloud-edge continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' We also propose a  RIC Orchestrator (RIC-O) to deploy (or place) and monitor the Near-RT RIC components so that  4  they can meet the stringent latency requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The RIC-O employs optimization to deploy  the Near-RT RIC components across the cloud-edge continuum while keeping the overall cost  as low as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Edge computing nodes often present limited resources and are expensive  compared to cloud computing nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' While performance-critical components of the Near-RT  RIC platform and certain xApps should run at the edge, other components can run on cloud  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Furthermore, RIC-O employs a fast and efficient strategy to react dynamically to sudden  changes and redeploy components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' We also explore the flexibility of the O-RAN architecture to  introduce a proposal that replicates some specific components of the Near-RT RIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' We evaluate  the proposal through analytical modeling and real-world experiments in an extended Kubernetes  deployment that runs the RIC-O and the disaggregated Near-RT RIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  The main contributions of this work can be summarized as follows:  Proposes a disaggregation strategy to place the Near-RT RIC components across the cloud-  edge continuum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Formalizes the problem of minimizing the overall cost of the placement of Near-RT RIC  components while ensuring the latency-sensitive control loop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Proposes a hybrid approach that combines heuristic and optimal strategies to quickly provide  cost-efficient solutions for placing Near-RT RIC components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Provides performance evaluation results to illustrate the advantage of the proposed approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  All implementations are publicly available, including the source code*, thus, allowing the  work reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Section II presents an overview of the O-RAN architecture, particularly of the Near-RT  RIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Section III describes the related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Section IV provides the system model, problem  formulation, and the optimal and heuristic strategies for the placement of the Near-RT RIC  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Section V introduces the RIC orchestrator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', the proposed architecture that allows  efficient placement of Near-RT RIC components and clustering of RAN nodes across the cloud-  edge continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The implementation of the RIC orchestrator is described in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Real-  world experiments and an analytical performance evaluation are presented in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Finally,  Section VIII concludes the article and discusses future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='com/LABORA-INF-UFG/paper-GGAMECK-2023  5  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' BACKGROUND  The O-RAN architecture defines two RICs responsible for controlling and managing RAN  nodes on different time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' While the Non-RT RIC is in charge of tasks that can present a  latency above 1s, the Near-RT RIC is responsible for tasks that demand a latency between 10ms  and 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Figure 1 illustrates the O-RAN architecture highlighting the Near-RT RIC internal compo-  nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Several of those components are not latency-sensitive, such as the Management Services,  the xApp Manager, the Subscription Manager, and A1 and O1 terminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' However, E2  Nodes, E2 termination (E2T), xApps, and Shared Layers can be sensitive to latency and require  correct placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The xApp Manager provides a flexible way for deploying and managing the  near real-time applications (xApps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The xApps are applications that run services responsible  for improving the performance of the RAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Each xApp runs a well-defined function to access,  control, and monitor E2 Nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' While some xApps may implement latency-sensitive functions,  other xApps can also tolerate latencies up to 1s depending on the tasks they execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Moreover,  xApps can provide services to other xApps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The Subscription Manager is responsible for  providing E2 subscriptions from xApps to E2 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The Management Services includes several  services provided by the Near-RT RIC to the xApps and E2 nodes, such as routing management,  alarm notification service, logging, and E2 node management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  A1  O1  E2T  Messaging Infrastructure  Termination  xAppA xAppB  xAppZ  …  Subscription  Manager  xApp  Manager  Management  Services  …  R-NIB, UE-NIB, …  Service Management and Orchestration (SMO)  Near-RT RIC Internal Architecture  Non-RT RIC  E2 Nodes (RAN)  E2AP (Insert)  E2AP (Control)  O1  Shared Layers  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' O-RAN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  The A1 termination also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 1 provides an interface between Non-RT RIC and  Near-RT RIC to exchange information about policies and machine learning models that are exe-  6  cuted by xApps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The O1 termination represents the interface between the Service Management  and Orchestration (SMO) framework with network functions and E2 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' This termination  enables the SMO to execute Fault, Configuration, Accounting, Policy, and Security (FCAPS)  operations on those network functions and E2 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' All O-RAN components are expected to  support the O1 termination when exchanging messages with SMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The SMO is responsible for  managing and orchestrating the entire RAN, relying on the Non-RT RIC for RAN optimization,  and the O1 termination for RAN instrumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Moreover, SMO is in charge of managing and or-  chestrating any cloud infrastructure in use [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The E2T is responsible for connecting the Near-RT  RIC with one or more E2 Nodes using E2 Application Protocol (E2AP) and E2 Service Model  (E2SM) which are described further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The Messaging Infrastructure element corresponds to  the underlying messaging subsystem employed by all the Near-RT RIC components to exchange  information [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The Shared Layers correspond to the Shared Data Layer (SDL) and Shared  Time-Series Layer (STSL), which are high-speed interfaces for accessing shared data storage  by several stateless Near-RT RIC components and xApps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' R-NIB and UE-NIB are examples of  databases that store network information for RAN and User Equipment (UE), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  According to the O-RAN specifications [4], RAN nodes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', DUs, CUs, or O-RAN-compliant  LTE eNBs) are E2 Nodes controlled through the E2 interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' This interface allows the Near-  RT RIC to control procedures and functionalities of those nodes and is logically organized  into two parts: E2AP and E2SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' E2AP enables the communication between the Near-RT RIC  and E2 nodes, which provide four services to the Near-RT RIC: Report, Insert, Control, and  Policy [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The Report service allows xApps to subscribe to E2 nodes to receive information  about specific RAN events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The Insert service also provides RAN information to xApps, but  also allows xApps to configure E2 nodes through control messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The Control service allows  the Near-RT RIC and xApps to send control messages to E2 nodes, which can initiate new  procedures or resume a previously suspended procedure associated with that E2 node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Such a  procedure should specify exactly and completely the functional behavior of a given E2 node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Finally, the Near-RT RIC uses the Policy service to establish policy-driven monitoring and control  the behavior of the corresponding E2 node [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' These services can be combined in different  ways to implement E2SMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' An E2SM [9] can be described as a contract between xApps and  the RAN functions on E2 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Each RAN function exposed by a given E2SM allows the  Near-RT RIC to monitor, suspend, stop, override or even control the behavior of the RAN on  that E2 node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Therefore, xApps and their corresponding E2 nodes must implement the same  7  E2SM definitions to communicate and control the RAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  The E2AP Insert service is the most latency-sensitive loop of the Near-RT RIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' This loop  starts when a given E2 node sends a message to the E2T using the E2AP (Insert) service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The  E2T, in its turn, delivers that message to the corresponding xApp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' This xApp then processes the  message and sends back the corresponding reply using the E2AP Control service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Moreover,  the control loop can involve access to Shared Layers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', SDL or STSL), it may also have to  access one or more databases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', R-NIB and UE-NIB) and even other xApps, as illustrated in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' We note that in some use cases, such as the O-RAN Signaling Storm Protection [11], it  is essential to deliver the RIC control message back to the E2 node within the 10ms threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  In this example use case, the E2 node suspends the current procedure execution and waits for  the corresponding reply from an E2AP (Control) service to resume its operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Although an  E2 node can apply a default action when the 10ms threshold is exceeded, this usually implies  applying further optimization procedures and performance penalties for the RAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' In this work,  we propose to disaggregate and distribute the Near-RT RIC components to guarantee the latency  restrictions of the time-sensitive xApps, which execute under a control loop limit of 10ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  The O-RAN Software Community (OSC) provides an open-source reference implementa-  tion [12] of the Near-RT RIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' This implementation also functions as a baseline that allows the  community to explore alternative deployments, including, for instance, replication, among other  innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The SMO framework must be particularly aware of any alternative configuration  so that it can dynamically (1) detect any latency violation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', latency above the limit in the  critical loop);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' (2) redeploy components to solve the violations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' (3) re-orchestrate the Near-RT  RIC components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' and, (4) reconfigure all involved elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 1, the SMO  communicates with all RAN elements, including the E2 Nodes, through the O1 interface that  offers the necessary means to reconfigure and reorganize the cluster of E2 nodes that each E2T  and xApps are responsible for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Current SMOs do not fully implement all tasks in a coordinated  manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' We claim one of the contributions of the current work is to present an SMO extension that  can provide all necessary capabilities to accomplish those tasks coherently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Another contribution  is related to task (2), which comprises the resource allocation problem, for which we formulate  and propose optimal and heuristic strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  8  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' RELATED WORK  This section presents an overview of related work that also has a focus on the RIC architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  In a small RAN with a few dozen or less E2 nodes, it may be possible to deploy a single Near-RT  RIC that satisfies the latency-sensitive control loop of the corresponding xApps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' However, larger  RANs with hundreds or even thousands of E2 nodes demand a different approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Dryjanski and  Kliks [7] present two options for implementing the Near-RT RIC: centralized and distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  In the centralized option, every E2 node (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', the whole gNB or eNB) is handled by the same  and only Near-RT RIC, which can take unified decisions for an individual E2 node and globally  optimize operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' In the distributed option, each E2 node type (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', O-CU, O-DU, or O-eNB)  is handled by a specialized instance of the Near-RT RIC that allows optimizing these individual  types of E2 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The authors discuss the impact of these two implementation options in the  design of the E2 interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' However, they do not tackle the distribution and replication of the  Near-RT RIC components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Singh and Nguyen [13] propose a framework called O-RANFed to deploy and optimize a set  of Federated Learning (FL) tasks that provide 5G slicing services in the context of the O-RAN  specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' To be precise: the authors introduce a theoretical model of the RIC architecture  with support for FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Moreover, the authors present an optimization model for the problem of  local learning selection and resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The performance of FL improves with modeling  and training done in every iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' However, some details of the proposed FL-supporting RIC  architecture are missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Huff, Hiltunen, and Duarte [14] discuss and evaluate techniques to make the RIC fault-tolerant  while preserving high scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The fundamental assumption of this work is that traditional  replication techniques cannot sustain high throughput and low latency as required by RAN  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The authors propose techniques that use state partitioning, partial replication, and fast  re-route with role awareness to decrease the overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Moreover, the authors implemented the  fault tolerance techniques as a library called RFT (RIC Fault Tolerance) considering a distributed  RIC, but do not deal with the problem of the placement of RIC components in a disaggregated  virtual infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  D’Oro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' [8] introduced a novel orchestration framework called OrchestRAN for the Non-  RT RIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' OrchestRAN allows network operators to specify high-level control and inference  objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The orchestrator defines the optimal set of data-driven algorithms and their execution  9  locations to achieve intents specified by the network operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The work assumes that a complete  instance of the Near-RT RIC is deployed to control each cluster of E2 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Moreover, the  authors show that the intelligence orchestration problem in O-RAN is NP-hard and design low-  complexity approaches to support real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' A prototype of OrchestRAN was  implemented and tested at scale on Colosseum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', the world’s largest wireless network emulator  with hardware in the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Schimidt, Irazabal, and Nikaein [15] presented FlexRIC, a software development kit to build  specialized service-oriented controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' FlexRIC has a centralized modular architecture with  a minimal footprint and is designed for extensibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The authors validate FlexRIC building  implementations for two use cases: (i) a recursive RAN controller to allow multiple tenants  to concurrently control and operate their services in a shared infrastructure over heterogeneous  5G networks, and (ii) an SD-RAN controller supplying programmability for RAN slicing with  multi-radio technology, and flow-based traffic control aiming at low-latency communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Balasubramanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' [16] disaggregate the traditional monolithic control plane RAN ar-  chitecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The authors introduce a Near-RT RIC platform that decouples the control and data  planes of the RAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The motivation of the project is to work towards intelligent and continuously  evolving radio networks by fostering network openness and empowering network intelligence  with AI-enabled applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The authors provide functional and software architecture of the  Near-RT RIC and discuss related design challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Moreover, they elaborate on how the Near-  RT RIC can enable network optimization in 5G for the dual-connectivity use case using machine  learning control loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' In this context, the Near-RT RIC architecture design is generic, providing  several options for its implementation and deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Other works focus on developing xApps and rApps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' For example, Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' [17] [18] propose  an intelligent user access control scheme with Deep Reinforcement Learning (DRL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' A federated  DRL-based method is proposed with a global model server running on RIC that updates the  distributed deep Q-networks (DQNs) parameters to optimize the performance of DQNs trained  by UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Johnson, Maas, and Van Der Merwe [19] introduce NexRAN, a use case of the Powder  mobile and wireless research platform, allowing closed-loop control of an approach for O-RAN  slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' O-RAN slicing, in this case, is implemented in the srsRAN open-source mobility stack  and is exposed through a custom service model to xApps, executing on a Near-RT RIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' SYSTEM MODEL, PROBLEM FORMULATION,  10  AND THE PROPOSED STRATEGY  In the following, we describe the system model, formulate the problem, and describe the  optimal and heuristic strategies proposed to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' System Model  We assume a Radio Access Network that follows the O-RAN specifications and is composed  of a set N = {n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', n|N|} of E2 nodes and a set C = {c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', c|C|} of edge computing  nodes (CNs), where each CN is a host consisting of a general purpose hardware characterized  by its processing capability (cProc  m  ), memory (cMem  m  ), and storage (cSto  m ) resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Moreover, we  consider a node c0 representing a cloud computing infrastructure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', a large data center) with  unlimited processing, memory, and storage resources but positioned far from the E2 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' We  propose disaggregating the Near-RT RIC so that different components can run on various nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  The node c0 and each CN cm ∈ C can run any disaggregated RIC function as long as it does  not exceed its resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' To represent the overlay communication network among all nodes, we  define a graph G = (V, E) with V = VC ∪N representing the nodes (where VC = {c0}∪C), and  E = {eij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' vi, vj ∈ V} represent the set of links, where each link is characterized by its latency  eLat  i,j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' If vi = vj, then eLat  i,j = 0, which is relevant for different components running on the same  computing node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 2, c0 and each CN cm ∈ C can communicate with each  other, while each E2 node ni ∈ N can communicate with c0 or any CN cm ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Therefore,  the paths starting at E2 nodes and ending at computing nodes (that run RIC components) are  defined over this graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Cloud node  Edge computing node  E2 node  n2  n1  n|N|  c1  c2  c|C|  c0  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Overlay communication network among E2 nodes and CNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  We assume that the Near-RT RIC can be disaggregated and distributed into five main groups  of components: Near-RT RIC Management (RIC Man), E2T, SDL/STSL, NIBs, and xApps, as  11  illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Each E2 node ni ∈ N is connected to an E2T component, which servers a set  of xApps A = {xApp1, xApp2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', xApp|A|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' While the control loop of a latency-sensitive xApp  must be monitored per E2 node (ni ∈ N), we are also interested in creating clusters of E2 nodes  to minimize the replication of Near-RT RIC components, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The red lines  with arrows in this figure represent the control loop that each latency-sensitive xApp establishes  with each E2 node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Given the computing nodes where the Near-RT RIC components are running  and the graph from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 2, it is possible to compute the round-trip latency starting from an E2  node ni ∈ N, going through an E2T until an xApp (and possibly other RIC components), and  back by the same path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  For each E2 node ni ∈ N, there are four atomic (or indivisible) groups: RIC Man, E2T,  SDL/STSL, and NIBs, which the placement is represented by a tuple p = (r, t, s, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' We also  employ the following auxiliary variables: Rcm = {0, 1} that indicates if CN cm ∈ VC runs  at least an instance of RIC Man, Tcm = {0, 1} that indicates if CN cm ∈ VC runs at least  an instance of E2T, Scm = {0, 1} that indicates if CN cm ∈ VC runs at least an instance of  SDL/STSL, and Dcm = {0, 1} that indicates if CN cm ∈ VC hosts at least one NIB instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' For  example, the tuple p = (c0, c1, c1, c1) represents RIC Man running on c0 and the other atomic  groups running on c1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', Rc0 = 1, Tc1 = 1, Sc1 = 1, and Dc1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Finally, we define a set  of configurations P = {p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', p|P|} that lists all combinations of atomic groups running in  the available computing nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' However, the group of xApps can be further disaggregated and  distributed into multiple computing nodes and, therefore, is not part of this configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content="  E2TA  xAppA  xAppZ  …  Near-RT  RIC Management  SDL/STSL  NIBs  Near-RT RIC  Summarized  Internal Architecture  E2 Nodes cluster A  E2TZ  xAppA'  xAppZ'  …  SDL/STSL  NIBs  …  E2 Nodes cluster Z  …  Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Summarized version of the Near-RT RIC architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  12  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Problem Formulation  The objective function minimizes the total cost to run a disaggregated and distributed Near-  RT RIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The cost has two parts, one fixed and another variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The fixed cost corresponds to  activating or leasing a CN, which is paid independently of the number of consumed resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  While the variable cost depends on the number of Near-RT RIC components running on a CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Each computing node may be assigned a fixed cost, a variable cost, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Let the decision variable xp  i ∈ {0, 1} represent the choice of configuration p ∈ P to serve E2  node ni ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Let U p  cm = {0, 1} indicate if CN cm ∈ VC is used in configuration p ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The  activation (or leasing) of CN cm ∈ VC is given by:  Ucm =  �  �  �  �  �  �  �  1,  if �  ni∈N  �  p∈P  (xp  i U p  cm) > 0  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  (1)  The fixed cost is defined as:  ΦFix =  �  cm∈VC  (UcmF(cm)),  (2)  where F(cm) ∈ R represents the cost to activate CN cm ∈ VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' While the fixed cost may be less  common in cloud-native infrastructures, it is included in the model to make it flexible enough  to represent configurations such as bare-metal services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Let the decision variable ya  i,m ∈ {0, 1} represent the choice of xApp a ∈ A, running on  cm ∈ VC, to control the E2 node ni ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The utilization of CN cm ∈ VC to run xApp a ∈ A is  given by:  Aa  cm =  �  �  �  �  �  �  �  1,  if �  ni∈N  ya  i,m > 0  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  (3)  The variable cost is defined as:  ΦV ar =  �  cm∈VC  �  RcmCR(cm) + TcmCE(cm) + ScmCS(cm)  + DcmCD(cm)  �  +  �  a∈A  Aa  cmCAa(cm),  (4)  where CR(cm), CE(cm), CS(cm), CD(cm), and CAa(cm) represent the cost of running RIC Man,  E2T, SDL/STSL, NIBs, and xApp a ∈ A, respectively, on a given CN cm ∈ VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  13  The objective function of minimizing the total cost is finally defined as:  minimize  ΦFix + ΦV ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  (5)  Next, we present the constraints of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' For each E2 node ni ∈ N, exactly a single  configuration p ∈ P must be selected:  �  p∈P  xp  i = 1,  ∀ni ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  (6)  The latency-sensitive control loop has two mandatory segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The latency of the first one  is measured between E2 node ni ∈ N and CN cm ∈ VC, which runs the corresponding E2T:  LcT  i  =  �  cm∈VC  (δ(xp  i )TcmeLat  ni,cm),  where δ(xp  i ) = {0, 1} indicates if CN cm ∈ VC runs the E2T that controls the E2 node ni ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Concerning the second segment of the latency-sensitive control loop, its latency is measured  between CN cm ∈ VC that runs E2T (associated with the E2 node ni ∈ N) and CN cm′ ∈ VC  that runs xApp a ∈ A (responsible for controlling the same E2 node ni ∈ N):  LcT ,cA  i  =  �  cm,cm′∈VC  (δ(xp  i , ya  i,m′)Aa  cm′eLat  cm,cm′),  where δ(xp  i , ya  i,m′) = {0, 1} indicates if CNs cm, cm′ ∈ VC run, respectively, E2T and xApp  a ∈ A responsible for controlling the E2 node ni ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  In a latency-sensitive control loop, however, an xApp a ∈ A (running on CN cm ∈ VC)  may need to access an SDL/STSL (running on CN cm′ ∈ VC) to reach a NIB (running in CN  cm′′ ∈ VC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' This communication has the following latency:  LcA,cS,cD  i  = ψ(a)  �  cm,cm′,cm′′∈VC  �  δ(ya  i,m, xp  i )Scm′eLat  cm,cm′  +δ′(ya  i,m, xp  i )Dcm′′eLat  cm′,cm′′  �  ,  where ψ(a) = {0, 1} indicates if xApp a ∈ A needs to access SDL/STSL and NIB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' δ(ya  i,m, xp  i ) =  {0, 1} indicates if CNs cm, cm′ ∈ VC run, respectively, xApp a ∈ A and SDL/STSL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' and  δ′(ya  i,m, xp  i ) = {0, 1} indicates if CNs cm, cm′′ ∈ VC run, respectively, xApp a ∈ A and NIB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  These components are related to E2 node ni ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  14  Additionally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' in a latency-sensitive control loop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' an xApp a ∈ A (running on CN cm ∈ VC)  may need to interact with other xApp a′ ∈ A (running on CN cm′ ∈ VC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' which generates the  following latency:  L  cA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='cA′  i  = ψ′(a)  �  cm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='cm′∈VC  (δ(ya  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' ya′  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='m′)Aa′  cm′eLat  cm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='cm′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  where ψ′(a) = {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 1} indicates if xApp a ∈ A needs to communicate with another xApp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' and  δ(ya  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' ya  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='m′) = {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 1} indicates if CNs cm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' cm′ ∈ VC run,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' xApps a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' a′ ∈ A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' which  control the E2 node ni ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Considering the potential chain of xApps Ach ⊆ A involved in the control loop of a certain  xApp a ∈ A, its total latency is the sum of the four previous equations and cannot exceed the  threshold ρa:  LcT  i  + LcT ,cA  i  + LcA,cS,cD  i  +  (7)  �  a,a′∈Ach  �  L  cA,cA′  i  + L  cA′,cS,cD  i  �  ≤ ρa, ∀ni ∈ N, ∀a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  The Near-RT RIC components running on a CN cm ∈ VC must not exceed its processing  capability (cProc  m  ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' memory (cMem  m  ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' and storage (cSto  m ) resources:  RcmRProc + TcmT Proc + ScmSProc + DcmDProc+  �  a∈Acm⊆A  (Aa  cmAa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Proc) ≤ cProc  m  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  ∀cm ∈ VC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  (8)  RcmRMem + TcmT Mem + ScmSMem + DcmDMem+  �  a∈Acm⊆A  (Aa  cmAa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Mem) ≤ cMem  m  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  ∀cm ∈ VC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  (9)  RcmRSto + TcmT Sto + ScmSSto + DcmDSto+  �  a∈Acm⊆A  (Aa  cmAa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Sto) ≤ cSto  m ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  ∀cm ∈ VC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  (10)  where R•,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' T •,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' S•,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' D•,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' and Aa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='• represent processing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' memory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' and storage demands for RIC Man,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  E2T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' SDL/STSL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' NIBs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' and xApp a ∈ Acm ⊆ A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  As discussed in [20], the problem represented by Equation (5) is NP-Complete because it  corresponds to a mixed-integer quadratic programming (MIQP) decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The proof shows that  there is a polynomial-length certificate for yes-instances of the MIQP decision so that the MIQP  decision is in NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Moreover, the proof that the MIQP decision is NP-complete consists of  15  providing a polynomial-time reduction of the problem for determining whether there is a cut of  cardinality at least k in a graph G (which is NP-complete) to an instance of the MIQP decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Heuristic Strategy  MIQP problems such as the one represent by Equation (5) can be optimally solved using  traditional solvers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', IBM ILOG CPLEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' However, this problem has a large number of  decision variables that can be estimated as (|N|·|VC|)comp+(|N|·|VC|)|A|, where comp represents  the number of Near-RT RIC components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' For example, a configuration with |N| = 100 E2 nodes,  comp = 4, |VC| = 5 computing nodes, and |A| = 5 xApps involves more than 3·1013 optimization  variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Therefore, due to the computational time and the number of resources required, which  are extremely high, solving the problem optimally is impractical, even for small instances of the  problem, as we present in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  To deal with large instances of the problem and quickly provide solutions for RIC-O re-  orchestration requests, the efficient heuristic strategy described as Algorithm 1 is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The  heuristic starts with a feasible solution and then seeks to improve the cost efficiency through a  greedy strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The heuristic takes as input the same parameters used in the optimal strategy and  produces as output the placement of the Near-RT RIC components across CNs, including xApps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  The output is represented by Places[ni, ‘Near-RT RIC component’], where ni ∈ N identifies the  E2 node that is associated with a ‘Near-RT RIC component’, which can be RIC Man (Ri), E2T  (Ti), SDL/STSL (Si), NIBs (Di), or xApp ai ∈ Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Moreover, let (Zi, ⪯) = Ti ∪ Ai ∪ Si ∪ Di  be the ordered set of components involved in the latency-sensitive control loop, as described in  Section IV-A and illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' We present additional details in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Initially, for each ni ∈ N, the heuristic strategy seeks to place every Near-RT RIC component  ζi ∈ (Zi, ⪯) at the closest CN cm ∈ VC with enough available resources, as shown in Algorithm 1  (lines 1–3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' This corresponds to the best solution regarding latency, but it ignores the cost of CNs  and causes unnecessary replication of Near-RT RIC components, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', E2T and xApps, which  also increases cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Next (lines 4, 5), in CNcost[ˆζi], the algorithm sorts in decreasing order of  cost the list of CNs that are available for running each Near-RT RIC component ˆζi ∈ Ri∪(Zi, ⪯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  RIC Man is included because it is a Near-RT RIC component, despite of not being involved in  the latency-sensitive control loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The last part of the algorithm (lines 6–9) focuses on improving  cost efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' First (line 7), the RIC Man component is placed on the cheapest CN ni ∈ N  that has enough available resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' RIC Man tends to be placed on c0 since this tends to be  16  Algorithm 1: Heuristic strategy  Input : The same input of the optimal strategy  Output: Placement of the Near-RT RIC components  1 forall ni ∈ N do  2  forall ζi ∈ (Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' ⪯) do  3  Places[ni,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' ζi] ← closestCN(ni,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' ζi)  4 forall ˆζi ∈ Ri ∪ (Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' ⪯) do  5  CNcost[ˆζi] ← sortByDecreasingCost(ˆζi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='VC)  6 forall ni ∈ N do  7  Places[ni,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Ri] ← rePlace(Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' CNcost[Ri])  8  forall ζi ∈ (Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' ⪯) do  9  Places[ni,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' ζi] ← rePlace(ζi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' CNcost[ζi])  10 return Places[•,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' •]  the cheapest CN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' generally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' function rePlace determines that node has enough available  resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Moreover, a single RIC Man is probably able to serve all E2 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Last (lines 8,  9), the algorithm seeks to find the cheapest CN ni ∈ N for each Near-RT RIC component  ζi ∈ (Zi, ⪯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Therefore, in addition to checking CN resource availability, function rePlace also  checks compliance with the latency-sensitive control loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' These final steps of the heuristic also  tend to reduce the number of instances of Near-RT RIC components because a single component  may be able to ensure the latency is within the required threshold for multiple E2 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  The proposed heuristic strategy solves the problem in polynomial time and produces sat-  isfactory results regarding the placement of the Near-RT RIC components, as we present in  Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' To be precise, the asymptotic complexity of the Algorithm 1 is O((comp + |A|)2 ·  |N| · |VC|), which represents the placement of all Near-RT RIC components that serve all E2  nodes, considering, in the worst case, all available CNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' RIC ORCHESTRATOR  This section describes the architecture of the RIC-O which is responsible for the dynamic  and efficient placement of the disaggregated Near-RT RIC components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The architecture of the  RIC-O is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 4, along with some of the RIC components required to describe the  17  orchestrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' RIC-O was designed considering components that run in both Non-RT RIC and  Near-RT RIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The RIC-O Optimizer and RIC-O Deployer run on the Non-RT RIC, while  the RIC-O Triggers execute on the Metrics Server of the Near-RT RIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The Monitoring  System is in charge of monitoring resource usage of the Near-RT RIC components, E2 Nodes,  and O-Cloud infrastructure, in addition to throwing alerts whenever it detects threshold violations  in the latency-sensitive control loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Near-RT RIC  Near-RT RIC  Management  Service Management and Orchestration (SMO)  Non-RT RIC  E2 Nodes (RAN)  SDL/STSL  RIC-O Optimizer  RIC-O Deployer  O2  O-Cloud  NIBs  NIBs  xApps  xApps  xApp  E2  O1  A1  Monitoring System  Alert Manager  Metrics Server  RIC-O Triggers  TSDB  xApps  xApps  E2T  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' RIC-O architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  The Metrics Server is a monitoring module that collects metrics information from all  xApp and E2T instances running on the Near-RT RIC platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Example metrics include CPU  utilization, memory usage, and the communication latency to monitored instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The latency  to access SDL/STSL shared layers and databases are also collected by the Metrics Server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  All metrics are stored in a time-series database (TSDB) for further processing and reporting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  The RIC-O Triggers component consists of predefined alert rules (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', the latency is above a  given threshold) upon the occurrence of which an alert is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' As the Metrics Server  detects some metric violation that is associated with some trigger, it notifies the Alert Manager  that a metric has been violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The Alert Manager, in its turn, sends a message to the RIC-O  18  Optimizer requesting a new round of optimization to reconfigure the system so that it can satisfy  the metric that has been violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  When a request is delivered to the RIC-O Optimizer, it fetches the current metric measure-  ments from the Metrics Server, executing the optimization strategies described in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  The heuristic and optimal optimization strategies run in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The heuristic strategy should  find a new solution quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' However, the optimal strategy is executed even if the heuristic  completes earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The solution generated by the optimal strategy is only applied if is superior to  the heuristic and if it becomes available before a new notification of metric violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The RIC-O  Optimizer sends the configuration of the computed solution to the RIC-O Deployer, which  redeploys the Near-RT RIC components accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The redeployment is only executed if the  RIC-O Optimizer finds a solution different from the previous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Lastly, the RIC-O Deployer  uses the standardized O2 interface defined by O-RAN to exchange messages with the O-Cloud  element to effectively apply the new placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' IMPLEMENTATION  We implemented a proof-of-concept prototype of RIC-O to validate and evaluate our proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  The prototype is described in this section, along with the testbed used for its evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' RIC-O  Optimizer and RIC-O Deployer were implemented using the Python language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The Monitoring  System component comprises the monitoring subsystem of the Near-RT RIC platform [21] that  runs on Kubernetes (K8S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The Near-RT RIC uses Prometheus to collect metrics, and Metrics  Server corresponds to the main Prometheus server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Metrics are collected at the pod and cluster  levels and stored as time series data in the internal Prometheus database (TSDB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Prometheus  was a natural choice to collect metrics in our proposal since it is the native solution of K8S for  data collection and is part of the Near-RT RIC platform from the O-RAN SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  We also employed the Alert Manager component from Prometheus to handle alerts issued by  different client applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', Metrics Server).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Alert handling may include deduplicating,  grouping, silencing, inhibiting, and routing alerts to other endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The Alert Manager notifies  RIC-O Optimizer of the need for starting a new optimization round when a given RIC-O trigger  detects a latency threshold has been violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The RIC-O Triggers element was implemented  using the Prometheus Query Language (PromQL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' In this context, our prototype monitors CPU,  memory, and network latency from K8S system pods, Near-RT RIC pods, and K8S nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  One of the monitored metrics is the latency-sensitive control loop of the Near-RT RIC, which  19  cannot surpass a given threshold (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', above 10ms for certain control loops).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Upon reaching this  threshold, a trigger launches the RIC-O Optimizer to compute a new placement solution for  the Near-RT RIC components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' After the optimization is computed, a notification is delivered to  RIC-O Deployer, which redeploys the Near-RT RIC components based on the outcome of the  RIC-O Optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Figure 5 shows the message exchange between the RIC-O components, Monitoring System,  Near-RT RIC components, and the O-Cloud element while orchestrating the optimization and  deployment tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Initially, the deployment is executed as a response to the original request  coming from the network operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Subsequently, a re-deploy is executed only after the Alert  Manager sends a notification to the RIC-O Optimizer due to a latency violation in the Near-RT  RIC control loop or due to the unavailability of required computing resources in the O-Cloud  element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Notify RIC (Re)deployment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Send metrics data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Request metrics data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='SMO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='RIC-O  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Deployer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='RIC-O  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Optimizer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Near-RT RIC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Components  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='O-Cloud  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Alert  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Manager  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Deploy (Re)allocation Plan  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Instantiate RIC Components  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Notify Instantiation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Deallocate RIC Components  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Notify Deallocation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Run Optimization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Monitoring  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='System  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Send metrics data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Send metrics data from RIC and O-Cloud  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Only on  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Redeploying  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='For  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Each  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Demand Reallocation Plan  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Request metrics data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Request metrics data from RIC and O-Cloud  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Send (Re)allocation Plan  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Monitoring  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Loop  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Messages exchanged for Near-RT RIC optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  The Monitoring System shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 5 collects metrics regularly from the Near-RT RIC  Components and the O-Cloud element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' These metrics include CPU and memory usage of  20  Near-RT RIC pods, K8S nodes, and their communication latency (messages 1, 2, 3, and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  The Monitoring Loop is executed regularly at every second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Once Metrics Server notifies  Alert Manager that a given RIC-O trigger has been violated, message 5 is delivered to RIC-O  Optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Afterward, the RIC-O Optimizer starts a reallocation plan to compute the placement of the  Near-RT RIC Components that run on the O-Cloud element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' RIC-O Optimizer then requests  metrics information that includes the available computing resources of K8S pods, CNs, and the  communication latency between them (messages 6 and 7), which is the information required  as input for the optimization strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Next, the heuristic and optimal strategies are initiated  and run in parallel (message 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Once an outcome from the optimization strategies becomes  available, it is sent to the RIC-O Deployer in message 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Eventually, if there is enough time  for computing the optimal solution, two outcomes are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The RIC-O Deployer, in turn,  starts deploying the new (re)allocation plan in O-Cloud through the O2 interface defined by  O-RAN (message 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Near-RT RIC components are instantiated by O-Cloud, which is notified  when all the components are deployed (messages 11 and 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' In the particular case when the  Near-RT RIC components were deployed previously, O-Cloud releases the resources of the Near-  RT RIC components that are no longer required (messages 13 and 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Finally, O-Cloud notifies  the RIC-O Deployer that the deployment has finished (message 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  We also implemented an extended xApp and an E2 simulator (E2Sim), both coded in C++  language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Our implementation is based on the open-source versions of the Bouncer xApp* and  the E2Sim† from the O-RAN SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Moreover, we included new features and updated the service  models of both the Bouncer xApp and E2Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Originally, the Bouncer and E2Sim employed the  Key Performance Measurement service model (E2SM-KPM) to expose performance measure-  ments of logical functions running on E2 Nodes [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' However, E2SM-KPM does not support  RAN control through the E2AP Control service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Therefore, we implemented the E2SM-RC  (RAN Control) [23] service model, which provides RAN control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Figure 6 shows the message  exchange between the Bouncer xApp, E2Sim, and Near-RT RIC components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The goal is to  monitor the latency-sensitive control loop between the E2 nodes and the corresponding xApp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  We assume that all the Near-RT RIC components and E2Sim are already set up and running  https://gerrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='o-ran-sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='org/r/admin/repos/ric-app/bouncer  †https://gerrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='o-ran-sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='org/r/admin/repos/sim/e2-interface  21  when Bouncer xApp is deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Control  Loop  Near-RT RIC Management  loop  Registration Request  Registration Response  REST Subscription Request  RIC Subscription Response  RIC Subscription Request  Subscription  Manager  Routing  Manager  xApp  Manager  E2T  E2Sim  REST Subscription Response  Route Create  E2 Subscription Request  E2 Subscription Response  REST Subscription Notification  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='3  4  6  5  7  8  Bouncer  xApp List Request  RIC Insert Request  RIC Insert Request  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='2  9  10  RIC Control  RIC Control  11  12  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='4  xApp List Response  Route Create  a) Update Routes  b) Update Routes  a) Update Routes  b) Update Routes  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='4  par  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Messages exchanged for xApp deployment and execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  When Bouncer xApp initializes, it registers itself with the Near-RT RIC platform by sending  a registration message to xApp Manager, which sends back a reply indicating the outcome  (messages 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' In the meantime, the Routing Manager periodically retrieves the list  of xApps registered in xApp Manager to set up the corresponding routing rules in the Near-RT  RIC platform (messages 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Moreover, the Routing Manager builds a routing table  for all registered xApps (message 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='3) and then distributes it to Bouncer and E2T (messages  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='4a and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' These routes are required to allow transparent and dynamic message delivery to  the endpoints concerned in the communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' During the registration, the Bouncer xApp also  22  subscribes to Subscription Manager to receive messages from the E2AP Insert service from  the RAN (messages 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Upon receiving the subscription request, the Subscription  Manager notifies Routing Manager to create the corresponding routing rules to deliver those  messages to Bouncer xApp (message 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Next, the Routing Manager distributes the generated  routing rules to the respective endpoints (messages 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='4a and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Additionally, the Subscription Manager issues a new subscription request (message 4) to  the E2T, which implements the E2 interface to communicate with E2 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The E2T then  deals with the subscription communication with the E2Sim (messages 5 and 6) according to the  subscription that the Bouncer xApp has generated in message 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The communication between  E2T and E2Sim is established through Stream Control Transmission Protocol (SCTP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Messages  7 and 8 notify Subscription Manager and Bouncer xApp about the subscription outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' At  this point, Bouncer xApp is fully deployed in the Near-RT RIC platform, and communication  with E2Sim can take place through the E2T component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  The Control Loop block in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 6 illustrates the latency-sensitive control loop of 10ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' When  a given E2 node issues a RIC Insert Request message to the Near-RT RIC, the E2 node suspends  its execution and waits for the corresponding reply message (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', RIC Control) to resume its  operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Messages 9-12 illustrate this control loop that needs to complete within the 10ms  threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' If no RIC Control message is delivered within this threshold time, a default action is  executed by the E2 node, which usually implies penalties in the optimization and performance of  RAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Moreover, we implemented the capability of computing the control loop latency between  E2Sim and Bouncer xApp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' In this case, E2Sim collects the timestamps on sending message 9 and  receiving message 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The latency-sensitive control loop is exported to the Prometheus metrics  collector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' EVALUATION  This section presents a performance evaluation of RIC-O both from a theoretical perspective,  using analytical modeling (in Subsection VII-A), and from a practical perspective, employing  real-world experiments (in Subsection VII-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The first part of the evaluation focuses on scala-  bility, while the second mainly deals with timing issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  23  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Analytical modeling  In this part of the evaluation, we assume as RAN topology a next-generation hierarchical  network with 512 E2 nodes organized in three main tiers [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The top one (Tier-1) is closest  to the core and so to the cloud node (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', c0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The one-way latency (in ms) between the tiers  is a uniform random choice from the set {1, 2, 2, 3, 3}, and the one-way latency from Tier-1  to c0 is 4ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' We vary the number of edge computing nodes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', cm ∈ C) along the network  tiers and, for each number of CNs, extract an overlay topology equivalent to the one shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' There is always a cloud node, and all others are edge computing nodes distributed  from bottom to top in the hierarchical RAN network as the number of CNs increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The cloud  node has no fixed cost and presents the following variable cost for Near-RT RIC components:  two (RIC Man, E2T), one (SDL/STSL, NIBs), and one (xApp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' There are two xApps, and both  of them access the database, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', SDL/STSL and NIBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Additional information about underlay  RAN topology, and Near-RT RIC is presented in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Parameters  Tier-1  Tier-2  Tier-3  Number of E2 nodes  5  20  487  CN fixed cost  10  20  30  CN variable cost: (RIC Man, E2T),  (SDL/STSL, NIBs), (xApps)  4, 2, 1  8, 4, 2  16, 8, 4  cP roc  m  (cores)  32  16  8  cMem  m  (GB)  64  32  16  cSto  m  (GB)  256  256  256  Near-RT RIC requirements  Proc  Mem  Sto  RIC Man  4  8  4  E2T  2  4  2  SDL/STSL  2  4  1  NIBs  1  2  50  xApps  1  2  1  TABLE I  PARAMETERS OF THE UNDERLAY RAN TOPOLOGY AND REQUIREMENTS OF THE NEAR-RT RIC COMPONENTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  We first investigate to which extent the complexity of the exact optimization model (Sub-  section IV-B) limits its scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Although our heuristic strategy (Subsection IV-C) is very  efficient, it is necessary to investigate the quality of its solutions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', how close to optimal those  solutions tend to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Therefore, we evaluate and compare the optimal and heuristic strategies  24  regarding computing time and objective function as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The computing time is  related to the scalability of each strategy, and the objective function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', the total cost, represents  the quality of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Moreover, the metrics vary as a function of the number of CNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Comparison between the optimal and heuristic strategies regarding computing time and total cost (objective function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  We can observe in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 7 that the optimal strategy scales poorly, being unable to find a solution  for a RAN with 12 CNs even after running for an entire day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Therefore, the optimal strategy  is rarely helpful for large real-world instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Nevertheless, the heuristic strategy scales very  well and finds a solution for 512 CNs in less than 10 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Concerning the quality of the  solutions, as expected, the heuristic strategy cannot always find the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' However,  a decreasing trend can be discerned in the objective function that suggests the heuristic strategy  is in the right direction, including in scenarios with a large number of CNs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', 512 CNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' In  those cases, the cheapest edge computing nodes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', those in Tier-1) are readily available, and  the solution improves noticeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  We further investigate the optimization strategies by evaluating the number of instances of the  main Near-RT RIC components involved in the latency-sensitive control loop, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', xApps and  E2T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' As presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 8, our heuristic strategy finds the best (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', the smallest) number of E2T  instances, matching the result obtained by the optimal strategy when this is available and keeping  it at the minimum (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', one instance) as the number of CNs increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Moreover, the heuristic  strategy finds some non-optimal solutions for the number of xApps instances, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', with four  108  Heuristic  106  Optimal  104  102  100  2  4  6  8  10  12  16  32  64  128  256  512  CNs (#)Objective Function (#)  170  130  90  50  10  2  4  6  8  10  12  16  32  64  128  256  512  CNs (#)25  2  4  6  8  10  12  16  32  64  128  256  512  CNs (#)  0  1  2  3  4  5  Instances (#)  Heuristic  Optimal  xApps  xApps  E2T  E2T  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Comparison between optimal and heuristic strategies regarding the number of xApps and E2T instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  and six CNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' This behavior is expected from our heuristic strategy because it prioritizes Near-  RT RIC components involved in the latency-sensitive control loop in the following order: E2T,  xApps, SDL/STSL, and NIBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Since latency optimization and cost optimization are performed  in this order and without revisiting previous decisions for improvement, the heuristic strategy  potentially finds non-optimal solutions involving xApps, SDL/STSL, or NIBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Real-world experiments  In this part of the evaluation, we run the experiments in a scenario with five CNs, which are  virtual machines (VMs) with the following configuration: 4 vCPUs, 8 GB RAM, and 50 GB  of the virtual disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' One CN represents the cloud node (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', c0), and the others represent the  edge computing nodes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', cm ∈ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' These CNs are worker nodes in a K8S cluster managed  by a master node running a sixth VM with the following configuration: eight vCPUs, 16 GB  RAM, and 100 GB of the virtual disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' All VMs are hosted on a DELL PowerEdge M610 server  with four Intel Xeon X5660 processors and 192 GB RAM, which runs VMware ESXi 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='7 as the  hypervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Additional details about the software tools employed in the experiments are available  in the public repository of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' We also employed an E2 simulator to represent four E2  nodes that must be served by the Near-RT RIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Overlay and underlay topologies are the same  in this part of the evaluation and correspond to the one presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  To illustrate the orchestration capabilities of RIC-O, we designed two scenarios in which the  latency-sensitive control loop is disrupted and show how our proposal acts to bring the Near-RT  RIC back to normal operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' In the first scenario, RIC-O must deal with a sudden and high  increase in the latency of the path used to serve a certain E2 node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The second scenario is  more challenging because RIC-O needs to deal with a CN that becomes unavailable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', any  26  latency-sensitive control loop involving this CN disappears since the Near-RT RIC components  running in it suddenly become inaccessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  In a real-world RAN, the latency between a pair of nodes may change due to a (re)route  decision in the underlying network, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Since our underlay network matches the overlay  one, we emulate the sudden increase in the latency between the E2 node and its corresponding  E2T by reconfiguring the latency in the virtual link connecting these nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Figure 9 illustrates the  main events occurring along the time in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' This figure shows the status of the control  loop between each E2 node and its corresponding xApp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' In addition, the figure presents the CPU  utilization of some essential software components (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', RIC-O Deployer, RIC-O Optimizer,  xApps, and E2T), which helps keep track of the actions performed by RIC-O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Control loop  failure  Control loop  satisfied  Start  redeploy  Optimization  trigger  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Reaction to a sudden violation of requirements of the latency-sensitive control loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 9, the first scenario is initially in a fully operational state, and the  latency-sensitive control loop of each E2 node is satisfied by the Near-RT RIC thanks to the  initial orchestration defined by RIC-O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Then, at time instant 150s, the latency of the control loop  from E2 node 4 increases suddenly and remains persistently above 10ms, as indicated by event  Control loop failure in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' After 10 seconds, the Monitoring System considers that the  event is a consistent control loop violation and notifies the RIC-O Optimizer to compute a  new solution, at time instant 160s, as indicated by the event Optimization trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The heuristic  strategy of the RIC-O Optimizer quickly finds a solution and requests RIC-O Deployer to apply  Control loop (ms)  E2Node 1  102  E2Node 2  E2Node 3  E2Node 4  101  M  20  40  60  80  100  120  140  160  180  200  Time (s)101  CPU (milicores)  Deployer  Optimizer  xApps  E2T  100  10-1  10-2  10  =  1027  this new placement nearly 5 seconds later, as indicated by the event Start redeploy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The RIC-O  Optimizer keeps running the optimal strategy thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Finally, the redeploy of the Near-RT RIC  components and reconfiguration of E2 nodes completes at time instant 200s, as indicated by  event Control loop satisfied, when the latency-sensitive control loop is again limited to 10ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Figure 9, and also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 10 show a few measurements of the control loop that go above 10ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  This behavior is related to the underlying operating system and virtualization platform (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=',  hypervisor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' A tight threshold of 10ms is on such a sensitive scale that even a traditional process  scheduler may sometimes cause a small variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Since fine-tuning those systems are out of the  scope of this work, we configured the Monitoring System to report only persistent violations  of the 10ms threshold in latency-sensitive control loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  The second scenario, in which a CN suddenly crashes, may either represent a software or  hardware failure, or network outage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' We emulate this behavior by abruptly forcing a shutdown  of the VM running the CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Figure 10 illustrates the main events occurring along the time in  this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 9, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 10 also shows the status of the control loop between each  E2 node and its corresponding xApp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' However, we have not identified relevant information that  justified presenting measurements related to Near-RT RIC nor RIC-O components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  CN down  Finish  redeploy  Start  redeploy  Optimization  trigger  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Reaction to sudden unavailability of CN under use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 10, the second scenario also starts from a fully operational state, where  the latency-sensitive control loops of all E2 nodes are satisfied by the Near-RT RIC thanks to the  initial orchestration defined by RIC-O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' The CN running Near-RT RIC components responsible  for serving the E2 node 4 suddenly becomes unavailable, as indicated by the event CN down  in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' In this case, the latency-sensitive control loop of this E2 node is totally disrupted,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=', there are no more measurements of the control loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' After 50 seconds, the Monitoring  System detects the problem and notifies the RIC-O Optimizer to compute a new solution, as  (sw) dooj  E2Node 1  102  E2Node 2  E2Node 3  E2Node 4  101  Control  AMMA  100  20  40  60  80  100  120  140  160  180  Time (s)28  indicated by the event Optimization trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' This time interval for reporting the problem may  seem long, but it is the default K8S policy for detecting worker node unavailability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Again,  after nearly 5 seconds, the heuristic strategy of RIC-O Optimizer finds a solution and requests  RIC-O Deployer to apply the new placement, as indicated by the event Start redeploy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Finally,  the redeployment of the Near-RT RIC components finishes, and the latency-sensitive control loop  of E2 node 4 is reestablished at time instant 130s, as indicated by the event Finish redeploy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' CONCLUSION  In this work, we claim that the most efficient approach to ensure that the Near-RT RIC is able to  deal with a large real-world RAN is by disaggregating and distributing its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' However,  it is necessary to orchestrate these components properly across the edge-cloud continuum to  ensure the latency-sensitive control loop that involves xApps running over the Near-RT RIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  Therefore, we propose the RIC orchestrator (RIC-O) that employs a hybrid strategy combining  optimal and heuristic strategies for efficient placement of the Near-RT RIC components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' We  evaluate RIC-O through analytical modeling and real-world experiments to illustrate its properties  and benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Future work includes the investigation of machine learning-based time series  forecasting for improving the orchestration of Near-RT RIC components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Moreover, the adoption  of the O-RAN architecture by large-scale production networks will provide additional data on  its usage, thus allowing extensive investigation of novel optimization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Finally, we  envision the potential of the RIC-O being adopted as part of the Non-RT RIC, running as an  rApp, which will demand further experiments and software integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  ACKNOWLEDGMENT  This work was conducted with partial financial support from the National Council for Scientific  and Technological Development (CNPq) under Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 405111/2021-5 and 130555/2019-3  and from the Coordination for the Improvement of Higher Education Personnel (CAPES) -  Finance Code 001, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' Moreover, RNP partially supported the work, with resources from  MCTIC, Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content=' 01245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='010604/2020-14, under the 6G Mobile Communications Systems  project and by MCTIC/CGI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='br/S˜ao Paulo Research Foundation (FAPESP) through the Project  SAMURAI - Smart 5G Core And MUltiRAn Integration under Grant 2020/05127-2 and Pro-  grammability, ORchestration and VIRtualization of 5G Networks (PORVIR-5G) under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
+page_content='  2020/05182-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE0T4oBgHgl3EQf6QIN/content/2301.02760v1.pdf'}
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+arXiv:2301.00205v1  [math.AP]  31 Dec 2022
+Gevrey-class-3 regularity of the linearised hyperbolic
+Prandtl system on a strip
+Francesco De Anna 1, Joshua Kortum 2, Stefano Scrobogna 3
+1 Institute of Mathematics, University of W¨urzburg, Germany
+email: francesco.deanna@mathematik.uni-wuerzburg.de
+2 Institute of Mathematics, University of W¨urzburg, Germany
+email: joshua.kortum@mathematik.uni-wuerzburg.de
+3 Dipartimento di Matematica e Geoscienze, Universit`a degli Studi di Trieste, Trieste, Italy,
+email: stefano.scrobogna@units.it
+Abstract
+In the present paper, we address a physically-meaningful extension of the linearised Prandtl equations
+around a shear flow. Without any structural assumption, it is well-known that the optimal regularity
+of Prandtl is given by the class Gevrey 2 along the horizontal direction. The goal of this paper is to
+overcome this barrier, by dealing with the linearisation of the so-called hyperbolic Prandtl equations in
+a strip domain. We prove that the local well-posedness around a general shear flow Ush ∈ W 3,∞(0, 1)
+holds true, with solutions that are Gevrey class 3 in the horizontal direction.
+1
+Introduction
+The main concern of this paper is to set up a rigorous well-posedness theory for the following extension
+of the linearised and reduced Prandtl equations on a thin strip:
+
+
+
+
+
+
+
+
+
+
+
+(∂t + 1)
+�
+∂tu + Ush∂xu + v U ′
+sh
+�
+− ∂2
+yu = 0,
+(t, x, y) ∈ (0, T) × T × (0, 1),
+∂xu + ∂yv = 0
+(0, T) × T × (0, 1),
+(u, ut)|t=0 = (uin, ut,in)
+T × (0, 1),
+u|y=0 = 0,
+v|y=0 = 0
+(0, T) × T.
+(1.1)
+In this system, the unknown is the horizontal component u = u(t, x, y) of the velocity field (u, v)T :
+(0, T) × T × (0, 1) → R2, while the vertical component v = v(t, x, y) is determined by the divergence-free
+condition ∂xu + ∂yv = 0 and by the Dirichlet relation v|y=0 = 0, which formally imply
+v(t, x, y) = −∂x
+� y
+0
+u(t, x, z)dz,
+(t, x, y) ∈ (0, T) × T × (0, 1).
+The function Ush = Ush(y) depends uniquely upon the vertical variable y ∈ (0, 1) and describes a shear
+flow (Ush(y), 0)T , around which the original equations have been linearised. System (1.1) arises indeed
+from a meaningful extension of the classical Prandtl equations, known as hyperbolic Prandtl equations[26].
+With our analysis, we aim to show how System (1.1) might be particularly desirable, in order to overcome
+certain analytic barriers that are typical of the classical Prandtl theory. More precisely, System (1.1)
+is amenable in terms of solutions that have regularity Gevrey-class 3 along the vertical variable x ∈ T,
+overcoming therefore the well-known barrier of Gevrey-class 2 of the Prandtl theory. Details about this
+statement and our main result are presented starting from Section 1.4.
+First, we shall provide some
+background on the origin of this model.
+1
+
+1.1
+The Prandtl equations and the barrier of Gevrey-class 2
+In order to understand the major characteristics of (1.1), we shall briefly overview the original model of
+Prandtl, that was introduced during a 10 minutes presentation of the 1904 Third International Mathe-
+matics Congress in Heidelberg [32]. Such short presentation has scientifically impacted many disciplines,
+so much that nowadays the field of aerodynamics is still shaped by his fundamental idea: in order to
+describe the inviscid limit of an incompressible fluid in a region close to a solid surface (where dissipative
+forces are predominant), one would rather consider the velocity of the fluid in terms of rescaled variables,
+which concentrate the dissipative effects in a thin region close to the boundary. This procedure provides
+a “split” in the behaviour of the flow:
+• within the bulk, the hydrodynamics is dominated by the incompressible Euler equations with no-
+penetration boundary conditions,
+• on a neighbourhood of the boundary, a corrective term (the so-called boundary layer) provided by
+the Prandtl equations restores the natural no-slip boundary conditions for viscous flows.
+One of the simplest forms of the (non-linear) Prandtl equations in two dimensions is given by
+
+
+
+
+
+
+
+
+
+
+
+
+
+∂tu + u∂xu + v∂yu − ∂2
+yu = ∂tuE + uE∂xuE,
+(t, x, y) ∈(0, T) × X × (0, +∞),
+∂xu + ∂yv = 0
+(0, T) × X × (0, +∞),
+(u, v)|y=0 = 0
+lim
+y→+∞ u = uE
+(0, T) × X,
+ut|=0 = uin
+X × (0, +∞),
+(1.2)
+where x ∈ X describes the (local) arc-length parametrisation of the solid surface (usually in the math-
+ematical community X = T or X = R), while uE = uE(t, x) is determined by the solution of the Euler
+equation in the bulk of the flow, when approaching the boundary.
+The analysis of (1.2) has received from the mathematical community numerous investigations during the
+past decades. Although the Prandtl equations are classical, their applications are rather narrowed because
+of the particular unstable nature of the underlying solutions. These instabilities are nowadays moderately
+well understood and relate mainly to separation phenomena (appearance of reversed flow in the boundary
+layers).
+The first rigorous mathematical study addressing the well-posedness of the Prandtl equations (1.2) was
+performed in the book of Oleinik and Samokhin [23] in the case of so-called monotonic initial data (namely
+initial velocity uin in (1.2), satisfying ∂yuin > 0). Roughly speaking, the Olenik’s monotonicity prevents
+the mentioned flow separation, at least locally in time. This allows to recast the velocity field through a
+meaningful transformation (known as Crocco transformation), providing a solid ground to the local-in-
+time well posedness of (1.2) within function spaces typical of hydrodynamics, such as Sobolev ones. For
+more details on the Olenik’s monotonicity, we refer the reader to the more recent result [2], in which the
+authors construct local-in-time solutions via a Nash-Moser argument. See also [21] for a proof performed
+purely by energy methods.
+For initial data lacking monotonicity, the well-posedness becomes much more involved and one has to
+consider function spaces that control infinite derivatives of the solutions.
+This was addressed in the
+celebrated result [31] of Caflisch and Sammartino, where the authors dealt with non-monotonic initial
+data uin that are analytic in the variable x ∈ X = R. In the framework of a periodic variable x ∈ X = T,
+analytic initial data can be easily understood through their Fourier series, under a strong localisation of
+the frequencies:
+uin(x, y) =
+�
+k∈Z
+uin,k(y)eikx,
+where the modes uin,k(y) decays exponentially as uin,k(y) ∼ e−a|k|, for some radius of analyticity a > 0.
+This type of initial data are however extremely regular and with reduced applications to real phenomena.
+For this reason, an increasing number of works were devoted to relax this framework.
+2
+
+The first breakthrough was provided in [14] by G´erard-Varet and Masmoudi, where the authors showed
+that the Prandtl system is actually locally well-posed for data that are Gevrey-class 7/4 in the x-variable.
+Roughly speaking, an initial data uin is Gevrey-class m along x ∈ T, with m > 1, if the modes uin,k(y)
+decay exponentially as uin,k(y) ∼ e−a|k|1/m, for a suitable radius a > 0. Already in [14], however, the
+authors remarked that the Gevrey-class 7/4 was unlikely to be optimal and that further insights from
+numerics suggested rather a threshold of Gevrey-class 2 (i.e. uin,k(y) ∼ e−a|k|1/2).
+Eventually, this remark was mathematically formalised and a first result in this direction was attained
+in [20], assuming that the velocity (u, v)T in (1.2) is a small perturbation of a suitable shear flow, which
+satisfies a non-degenerate condition (for details cf. Assumption 1.1 in [20]).
+The breakthrough of Gevrey 2 was however achieved by Gerard-Varet and Dietert in [11], where the
+authors developed a robust local well-posedness theory without any structural assumption on the flow
+(such as monotonicity or critical points). Their result was based on a meaningful change of state variable,
+from which the present work takes substantial inspiration.
+From the work of Gerard-Varet and Dietert followed a variety of questions, in particular to determine
+whether the Gevrey-class 2 was optimal for the well-posedness of System (1.2) or if further insights
+would have led to weaker regularities. Surprisingly, Gerard-Varet and Dormy overturned any possibility
+of improvement, providing indeed a negative answer to this open problem. In their seminal result [13],
+the authors showed that already at the level of the linearised equations around a shear flow (Ush(y), 0),
+namely replacing the first equation of (1.2) with
+∂tu + Ush∂xu + v U ′
+sh − ∂2
+yu = 0,
+(t, x, y) ∈ (0, T) × T × (0, +∞),
+(1.3)
+the linear propagator of regularity is unbounded in Gevrey-class higher than 2. Roughly speaking, the
+authors showed the existence of solutions, whose modes in the frequencies k ∈ Z experience an exponential
+growth with rate |k|1/2. In general, this growth could be counteracted only by Gevrey-2 initial data,
+precluding any room for improvement. In other words, this was the first encounter with the barrier of
+Gevrey 2: the linearised Prandtl equation is ill-posed within any larger setting.
+We refer the reader to the works [10, 12, 17, 18], as well, which concern further instabilities of the Prandtl
+equations.
+1.2
+The hydrostatic approximation
+When the vertical variable y is bounded, for instance with y ∈ (0, 1) (as in our System (1.1)), a different
+type of equations has adequately found a mathematical relevance, namely the so-called hydrostatic ap-
+proximation of Navier-Stokes/Prandtl. In two dimensions, these equations are a reminiscent of Prandtl
+and take the form
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+∂tu + u∂xu + v∂yu − ∂2
+yu + ∂xp = 0,
+(t, x, y) ∈(0, T) × X × (0, 1),
+∂yp = 0,
+(0, T) × X × (0, 1),
+∂xu + ∂yv = 0
+(0, T) × X × (0, 1),
+(u, v)|y=0,1 = 0
+(0, T) × X,
+ut|=0 = uin
+X × (0, 1).
+(1.4)
+This model is significant in several phenomena of atmospheric science and can be derived from the so-
+called primitive equations. Beside the vertical domain, System (1.4) inherently differs from (1.2) in the
+boundary conditions of v. In (1.4) v is null in both y = 0, 1, whereas in (1.2) v has homogeneous condition
+only in y = 0 (without any assumption for y → +∞). Because of this, the pressure p in the hydrostatic
+approximation (1.4) is non-trivial and can be interpreted as a Lagrangian multiplier associated to the
+constraint v|y=1 = 0. We refer the reader to the works [3, 4, 15, 16, 21, 22, 27, 34] and as well the
+interesting result in [30] in which the author proves that, contrarily to what happens for the Prandtl
+3
+
+equations, the presence of an inflexion point may trigger high-frequencies instabilities in the linearization
+of eq. (1.4) around a shear flow, i.e. eq. (1.4) in which the first equation is substituted by eq. (1.3). We
+want to highlight that, to the best of our knowledge, the best regularity result for eq. (1.4) is provided
+in [15, 33] for 9/8–Gevrey data under an additional convexity assumption. Hence the optimal stability
+vs. instability question is still an open question for the system (1.4), contrarily to what is known for the
+Prandtl system (1.2).
+Remark 1.1. This paper addresses the well-posedness of (1.1) within y ∈ (0, 1), nevertheless our intent is
+to provide insights about an extension of the Prandtl equations (1.2) (for which we know that the barrier is
+Gevrey 2) rather than the hydrostatic ones in (1.4). Dealing with the pressure and homogeneous Dirichlet
+conditions on v is beyond our interest (certainly, with the pressure, the problem would be much more
+involved). We address a bounded vertical domain y ∈ (0, 1) uniquely for the sake of a clear presentation
+of our analysis. To the best of our knowledge, our work is indeed the first to overcome the barrier of
+Gevrey 2 for a meaningful extension of Prandtl. We infer that a similar result can be achieved in the
+classical domain (t, x, y) ∈ (0, T) × T × (0, ∞), making use of a related ansatz on function spaces with
+weighted norms in the vertical direction.
+1.3
+The Cattaneo’s law on the hydrostatic approximation
+Besides the barrier of Gevrey 2, a more physical drawback of Systems (1.2) and (1.4) can be found at
+the level of the Navier-Stokes equations (from which (1.2) and (1.4) are indeed asymptotically derived),
+because of the so-called infinite propagation speed of the velocity field (any local variation of the velocity
+field perturbs immediately the flow in all the domain). To avoid this scenario (which may be occasionally
+unsatisfactory, especially in the hydrodynamics of fluids at large scale), a suitable hyperbolic extension
+of Navier-Stokes has found growth in popularity in the mathematical community (cf. [1, 5, 9, 24, 26, 28,
+29]). At a first glance, this extension seems to introduce obstacles, for instance it enlarges the hydrostatic
+equations (1.4) into
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+�
+τ∂t + 1
+��
+∂tu + u∂xu + v∂yu
+�
+− ∂2
+yu + ∂xp = 0,
+(0, T) × X × (0, 1),
+∂yp = 0
+(0, T) × X × (0, 1),
+∂xu + ∂yv = 0
+(0, T) × X × (0, 1),
+(u, v)|y=0,1 = 0
+(0, T) × X,
+(u, ut)|t=0 = (uin, ut,in)
+X × (0, 1),
+,
+(1.5)
+where τ > 0 is a meaningful parameter, konwn as relaxation time. Once more, the pressure p in (1.5) is
+uniquely due to v|y=1 = 0 and would vanish when relaxing this constraint (as in our model (1.1)).
+System (1.5) arises (at least formally) from the inviscid limit of the Navier-Stokes equations, whose Cauchy
+stress tensor is “delayed” through a first-order Taylor expansion:
+S(t + τ, ·) ≈ S(t, ·) + τ∂tS(t, ·) = ν ∇u(t, ·) + ∇u(t, ·)T
+2
+(we refer to [5] for more details). This relation was introduced in fluid-dynamics by Carrassi and Morro [6],
+inspired by the celebrated work of Cattaneo [7, 8] on heat diffusion.
+Despite its relevance, the well-posedness theory of System (1.5) is unfortunately much less understood.
+In [26], the authors considered τ = 1 and neglected the term ∂t(u∂xu + v∂yu) in the first equation. By
+exploiting a similar technique as the one used in [25], they showed that small initial data in Gevrey 2
+generate global-in-time solutions. Moreover, in this framework, they justified the asymptotic limit of the
+Navier-Stokes under Cattaneo’s law towards the solutions of (1.5), when the viscosity is vanishing.
+The main goal of this paper is to show that one can potentially overcome the barrier of Gevrey 2,
+when dealing with the relevant extensions (1.5) of the classical Prandtl equation. We indicate accurately
+4
+
+this principle on the linearised equation (1.1) of System (1.5) around a shear flow (Ush(y), 0), when the
+constraint v|y=1 = 0 is relaxed (hence no pressure is involded, as for Prandtl). We establish that this
+model is indeed well-posed locally-in-time, when the initial data are Gevrey-class 3 in x (thus less regular
+than Gevrey 2) and Sobolev in y. We provide also some remarks about the non-linear system in Section 3.
+1.4
+Preliminaries and statement of the main result
+To formalise our statement, we shall briefly recall the formal definition of Gevrey functions, that we will
+use throughout our analysis.
+Definition 1.2. Let σ > 0 and m ≥ 1.
+We define the Banach space Gm
+σ,xL2
+y = Gm
+σ (T, L2(0, 1))
+(resp. Gm
+σ,xH1
+0,y = Gm
+σ (T, H1
+0(0, 1))) as all integrable functions f ∈ L1(T × (0, 1)) satisfying:
+• Each coefficient fk : (0, 1) → R of the Fourier transform in the x-variable
+fk(y) = 1
+2π
+�
+T
+f(x, y)e−ikxdx,
+y ∈ (0, 1),
+belongs to L2(0, 1) (resp. H1
+0(0, 1)).
+• The sequences of norms (∥fk∥L2)k∈Z (resp. (∥∂yfk∥L2)k∈Z) decays exponentially as e−σ|k|1/m at high
+frequencies.
+More precisely, f ∈ L1(T×(0, 1)) belongs to Gm
+σ,xL2
+y (resp. Gm
+σ,xH1
+0,y), if the following norm is indeed finite:
+∥f∥Gm
+σ,xL2y :=
+���eσ|k|
+1
+m ∥fk∥L2(0,1)
+���
+ℓ∞(Z) = sup
+k∈Z
+�
+eσ|k|
+1
+m
+� � 1
+0
+|fk(y)|2dy
+� 1
+2 �
+< +∞,
+(1.6)
+(resp. ∥f∥Gm
+σ,xH1
+0,y := ∥∂yf∥Gm
+σ,xL2y < +∞).
+Function spaces with Gevrey regularity are rather standard, especially in the mathematical treatment of
+the Prandtl equations. Indeed, by strongly localising the frequencies, one copes with the major instabilities
+of the underlying solutions. When f = f(x) depends uniquely upon x ∈ T, however, the definition of
+the Gevrey norm in (1.6) may vary in terms of the preferred analytical tools. Among the most relevant
+norms, we mention for instance
+� �
+k∈Z
+e2σ|k|
+1
+m |fk|2
+� 1
+2
+,
+sup
+n∈N
+� σn
+(n!)m ∥∂n
+xf∥L∞(T)
+�
+,
+which are somehow equivalent to (1.6), for positive radii close to σ.
+Definition 1.3. Let T denote a lifespan in (0, +∞] and η : (0, T) → (0, +∞) be a positive continuous
+non-increasing function, representing the time-evolution of the Gevrey radius of regularity. We say that
+a function f belongs to Lp(0, T; Gm
+η(t),xL2
+y) (resp. Lp(0, T; Gm
+η(t),xH1
+0,y)), for a fixed 1 ≤ p ≤ ∞, if
+• f belongs to Lp(0, T; L1(T × (0, 1))),
+• f(t) belongs to Gm
+η(t),xL2
+y (resp. Gm
+η(t),xH1
+0,y), for almost any t ∈ (0, T),
+• the function t ∈ (0, T) → ∥f(t)∥Gm
+η(t),xL2y (resp. t ∈ (0, T) → ∥f(t)∥Gm
+η(t),xH1
+0,y) belongs to Lp(0, T).
+The function space being set up, the main goal of this paper is to establish the local-in-time existence of
+solutions for the linearised equation (1.1), whose initial data uin and ut,in are indeed Gevrey-class 3 in the
+horizontal variable, as described by Definition 1.2.
+5
+
+Theorem 1.4. Assume that the shear flow y ∈ (0, 1) �→ Ush(y) is in W 3,∞(0, 1), while the initial data
+uin, ut,in : T × (0, 1) → R are in G3
+σ,xH1
+0,y and G3
+σ,xL2
+y, respectively, for a positive σ > 0. Denoting by
+Tσ > 0 the lifespan
+Tσ := sup
+�
+t > 0
+such that
+σ
+8 − 2
+5
+6
+�
+∥U ′′′
+sh∥L∞ + 2∥U ′′
+sh∥L∞
+� 1
+3 e
+t
+3t > 0
+�
+∈ (0, +∞]
+(1.7)
+and by β, γ : [0, Tσ) → (0, +∞) the following time-dependent radii of Gevrey-class regularity
+β(t) := σ
+4 − 2
+5
+6
+�
+∥U ′′′
+sh∥L∞ + 2∥U ′′
+sh∥L∞
+� 1
+3e
+t
+3t > 0,
+γ(t) := β(t) − σ
+8 > 0,
+(1.8)
+then the linearised system (1.1) admits a unique weak solution u : [0, Tσ) × T × (0, 1) → R in the function
+space
+u ∈ L∞(0, Tσ; G3
+β(t),xH1
+0,y)
+with
+∂tu ∈ L∞(0, Tσ; G3
+γ(t),xL2
+y).
+(1.9)
+Furthermore, the following estimate holds true at any time t ∈ [0, Tσ):
+∥u(t)∥G3
+β(t),xH1
+0,y + ∥∂tu(t)∥G3
+γ(t),xL2y ≤ Dσ(Ush)(1 + t)5et�
+∥uin∥G3σ,xH1
+0,y + ∥ut,in∥G3σ,xL2y
+�
+,
+(1.10)
+for a suitable constant Dσ(Ush) > 0, which depends uniquely upon σ > 0 and the W 3,∞-norm of Ush.
+Before presenting the major novelties and implications of this result, some remarks on the statement are
+here in order.
+The solutions of Theorem 1.4 are weak because of the regularity of the initial data along the vertical
+direction y ∈ (0, 1). This regularity comes from the underlying a-priori energy of the System (1.1)
+1
+2
+d
+dt
+�
+∥(∂t + 1)u∥2
+L2 + ∥∂yu∥2
+L2
+�
++ ∥∂yu∥2
+L2 = 0.
+Although the solution u and its derivative ∂tu are in L∞ with respect to the time variable, we write the
+estimate (1.10) at any t ∈ [0, Tσ) (instead of “for a.e. t ∈ (0, Tσ)”). This is due to the fact that both u
+and ∂tu admit a continuous representative in the following space: for any finite T ∈ (0, Tσ) and for any
+fixed radius µ ∈ (0, β(T)) (resp. µ ∈ (0, γ(T))), t ∈ [0, T] → u(t) belongs to C([0, T], G3
+µ,xH1
+0,y) (resp. ∂tu
+belongs to C([0, T], G3
+µ,xH1
+0,y)). Writing directly the expression u ∈ C([0, Tσ), G3
+β(t),xH1
+0,y) in (1.9) would be
+somehow incomplete without a proper clarification, since the norm of G3
+β(t),xH1
+0,y does progress in time.
+Certainly one may define this continuity in terms of topology, however this would just complicate the
+readability of the paper. We maintain therefore the function space of (1.9) and the estimate (1.10) at any
+time t ∈ [0, Tσ).
+The solutions provided by Theorem 1.4 are (in general) only local-in-time, although system (1.1) is linear
+in its state variables. Indeed, as for the classical Prandtl equations, the high regularity of the initial data
+uin, ut,in is persistently eroded by the term vU ′
+sh in (1.1) (and in our extension of Prandtl with Cattaneo
+also by ∂tvU ′
+sh), together with the viscous operator −∂2
+yu in y ∈ (0, 1) (cf. [13]). This aspect is here
+reflected by the decaying of the Gevrey radii β and γ in (1.8), as time t > 0 approaches the lifespan Tσ.
+The radii β(0) and γ(0) in (1.8) at initial time t = 0 correspond to σ/4 and σ/8, respectively. One would
+rather expect them to coincide with the radius σ > 0 of the initial data. This gap is merely an artifact of
+our analysis, since we also aim to determine an explicit (and readable) form of the constant Ds(Ush) > 0
+in (1.10). To this end, we make use of a given amount of the exponential decay in e−σ|k|1/3 (for instance
+the missing e−3σ|k|1/3/4 between e−γ(0)|k|1/3 and e−σ|k|1/3), in order to absorb the contribution of certain
+terms, which arise from vU ′
+sh and ∂tvU ′
+sh (cf. for instance (2.38)). Accordingly, we may explicitely set
+Dσ(Ush) > 0 in (1.10) as
+Dσ(Ush) := 104 max{1, 12/σ}15(1 + ∥Ush∥L∞ + ∥U ′
+sh∥L∞ + ∥U ′′
+sh∥L∞ + ∥U ′′′
+sh∥L∞)3.
+(1.11)
+6
+
+This arrangement is certainly far from being sharp. We may for instance build our Gevrey-class-3 solution
+with radii of regularity
+βε(t) = γε(t) := σ − ε − 2
+5
+6
+�
+∥U ′′′
+sh∥L∞ + 2∥U ′′
+sh∥L∞
+� 1
+3e
+t
+3 t,
+for any small ε > 0. This definition would nevertheless complicate the constant Dσ(Ush) (behaving now
+also like 1/ε). For the sake of simple presentation, we do not pursue this direction and we simply remark
+that as long as uin, ut,in are in Gm
+σ,xL2
+y and Gm
+σ,xH1
+0,y, respectively, then they are also in Gm
+σ/4,xL2
+y and
+Gm
+σ/8,xH1
+0,y.
+1.5
+Novelty and implications
+Let us highlight the novelties and consequences of Theorem 1.4 and the discussion in this work. Indeed,
+the improved Gevrey-3 well-posedness of Theorem 1.4 of System (1.1) is far from being trivial. Standard
+considerations of the (linearized as well as non-linear) hyperbolic Prandtl system yield to a well-posedness
+within Gevrey-class 2, at best (see e.g. [19]). We show that a suitable cancellation mechanism is inherent
+to (1.1), which is similar to the one presented in [11]. Contrary to the classical Prandtl equations, however,
+the hyperbolic behaviour of (1.1) unlocks more refined estimates, that endow the mentioned improvements
+of Gevrey-class 3.
+In Section 2 of [11], Dietert and Gerard-Varet provided a rather clear intuition on why the well-posedness
+result of Gevrey 2 holds true for the linearised Prandtl equations (later on, their result further address
+the non-linear system). In order to successfully simplify the comprehension of their idea, they made use
+of calculations involving the Laplace transform on the time variable. Then, potentially, some algebraic
+calculations relating the Laplace variable (in time) with the Fourier variable (in space, along the horizontal
+direction) allowed to deduce the correct regularity of solutions, since they showed a possible behaviour of
+the associated semigroup on the linearised system. The downside of this approach consists, however, in
+the fact that the argument seems to lack some final implementation. The authors indeed derived certain
+a-priori estimates in the Laplace variable, however the inverse Laplace transform does not commute with
+norms1, thus this estimates could not be transferred formally to the original solution. Our approach differ
+with the one related to the Laplace transform in [11]. In particular, Lemma 2.1 provides a simple, yet
+very useful tool (an improved Gronwall estimate) to infer Gevrey-estimates by energy estimates (we refer
+to Section 2 for more details).
+Furthermore, we give detailed bounds on regularity, life span and explicit quantitative dependence on
+the shear flow Ush (cf. (1.8)). For example, if U ′′
+sh = 0, Theorem 1.4 shows the global well-posedness of
+(1.1), being in correspondence with the results for the classical Prandtl equations with monotonic data
+(cf. [23]).
+Additionally, we give a detailed discussion on possible improved well-posedness results for the nonlinear
+hyperbolic Prandtl system (1.5). In particular, our work shows that one cannot rely on further simplifica-
+tions of (1.5) in order to achieve existence results beyond the expected Gevrey 2 class (see, e.g., [19] and
+[26]). We refer to Section 3 for a consideration of three possible nonlinear variants with their advantages
+and drawbacks in terms of regularity propagation.
+Finally, a short summary of the remaining parts.
+The beginning of Section 2 contains an extended
+overview of the proof of Theorem 1.4 which is split into six parts. In the aforementioned Section 3, the
+discussion of possible extensions of the arguments to the nonlinear system is provided.
+1Of course, the actual argument in [11] for the nonlinear system is consistent and rigorous. The aim of the authors in
+Section 2 was to provide a clear understanding.
+7
+
+2
+Proof of Theorem 1.4
+We state first the general principles that we set as the basis of our analysis, and we postpone the details
+of our proof to the remaining paragraphs.
+Our approach is grounded in a similar ansatz as the one developed by Dietert and G´erard-Varet for the
+linearised system of the classical Prandtl equations (cf. Section 2 of [11], outline of the strategy).
+In Section 2.2, we indeed use the Fourier transform along the variable x ∈ T, in order to address the
+behaviour of each mode uk : (0, T) × (0, 1) → R of the velocity field
+u(t, x) =
+�
+k∈Z
+uk(t, y)eikx,
+at any frequency k ∈ Z. Regrettably, the equation of uk (cf. (2.8)) is incapable to derive alone a better
+stability than analytic (initial data far more regular then Gevrey 3). A further development is therefore
+necessary, in order to overcome this first barrier.
+On this account, following [11], we introduce a new state variable ψk : (0, T) × (0, 1) → R in Section 2.2,
+which depends on uk (or rather upon the corresponding stream function Φk, uk = ∂yΦk, cf. (2.17)). Our
+main objective is indeed to asses ψk, in order to remove the (problematic) terms due to vU ′
+sh and ∂tvU ′
+sh.
+These terms preclude indeed an analysis beyond analytic, thus, by eliminating them, we determine a new
+form of the main equation (written now in terms of ψk, cf. (2.20))
+�
+(∂t + 1)(∂t + ikUsh) − ∂2
+y
+�2∂yψk = ik[U ′
+sh, ∂2
+y]
+�
+(∂t + 1)ψk
+�
+,
+(2.1)
+which shall eventually facilitate our analysis in Gevrey-class 3.
+Following (2.1), our approach begins to inherently diverge with respect to the one of Dietert and G´erard-
+Varet in [11]. We avoid entirely their ansatz on the Laplace transform in time, since (despite its clearness)
+it would lead to the difficulties mentioned in Section 1.5. Contrarily, we develop our analysis around a
+specific “weighted” version of the Gronwall’s lemma, which plays somehow the role of cornerstone for our
+entire proof. Its statement is thus the first that we present in Section 2.1 (cf. Lemma 2.1).
+To be more specific, we take advantage of (2.1), in order to determine a certain meaningful estimate on
+the derivatives ∂2
+yψk and (∂t + 1)ψk, as described in details in Proposition 2.3. This estimate can be
+expressed essentially as
+∥ψk(t)∥ ≤ gk(t) + C(t)|k|
+� t
+0
+(t − s)2∥ψk(s)∥ds,
+(2.2)
+where t ∈ (0, T) → ∥ψk(t)∥ represents the L2-norms (in y ∈ (0, 1)) of ∂2
+yψk and (∂t + 1)ψk (for the
+complete version, we refer to Proposition 2.3). The function gk and C in (2.2) are non-decreasing, while
+the integral is expressed also in terms of a “weight” in time: the kernel (t − s)2.
+The kernel (t−s)2 in (2.2) unlocks the regularity Gevrey-class 3, for the derivatives ∂2
+yψk and (∂t+1)ψk. To
+grasp this principle, we shall first remark that, in its absence (thus within a standard Gronwall inequality),
+we may at best derive an estimate of the form ∥ψk(t)∥ ≤ gk(0) exp(tC(t)|k|), where the modes ψk growth
+exponentially as |k| (the setting of analytic solutions). The presence of the kernel provides us however
+better information: because of the weighted Gronwall inequality in Lemma 2.1, the norm ∥ψk(t)∥ can at
+worst growth as gk(0) exp( 3�
+tC(t)|k|), i.e. exponentially as |k|1/3, the framework of Gevrey-class 3.
+The remaining sections are devoted to transfer the aforementioned estimate of ∂2
+yψk and (∂t + 1)ψk first
+to ψk (cf. Lemma 2.4) and secondly to uk (cf Proposition 2.5). Proposition 2.5 is moreover essential to
+determine the final solution u of (1.1), which is Gevrey-class 3 in x ∈ T. Furthermore, this result provides
+the final estimate (1.10) on the Gevrey-norm of u at any time t ∈ [0, Tσ), with also the corresponding
+constant Dσ(Ush) in (1.11).
+Summarising, the forthcoming sections are structured as follows:
+• Section 2.1 and Lemma 2.1 are devoted to the proof of the “weighted” Gronwall’s inequality.
+8
+
+• In Section 2.2 we introduce the new state variable ψk and derive the corresponding equation (2.1).
+• In Section 2.3, we first state the main inequality (2.2) in Proposition 2.3 (whose proof is postponed
+to Section 2.5 and Section 2.6). We furthermore transfer the Gevrey estimates of ∂2
+yψk and (∂t+1)ψk
+to ψk in Lemma 2.4.
+• In Section 2.4, with Proposition 2.5, we transfer the Gevrey estimates to uk and build our final
+solution u of System (1.1). To conclude the proof of the main Theorem 1.4, we determine moreover
+the estimate (1.10) on the Gevrey norm of the solution.
+• Finally, Section 2.5 and Section 2.6 are devoted to the proof of Proposition 2.3 and the main
+inequality (2.2).
+2.1
+A weighted Gronwall inequality
+One of the main ingredient used to prove Theorem 1.4 is the following Gronwall-type Lemma. It asserts
+that any non-negative function, which satisfies a “weighted” Gronwall’s inequality proportional to a
+suitable time-dependent function λ(t)3, can not grow up exponentially faster than λ(t)t.
+Lemma 2.1. Let T > 0 and f : [0, T) → [0, ∞) be a non-negative continuous function, satisfying
+f(t) ≤ g(t) + λ(t)3
+2
+� t
+0
+(t − s)2f(s)ds,
+(2.3)
+for two continuous functions λ, g : [0, T) → [0, ∞), that are non-negative and non-decreasing. Then the
+following inequality holds true at any time t ∈ [0, T):
+f(t) ≤ g(t)
+�
+1 + (λ(t)t)3
+6
+�
+eλ(t)t.
+(2.4)
+Proof. We begin with by defining ω in C3([0, T)) as the following integral: (2.3) as
+ω(t) := 1
+2
+� t
+0
+(t − s)2f(s)ds =
+� t
+0
+f(s)
+� t
+s
+� t
+τ
+drdτds =
+� t
+0
+� r
+0
+� τ
+0
+f(s)dsdτdr.
+The function ω is everywhere non-negative in [0, T). Furthermore, at t = 0, ω(0) and its derivatives
+ω′(0), ω′′(0) are all identically null. We next write inequality (2.3) in terms of ω:
+ω′′′(t) ≤ g(t) + λ(t)3ω(t),
+for all
+t ∈ [0, T).
+(2.5)
+Hence, we fix a general time ˜t ∈ (0, T) and we momentarily consider only values of t within [0, ˜t]. We
+multiply equation (2.5) with e−λ(˜t)t (where λ(˜t) is fixed and plays momentarily the role of a constant).
+By means of standard calculations on the derivatives, we gather that
+ω′′′(t)e−λ(˜t)t = d3
+dt3
+�
+ω(t)e−λ(˜t)t�
++3 d2
+dt2
+�
+λ(˜t)(ω(t)e−λ(˜t)t�
++3 d
+dt
+�
+λ(˜t)2ω(t)e−λ(˜t)t�
++λ(˜t)3ω(t)e−λ(˜t)t
+≤
+�
+g(t) + λ(t)3ω(t)
+�
+e−λ(˜t)t ≤ g(t)e−λ(˜t)t + λ(˜t)3ω(t)e−λ(˜t)t.
+(2.6)
+In the last inequality, we have used that λ is non-decreasing and non-negative, ω ≥ 0 and that t < ˜t. We
+shall now remark that the term λ(˜t)3ω(t)e−λ(˜t)t cancel out and the left-hand side of (2.6) is hence left
+with only time derivatives. We are hence in the condition to integrate (2.6) along [0, t], to gather that
+d2
+dt2
+�
+ω(t)e−λ(˜t)t�
++ 3λ(˜t) d
+dt
+�
+ω(t)e−λ(˜t)t�
++ 3λ(˜t)2ω(t)e−λ(˜t)t ≤
+� t
+0
+g(s)e−λ(˜t)sds ≤
+� t
+0
+g(s)ds.
+9
+
+We can drop the term 3λ(˜t)2ω(t)e−λ(˜t)t at the left-hand side (since it is positive) and integrate once more
+along the interval [0, t], for a general t ∈ [0, ˜t]:
+d
+dt
+�
+ω(t)e−λ(˜t)t�
++ 3λ(˜t)ω(t)e−λ(˜t)t ≤
+� t
+0
+� s
+0
+g(z)dz =
+� t
+0
+(t − z)g(z)dz.
+Since both λ(˜t) and ω(t) are positive, we can drop 3λ(˜t)ω(t)e−λ(˜t)t and integrate a final time along (0, t):
+ω(t)e−λ(˜t)t ≤
+� t
+0
+� s
+0
+(s − z)g(z)dzds =
+� t
+0
+g(z)(t − z)2
+2
+dz
+⇒
+ω(t) ≤ eλ(˜t)t
+� t
+0
+g(z)(t − z)2
+2
+dz.
+We are now in the condition to combine the last relation in t = ˜t together with (2.3), which ensures that
+f(˜t) ≤ g(˜t) + eλ(˜t)˜tλ(˜t)3
+� ˜t
+0
+g(s)(˜t − s)2
+2
+ds
+≤ g(˜t) + eλ(˜t)˜tλ(˜t)3g(˜t)
+� ˜t
+0
+(˜t − s)2
+2
+ds ≤ g(˜t)
+�
+1 + λ(˜t)3˜t3
+6
+�
+eλ(˜t)˜t.
+Re-denoting ˜t = t and from its arbitrariness in (0, T), we finally achieve inequality (2.4) (also remarking
+that (2.4) is trivially satisfied in t = 0). This concludes the proof of the lemma.
+2.2
+The stream function formulation
+In this section, we begin developing our analysis of System (1.1) and we first decompose the corresponding
+equations in terms of several Fourier coefficients uk : (t, y) ∈ (0, Tσ) × (0, 1) → R of the velocity field u,
+at any frequency k ∈ Z. The lifespan Tσ > 0 (denoted by T in (1.1)) shall be considered from now on as
+in (1.7) of Theorem 1.4, nevertheless its form will play a major role only starting from Section 2.3.
+Eventually, we will build the final solution (u, v) of (1.1), by invoking the Fourier Series with respect to the
+variable x ∈ T and the divergence-free condition ∂xu+∂yv = 0 (which at any frequency is ikuk +∂yvk = 0)
+u(t, x, y) =
+�
+k∈Z
+uk(t, y)eikx,
+uk(t, y) := 1
+2π
+�
+T
+u(t, x, y)e−ixkdx,
+v(t, x, y) =
+�
+k∈Z
+vk(t, y)eikx,
+vk(t, y) := −ik
+� y
+0
+uk(t, z)dz,
+(2.7)
+however we shall first determine some uniform estimates on (uk)k∈Z, in order to provide a sense of the
+series above. Hence, we begin with by considering System (1.1) rather as a family of PDEs in the variables
+(t, y) ∈ (0, Tσ) × (0, 1), which depend upon each frequency k ∈ Z:
+
+
+
+
+
+
+
+
+
+
+
+∂2
+t uk + ikUsh∂tuk + U ′
+sh∂tvk + ∂tuk + ikUshuk + vkU ′
+sh − ∂2
+yuk = 0
+(0, Tσ) × (0, 1),
+ikuk + ∂yvk = 0
+(0, Tσ) × (0, 1),
+(uk, ∂tuk)|t=0 = (uin,k, ut,in,k)
+(0, 1),
+(uk, vk)|y=0,1 = (0, 0)
+(0, Tσ).
+(2.8)
+The second equation ikuk + ∂yvk = 0 and the boundary conditions vk|y=0 = 0 allow to interpret System
+(2.8) only on the state variable uk, since the vertical component vk is explicitly determined by (2.7). The
+initial data uin,k and ut,in,k are in H1
+0(0, 1) and L2(0, 1), respectively, since uin and ut,in are in G3
+σ,xH1
+0,y
+and G3
+σ,xL2
+0,y, as described by Definition 1.2.
+We hence state the following result about the existence and uniqueness of solutions for System (2.8).
+10
+
+Proposition 2.2. For any fixed frequency k ∈ Z and any initial data (uin,k, ut,in,k) in H1
+0(0, 1) × L2(0, 1)
+there exists a unique solution uk : [0, T) × (0, 1) → R of (2.8), which belongs to
+(uk, ∂yuk) ∈ C([0, T], H1
+0),
+∂tuk ∈ L2(0, T; L2),
+(2.9)
+for any real time T > 0.
+We shall here remark that the real T > 0 may not correpond to Tσ, since the lifespan in (1.7) may be
+Tσ = +∞ (for shear flow with U ′′
+sh ≡ U ′′′
+sh ≡ 0). In this case, we shall always treat uk as in (2.9), for any
+0 < T < Tσ = +∞. Contrarily, if Tσ < +∞ (which is satisfied for more general Ush), then we replace T
+in (2.9) directly with Tσ.
+Proof. For the sake of simplicity, we provide here only a sketch, since the result can be shown through
+standard arguments on linear PDEs. We remark indeed that Equation (2.8) can be written as a 1D
+damped wave equation with Dirichelet boundary conditions
+(□ + ∂t) uk = Fk,
+□ = ∂2
+t − ∂2
+y,
+uk|y=0,1 = 0,
+(2.10)
+and forcing term Fk, which depends linearly on uk and is given by
+Fk = −
+�
+ikUsh∂tuk − ikU ′
+sh∂t
+� y
+0
+uk(t, z)dz + ikUshuk − ikU ′
+sh
+� y
+0
+uk(t, z)dz
+�
+.
+Making use of a standard computation combined with the Poincar´e inequality, we infer that Fk satisfies
+∥Fk∥H1
+0 ≤ C|k| ∥Ush∥W 3,∞
+�
+∥∂tuk∥H1
+0 + ∥∂yuk∥H1
+0
+�
+,
+(2.11)
+for a suitable positive constant C > 0. We can hence apply a standard Galerkin method to deduce the
+existence of a unique solution within the function space of (2.9).
+We come back now to System (2.8). Since uk(t, ·) is divergent free, it can be written in terms of a stream
+function Φk = Φk(t, y), which is in C([0, T], H2) with ∂tΦk ∈ L2(0, T; H1(0, 1)), for any real T ≤ Tσ.
+Furthermore, because ikΦk = vk, the function Φk is identically null in y = 0 (in the sense of trace),
+therefore
+uk(t, y) = ∂yΦk(t, y)
+�
+i.e. Φk(t, y) :=
+� y
+0
+uk(t, z)dz
+�
+⇒
+vk(t, y) = −ikΦk(t, y).
+(2.12)
+The initial data of Φk at any y ∈ (0, 1) are given by
+Φin,k(y) :=
+� y
+0
+uin,k(z)dz,
+Φt,in,k(y) :=
+� y
+0
+ut,in,k(z)dz,
+which ensures Φin,k ∈ H2(0, 1) ∩ H1
+0(0, 1) and Φt,in,k ∈ H1(0, 1)(remark that Φin,k(1) = 0, since uin,k is
+average free).
+From (2.8), we deduce that Φk is solution in (0, Tσ) × (0, 1) of the following system:
+
+
+
+
+
+∂2
+t ∂yΦk+ikUsh(y)∂t∂yΦk−ikU ′
+sh(y)∂tΦk+∂t∂yΦk+Ush(y)ik∂yΦk−ikΦkU ′
+sh(y) − ∂3
+yΦk = 0,
+(Φk, ∂tΦk)|t=0 = (Φin,k, Φt,in,k)
+(∂yΦk)|y=0,1 = 0,
+Φk|y=0 = 0.
+(2.13)
+We next provide some heuristics on the L2-estimates satisfied by Φk and show that, unfortunately, System
+(2.13) (as it is written) may not prevent the stream function Φk to exponentially growth as exp(a|k|), for
+11
+
+some positive a > 0. To this end, we first isolate the linear operator in equation (2.13) that affects only
+the derivative ∂yΦk and transfer the remaining terms in Φk on the right-hand side of the identity:
+((∂t + 1)(∂t + ikUsh) − ∂2
+y)∂yΦk = (∂t + 1)ikU ′
+shΦk.
+(2.14)
+We will shortly see that the operator on the left-hand side of (2.14) is crucial for our next analysis (in
+particular, to define a new state variable ψk in (2.17)). We first outline, however, that, in the current form,
+equation (2.14) is still ineffective and does not predict the crucial Gevrey-3 regularity of our solutions.
+Indeed, we infer that a standard energy approach would provide (at best) an L2-estimate of ∂yΦk = uk
+of the form
+1
+2
+d
+dt
+�
+∥(∂t + 1)∂yΦk∥2
+L2 + ∥∂t∂yΦk∥2
+L2 + 2∥∂2
+yΦk∥2
+L2
+�
+≤ C|k|
+�
+∥(∂t + 1)∂yΦk∥2
+L2 + ∥∂t∂yΦk∥2
+L2
+�
+,
+(2.15)
+for a suitable positive constant C, which is also calibrated with the following Poincar´e-type inequality of
+Φk in the domain y ∈ (0, 1):
+∥Φk(t)∥L2 =
+� � 1
+0
+|Φk(t, y)|2dy
+� 1
+2
+=
+� � 1
+0
+���
+� y
+0
+∂yΦk(t, z)dz
+���
+2
+dy
+� 1
+2
+≤
+� � 1
+0
+y
+� y
+0
+���∂yΦk(t, z)
+���
+2
+dzdy
+� 1
+2
+≤ 1
+2
+� � 1
+0
+���∂yΦk(t, z)
+���
+2
+dz
+� 1
+2
+≤ ∥∂yΦk(t)∥L2.
+(2.16)
+Hence, roughly speaking, in this regime the L2-norms of ∂yΦk and ∂t∂yΦk would growth exponentially
+as ec|k|(∥∂yΦin,k∥L2 + ∥∂t∂yΦin,k∥L2), a setting which is typical of analytic solutions (which are of course
+much more regular than any Gevrey-class m, m > 1). To achieve the Gevrey regularity, we shall therefore
+perform a further development. To this end, we introduce a new state variable ψk : [0, Tσ) × (0, 1) → R.
+Following the approach used in [11] for the classical Prandtl equation, ψk = ψk(t, y) is chosen in a form
+that gets rid of the terms ikU ′
+sh∂tΦk and ikU ′
+shΦk at the left-hand side of (2.13). More precisely, we define
+ψk as the unique solution in L∞(0, T; H2) ∩ L2(0, T; H2) of the following PDE:
+
+
+
+
+
+((∂t + 1)(∂t + ikUsh) − ∂2
+y)ψk = Φk
+(0, T) × (0, 1),
+(ψk, ∂tψk)|t=0 = (0, 0)
+(0, 1),
+ψk|y=0,1 = 0
+(0, T).
+(2.17)
+The most compelling reason for this definition is a meaningful cancellation that occurs when coupling
+(2.17) together with (2.14). More precisely, equation (2.14) implies that ψk satisfies
+((∂t + 1)(∂t + ikUsh) − ∂2
+y)∂y((∂t + 1)(∂t + ikUsh) − ∂2
+y)ψk = ikU ′
+sh(∂t + 1)Φk,
+which is
+�
+(∂t +1)(∂t +ikUsh)−∂2
+y
+�2∂yψk +
+�
+(∂t +1)(∂t +ikUsh)−∂2
+y
+��
+ikU ′
+sh(∂t +1)ψk
+�
+= ikU ′
+sh(∂t +1)Φk. (2.18)
+The second term at left-hand side of (2.18) almost coincides with ikU ′
+sh(∂t+1)Φk at the right-hand side. To
+complete the aforementioned cancellation, we first invoke the commutator ik[U ′
+sh, ∂2
+y]ψk = ikU ′
+sh∂2
+y(ψk) −
+ik∂2
+y(U ′
+shψk), in order to write the second term in (2.18) as
+�
+(∂t +1)(∂t +ikUsh)−∂2
+y
+��
+ikU ′
+sh(∂t +1)ψk
+�
+= ikU ′
+sh
+�
+(∂t +1)(∂t +ikUsh)−∂2
+y
+��
+(∂t +1)ψk
+�
+−ik[U ′
+sh, ∂2
+y]ψk.
+Finally, we plug this identity into (2.18), to gather
+�
+(∂t + 1)(∂t + ikUsh) − ∂2
+y
+�2∂yψk + ikU ′
+sh(∂t + 1)
+� �
+(∂t + 1)(∂t + ikUsh) − ∂2
+y
+�
+ψk
+�
+��
+�
+=Φk
+�
++
+−[ikU ′
+sh(y), ∂2
+y]
+�
+(∂t + 1)ψk
+�
+= ikU ′
+sh(∂t + 1)Φk.
+(2.19)
+12
+
+Recalling that ψk satisfies (2.17), we remark that ikU ′
+sh(∂t+1)Φk appears both on the left- and right-hand
+sides of (2.19). We thus obtain the following final form of the ψk-equation:
+�
+(∂t + 1)(∂t + ikUsh) − ∂2
+y
+�2∂yψk = ik[U ′
+sh, ∂2
+y]
+�
+(∂t + 1)ψk
+�
+,
+(2.20)
+which was indeed claimed at the beginning in (2.1). We shall remark that (2.20) still presents a forcing
+term ik[U ′
+sh, ∂2
+y]
+�
+(∂t + 1)ψk
+�
+, which growths linearly like |k| at high frequencies |k| ≫ 1 (similarly as
+ikU ′
+sh(∂t +1)Φk in (2.14)). Nonetheless, the operator ((∂t +1)(∂t +ikUsh)−∂2
+y)2 on ∂yψk has now doubled
+in order (in comparison with just ((∂t + 1)(∂t + ikUsh) − ∂2
+y) in (2.14)). This will unlock more-refined
+estimates on ∂yψk (and its derivatives) than the ones in (2.15) for ∂yΦk. We formalise these heuristics in
+the next sections.
+2.3
+Reaching Gevrey-class 3
+In the forthcoming analysis, we illustrate how the derived equation (2.20) succeeds in enabling an L2-
+estimate of Gevrey-3 type to the new state variable ∂yψk, as well its time derivative ∂t∂yψk. This estimate
+is a consequence of the following statement, that we set as the basis of our development. It guarantees
+that the functions (∂t + 1)∂yψk and ∂2
+yψk satisfy an improved Gronwall-type inequality, as described by
+Lemma 2.1.
+Proposition 2.3. The following estimate on the functions (∂t + 1)∂yψk and ∂2
+yψk holds true, for any
+frequency k ∈ Z and at any time t ∈ (0, Tσ):
+sup
+s∈[0,t]
+���(∂t + 1)∂yψk(s)
+��
+L2 +
+��∂2
+yψk(s)
+��
+L2
+�
+≤ gk(t) + λk(t)3
+2
+� t
+0
+(t − s)2 sup
+τ∈[0,s]
+���(∂t + 1)∂yψk(τ)
+��
+L2 +
+��∂2
+yψk(τ)
+��
+L2
+�
+ds.
+(2.21)
+The functions gk, λk are increasing in time and depend uniquely on k ∈ Z, the shear flow Ush and the
+initial data (uin,k, ut,in,k, Φin,k). More precisely, gk, λk are defined by
+gk(t) := 4t
+�
+|k|
+�
+∥U ′
+sh∥L∞∥Φin,k∥L2 + ∥Ush∥L∞∥uin,k∥L2
+�
++ ∥ut,in,k∥L2 + (3 +
+√
+2)∥uin,k∥L2
+�
+λk(t) := 2
+5
+6 |k|
+1
+3
+�
+∥U ′′′
+sh∥L∞ + 2∥U ′′
+sh∥L∞
+� 1
+3 e
+t
+3.
+(2.22)
+for k ∈ Z and t ∈ (0, Tσ).
+Since the proof of this Proposition is rather technical, we postpone it to Section 2.5 and we focus the next
+paragraphs on the remaining steps to prove Theorem 1.4.
+Let us observe that inequality (2.21) encompasses the form (2.3) of Lemma 2.1. This ensures therefore
+that the following improved Gronwall’s inequality holds true for any t ∈ [0, Tσ):
+��(∂t + 1)∂yψk(t)
+��
+L2 +
+��∂2
+yψk(t)
+��
+L2 ≤ sup
+s∈[0,t]
+��(∂t + 1)∂yψk(s)
+��
+L2 ≤ gk(t)
+�
+1 + (λk(t)t)3
+6
+�
+eλk(t)t.
+We have essentially achieved the claimed regularity of Gevrey 3, at least for (∂t + 1)∂yψk and ∂2
+yψk, since
+the definitions of gk and λk in (2.22) imply that
+sup
+s∈[0,t]
+���(∂t+1)∂yψk(s)
+��
+L2+
+��∂2
+yψk(s)
+��
+L2
+�
+≤4t
+�
+|k|
+�
+∥U ′
+sh∥L∞∥Φin,k∥L2 + ∥Ush∥L∞∥uin,k∥L2
+�
++∥ut,in,k∥L2+
++(3+
+√
+2)∥uin,k∥L2
+��
+1+
+2
+√
+2|k|
+�
+∥U ′′′
+sh∥L∞ + 2∥U ′′
+sh∥L∞
+�
+ett3
+3
+�
+exp
+�
+|k|
+1
+32
+5
+6
+�
+∥U ′′′
+sh∥L∞+2∥U ′′
+sh∥L∞
+� 1
+3 e
+t
+3t
+�
+.
+(2.23)
+13
+
+In particular, the L2-norm increases exponentially at worst as |k|1/3, which we can counteract (at least
+locally in time), by imposing that the initial data exponentially decay with the same order. To formalise
+this principle, we shall however first transfer all frequencies |k| of (2.23) to the exponential function, as
+well as transfer these Gevrey-3 estimates also to ψk, ∂yψk and ∂t∂yψk (instead of just (∂t + 1)∂yψk and
+∂2
+yψk). We cope with this issue in the following lemma.
+Lemma 2.4. Assume that the sequence of initial data (uin,k, ut,in,k, Φin,k)k∈Z satisfies
+sup
+k∈Z
+�
+eσ|k|
+1
+3 �
+∥uin,k∥L2 + ∥ut,in,k∥L2 + ∥Φin,k∥L2
+��
+< +∞,
+(2.24)
+for a given radius σ > 0. Let α : [0, Tσ) → R+ be defined by
+α(t) := σ
+2 − 2
+5
+6
+�
+∥U ′′′
+sh∥L∞ + 2∥U ′′
+sh∥L∞
+� 1
+3 e
+t
+3 t ≥ 0.
+(2.25)
+Then the sequence (ψk)k∈N generates a function ψ : [0, Tσ) × T × (0, 1) → R through the Fourier series
+ψ(t, x, y) =
+�
+k∈Z
+ψk(t, y)eikx,
+(t, x, y) ∈ [0, Tσ) × T × (0, 1)
+(2.26)
+such that ψ, ∂yψ and ∂tψ are all in L∞(0, Tδ; G3
+α(t),xH1
+0,y). In particular, the following estimate on the
+norms holds true at any time t ∈ [0, Tσ):
+∥ψ(t)∥G3
+α(t),xH1
+0,y+∥∂yψ(t)∥G3
+α(t),xH1
+0,y+∥∂2
+tyψ(t)∥G3
+α(t),xH1
+0,y+∥∂2
+yψ(t)∥G3
+α(t),xH1
+0,y=
+sup
+k∈Z
+�
+eα(t)|k|
+1
+3��ψk(t)
+��
+L2
+�
++sup
+k∈Z
+�
+eα(t)|k|
+1
+3��∂yψk(t)
+��
+L2
+�
++sup
+k∈Z
+�
+eα(t)|k|
+1
+3��∂2
+tyψk(t)
+��
+L2
+�
++sup
+k∈Z
+�
+eα(t)|k|
+1
+3��∂2
+yψk(t)
+��
+L2
+�
+≤ Cσ(t)
+�
+1+∥Ush∥L∞ +∥U ′
+sh∥L∞ +∥U ′′
+sh∥L∞ +∥U ′′′
+sh∥L∞
+�2
+sup
+k∈Z
+�
+eσ|k|
+1
+3 �
+∥Φin,k∥L2 +∥uin,k∥L2+∥ut,in,k∥L2
+��
+,
+(2.27)
+where Cσ(t) = 170 · max{1, 12/σ}6t(1 + t)3et.
+Proof. If the sequence (ψk)k∈Z and its derivatives satisfy the corresponding inequality (2.27), then it
+generates trivially a function ψ : [0, Tδ) × T × (0, 1) → R as described in (2.26) , since the series converges
+strongly in L∞(0, Tσ; L2(T × (0, 1)) and the limit has explicit Fourier coefficients given by (ψk)k∈Z. Our
+main objective is therefore to prove uniquely the inequality (2.27) in the coefficient (ψk)k∈Z.
+We first show that the function (∂t + 1)∂yψk and ∂2
+yψk satisfies a similar inequality, namely
+eα(t)|k|
+1
+3 ���(∂t + 1)∂yψk(t)
+��
+L2 +
+��∂2
+yψk(t)
+��
+L2
+�
+≤ Cσ(t)
+5
+�
+1 + ∥Ush∥L∞+
++ ∥U ′
+sh∥L∞ + ∥U ′′
+sh∥L∞ + ∥U ′′′
+sh∥L∞
+�2
+sup
+˜k∈Z
+�
+eσ|˜k|
+1
+3 �
+∥Φin,˜k∥L2 +∥uin,˜k∥L2+∥ut,in,˜k∥L2
+��
+,
+(2.28)
+for any k ∈ Z and t ∈ [0, Tσ). When k = 0, then (2.28) is essentially a direct consequence of (2.23), which
+implies in particular
+��(∂t + 1)∂yψ0(t)
+��
+L2 +
+��∂2
+yψ0(t)
+��
+L2 ≤ 4t
+�
+∥uin,0∥L2 + ∥ut,in,0∥L2
+�
+.
+We turn our attention therefore to |k| ≥ 1, so that (2.23) yields
+sup
+s∈[0,t]
+���(∂t+1)∂yψk(s)
+��
+L2+
+��∂2
+yψk(s)
+��
+L2
+�
+≤ 4t(3 +
+√
+2)
+�
+1+∥U ′
+sh∥L∞ +∥Ush∥L∞
+��
+∥Φin,k∥L2+∥uin,k∥L2+
++∥ut,in,k∥L2
+�4
+√
+2
+3 et(1 + t)3�
+1 + ∥U ′′′
+sh∥L∞ + ∥U ′′
+sh∥L∞
+�
+|k|2 exp
+�
+2
+5
+6 |k|
+1
+3
+�
+∥U ′′′
+sh∥L∞ + 2∥U ′′
+sh∥L∞
+� 1
+3 e
+t
+3 t
+�
+.
+14
+
+Hence, we collect all terms of the shear flow Ush within a single parenthesis, we extrapolate eσ|k|1/3 in
+front of the initial data and we choose s = t on the supremum at the left-hand side
+��(∂t + 1)∂yψk(t)
+��
+L2 +
+��∂2
+yψk(t)
+��
+L2 ≤ 34t(1 + t)3et�
+1 + ∥Ush∥L∞ + ∥U ′
+sh∥L∞+
++∥U ′′
+sh∥L∞ + ∥U ′′′
+sh∥L∞�2
+�
+sup
+˜k∈Z
+�
+eσ|˜k|
+1
+3 �
+∥Φin,˜k∥L2+∥uin,˜k∥L2 + ∥ut,in,˜k∥L2
+���
+×
+× |k|2 exp
+�
+− σ|k|
+1
+3 +2
+1
+3 |k|
+1
+3
+�
+∥U ′′′
+sh∥L∞ +2∥U ′′
+sh∥L∞
+� 1
+3e
+t
+3 t
+�
+.
+Since one has a ≤ ea, for any positive real number a > 0, we can bound the term |k|2, by means of
+|k|2 =
+�
+|k|
+1
+3 �6 =
+�12
+σ
+�6� σ
+12|k|
+1
+3
+�6
+≤
+�12
+σ
+�6�
+e
+σ
+12 |k|
+1
+3 �6
+=
+�12
+σ
+�6
+e
+σ
+2 |k|
+1
+3 .
+Hence, remarking that 34(12/σ)6t(1 + t)3et ≤ Cσ(t)/5, we deduce that
+��(∂t + 1)∂yψk(t)
+��
+L2 +
+��∂2
+yψk(t)
+��
+L2 ≤ Cσ(t)
+4
+�
+1 + ∥Ush∥L∞ + ∥U ′
+sh∥L∞ + ∥U ′′
+sh∥L∞ + ∥U ′′′
+sh∥L∞�2×
+×
+�
+sup
+˜k∈Z
+�
+eσ|˜k|
+1
+3 �
+∥Φin,˜k∥L2 + ∥uin,˜k∥L2 + ∥ut,in,˜k∥L2
+���
+exp
+�
+−
+�σ
+2 −2
+1
+3
+�
+∥U ′′′
+sh∥L∞ +2∥U ′′
+sh∥L∞
+� 1
+3e
+t
+3 t
+�
+�
+��
+�
+=α(t)
+|k|
+1
+3
+�
+,
+which coincides with (2.28). We now transfer the estimate (2.28) directly to the functions ψk, ∂yψk and
+∂t∂yψk in (2.27). We begin with by developing ∂yψk through
+∂yψk(t, y) = e−tet∂yψk(t, y) = e−t
+� t
+0
+∂s(es∂yψk(s, y))ds =
+� t
+0
+es−t�
+(∂t + 1)∂yψk
+�
+(s, y)ds,
+for any (t, y) ∈ (0, Tσ) × (0, 1). Hence, by taking the L2-norm on both left and right-hand sides and
+keeping in mind that α is decreasing in time, we remark that
+eα(t)|k|
+1
+3 ∥∂yψk(t)∥L2 ≤ eα(t)|k|
+1
+3
+� t
+0
+es−t∥(∂t + 1)∂yψk(s)∥L2ds ≤
+� t
+0
+es−teα(s)|k|
+1
+3 ∥(∂t + 1)∂yψk(s)∥L2ds
+≤ Cσ(t)
+5
+�
+1+∥Ush∥L∞+∥U ′
+sh∥L∞ +∥U ′′
+sh∥L∞ +∥U ′′′
+sh∥L∞
+�2
+sup
+˜k∈Z
+�
+eσ|˜k|
+1
+3 �
+∥Φin,˜k∥L2 +∥uin,˜k∥L2+∥ut,in,˜k∥L2
+��
+,
+(2.29)
+where we have used
+� t
+0 es−tds = 1 − e−t < 1. Invoking the Poincar´e inequality ∥ψk∥L2 ≤ ∥∂yψk∥L2, it is
+easy at this stage to check that
+sup
+k∈Z
+�
+eα(t)|k|
+1
+3��ψk(t)
+��
+L2
+�
++sup
+k∈Z
+�
+eα(t)|k|
+1
+3��∂yψk(t)
+��
+L2
+�
++sup
+k∈Z
+�
+eα(t)|k|
+1
+3��∂2
+tyψk(t)
+��
+L2
+�
++sup
+k∈Z
+�
+eα(t)|k|
+1
+3��∂2
+yψk(t)
+��
+L2
+�
+≤ 2 sup
+k∈Z
+�
+eα(t)|k|
+1
+3 ��∂yψk(t)
+��
+L2
+�
++ sup
+k∈Z
+�
+eα(t)|k|
+1
+3 ��∂2
+tyψk(t)
+��
+L2
+�
++ sup
+k∈Z
+�
+eα(t)|k|
+1
+3 ��∂2
+yψk(t)
+��
+L2
+�
+≤ 3 sup
+k∈Z
+�
+eα(t)|k|
+1
+3 ��∂yψk(t)
+��
+L2
+�
++ sup
+k∈Z
+�
+eα(t)|k|
+1
+3 ��(∂t + 1)∂yψk(t)
+��
+L2
+�
++ sup
+k∈Z
+�
+eα(t)|k|
+1
+3 ��∂2
+yψk(t)
+��
+L2
+�
+.
+We finally couple the last inequality together with (2.28) and (2.29), which finally implies the estimate
+(2.27). This concludes the proof of the lemma.
+2.4
+Transferring Gevrey 3 to the velocity field
+We shall now transfer the Gevrey regularity from the function ψ of Lemma 2.4 to a solution u of the
+original hyperbolic Prandtl equation (1.5).
+15
+
+Proposition 2.5. Assume that uin is in G3
+σ,xH1
+0,y, while ut,in is in G3
+x,σL2
+y, for a given σ > 0.
+Let
+Tσ > 0 and β, γ : [0, Tσ) → R be as in Theorem 1.4. Then the sequence (uk)k∈N generates a function
+u : [0, Tσ) × T × (0, 1) → R through the inverse Fourier transform
+u(t, x, y) =
+�
+k∈Z
+uk(t, y)eikx,
+(t, x, y) ∈ [0, Tσ) × T × (0, 1),
+(2.30)
+such that u ∈ L∞(0, Tσ; Gβ(t),xH1
+0,y) and ∂tu ∈ L∞(0, Tσ; Gγ(t),xL2
+y). In particular, the following estimate
+holds true at any time t ∈ [0, Tσ):
+∥u(t)∥G3
+β(t),xL2y + ∥∂yu(t)∥G3
+β(t),xL2y + ∥∂tu(t)∥G3
+γ(t),xL2y
+= sup
+k∈Z
+�
+eβ(t)|k|
+1
+3 ��uk(t)
+��
+L2
+�
++ sup
+k∈Z
+�
+eβ(t)|k|
+1
+3 ��∂yuk(t)
+��
+L2
+�
++ sup
+k∈Z
+�
+eγ(t)|k|
+1
+3 ��∂tuk(t)
+��
+L2
+�
+≤ ˜Cσ(t)
+�
+1 + ∥Ush∥L∞ + ∥U ′
+sh∥L∞ + ∥U ′′
+sh∥L∞ + ∥U ′′′
+sh∥L∞
+�3�
+∥uin∥G3σ,xH1
+0,y + ∥ut,in∥G3σ,xL2y
+�
+,
+(2.31)
+where ˜Cσ(t) = 104 max{1, 12/σ}15(1 + t)5et.
+Proof. Similarly as in Lemma 2.4, we focus this entire proof to show the estimate (2.31) on the sequence
+(uk)k∈Z. The solution u is then automatically determined by the Fourier series (2.30).
+We fix momentarily the frequency k ∈ Z. If the initial data (uin,k, uin,k,t) are identically null, then the
+solution uk of (2.8) is identically null (since the equation in (2.8) for a fixed k ∈ Z is linear, hyperbolic
+and damped) and the inequality (2.31) is automatically satisfied. We shall thus focus this proof to the
+case (uin,k, uin,k,t) ̸= (0, 0).
+We begin with by setting the function fk :=
+�
+(∂t +1)(∂t +ikUsh)−∂2
+y
+�
+∂yψk. Thanks to (2.17), fk satisfies
+fk(t, y) = ikU ′
+sh(y)(∂t + 1)ψk(t, y) + ∂yΦk(t, y) = ikU ′
+sh(y)(∂t + 1)ψk(t, y) + uk(t, y),
+(2.32)
+for any (t, y) ∈ [0, Tσ) × (0, 1). Because of the boundary conditions on ψk and uk, the function fk fulfills
+homogeneous Dirichlet conditions fk|y=0,1 = 0. Furthermore its initial data are determined by
+fk(0, y) = uin,k(y)
+∂tfk(0, y) = ikU ′
+sh(y)∂2
+t ψk(0, y) + ut,in,k(y) = ikU ′
+sh(y)Φin,k(y) + ut,in,k(y)
+(2.33)
+for any y ∈ (0, 1). Thanks to identity (2.20), we remark moreover that fk is also solution of
+�
+(∂t + 1)(∂t + ikUsh) − ∂2
+y
+�
+fk = [ikU ′
+sh, ∂2
+y]
+�
+(∂t + 1)ψk
+�
+.
+(2.34)
+Starting from (2.34), we aim to develop some suitable estimates on the L2-norm of (∂t + 1)fk and ∂yfk,
+which we will eventually transfer to uk, ∂yuk and ∂tuk, making use of (2.32). We shall observe that we
+have now a complete control on the right-hand side of (2.34), because of the uniform estimates given by
+Lemma 2.4.
+We multiply (2.34) with the conjugate (∂t + 1)fk, then we integrate along (0, t)×(0, 1) for a time t ∈ [0, Tσ)
+and finally we calculate the real part Re of the result. This leads in particular to
+1
+2
+��(∂t + 1)fk(t)
+��2
+L2 + 1
+2
+��∂yfk(t)
+��2
+L2 +
+� t
+0
+��∂yfk(s)
+��2
+L2ds = 1
+2
+��uin,k + ikU ′
+shΦin,k + ut,in,k
+��2
+L2+
++1
+2
+��∂yuin,k
+��2
+L2 + Re
+�
+ik
+� t
+0
+� 1
+0
+[ikU ′
+sh, ∂2
+y]
+�
+(∂t + 1)ψk
+�
+(∂t + 1)fkdyds
+�
+,
+(2.35)
+where we have used the initial conditions in (2.33), as well as
+Re
+�
+ik
+� t
+0
+� 1
+0
+Ush(y)|(∂t + 1)fk(s, y)|2dyds
+�
+= 0.
+16
+
+Multiplying (2.35) by 2 and applying the Cauchy-Schwarz inequality on the last integral, we obtain
+��(∂t + 1)fk(t)
+��2
+L2 +
+��∂yfk(t)
+��2
+L2 + 2
+� t
+0
+��∂yfk(s)
+��2
+L2ds ≤
+��uin,k + ikU ′
+shΦin,k + ut,in,k
+��2
+L2+
++∥∂yuin,k∥2
+L2 + 2|k|
+� t
+0
+��[U ′
+sh, ∂2
+y]
+�
+(∂t + 1)ψk(s)
+���
+L2∥(∂t + 1)fk(s)∥L2ds.
+(2.36)
+We next address the last integral in (2.35) with the commutator [U ′
+sh, ∂2
+y]((∂t + 1)ψk). First, we remark
+��[U ′
+sh, ∂2
+y]
+�
+(∂t + 1)ψk(s)
+���
+L2 =
+��U ′
+sh∂2
+y
+�
+(∂t + 1)ψk(s)
+�
+− ∂2
+y
+�
+U ′
+sh(∂t + 1)ψk(s)
+���
+L2
+=
+��U ′′′
+sh(∂t + 1)ψk(s) + 2U ′′
+sh(∂t + 1)∂yψk(s)
+��
+L2
+≤ ∥U ′′′
+sh∥L∞∥(∂t + 1)ψk(s)∥L2 + 2∥U ′′
+sh∥L∞∥(∂t + 1)∂yψk(s)∥L2
+≤
+�
+∥U ′′′
+sh∥L∞ + 2∥U ′′
+sh∥L∞�
+∥(∂t + 1)∂yψk(s)∥L2
+Hence, we plug this last inequality into (2.36), we take the Supermum within the time interval [0, t] and
+finally we divide the result by sups∈[0,t](∥(∂t + 1)fk(s)∥2
+L2 + ∥(∂t + 1)fk(s)∥2
+L2)1/2 (which is not null, since
+the initial data are not all zero). This leads to
+sup
+s∈[0,t]
+�
+∥(∂t + 1)fk(s)∥L2 + ∥∂yfk(s)∥L2
+�
+≤
+√
+2 sup
+s∈[0,t]
+�
+∥(∂t + 1)fk(s)∥2
+L2 + ∥∂yfk(s)∥2
+L2
+� 1
+2
+≤
+√
+2
+sups∈[0,t]
+�
+∥(∂t + 1)fk(s)∥2
+L2 + ∥∂yfk(s)∥2
+L2
+�
+sups∈[0,t]
+�
+∥(∂t + 1)fk(s)∥2
+L2 + ∥∂yfk(s)∥2
+L2
+� 1
+2
+≤
+√
+2
+��uin,k + ikU ′
+shΦin,k + ut,in,k
+��2
+L2 + ∥∂yuin,k∥2
+L2
+sups∈[0,t]
+�
+∥(∂t + 1)fk(s)∥2
+L2 + ∥∂yfk(s)∥2
+L2
+� 1
+2
++
++ 2
+√
+2
+�
+∥U ′′′
+sh∥L∞ + 2∥U ′′
+sh∥L∞�
+|k|
+� t
+0
+∥(∂t + 1)∂yψk(s)∥L2
+∥(∂t + 1)fk(s)∥L2
+sups∈[0,t]
+�
+∥(∂t + 1)fk(s)∥2
+L2 + ∥∂yfk(s)∥2
+L2
+� 1
+2
+ds.
+At s = 0, the functions (∂t + 1)fk(s) and ∂yfk(s) coincide with uin,k + ikU ′
+shΦin,k + ut,in,k and ∂yuin,k,
+respectively. We deduce hence that
+sup
+s∈[0,t]
+�
+∥(∂t + 1)fk(s)∥L2 + ∥∂yfk(s)∥L2
+�
+≤
+√
+2
+���uin,k + ikU ′
+shΦin,k + ut,in,k
+��2
+L2 + ∥∂yuin,k∥2
+L2
+� 1
+2 +
++2
+√
+2
+�
+∥U ′′′
+sh∥L∞ +2∥U ′′
+sh∥L∞�
+|k|
+� t
+0
+∥(∂t+1)∂yψk(s)∥L2ds ≤
+√
+2
+�
+∥uin,k∥L2+|k|∥U ′
+sh∥L∞∥Φin,k∥L2+
++∥ut,in,k∥L2 + ∥∂yuin,k∥L2
+�
++ 2
+√
+2
+�
+∥U ′′′
+sh∥L∞ + 2∥U ′′
+sh∥L∞�
+|k|
+� t
+0
+∥(∂t + 1)∂yψk(s)∥L2ds.
+(2.37)
+We are now in the condition to reveal the uniform estimates of the Gevrey-three regularity on the sequences
+((∂t + 1)fk)k∈Z and (∂yfk)k∈Z, by establishing the corresponding exponential growth on the modes |k|.
+To this end, we take advantage of Lemma 2.4, so that the last integral in (2.37) enables
+� t
+0
+∥(∂t + 1)∂yψk(s)∥L2ds ≤
+� � t
+0
+Cσ(s)e−α(s)|k|
+1
+3 ds
+��
+1 + ∥Ush∥L∞ + ∥U ′
+sh∥L∞ + ∥U ′′
+sh∥L∞+
++∥U ′′′
+sh∥L∞
+�2
+sup
+˜k∈Z
+�
+eσ|˜k|
+1
+3 �
+∥Φin,˜k∥L2 +∥uin,˜k∥L2+∥ut,in,˜k∥L2+∥∂yuin,˜k∥L2
+��
+,
+17
+
+where we recall that Cσ in Lemma 2.4 is defined as Cσ(t) = 170 max{1, 12/σ}6t(1+t)3et, while the radius
+α(t) = σ/2 − 2
+5
+6(∥U ′′′
+sh∥L∞ + 2∥U ′′
+sh∥L∞)
+1
+3et/3t (which satisfies α(t) ≥ 7σ/8 > 0, for any t ∈ [0, Tσ)). Since
+s ∈ [0, t] → Cσ(s)e−α(s)|k|
+1
+3 is a non-decreasing function, it can be bound by its value at s = t, so that
+� t
+0
+∥(∂t + 1)∂yψk(s)∥L2ds ≤ tCσ(t)e−α(t)|k|
+1
+3 �
+1 + ∥Ush∥L∞ + ∥U ′
+sh∥L∞ + ∥U ′′
+sh∥L∞+
++∥U ′′′
+sh∥L∞
+�2
+sup
+˜k∈Z
+�
+eσ|˜k|
+1
+3 �
+∥Φin,˜k∥L2+∥uin,˜k∥L2+∥ut,in,˜k∥L2+∥∂yuin,˜k∥L2
+��
+.
+We thus couple this last inequality together with (2.37), which guarantees
+∥(∂t + 1)fk(t)∥L2 + ∥∂yfk(t)∥L2 ≤
+√
+2
+�
+∥uin,k∥L2 + |k|∥U ′
+sh∥L∞∥Φin,k∥L2 + ∥ut,in,k∥L2 + ∥∂yuin,k∥L2
+�
++
++4
+√
+2
+�
+∥U ′′′
+sh∥L∞ + ∥U ′′
+sh∥L∞�
+|k|tCσ(t)e−α(t)|k|
+1
+3 �
+1 + ∥Ush∥L∞ + ∥U ′
+sh∥L∞+
++∥U ′′
+sh∥L∞ + ∥U ′′′
+sh∥L∞
+�2
+sup
+˜k∈Z
+�
+eσ|˜k|
+1
+3 �
+∥Φin,˜k∥L2+∥uin,˜k∥L2+∥ut,in,˜k∥L2 +∥∂yuin,k∥L2
+��
+.
+Finally, to obtain our uniform estimates, we multiply both left and right-hand sides with eβ(t)|k|1/3 (where
+β(t) = α(t) − σ/4 is defined in (1.8)), to obtain
+eβ(t)|k|
+1
+3 �
+∥(∂t + 1)fk(t)∥L2 + ∥∂yfk(t)∥L2
+�
+≤
+√
+2eβ(t)|k|
+1
+3 �
+∥uin,k∥L2 + |k|∥U ′
+sh∥L∞∥Φin,k∥L2 + ∥ut,in,k∥L2+
++∥∂yuin,k∥L2
+�
++ 4
+√
+2e− σ
+4 |k|
+1
+3 �
+∥U ′′′
+sh∥L∞ + ∥U ′′
+sh∥L∞�
+|k|tCσ(t)
+�
+1 + ∥Ush∥L∞ + ∥U ′
+sh∥L∞+
++∥U ′′
+sh∥L∞ + ∥U ′′′
+sh∥L∞
+�2
+sup
+˜k∈Z
+�
+eσ|˜k|
+1
+3 �
+∥Φin,˜k∥L2+∥uin,˜k∥L2+∥ut,in,˜k∥L2 +∥∂yuin,k∥L2
+��
+.
+The left-hand side already reveals the Gevrey-three norm on ((∂t + 1)fk)k∈Z and (∂yfk)k∈Z. We shall
+however provide a uniform estimate of the right-hand side, with respect to the frequencies k ∈ Z. To this
+end, we first observe that
+|k| =
+� 4
+σ
+�3�σ
+4 |k|
+1
+3
+�3
+≤
+� 4
+σ
+�3
+e
+3σ
+4 |k|
+1
+3 ,
+|k| =
+�12
+σ
+�3� σ
+12|k|
+1
+3
+�3
+≤
+�12
+σ
+�3
+e
+σ
+4 |k|
+1
+3 .
+(2.38)
+Therefore, since 1 ≤ max{1, 4/σ}3e3σ|k|1/3/4, we gather that
+eβ(t)|k|
+1
+3 �
+∥(∂t+1)fk(t)∥L2 +∥∂yfk(t)∥L2
+�
+≤
+√
+2e
+�
+β(t)+ 3σ
+4
+�
+|k|
+1
+3 max
+�
+1, 4
+σ
+�3�
+∥uin,k∥L2+∥U ′
+sh∥L∞∥Φin,k∥L2+
++∥ut,in,k∥L2 + ∥∂yuin,k∥L2
+�
++ 4
+√
+2e− σ
+4 |k|
+1
+3 �
+∥U ′′′
+sh∥L∞ + ∥U ′′
+sh∥L∞��12
+σ
+�3
+e
+σ
+4 |k|
+1
+3 tCσ(t)
+�
+1 + ∥Ush∥L∞+
++∥U ′
+sh∥L∞ + ∥U ′′
+sh∥L∞ + ∥U ′′′
+sh∥L∞
+�2
+sup
+˜k∈Z
+�
+eσ|˜k|
+1
+3 �
+∥Φin,˜k∥L2+∥uin,˜k∥L2 +∥ut,in,˜k∥L2+∥∂yuin,˜k∥L2
+��
+.
+This provides the required uniform estimate in k ∈ Z, since 3σ/4 + β(t) = α(t) + σ/2 ≤ σ for any
+t ∈ [0, Tσ), hence
+eβ(t)|k|
+1
+3 �
+∥(∂t + 1)fk(t)∥L2 + ∥∂yfk(t)∥L2
+�
+≤ C2,σ(t)
+�
+1 + ∥Ush∥L∞ + ∥U ′
+sh∥L∞ + ∥U ′′
+sh∥L∞+
++∥U ′′′
+sh∥L∞
+�3
+sup
+˜k∈Z
+�
+eσ|˜k|
+1
+3 �
+∥Φin,˜k∥L2 + ∥uin,˜k∥L2 + ∥ut,in,˜k∥L2 + ∥∂yuin,˜k∥L2
+��
+,
+(2.39)
+18
+
+where the function t ∈ (0, Tσ) → C2,σ(t) is now defined by
+C2,σ(t) :=
+√
+2 max
+�
+1, 4
+σ
+�3
++ 4
+√
+2
+�12
+σ
+�3
+tCσ(t)
+=
+√
+2 max
+�
+1, 4
+σ
+�3
++ 4
+√
+2
+�12
+σ
+�3
+170 · max
+�
+1, 12
+σ
+�6
+t2(1 + t)3et.
+(2.40)
+We now take advantage of (2.39), in order to transfer the corresponding Gevrey-three estimates to the
+sequences (uk)k∈Z, (∂yuk)k∈Z and (∂tuk)k∈Z. We begin with ∂yuk and we invoke identity (2.32), which
+ensures that ∂yuk = ∂yfk − ikU ′′
+sh(∂t + 1)ψk − ikU ′
+sh(∂t + 1)∂yψk. Hence
+eβ(t)|k|
+1
+3 ∥∂yuk(t)∥L2 ≤ eβ(t)|k|
+1
+3 �
+∥∂yfk∥L2 + |k|∥U ′
+sh∥L∞∥(∂t + 1)∂yψk∥L2 + |k|∥U ′′
+sh∥L∞∥(∂t + 1)ψk∥L2
+�
+,
+which we couple together with (2.38), the Poincar´e inequality ∥(∂t + 1)ψk∥L2 ≤ ∥(∂t + 1)∂yψk∥L2 and the
+relation α(t) = β(t) + σ/4, to obtain
+eβ(t)|k|
+1
+3 ∥∂yuk(t)∥L2 ≤ eβ(t)|k|
+1
+3
+�
+∥∂yfk∥L2 +
+�12
+σ
+�3
+e
+σ
+4 |k|
+1
+3 �
+∥U ′
+sh∥L∞ + ∥U ′′
+sh∥L∞�
+∥(∂t + 1)∂yψk∥L2
+�
+≤ eβ(t)|k|
+1
+3 ∥∂yfk∥L2 +
+�12
+σ
+�3
+eα(t)|k|
+1
+3 �
+∥U ′
+sh∥L∞ + ∥U ′′
+sh∥L∞�
+∥(∂t + 1)∂yψk∥L2.
+Thanks to (2.39) and the Poincar´e inequality ∥uk∥L2 ≤ ∥∂yuk∥L2, we establish therefore the uniform
+estimate
+eβ(t)|k|
+1
+3 �
+∥uk(t)∥L2 + ∥∂yuk(t)∥L2
+�
+≤
+�
+C2,σ(t) +
+�12
+σ
+�3
+Cσ(t)
+��
+1 + ∥Ush∥L∞ + ∥U ′
+sh∥L∞+
++∥U ′′
+sh∥L∞ + ∥U ′′′
+sh∥L∞
+�3
+sup
+˜k∈Z
+�
+eσ|˜k|
+1
+3 �
+∥Φin,˜k∥L2 + ∥uin,˜k∥L2 + ∥ut,in,˜k∥L2 + ∥∂yuin,˜k∥L2
+��
+.
+(2.41)
+Next, we aim to address the sequence of the time derivative (∂tuk)k∈Z. We invoke once more the relation
+uk = fk − ikU ′
+sh(∂t + 1)ψk in (2.32) and we decompose ∂tuk as follows:
+∂tuk = ∂tfk − ikU ′
+sh(∂t + 1)∂tψk
+= (∂t + 1)fk − fk − ikU ′
+sh
+�
+(∂t + ikUsh)(∂t + 1) − ∂2
+y
+�
+ψk − k2U ′
+shUsh(∂t + 1)ψk − ikU ′
+sh∂2
+yψk.
+Recalling that ((∂t + ikUsh)(∂t + 1) − ∂2
+y)ψk = Φk from (2.17), we gather that
+∂tuk = (∂t + 1)fk − fk − ikU ′
+shΦk − k2U ′
+shUsh(∂t + 1)ψk − ikU ′
+sh∂2
+yψk.
+A straightforward calculation leads hence to the estimate
+eγ(t)|k|
+1
+3 ∥∂tuk(t)∥L2 ≤ eγ(t)|k|
+1
+3
+�
+∥(∂t + 1)fk(t)∥L2 + ∥fk(t)∥L2 + |k|∥U ′
+sh∥L∞∥Φk(t)∥L2+
++ k2∥U ′
+sh∥L∞∥Ush∥L∞∥(∂t + 1)ψk(t)∥L2 + |k|∥U ′
+sh∥L∞∥∂2
+yψk(t)∥L2
+�
+.
+We now remark that γ(t) ≤ β(t), for any t ∈ R. Thus, making use of the Poincar´e inequalities given by
+∥fk(t)∥L2 ≤ ∥∂yfk(t)∥L2 and ∥Φk(t)∥L2 ≤ ∥∂yΦk(t)∥L2 = ∥uk∥L2, as well as recalling (2.38) together with
+k2 =
+�24
+σ
+�6� σ
+24|k|
+1
+3
+�6
+≤
+�24
+σ
+�6�
+e
+σ
+24 |k|
+1
+3 �6
+= 64
+�12
+σ
+�6
+e
+σ
+4 |k|
+1
+3 ,
+19
+
+we finally obtain
+eγ(t)|k|
+1
+3 ∥∂tuk(t)∥L2 ≤ eβ(t)|k|
+1
+3 �
+∥(∂t + 1)fk(t)∥L2 + ∥∂yfk(t)∥L2
+�
++
+�12
+σ
+�3
+∥U ′
+sh∥L∞eβ(t)|k|
+1
+3 ∥uk(t)∥L2+
++ 64
+�12
+σ
+�6
+∥U ′
+sh∥L∞∥Ush∥L∞eα(t)|k|
+1
+3 ∥(∂t + 1)∂yψk(t)∥L2 +
+�12
+σ
+�3
+∥U ′
+sh∥L∞eα(t)|k|
+1
+3 ∥∂2
+yψk(t)∥L2.
+We hence plug (2.28), (2.39) and (2.41) into this last relation, to gather
+eγ(t)|k|
+1
+3 ∥∂tuk(t)∥L2 ≤
+�
+C2,σ(t)
+� �� �
+from all fk
++
+�12
+σ
+�3 �
+C2,σ(t) +
+�12
+σ
+�3
+Cσ(t)
+�
+�
+��
+�
+from uk
++64
+�12
+σ
+�6 Cσ(t)
+5
+� �� �
+from (∂t+1)∂yψk
++
+�12
+σ
+�3 Cσ(t)
+5
+� �� �
+from ∂2yψk
+��
+1+
++∥Ush∥L∞ + ∥U ′
+sh∥L∞ + ∥U ′′
+sh∥L∞ + ∥U ′′′
+sh∥L∞
+�4
+sup
+˜k∈Z
+�
+eσ|˜k|
+1
+3 �
+∥uin,˜k∥L2 + ∥ut,in,˜k∥L2 + ∥∂yuin,˜k∥L2
+��
+.
+To simplify the summation of the terms depending on Cσ and C2,σ, we make use of their definitions in
+Lemma 2.4 and (2.40), so that
+C2,σ(t) +
+�12
+σ
+�3�
+C2,σ(t) +
+�12
+σ
+�3
+Cσ(t)
+�
++ 64
+�12
+σ
+�6 Cσ(t)
+5
++
+�12
+σ
+�3 Cσ(t)
+5
+≤ max
+�
+1, 12
+σ
+�6�
+2C2,σ(t) + 14Cσ(t)
+�
+≤ 5000 max
+�
+1, 12
+σ
+�15
+(1 + t)5et =
+˜Cσ(t)
+2
+.
+This provides indeed the following estimate on ∂tuk
+eγ(t)|k|
+1
+3 ∥∂tuk(t)∥L2 ≤
+˜Cσ(t)
+2
+�
+1 + ∥Ush∥L∞ + ∥U ′
+sh∥L∞+
++ ∥U ′′
+sh∥L∞ + ∥U ′′′
+sh∥L∞
+�4
+sup
+˜k∈Z
+�
+eσ|˜k|
+1
+3 �
+∥uin,˜k∥L2 + ∥ut,in,˜k∥L2 + ∥∂yuin,˜k∥L2
+��
+,
+(2.42)
+which together with (2.41) (and the fact that C2,σ(t) + (12/σ)3Cσ(t) ≤ ˜C(t)/2) imply finally the claimed
+inequality (2.31). This concludes the proof of Proposition 2.5.
+2.5
+A suitable test function
+In order to conclude the proof of Theorem 1.4, we need to establish Proposition 2.3 about the uniform
+estimate (2.21) on (∂t + 1)∂yψk and ∂2
+yψk. In the present section we establish a suitable test function
+ωτ,k for equation (2.20), that reveals some advantageous estimates, when analysing the L2-inner product
+between ωτ,k and equation (2.20). For a given positive time τ ∈ (0, Tσ), we consider ωτ,k as the unique
+solution of the following backward-in-time linear problem:
+
+
+
+
+
+((∂t − 1)(∂t + ikUsh) − ∂2
+y)ωτ,k = (∂t + 1)∂yψk
+(0, τ) × (0, 1),
+(ωτ,k, ∂tωτ,k)|t=τ = (0, 0)
+(0, 1),
+ωτ,k|y=0,1 = 0
+(0, τ).
+(2.43)
+With the next lemma, we determine the relations between certain meaningful norms of ωτ,k and the ones
+of (∂t + 1)∂yψk.
+Lemma 2.6. The solution ωτ,k of (2.43) satisfies at any time t ∈ [0, τ]
+sup
+s∈(t,τ)
+∥(∂t − 1)ωτ,k(s)∥L2 ≤ 2
+� τ
+t
+∥(∂t + 1)∂yψk(s)∥L2ds,
+sup
+s∈(t,τ)
+∥∂yωτ,k(s)∥L2 ≤
+√
+2
+� τ
+t
+∥(∂t + 1)∂yψk(s)∥L2ds,
+sup
+s∈(t,τ)
+∥ωτ,k(s)∥L2 ≤ 2eτ−t
+� τ
+t
+(s − t)∥(∂t + 1)∂yψk(s)∥L2ds.
+(2.44)
+20
+
+Proof. We multiply the first equation in (2.43) with the complex conjugate (∂t − 1)ωτ,k. Hence, for a given
+time ˜t ∈ (0, τ), we integrate the achieved identity within the domain (˜t, τ) × (0, 1) and we extrapolate the
+corresponding real part:
+−1
+2∥(∂t − 1)ωτ,k(˜t)∥2
+L2 − 1
+2∥∂yωτ,k(˜t)∥2
+L2 −
+� τ
+˜t
+∥∂yωτ,k(s)∥2
+L2ds
+=
+� τ
+˜t
+� 1
+0
+Re
+�
+(∂t + 1)∂yψk · (∂t − 1)ωτ,k
+�
+(s, y)dyds.
+(2.45)
+We multiply (2.45) by −1 and we take the Supremum of within ˜t ∈ (t, T), for a fixed t ∈ (0, τ). Thanks
+to Cauchy-Schwarz, we hence establish that
+sup
+˜t∈(t,τ)
+∥(∂t − 1)ωτ,k(˜t)∥2
+L2 ≤ 2
+� τ
+t
+∥(∂t + 1)∂yψk(s)∥L2∥(∂t − 1)ωτ,k(s)∥L2ds
+≤ 2
+� τ
+t
+∥(∂t + 1)∂yψk(s)∥L2ds sup
+s∈(t,τ)
+∥(∂t − 1)ωτ,k(s)∥L2.
+This corresponds to the first inequality of (2.44). Next, we deal with the norm ∥∂yωτ,k∥L2 in (2.44). By
+invoking (2.45), we have first
+sup
+s∈(t,T)
+∥∂yωτ,k(s)∥L2 ≤
+� � τ
+t
+∥(∂t + 1)∂yψk(s)∥L2ds sup
+s∈(t,T)
+∥(∂t − 1)ωτ,k(s)∥L2
+� 1
+2
+.
+The result is thus obtained by invoking the first inequality of (2.44). To conclude the proof, we deal now
+with the last inequality of (2.44). Since ωτ,k is null at t = τ, we have
+∥ωτ,k(t)∥L2 =
+��� −
+� τ
+t
+∂tωτ,k(s)ds
+���
+L2 ≤
+� τ
+t
+∥∂tωτ,k(s)∥L2ds
+≤
+� τ
+t
+∥(∂t − 1)ωτ,k(s)∥L2ds +
+� τ
+t
+∥ωτ,k(s)∥L2ds.
+Furthermore, the first estimate in (2.44) guarantees that
+∥ωτ,k(t)∥L2 ≤ 2
+� τ
+t
+� τ
+s
+∥(∂t + 1)∂yψk(z)∥L2dzds +
+� τ
+t
+∥ωτ,k(s)∥L2ds.
+The result is then achieved by applying the Gronwall’s lemma:
+∥ωτ,k(t)∥L2 ≤ 2eτ−t
+� τ
+t
+� τ
+s
+∥(∂t + 1)∂yψk(z)∥L2ds.
+This concludes the proof of Lemma 2.6.
+2.6
+Proof of Proposition 2.3
+This section is devoted to the proof of Proposition 2.3, which is based on the specific test function ωτ,k,
+introduced in Section 2.5. We begin with, by recalling system (2.20) for the evolution of ∂yψk:
+�
+(∂t + 1)(∂t + ikUsh) − ∂2
+y
+�2∂yψk = [ikU ′
+sh, ∂2
+y]
+�
+(∂t + 1)ψk
+�
+,
+(2.46)
+with initial data ∂tψk|t=0 = ψk|t=0 = 0 and boundary conditions ψk|y=0,1 = 0. Next, we fix a general time
+τ ∈ (0, Tσ) and we multiply the equation (2.46) with the conjugate ωτ,k of the test function defined in
+(2.43). By integrating the result along (0, τ) × (0, 1), we obtain the following identity:
+� τ
+0
+� 1
+0
+��
+(∂t + 1)(∂t + ikUsh(y)) − ∂2
+y
+�2∂yψk
+�
+(t, y)ωτ,k(t, y)dydt
+=
+� τ
+0
+� 1
+0
+[ikU ′
+sh(y), ∂2
+y]
+�
+(∂t + 1)ψk(t, y)
+�
+ωτ,k(t, y)dydt.
+(2.47)
+21
+
+We aim therefore to integrate by parts the integral at the left-hand side. To this end, we first develop the
+operator (∂t + 1)(∂t + ikUsh(y)) − ∂2
+y into ∂2
+t + (1 + ikUsh(y))∂t + ikUsh(y) − ∂2
+y, which localises the order
+of each derivative. Hence, by considering momentarily the derivative ∂2
+t of second order, we gather
+� τ
+0
+� 1
+0
+�
+∂2
+t
+�
+(∂t + 1)(∂t + ikUsh(y)) − ∂2
+y
+�
+∂yψk
+�
+(t, y)ωτ,k(t, y)dydt
+=
+� τ
+0
+� 1
+0
+��
+(∂t + 1)(∂t + ikUsh(y)) − ∂2
+y
+�
+∂yψk
+�
+(t, y)∂2
+t ωτ,k(t, y)dydt+
++
+� 1
+0
+�
+∂t
+�
+(∂t + 1)(∂t + ikUsh(y)) − ∂2
+y
+�
+∂yψk
+�
+(0, y)ωτ,k(0, y)dy+
+−
+� 1
+0
+��
+(∂t + 1)(∂t + ikUsh(y)) − ∂2
+y
+�
+∂yψk
+�
+(0, y)∂tωτ,k(0, y)dy.
+(2.48)
+The last two integrals of (2.48) are set at t = 0 and can hence be recasted in terms of the initial data of
+the velocity field uin,k, ∂tuin,k and of the stream function Φin,k. Indeed, recalling that ψk is also solution
+of (2.17), we remark that the second integrand at the right-hand side of (2.48) satisfies
+�
+∂t((∂t + 1)(∂t + ikUsh) − ∂2
+y)∂yψk
+�
+(0, y) = ikU ′
+sh(y)
+�
+∂2
+t ψk(0, y) + ∂tψk(0, y)
+�
++ ∂t∂yΦk(0, y),
+for any y ∈ (0, 1). This can be simplified further, since ∂t∂yΦk(0, y) = ∂yΦt,in,k = ut,in,k, ∂tψk|t=0 = 0 and
+equation (2.17) implies that ∂2
+t ψk|t=0 = Φin,k. Thus
+�
+∂t((∂t + 1)(∂t + ikUsh) − ∂2
+y)∂yψk
+�
+(0, y) = ikU ′
+sh(y)Φin,k(y) + ut,in,k(y),
+(2.49)
+for any y ∈ (0, 1). An analogous approach leads moreover to the following identity for the third integrand
+at the right-hand side of (2.48):
+�
+((∂t + 1)(∂t + ikUsh) − ∂2
+y)∂yψk
+�
+(0, y) = ∂yΦk(0, y) = uin,k(y),
+y ∈ (0, 1).
+(2.50)
+Therefore, thanks to the relations (2.49) and (2.50), we can reformulate (2.48) as follows:
+� τ
+0
+� 1
+0
+�
+∂2
+t
+�
+(∂t + 1)(∂t + ikUsh(y)) − ∂2
+y
+�
+∂yψk
+�
+(t, y)ωτ,k(t, y)dydt
+=
+� τ
+0
+� 1
+0
+��
+(∂t + 1)(∂t + ikUsh(y)) − ∂2
+y
+�
+∂yψk
+�
+(t, y)∂2
+t ωτ,k(t, y)dydt+
++
+� 1
+0
+�
+ikU ′
+sh(y)Φin,k(y) + ut,in,k(y)
+�
+ωτ,k(0, y)dy −
+� 1
+0
+uin,k(y)∂tωτ,k(0, y)dy.
+(2.51)
+We now come back to our original identity (2.47) and we shall now integrate by parts the operator
+(1 + ikUsh)∂t, with a a first order derivative. As for (2.51), our aim is once more to recast the resulting
+integrals at t = 0 in terms of the initial data. A direct calculation guarantees that
+� τ
+0
+� 1
+0
+�
+(1 + ikUsh)∂t
+�
+(∂t + 1)(∂t + ikUsh) − ∂2
+y
+�
+∂yψk
+�
+(t, y)ωτ,k(t, y)dydt
+=
+� τ
+0
+� 1
+0
+��
+(∂t + 1)(∂t + ikUsh) − ∂2
+y
+�
+∂yψk
+�
+(t, y)(1 − ikUsh(y))∂tωτ,k(t, y)dydt+
++
+� 1
+0
+�
+(1 + ikUsh)
+�
+(∂t + 1)(∂t + ikUsh) − ∂2
+y
+�
+∂yψk
+�
+(0, y)ωτ,k(0, y)dy.
+(2.52)
+22
+
+Hence, recalling from (2.50) that [((1 + ikUsh)((∂t + 1)(∂t + ikUsh) − ∂2
+y)∂yψk](0, y) = uin,k(y), we obtain
+� τ
+0
+� 1
+0
+�
+(1 + ikUsh)∂t
+�
+(∂t + 1)(∂t + ikUsh)−∂2
+y
+�
+∂yψk
+�
+(t, y)ωτ,k(t, y)dydt
+=
+� τ
+0
+� 1
+0
+��
+(∂t + 1)(∂t + ikUsh(y)) − ∂2
+y
+�
+∂yψk
+�
+(t, y)(1 − ikUsh(y))∂tωτ,k(t, y)dydt+
++
+� 1
+0
+(1 + ikUsh(y))uin,k(y)ωτ,k(0, y)dy.
+(2.53)
+To conclude the integration by parts related to the operator ((∂t + 1)(∂t + ikUsh(y)) − ∂2
+y) in (2.47), we
+shall now treat −∂2
+y. Making use of the homogeneous conditions ωk|y=0,1 = 0 on the test function, we
+have that
+−
+� τ
+0
+� 1
+0
+�
+∂2
+y
+�
+(∂t + 1)(∂t + ikUsh(y)) − ∂2
+y
+�
+∂yψk
+�
+(t, y)ωτ,k(t, y)dydt
+=
+� τ
+0
+� 1
+0
+�
+∂y
+�
+(∂t + 1)(∂t + ikUsh(y)) − ∂2
+y
+�
+∂yψk
+�
+(t, y)∂yωτ,k(t, y)dydt.
+Now, recalling from (2.32) that ((∂t + 1)(∂t + ikUsh(y)) − ∂2
+y)∂yψk = fk = ikU ′
+sh(y)(∂t + 1)ψk + uk is null
+in y = 0, 1, we obtain
+−
+� τ
+0
+� 1
+0
+�
+∂2
+y
+�
+(∂t + 1)(∂t + ikUsh(y)) − ∂2
+y
+�
+∂yψk
+�
+(t, y)ωτ,k(t, y)dydt
+= −
+� τ
+0
+� 1
+0
+��
+(∂t + 1)(∂t + ikUsh(y)) − ∂2
+y
+�
+∂yψk
+�
+(t, y)∂2yωτ,k(t, y)dydt.
+(2.54)
+As final result, we couple the identities (2.51),(2.53) and (2.54), so that (2.47) can be recasted as
+� τ
+0
+� 1
+0
+�
+(∂t + 1)(∂t + ikUsh(y)) − ∂2
+y
+�
+∂yψk(t, y)
+�
+(∂t − 1)(∂t + ikUsh(y)) − ∂2y
+�
+ωτ,k(t, y)dydt+
++
+� 1
+0
+�
+ikU ′
+sh(y)Φin,k(y) + ut,in,k(y) + (1 + ikUsh(y))uin,k(y)
+�
+ωτ,k(0, y)dy−
+−
+� 1
+0
+uin,k(y)∂tωτ,k(0, y)dy =
+� τ
+0
+� 1
+0
+[ikU ′
+sh(y), ∂2
+y]
+�
+(∂t + 1)ψk(t, y)
+�
+ωτ,k(t, y)dydt.
+(2.55)
+Next, we make use of (2.55), in order to derive suitable estimates on the L2-norms of (∂t + 1)∂yψk and
+∂2
+yψk. These estimates shall not depend upon ωτ,k, hence we aim to get rid of this test function making
+use of Lemma 2.6 and system (2.43). By extrapolating the real part of (2.55), the first integral becomes
+Re
+� τ
+0
+� 1
+0
+�
+(∂t + 1)(∂t + ikUsh(y)) − ∂2
+y
+�
+∂yψk(t, y)(∂t + 1)∂yψk(t, y)dydt
+=
+� τ
+0
+� 1
+0
+Re
+�
+∂t(∂t + 1)∂yψk(t, y)(∂t + 1)∂yψk(t, y)
+�
+dydt+
++
+� τ
+0
+� 1
+0
+Re
+�
+ikUsh(y)∥(∂t + 1)∂yψk(t, y)∥2
+L2
+�
+�
+��
+�
+=0
+dydt +
+� τ
+0
+��∂2
+yψk(s)
+��2
+L2ds
+= 1
+2
+��(∂t + 1)∂yψk(τ)
+��2
+L2 + 1
+2
+��∂yψk(τ)
+��2
+L2 +
+� τ
+0
+��∂2
+yψk(s)
+��2
+L2ds,
+where we have used in the last identity the fact that ∂yψk and ∂t∂yψk are identically null at t = 0.
+Furthermore, when dealing with the real part of the second and third integrals in (2.55), we obtain
+����Re
+� 1
+0
+�
+ikU ′
+sh(y)Φin,k(y) + ut,in,k(y) + (1 + ikUsh(y))uin,k(y)
+�
+ωτ,k(0, y)dy
+����
+≤
+�
+|k|∥U ′
+sh∥L∞∥Φin,k∥L2 + ∥ut,in,k∥L2 +
+�
+1 + |k|∥Ush∥L∞�
+∥uin,k∥L2
+�
+∥ωτ,k(0)∥L2,
+(2.56)
+23
+
+as well as
+����Re
+� 1
+0
+uin,k(y)∂tωτ,k(0, y)dy
+���� ≤ ∥uin,k∥L2∥∂tωτ,k(0)∥L2.
+(2.57)
+Finally, the real part of the right-hand side in (2.55) fulfills
+����Re
+� τ
+0
+� 1
+0
+[ikU ′
+sh(y), ∂2
+y]
+�
+(∂t + 1)ψk(t, y)
+�
+ωτ,k(t, y)dydt
+����
+≤ |k|
+� τ
+0
+∥[U ′
+sh, ∂2
+y](∂t + 1)ψk(t)∥L2∥ωτ,k(t)∥L2dt
+≤ |k|
+� τ
+0
+∥[U ′
+sh, ∂2
+y](∂t + 1)ψk(t)∥L2∥ωτ,k(t)∥L2dt
+≤ |k|
+� τ
+0
+∥U ′′′
+sh(∂t + 1)ψk(t) + 2U ′′
+sh(∂t + 1)∂yψk(t)∥L2∥ωτ,k(t)∥L2dt
+≤ |k|
+� τ
+0
+�
+∥U ′′′
+sh∥L∞ + 2∥U ′′
+sh∥L∞
+�
+∥(∂t + 1)∂yψk(t)∥L2∥ωτ,k(t)∥L2dt,
+(2.58)
+where we have also made use of the Poincar´e inequality in y ∈ (0, 1): ∥(∂t+1)ψk∥L2 ≤ ∥(∂t+1)∂yψk∥L2.
+We can summarise hence our last estimates, by coupling (2.55) together with (2.56), (2.57) and (2.58).
+This guarantees that for any τ ∈ (0, Tσ)
+1
+2
+��(∂t + 1)∂yψk(τ)
+��2
+L2 + 1
+2
+��∂2
+yψk(τ)
+��2
+L2 +
+� τ
+0
+��∂2
+yψk(t)
+��2
+L2dt
+≤
+�
+|k|∥U ′
+sh∥L∞∥Φin,k∥L2 + ∥ut,in,k∥L2 +
+�
+1 + |k|∥Ush∥L∞�
+∥uin,k∥L2
+�
+∥ωτ,k(0)∥L2
++ ∥uin,k∥L2∥∂tωτ,k(0)∥L2 + |k|
+�
+∥U ′′′
+sh∥L∞ + 2∥U ′′
+sh∥L∞
+� � τ
+0
+∥(∂t + 1)∂yψk(t)∥L2∥ωτ,k(t)∥L2dt.
+(2.59)
+The right-hand side still depends upon the test function ωk,τ. We are however in the condition to get rid
+of that, by applying Lemma 2.6. This implies in particular (together with Poincar´e) that
+∥ωk,τ(0)∥L2 ≤ ∥∂yωk,τ(0)∥L2 ≤
+√
+2
+� τ
+0
+∥(∂t + 1)∂yψk(s)∥L2ds
+≤
+√
+2 τ sup
+s∈[0,τ]
+∥(∂t + 1)∂yψk(s)∥L2,
+∥∂tωk,τ(0)∥L2 ≤ ∥(∂t − 1)ωk,τ(0)∥L2 + ∥ωk,τ(0)∥L2 ≤ (2 +
+√
+2)
+� τ
+0
+∥(∂t + 1)∂yψk(s)∥L2ds
+≤
+√
+2(1 +
+√
+2)τ sup
+s∈[0,τ]
+∥(∂t + 1)∂yψk(s)∥L2,
+∥ωk(t)∥L2 ≤ eτ−t
+� τ
+t
+(s − t)∥(∂t + 1)∂yψk(s)∥L2ds
+≤ eτ
+� τ
+t
+(s − t)ds sup
+s∈[0,τ]
+∥(∂t + 1)∂yψk(s)∥L2 = eτ(τ − t)2
+2
+sup
+s∈[0,τ]
+∥(∂t + 1)∂yψk(s)∥L2.
+Thus, by taking now the supremum in τ ∈ (0, t) for a general t ∈ (0, Tσ) (and re-denoting the variables
+of integration), we can recast (2.59) uniquely in terms of the stream function ψk as follows:
+1
+2 sup
+s∈(0,t)
+���(∂t + 1)∂yψk(s)
+��2
+L2 +
+��∂2
+yψk(s)
+��2
+L2
+�
++
+� t
+0
+��∂2
+yψk(s)
+��2
+L2ds ≤
+sup
+s∈(0,t)
+∥(∂t+1)∂yψk(s)∥L2
+�√
+2
+�
+|k|∥U ′
+sh∥L∞∥Φin,k∥L2 + ∥ut,in,k∥L2 +
+�
+1 + |k|∥Ush∥L∞�
+∥uin,k∥L2
+�
+t+
++ 2(1 +
+√
+2)∥uin∥L2τ + |k|
+�
+∥U ′′′
+sh∥L∞ + 2∥U ′′
+sh∥L∞
+�
+et
+� t
+0
+∥(∂t + 1)∂yψk(s)∥L2(t − s)2ds
+�
+.
+24
+
+If sups∈(0,t) ∥(∂t+1)∂yψk(s)∥L2 = 0, then the main estimate (2.21) is automatically satisfied. On the other
+hand, in case this term is not identically null, we have that
+sup
+s∈(0,t)
+���(∂t + 1)∂yψk(s)
+��
+L2 +
+��∂2
+yψk(s)
+��
+L2
+�
+≤
+√
+2
+�
+sup
+s∈(0,t)
+���(∂t + 1)∂yψk(s)
+��2
+L2 +
+��∂2
+yψk(s)
+��2
+L2
+�� 1
+2
+≤
+2
+√
+2
+sups∈(0,t) ∥(∂t+1)∂yψk(s)∥L2
+1
+2 sup
+s∈(0,t)
+���(∂t + 1)∂yψk(s)
+��2
+L2 +
+��∂2
+yψk(s)
+��2
+L2
+�
+≤ 2
+√
+2
+�√
+2
+�
+|k|∥U ′
+sh∥L∞∥Φin,k∥L2 + ∥ut,in,k∥L2 +
+�
+1 + |k|∥Ush∥L∞�
+∥uin,k∥L2
+�
+t+
++ 2(1 +
+√
+2)∥uin,k∥L2t + |k|
+�
+∥U ′′′
+sh∥L∞ + 2∥U ′′
+sh∥L∞
+�
+et
+� t
+0
+∥(∂t + 1)∂yψk(s)∥L2(t − s)2ds
+�
+.
+We hence reorganise the last inequality into the following compact form:
+sup
+s∈[0,t]
+���(∂t + 1)∂yψk(s)
+��
+L2 +
+��∂2
+yψk(s)
+��
+L2
+�
+≤ gk(t) + λk(t)3
+2
+� t
+0
+(t − s)2 sup
+τ∈[0,s]
+∥(∂t + 1)∂yψk(τ)∥L2ds
+where the functions gk(t) and λk(t) are defined by means of
+gk(t) := 4t
+�
+|k|
+�
+∥U ′
+sh∥L∞∥Φin,k∥L2 + ∥Ush∥L∞∥uin,k∥L2
+�
++ ∥ut,in,k∥L2 + (3 +
+√
+2)∥uin,k∥L2
+�
+λk(t) := 2
+5
+6 |k|
+1
+3
+�
+∥U ′′′
+sh∥L∞ + 2∥U ′′
+sh∥L∞
+� 1
+3 e
+t
+3 .
+This last inequality corresponds to our claimed estimate (2.21). This concludes therefore the proof of
+Proposition 2.3.
+3
+Conclusion and remarks on the non-linear system
+In this section, we investigate why Theorem 1.4 cannot be proven for the nonlinear counterparts of
+(1.1) without further ado. Clearly, this is a consequence of the nonlinear structure but furthermore, the
+hyperbolic regime interferes with the known cancellation properties of the classical Prandtl/Navier-Stokes
+equations in anaggravating way.
+To begin with, we observe that several candidates exist for which Theorem 1.4 might hold true. Some
+of the represent simplifications of other formulations but, nevertheless, they contain drawbacks which
+cannot be dealt with easily. The simplest form of the nonlinear hyperbolic Prandtl equation consists of
+
+
+
+
+
+
+
+
+
+
+
+
+
+τ∂ttu + ∂tu + u∂xu + v∂yu − ∂2
+yu =
+�
+τ∂t + 1
+��
+∂tuE + uE∂xuE�
+,
+(0, T) × X × (0, +∞),
+∂xu + ∂yv = 0
+(0, T) × X × (0, +∞),
+(u, v)|y=0 = 0
+lim
+y→+∞ u = uE
+(0, T) × X,
+(u, ut)|t=0 = (uin, ut,in)
+X × (0, +∞),
+,
+(3.1)
+This system looks promising when trying to implement the strategy of [11]. However, besides the fact
+that the second time derivative produces difficulties (see below), a quick look at the linearization
+∂2
+t u + ∂tu + Ush(y)∂xu + v U ′
+sh(y) − ∂yyu = 0,
+on (0, T) × T × (0, 1)
+suffices to realize the Eigenvalues contain a positive real part in general. More precisely, solutions corre-
+sponding to a frequency k in x will behave like e
+√
+|k|t which restricts well-posedness theory to the Gevrey
+2 case. For a hyperbolic equation, this is expected and actually proven for (3.1) in [19] (see also [26]).
+25
+
+In conclusion, it is essential to maintain the convective structure of the hyperbolic Prandtl equations (as
+shown in Section 2). By Cattaneo’s law, it reads
+
+
+
+
+
+
+
+
+
+
+
+
+
+�
+τ∂t + 1
+��
+∂tu + u∂xu + v∂yu
+�
+− ∂2
+yu =
+�
+τ∂t + 1
+��
+∂tuE + uE∂xuE�
+,
+(0, T) × X × (0, +∞),
+∂xu + ∂yv = 0
+(0, T) × X × (0, +∞),
+(u, v)|y=0 = 0
+lim
+y→+∞ u = uE
+(0, T) × X,
+(u, ut)|t=0 = (uin, ut,in)
+X × (0, +∞),
+.
+(3.2)
+Unfortunately, the time derivative on the convective term brings several other difficulties with it. At first,
+note that one of the terms, ∂tu∂xu, competes with the damping mechanism for large values. Even for the
+hyperbolic Navier-Stokes equations, this circumstance poses a fundamental issue (see e.g. [29]).
+Secondly and much more inherent to the strategy followed in [11] and Section 2, the additional (second)
+time-derivatives produce corresponding terms terms on the right-hand side of the equation. Following
+Section 2.2, we realize that two commutators need to be evaluated in (2.18) and (2.20). In the classical
+Prandtl regime, the solenoidality of (u, v) and cancellation of curl-related terms enters the analysis, e.g.
+[∂y, ∂t + u∂x + v∂y − ∂yy]u = 0.
+None of these instances persist in the hyperbolic version. Instead, new terms arise like ∂xtv ∼ ∂xxtu.
+Confronted with a 4-order time derivative on the left-hand side, this does not present an improvement
+over the standard Gevrey 2 regularity result. A new, different (perhaps related) cancellation mechanism
+seems to be necessary but it is not clear how the system can be closed.
+At this point, we remark that the independence of the shear flow Ush on (t, x) is exploited heavily in
+Section 4. A third variant of (3.1), substituting the first equation by
+(τ∂tt + τu∂x∂t + ηv∂y∂t + ∂t + u∂x + v∂y − ∂yy)u = 0,
+bears similar problems, although the competing damping term ∂xu∂tu is not present here. However, the
+potential improvement of the above equation might lie in the transport structure related to the time-
+derivatives of u.
+In sum, we conjecture that Theorem 1.4 hints stability results around shear flows for the nonlinear
+hyperbolic equations in Gevrey 3 class. On the other hand, a general well-posedness theory for arbitrary
+initial data in G3 does not seem to be achievable without major novelties or improvements on the strategy
+followed in [11] and this work.
+Acknowledgment
+The author would like to thank Prof. M. Paicu for the several helpful advises on various technical issues
+examined in this Paper. The first author was partially supported by the Bavarian Funding Programme
+for the Initiation of International Projects (F¨orderkennzeichen: BayIntAn UWUE 2022 139). The third
+author was partially supported by GNAMPA and INDAM.
+Data Availability Statement
+Data sharing is not applicable to this article, since no datasets were generated or analysed during the
+current study.
+Conflict of interest
+The authors declare that they have no conflict of interest.
+26
+
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+
diff --git a/l9AyT4oBgHgl3EQfYfeT/content/tmp_files/load_file.txt b/l9AyT4oBgHgl3EQfYfeT/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf,len=1324
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='00205v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='AP]  31 Dec 2022  Gevrey-class-3 regularity of the linearised hyperbolic  Prandtl system on a strip  Francesco De Anna 1, Joshua Kortum 2, Stefano Scrobogna 3  1 Institute of Mathematics, University of W¨urzburg, Germany  email: francesco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='deanna@mathematik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='uni-wuerzburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='de  2 Institute of Mathematics, University of W¨urzburg, Germany  email: joshua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='kortum@mathematik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='uni-wuerzburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='de  3 Dipartimento di Matematica e Geoscienze, Universit`a degli Studi di Trieste, Trieste, Italy,  email: stefano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='scrobogna@units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='it  Abstract  In the present paper, we address a physically-meaningful extension of the linearised Prandtl equations  around a shear flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Without any structural assumption, it is well-known that the optimal regularity  of Prandtl is given by the class Gevrey 2 along the horizontal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' The goal of this paper is to  overcome this barrier, by dealing with the linearisation of the so-called hyperbolic Prandtl equations in  a strip domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We prove that the local well-posedness around a general shear flow Ush ∈ W 3,∞(0, 1)  holds true, with solutions that are Gevrey class 3 in the horizontal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  1  Introduction  The main concern of this paper is to set up a rigorous well-posedness theory for the following extension  of the linearised and reduced Prandtl equations on a thin strip:  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  (∂t + 1)  �  ∂tu + Ush∂xu + v U ′  sh  �  − ∂2  yu = 0,  (t, x, y) ∈ (0, T) × T × (0, 1),  ∂xu + ∂yv = 0  (0, T) × T × (0, 1),  (u, ut)|t=0 = (uin, ut,in)  T × (0, 1),  u|y=0 = 0,  v|y=0 = 0  (0, T) × T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1)  In this system, the unknown is the horizontal component u = u(t, x, y) of the velocity field (u, v)T :  (0, T) × T × (0, 1) → R2, while the vertical component v = v(t, x, y) is determined by the divergence-free  condition ∂xu + ∂yv = 0 and by the Dirichlet relation v|y=0 = 0, which formally imply  v(t, x, y) = −∂x  � y  0  u(t, x, z)dz,  (t, x, y) ∈ (0, T) × T × (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  The function Ush = Ush(y) depends uniquely upon the vertical variable y ∈ (0, 1) and describes a shear  flow (Ush(y), 0)T , around which the original equations have been linearised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1) arises indeed  from a meaningful extension of the classical Prandtl equations, known as hyperbolic Prandtl equations[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  With our analysis, we aim to show how System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1) might be particularly desirable, in order to overcome  certain analytic barriers that are typical of the classical Prandtl theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' More precisely, System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1)  is amenable in terms of solutions that have regularity Gevrey-class 3 along the vertical variable x ∈ T,  overcoming therefore the well-known barrier of Gevrey-class 2 of the Prandtl theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Details about this  statement and our main result are presented starting from Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  First, we shall provide some  background on the origin of this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1  The Prandtl equations and the barrier of Gevrey-class 2  In order to understand the major characteristics of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1), we shall briefly overview the original model of  Prandtl, that was introduced during a 10 minutes presentation of the 1904 Third International Mathe-  matics Congress in Heidelberg [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Such short presentation has scientifically impacted many disciplines,  so much that nowadays the field of aerodynamics is still shaped by his fundamental idea: in order to  describe the inviscid limit of an incompressible fluid in a region close to a solid surface (where dissipative  forces are predominant), one would rather consider the velocity of the fluid in terms of rescaled variables,  which concentrate the dissipative effects in a thin region close to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' This procedure provides  a “split” in the behaviour of the flow:  within the bulk, the hydrodynamics is dominated by the incompressible Euler equations with no-  penetration boundary conditions,  on a neighbourhood of the boundary, a corrective term (the so-called boundary layer) provided by  the Prandtl equations restores the natural no-slip boundary conditions for viscous flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  One of the simplest forms of the (non-linear) Prandtl equations in two dimensions is given by  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  ∂tu + u∂xu + v∂yu − ∂2  yu = ∂tuE + uE∂xuE,  (t, x, y) ∈(0, T) × X × (0, +∞),  ∂xu + ∂yv = 0  (0, T) × X × (0, +∞),  (u, v)|y=0 = 0  lim  y→+∞ u = uE  (0, T) × X,  ut|=0 = uin  X × (0, +∞),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2)  where x ∈ X describes the (local) arc-length parametrisation of the solid surface (usually in the math-  ematical community X = T or X = R), while uE = uE(t, x) is determined by the solution of the Euler  equation in the bulk of the flow, when approaching the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  The analysis of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2) has received from the mathematical community numerous investigations during the  past decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Although the Prandtl equations are classical, their applications are rather narrowed because  of the particular unstable nature of the underlying solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' These instabilities are nowadays moderately  well understood and relate mainly to separation phenomena (appearance of reversed flow in the boundary  layers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  The first rigorous mathematical study addressing the well-posedness of the Prandtl equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2) was  performed in the book of Oleinik and Samokhin [23] in the case of so-called monotonic initial data (namely  initial velocity uin in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2), satisfying ∂yuin > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Roughly speaking, the Olenik’s monotonicity prevents  the mentioned flow separation, at least locally in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' This allows to recast the velocity field through a  meaningful transformation (known as Crocco transformation), providing a solid ground to the local-in-  time well posedness of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2) within function spaces typical of hydrodynamics, such as Sobolev ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' For  more details on the Olenik’s monotonicity, we refer the reader to the more recent result [2], in which the  authors construct local-in-time solutions via a Nash-Moser argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' See also [21] for a proof performed  purely by energy methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  For initial data lacking monotonicity, the well-posedness becomes much more involved and one has to  consider function spaces that control infinite derivatives of the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  This was addressed in the  celebrated result [31] of Caflisch and Sammartino, where the authors dealt with non-monotonic initial  data uin that are analytic in the variable x ∈ X = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In the framework of a periodic variable x ∈ X = T,  analytic initial data can be easily understood through their Fourier series, under a strong localisation of  the frequencies:  uin(x, y) =  �  k∈Z  uin,k(y)eikx,  where the modes uin,k(y) decays exponentially as uin,k(y) ∼ e−a|k|, for some radius of analyticity a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  This type of initial data are however extremely regular and with reduced applications to real phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  For this reason, an increasing number of works were devoted to relax this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  2  The first breakthrough was provided in [14] by G´erard-Varet and Masmoudi, where the authors showed  that the Prandtl system is actually locally well-posed for data that are Gevrey-class 7/4 in the x-variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Roughly speaking, an initial data uin is Gevrey-class m along x ∈ T, with m > 1, if the modes uin,k(y)  decay exponentially as uin,k(y) ∼ e−a|k|1/m, for a suitable radius a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Already in [14], however, the  authors remarked that the Gevrey-class 7/4 was unlikely to be optimal and that further insights from  numerics suggested rather a threshold of Gevrey-class 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' uin,k(y) ∼ e−a|k|1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Eventually, this remark was mathematically formalised and a first result in this direction was attained  in [20], assuming that the velocity (u, v)T in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2) is a small perturbation of a suitable shear flow, which  satisfies a non-degenerate condition (for details cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1 in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  The breakthrough of Gevrey 2 was however achieved by Gerard-Varet and Dietert in [11], where the  authors developed a robust local well-posedness theory without any structural assumption on the flow  (such as monotonicity or critical points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Their result was based on a meaningful change of state variable,  from which the present work takes substantial inspiration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  From the work of Gerard-Varet and Dietert followed a variety of questions, in particular to determine  whether the Gevrey-class 2 was optimal for the well-posedness of System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2) or if further insights  would have led to weaker regularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Surprisingly, Gerard-Varet and Dormy overturned any possibility  of improvement, providing indeed a negative answer to this open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In their seminal result [13],  the authors showed that already at the level of the linearised equations around a shear flow (Ush(y), 0),  namely replacing the first equation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2) with  ∂tu + Ush∂xu + v U ′  sh − ∂2  yu = 0,  (t, x, y) ∈ (0, T) × T × (0, +∞),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3)  the linear propagator of regularity is unbounded in Gevrey-class higher than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Roughly speaking, the  authors showed the existence of solutions, whose modes in the frequencies k ∈ Z experience an exponential  growth with rate |k|1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In general, this growth could be counteracted only by Gevrey-2 initial data,  precluding any room for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In other words, this was the first encounter with the barrier of  Gevrey 2: the linearised Prandtl equation is ill-posed within any larger setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  We refer the reader to the works [10, 12, 17, 18], as well, which concern further instabilities of the Prandtl  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2  The hydrostatic approximation  When the vertical variable y is bounded, for instance with y ∈ (0, 1) (as in our System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1)), a different  type of equations has adequately found a mathematical relevance, namely the so-called hydrostatic ap-  proximation of Navier-Stokes/Prandtl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In two dimensions, these equations are a reminiscent of Prandtl  and take the form  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  ∂tu + u∂xu + v∂yu − ∂2  yu + ∂xp = 0,  (t, x, y) ∈(0, T) × X × (0, 1),  ∂yp = 0,  (0, T) × X × (0, 1),  ∂xu + ∂yv = 0  (0, T) × X × (0, 1),  (u, v)|y=0,1 = 0  (0, T) × X,  ut|=0 = uin  X × (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4)  This model is significant in several phenomena of atmospheric science and can be derived from the so-  called primitive equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Beside the vertical domain, System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4) inherently differs from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2) in the  boundary conditions of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4) v is null in both y = 0, 1, whereas in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2) v has homogeneous condition  only in y = 0 (without any assumption for y → +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Because of this, the pressure p in the hydrostatic  approximation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4) is non-trivial and can be interpreted as a Lagrangian multiplier associated to the  constraint v|y=1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We refer the reader to the works [3, 4, 15, 16, 21, 22, 27, 34] and as well the  interesting result in [30] in which the author proves that, contrarily to what happens for the Prandtl  3  equations, the presence of an inflexion point may trigger high-frequencies instabilities in the linearization  of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4) around a shear flow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4) in which the first equation is substituted by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We  want to highlight that, to the best of our knowledge, the best regularity result for eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4) is provided  in [15, 33] for 9/8–Gevrey data under an additional convexity assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Hence the optimal stability  vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' instability question is still an open question for the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4), contrarily to what is known for the  Prandtl system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' This paper addresses the well-posedness of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1) within y ∈ (0, 1), nevertheless our intent is  to provide insights about an extension of the Prandtl equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2) (for which we know that the barrier is  Gevrey 2) rather than the hydrostatic ones in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Dealing with the pressure and homogeneous Dirichlet  conditions on v is beyond our interest (certainly, with the pressure, the problem would be much more  involved).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We address a bounded vertical domain y ∈ (0, 1) uniquely for the sake of a clear presentation  of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' To the best of our knowledge, our work is indeed the first to overcome the barrier of  Gevrey 2 for a meaningful extension of Prandtl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We infer that a similar result can be achieved in the  classical domain (t, x, y) ∈ (0, T) × T × (0, ∞), making use of a related ansatz on function spaces with  weighted norms in the vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3  The Cattaneo’s law on the hydrostatic approximation  Besides the barrier of Gevrey 2, a more physical drawback of Systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4) can be found at  the level of the Navier-Stokes equations (from which (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4) are indeed asymptotically derived),  because of the so-called infinite propagation speed of the velocity field (any local variation of the velocity  field perturbs immediately the flow in all the domain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' To avoid this scenario (which may be occasionally  unsatisfactory, especially in the hydrodynamics of fluids at large scale), a suitable hyperbolic extension  of Navier-Stokes has found growth in popularity in the mathematical community (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' [1, 5, 9, 24, 26, 28,  29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' At a first glance, this extension seems to introduce obstacles, for instance it enlarges the hydrostatic  equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4) into  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  �  τ∂t + 1  ��  ∂tu + u∂xu + v∂yu  �  − ∂2  yu + ∂xp = 0,  (0, T) × X × (0, 1),  ∂yp = 0  (0, T) × X × (0, 1),  ∂xu + ∂yv = 0  (0, T) × X × (0, 1),  (u, v)|y=0,1 = 0  (0, T) × X,  (u, ut)|t=0 = (uin, ut,in)  X × (0, 1),  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5)  where τ > 0 is a meaningful parameter, konwn as relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Once more, the pressure p in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5) is  uniquely due to v|y=1 = 0 and would vanish when relaxing this constraint (as in our model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5) arises (at least formally) from the inviscid limit of the Navier-Stokes equations, whose Cauchy  stress tensor is “delayed” through a first-order Taylor expansion:  S(t + τ, ·) ≈ S(t, ·) + τ∂tS(t, ·) = ν ∇u(t, ·) + ∇u(t, ·)T  2  (we refer to [5] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' This relation was introduced in fluid-dynamics by Carrassi and Morro [6],  inspired by the celebrated work of Cattaneo [7, 8] on heat diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Despite its relevance, the well-posedness theory of System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5) is unfortunately much less understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  In [26], the authors considered τ = 1 and neglected the term ∂t(u∂xu + v∂yu) in the first equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' By  exploiting a similar technique as the one used in [25], they showed that small initial data in Gevrey 2  generate global-in-time solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Moreover, in this framework, they justified the asymptotic limit of the  Navier-Stokes under Cattaneo’s law towards the solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5), when the viscosity is vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  The main goal of this paper is to show that one can potentially overcome the barrier of Gevrey 2,  when dealing with the relevant extensions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5) of the classical Prandtl equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We indicate accurately  4  this principle on the linearised equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1) of System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5) around a shear flow (Ush(y), 0), when the  constraint v|y=1 = 0 is relaxed (hence no pressure is involded, as for Prandtl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We establish that this  model is indeed well-posed locally-in-time, when the initial data are Gevrey-class 3 in x (thus less regular  than Gevrey 2) and Sobolev in y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We provide also some remarks about the non-linear system in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4  Preliminaries and statement of the main result  To formalise our statement, we shall briefly recall the formal definition of Gevrey functions, that we will  use throughout our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Let σ > 0 and m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  We define the Banach space Gm  σ,xL2  y = Gm  σ (T, L2(0, 1))  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Gm  σ,xH1  0,y = Gm  σ (T, H1  0(0, 1))) as all integrable functions f ∈ L1(T × (0, 1)) satisfying:  Each coefficient fk : (0, 1) → R of the Fourier transform in the x-variable  fk(y) = 1  2π  �  T  f(x, y)e−ikxdx,  y ∈ (0, 1),  belongs to L2(0, 1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' H1  0(0, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  The sequences of norms (∥fk∥L2)k∈Z (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' (∥∂yfk∥L2)k∈Z) decays exponentially as e−σ|k|1/m at high  frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  More precisely, f ∈ L1(T×(0, 1)) belongs to Gm  σ,xL2  y (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Gm  σ,xH1  0,y), if the following norm is indeed finite:  ∥f∥Gm  σ,xL2y :=  ���eσ|k|  1  m ∥fk∥L2(0,1)  ���  ℓ∞(Z) = sup  k∈Z  �  eσ|k|  1  m  � � 1  0  |fk(y)|2dy  � 1  2 �  < +∞,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='6)  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' ∥f∥Gm  σ,xH1  0,y := ∥∂yf∥Gm  σ,xL2y < +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Function spaces with Gevrey regularity are rather standard, especially in the mathematical treatment of  the Prandtl equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Indeed, by strongly localising the frequencies, one copes with the major instabilities  of the underlying solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' When f = f(x) depends uniquely upon x ∈ T, however, the definition of  the Gevrey norm in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='6) may vary in terms of the preferred analytical tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Among the most relevant  norms, we mention for instance  � �  k∈Z  e2σ|k|  1  m |fk|2  � 1  2  ,  sup  n∈N  � σn  (n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' )m ∥∂n  xf∥L∞(T)  �  ,  which are somehow equivalent to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='6), for positive radii close to σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Let T denote a lifespan in (0, +∞] and η : (0, T) → (0, +∞) be a positive continuous  non-increasing function, representing the time-evolution of the Gevrey radius of regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We say that  a function f belongs to Lp(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Gm  η(t),xL2  y) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Lp(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Gm  η(t),xH1  0,y)), for a fixed 1 ≤ p ≤ ∞, if  f belongs to Lp(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' L1(T × (0, 1))),  f(t) belongs to Gm  η(t),xL2  y (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Gm  η(t),xH1  0,y), for almost any t ∈ (0, T),  the function t ∈ (0, T) → ∥f(t)∥Gm  η(t),xL2y (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' t ∈ (0, T) → ∥f(t)∥Gm  η(t),xH1  0,y) belongs to Lp(0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  The function space being set up, the main goal of this paper is to establish the local-in-time existence of  solutions for the linearised equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1), whose initial data uin and ut,in are indeed Gevrey-class 3 in the  horizontal variable, as described by Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  5  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Assume that the shear flow y ∈ (0, 1) �→ Ush(y) is in W 3,∞(0, 1), while the initial data  uin, ut,in : T × (0, 1) → R are in G3  σ,xH1  0,y and G3  σ,xL2  y, respectively, for a positive σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Denoting by  Tσ > 0 the lifespan  Tσ := sup  �  t > 0  such that  σ  8 − 2  5  6  �  ∥U ′′′  sh∥L∞ + 2∥U ′′  sh∥L∞  � 1  3 e  t  3t > 0  �  ∈ (0, +∞]  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='7)  and by β, γ : [0, Tσ) → (0, +∞) the following time-dependent radii of Gevrey-class regularity  β(t) := σ  4 − 2  5  6  �  ∥U ′′′  sh∥L∞ + 2∥U ′′  sh∥L∞  � 1  3e  t  3t > 0,  γ(t) := β(t) − σ  8 > 0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='8)  then the linearised system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1) admits a unique weak solution u : [0, Tσ) × T × (0, 1) → R in the function  space  u ∈ L∞(0, Tσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' G3  β(t),xH1  0,y)  with  ∂tu ∈ L∞(0, Tσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' G3  γ(t),xL2  y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='9)  Furthermore, the following estimate holds true at any time t ∈ [0, Tσ):  ∥u(t)∥G3  β(t),xH1  0,y + ∥∂tu(t)∥G3  γ(t),xL2y ≤ Dσ(Ush)(1 + t)5et�  ∥uin∥G3σ,xH1  0,y + ∥ut,in∥G3σ,xL2y  �  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='10)  for a suitable constant Dσ(Ush) > 0, which depends uniquely upon σ > 0 and the W 3,∞-norm of Ush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Before presenting the major novelties and implications of this result, some remarks on the statement are  here in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  The solutions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4 are weak because of the regularity of the initial data along the vertical  direction y ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' This regularity comes from the underlying a-priori energy of the System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1)  1  2  d  dt  �  ∥(∂t + 1)u∥2  L2 + ∥∂yu∥2  L2  �  + ∥∂yu∥2  L2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Although the solution u and its derivative ∂tu are in L∞ with respect to the time variable, we write the  estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='10) at any t ∈ [0, Tσ) (instead of “for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' t ∈ (0, Tσ)”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' This is due to the fact that both u  and ∂tu admit a continuous representative in the following space: for any finite T ∈ (0, Tσ) and for any  fixed radius µ ∈ (0, β(T)) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' µ ∈ (0, γ(T))), t ∈ [0, T] → u(t) belongs to C([0, T], G3  µ,xH1  0,y) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' ∂tu  belongs to C([0, T], G3  µ,xH1  0,y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Writing directly the expression u ∈ C([0, Tσ), G3  β(t),xH1  0,y) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='9) would be  somehow incomplete without a proper clarification, since the norm of G3  β(t),xH1  0,y does progress in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Certainly one may define this continuity in terms of topology, however this would just complicate the  readability of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We maintain therefore the function space of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='9) and the estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='10) at any  time t ∈ [0, Tσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  The solutions provided by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4 are (in general) only local-in-time, although system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1) is linear  in its state variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Indeed, as for the classical Prandtl equations, the high regularity of the initial data  uin, ut,in is persistently eroded by the term vU ′  sh in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1) (and in our extension of Prandtl with Cattaneo  also by ∂tvU ′  sh), together with the viscous operator −∂2  yu in y ∈ (0, 1) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' This aspect is here  reflected by the decaying of the Gevrey radii β and γ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='8), as time t > 0 approaches the lifespan Tσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  The radii β(0) and γ(0) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='8) at initial time t = 0 correspond to σ/4 and σ/8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' One would  rather expect them to coincide with the radius σ > 0 of the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' This gap is merely an artifact of  our analysis, since we also aim to determine an explicit (and readable) form of the constant Ds(Ush) > 0  in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' To this end, we make use of a given amount of the exponential decay in e−σ|k|1/3 (for instance  the missing e−3σ|k|1/3/4 between e−γ(0)|k|1/3 and e−σ|k|1/3), in order to absorb the contribution of certain  terms, which arise from vU ′  sh and ∂tvU ′  sh (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' for instance (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='38)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Accordingly, we may explicitely set  Dσ(Ush) > 0 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='10) as  Dσ(Ush) := 104 max{1, 12/σ}15(1 + ∥Ush∥L∞ + ∥U ′  sh∥L∞ + ∥U ′′  sh∥L∞ + ∥U ′′′  sh∥L∞)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='11)  6  This arrangement is certainly far from being sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We may for instance build our Gevrey-class-3 solution  with radii of regularity  βε(t) = γε(t) := σ − ε − 2  5  6  �  ∥U ′′′  sh∥L∞ + 2∥U ′′  sh∥L∞  � 1  3e  t  3 t,  for any small ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' This definition would nevertheless complicate the constant Dσ(Ush) (behaving now  also like 1/ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' For the sake of simple presentation, we do not pursue this direction and we simply remark  that as long as uin, ut,in are in Gm  σ,xL2  y and Gm  σ,xH1  0,y, respectively, then they are also in Gm  σ/4,xL2  y and  Gm  σ/8,xH1  0,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5  Novelty and implications  Let us highlight the novelties and consequences of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4 and the discussion in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Indeed,  the improved Gevrey-3 well-posedness of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4 of System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1) is far from being trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Standard  considerations of the (linearized as well as non-linear) hyperbolic Prandtl system yield to a well-posedness  within Gevrey-class 2, at best (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We show that a suitable cancellation mechanism is inherent  to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1), which is similar to the one presented in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Contrary to the classical Prandtl equations, however,  the hyperbolic behaviour of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1) unlocks more refined estimates, that endow the mentioned improvements  of Gevrey-class 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  In Section 2 of [11], Dietert and Gerard-Varet provided a rather clear intuition on why the well-posedness  result of Gevrey 2 holds true for the linearised Prandtl equations (later on, their result further address  the non-linear system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In order to successfully simplify the comprehension of their idea, they made use  of calculations involving the Laplace transform on the time variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Then, potentially, some algebraic  calculations relating the Laplace variable (in time) with the Fourier variable (in space, along the horizontal  direction) allowed to deduce the correct regularity of solutions, since they showed a possible behaviour of  the associated semigroup on the linearised system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' The downside of this approach consists, however, in  the fact that the argument seems to lack some final implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' The authors indeed derived certain  a-priori estimates in the Laplace variable, however the inverse Laplace transform does not commute with  norms1, thus this estimates could not be transferred formally to the original solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Our approach differ  with the one related to the Laplace transform in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In particular, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1 provides a simple, yet  very useful tool (an improved Gronwall estimate) to infer Gevrey-estimates by energy estimates (we refer  to Section 2 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Furthermore, we give detailed bounds on regularity, life span and explicit quantitative dependence on  the shear flow Ush (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' For example, if U ′′  sh = 0, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4 shows the global well-posedness of  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1), being in correspondence with the results for the classical Prandtl equations with monotonic data  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Additionally, we give a detailed discussion on possible improved well-posedness results for the nonlinear  hyperbolic Prandtl system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In particular, our work shows that one cannot rely on further simplifica-  tions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5) in order to achieve existence results beyond the expected Gevrey 2 class (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=', [19] and  [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We refer to Section 3 for a consideration of three possible nonlinear variants with their advantages  and drawbacks in terms of regularity propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Finally, a short summary of the remaining parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  The beginning of Section 2 contains an extended  overview of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4 which is split into six parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In the aforementioned Section 3, the  discussion of possible extensions of the arguments to the nonlinear system is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  1Of course, the actual argument in [11] for the nonlinear system is consistent and rigorous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' The aim of the authors in  Section 2 was to provide a clear understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  7  2  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4  We state first the general principles that we set as the basis of our analysis, and we postpone the details  of our proof to the remaining paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Our approach is grounded in a similar ansatz as the one developed by Dietert and G´erard-Varet for the  linearised system of the classical Prandtl equations (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Section 2 of [11], outline of the strategy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2, we indeed use the Fourier transform along the variable x ∈ T, in order to address the  behaviour of each mode uk : (0, T) × (0, 1) → R of the velocity field  u(t, x) =  �  k∈Z  uk(t, y)eikx,  at any frequency k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Regrettably, the equation of uk (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='8)) is incapable to derive alone a better  stability than analytic (initial data far more regular then Gevrey 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' A further development is therefore  necessary, in order to overcome this first barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  On this account, following [11], we introduce a new state variable ψk : (0, T) × (0, 1) → R in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2,  which depends on uk (or rather upon the corresponding stream function Φk, uk = ∂yΦk, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='17)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Our  main objective is indeed to asses ψk, in order to remove the (problematic) terms due to vU ′  sh and ∂tvU ′  sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  These terms preclude indeed an analysis beyond analytic, thus, by eliminating them, we determine a new  form of the main equation (written now in terms of ψk, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='20))  �  (∂t + 1)(∂t + ikUsh) − ∂2  y  �2∂yψk = ik[U ′  sh, ∂2  y]  �  (∂t + 1)ψk  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1)  which shall eventually facilitate our analysis in Gevrey-class 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Following (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1), our approach begins to inherently diverge with respect to the one of Dietert and G´erard-  Varet in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We avoid entirely their ansatz on the Laplace transform in time, since (despite its clearness)  it would lead to the difficulties mentioned in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Contrarily, we develop our analysis around a  specific “weighted” version of the Gronwall’s lemma, which plays somehow the role of cornerstone for our  entire proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Its statement is thus the first that we present in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  To be more specific, we take advantage of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1), in order to determine a certain meaningful estimate on  the derivatives ∂2  yψk and (∂t + 1)ψk, as described in details in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' This estimate can be  expressed essentially as  ∥ψk(t)∥ ≤ gk(t) + C(t)|k|  � t  0  (t − s)2∥ψk(s)∥ds,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2)  where t ∈ (0, T) → ∥ψk(t)∥ represents the L2-norms (in y ∈ (0, 1)) of ∂2  yψk and (∂t + 1)ψk (for the  complete version, we refer to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' The function gk and C in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2) are non-decreasing, while  the integral is expressed also in terms of a “weight” in time: the kernel (t − s)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  The kernel (t−s)2 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2) unlocks the regularity Gevrey-class 3, for the derivatives ∂2  yψk and (∂t+1)ψk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' To  grasp this principle, we shall first remark that, in its absence (thus within a standard Gronwall inequality),  we may at best derive an estimate of the form ∥ψk(t)∥ ≤ gk(0) exp(tC(t)|k|), where the modes ψk growth  exponentially as |k| (the setting of analytic solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' The presence of the kernel provides us however  better information: because of the weighted Gronwall inequality in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1, the norm ∥ψk(t)∥ can at  worst growth as gk(0) exp( 3�  tC(t)|k|), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' exponentially as |k|1/3, the framework of Gevrey-class 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  The remaining sections are devoted to transfer the aforementioned estimate of ∂2  yψk and (∂t + 1)ψk first  to ψk (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4) and secondly to uk (cf Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5 is moreover essential to  determine the final solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1), which is Gevrey-class 3 in x ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Furthermore, this result provides  the final estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='10) on the Gevrey-norm of u at any time t ∈ [0, Tσ), with also the corresponding  constant Dσ(Ush) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Summarising, the forthcoming sections are structured as follows:  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1 are devoted to the proof of the “weighted” Gronwall’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  8  In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2 we introduce the new state variable ψk and derive the corresponding equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3, we first state the main inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2) in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3 (whose proof is postponed  to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5 and Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We furthermore transfer the Gevrey estimates of ∂2  yψk and (∂t+1)ψk  to ψk in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4, with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5, we transfer the Gevrey estimates to uk and build our final  solution u of System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' To conclude the proof of the main Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4, we determine moreover  the estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='10) on the Gevrey norm of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Finally, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5 and Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='6 are devoted to the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3 and the main  inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1  A weighted Gronwall inequality  One of the main ingredient used to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4 is the following Gronwall-type Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' It asserts  that any non-negative function, which satisfies a “weighted” Gronwall’s inequality proportional to a  suitable time-dependent function λ(t)3, can not grow up exponentially faster than λ(t)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Let T > 0 and f : [0, T) → [0, ∞) be a non-negative continuous function, satisfying  f(t) ≤ g(t) + λ(t)3  2  � t  0  (t − s)2f(s)ds,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3)  for two continuous functions λ, g : [0, T) → [0, ∞), that are non-negative and non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Then the  following inequality holds true at any time t ∈ [0, T):  f(t) ≤ g(t)  �  1 + (λ(t)t)3  6  �  eλ(t)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We begin with by defining ω in C3([0, T)) as the following integral: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3) as  ω(t) := 1  2  � t  0  (t − s)2f(s)ds =  � t  0  f(s)  � t  s  � t  τ  drdτds =  � t  0  � r  0  � τ  0  f(s)dsdτdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  The function ω is everywhere non-negative in [0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Furthermore, at t = 0, ω(0) and its derivatives  ω′(0), ω′′(0) are all identically null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We next write inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3) in terms of ω:  ω′′′(t) ≤ g(t) + λ(t)3ω(t),  for all  t ∈ [0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5)  Hence, we fix a general time ˜t ∈ (0, T) and we momentarily consider only values of t within [0, ˜t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We  multiply equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5) with e−λ(˜t)t (where λ(˜t) is fixed and plays momentarily the role of a constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  By means of standard calculations on the derivatives, we gather that  ω′′′(t)e−λ(˜t)t = d3  dt3  �  ω(t)e−λ(˜t)t�  +3 d2  dt2  �  λ(˜t)(ω(t)e−λ(˜t)t�  +3 d  dt  �  λ(˜t)2ω(t)e−λ(˜t)t�  +λ(˜t)3ω(t)e−λ(˜t)t  ≤  �  g(t) + λ(t)3ω(t)  �  e−λ(˜t)t ≤ g(t)e−λ(˜t)t + λ(˜t)3ω(t)e−λ(˜t)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='6)  In the last inequality, we have used that λ is non-decreasing and non-negative, ω ≥ 0 and that t < ˜t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We  shall now remark that the term λ(˜t)3ω(t)e−λ(˜t)t cancel out and the left-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='6) is hence left  with only time derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We are hence in the condition to integrate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='6) along [0, t], to gather that  d2  dt2  �  ω(t)e−λ(˜t)t�  + 3λ(˜t) d  dt  �  ω(t)e−λ(˜t)t�  + 3λ(˜t)2ω(t)e−λ(˜t)t ≤  � t  0  g(s)e−λ(˜t)sds ≤  � t  0  g(s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  9  We can drop the term 3λ(˜t)2ω(t)e−λ(˜t)t at the left-hand side (since it is positive) and integrate once more  along the interval [0, t], for a general t ∈ [0, ˜t]:  d  dt  �  ω(t)e−λ(˜t)t�  + 3λ(˜t)ω(t)e−λ(˜t)t ≤  � t  0  � s  0  g(z)dz =  � t  0  (t − z)g(z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Since both λ(˜t) and ω(t) are positive, we can drop 3λ(˜t)ω(t)e−λ(˜t)t and integrate a final time along (0, t):  ω(t)e−λ(˜t)t ≤  � t  0  � s  0  (s − z)g(z)dzds =  � t  0  g(z)(t − z)2  2  dz  ⇒  ω(t) ≤ eλ(˜t)t  � t  0  g(z)(t − z)2  2  dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  We are now in the condition to combine the last relation in t = ˜t together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3), which ensures that  f(˜t) ≤ g(˜t) + eλ(˜t)˜tλ(˜t)3  � ˜t  0  g(s)(˜t − s)2  2  ds  ≤ g(˜t) + eλ(˜t)˜tλ(˜t)3g(˜t)  � ˜t  0  (˜t − s)2  2  ds ≤ g(˜t)  �  1 + λ(˜t)3˜t3  6  �  eλ(˜t)˜t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Re-denoting ˜t = t and from its arbitrariness in (0, T), we finally achieve inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4) (also remarking  that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4) is trivially satisfied in t = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' This concludes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2  The stream function formulation  In this section, we begin developing our analysis of System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1) and we first decompose the corresponding  equations in terms of several Fourier coefficients uk : (t, y) ∈ (0, Tσ) × (0, 1) → R of the velocity field u,  at any frequency k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' The lifespan Tσ > 0 (denoted by T in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1)) shall be considered from now on as  in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='7) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4, nevertheless its form will play a major role only starting from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Eventually, we will build the final solution (u, v) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1), by invoking the Fourier Series with respect to the  variable x ∈ T and the divergence-free condition ∂xu+∂yv = 0 (which at any frequency is ikuk +∂yvk = 0)  u(t, x, y) =  �  k∈Z  uk(t, y)eikx,  uk(t, y) := 1  2π  �  T  u(t, x, y)e−ixkdx,  v(t, x, y) =  �  k∈Z  vk(t, y)eikx,  vk(t, y) := −ik  � y  0  uk(t, z)dz,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='7)  however we shall first determine some uniform estimates on (uk)k∈Z, in order to provide a sense of the  series above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Hence, we begin with by considering System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1) rather as a family of PDEs in the variables  (t, y) ∈ (0, Tσ) × (0, 1), which depend upon each frequency k ∈ Z:  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  ∂2  t uk + ikUsh∂tuk + U ′  sh∂tvk + ∂tuk + ikUshuk + vkU ′  sh − ∂2  yuk = 0  (0, Tσ) × (0, 1),  ikuk + ∂yvk = 0  (0, Tσ) × (0, 1),  (uk, ∂tuk)|t=0 = (uin,k, ut,in,k)  (0, 1),  (uk, vk)|y=0,1 = (0, 0)  (0, Tσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='8)  The second equation ikuk + ∂yvk = 0 and the boundary conditions vk|y=0 = 0 allow to interpret System  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='8) only on the state variable uk, since the vertical component vk is explicitly determined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' The  initial data uin,k and ut,in,k are in H1  0(0, 1) and L2(0, 1), respectively, since uin and ut,in are in G3  σ,xH1  0,y  and G3  σ,xL2  0,y, as described by Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  We hence state the following result about the existence and uniqueness of solutions for System (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  10  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' For any fixed frequency k ∈ Z and any initial data (uin,k, ut,in,k) in H1  0(0, 1) × L2(0, 1)  there exists a unique solution uk : [0, T) × (0, 1) → R of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='8), which belongs to  (uk, ∂yuk) ∈ C([0, T], H1  0),  ∂tuk ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' L2),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='9)  for any real time T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  We shall here remark that the real T > 0 may not correpond to Tσ, since the lifespan in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='7) may be  Tσ = +∞ (for shear flow with U ′′  sh ≡ U ′′′  sh ≡ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In this case, we shall always treat uk as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='9), for any  0 < T < Tσ = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Contrarily, if Tσ < +∞ (which is satisfied for more general Ush), then we replace T  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='9) directly with Tσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' For the sake of simplicity, we provide here only a sketch, since the result can be shown through  standard arguments on linear PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We remark indeed that Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='8) can be written as a 1D  damped wave equation with Dirichelet boundary conditions  (□ + ∂t) uk = Fk,  □ = ∂2  t − ∂2  y,  uk|y=0,1 = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='10)  and forcing term Fk, which depends linearly on uk and is given by  Fk = −  �  ikUsh∂tuk − ikU ′  sh∂t  � y  0  uk(t, z)dz + ikUshuk − ikU ′  sh  � y  0  uk(t, z)dz  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Making use of a standard computation combined with the Poincar´e inequality, we infer that Fk satisfies  ∥Fk∥H1  0 ≤ C|k| ∥Ush∥W 3,∞  �  ∥∂tuk∥H1  0 + ∥∂yuk∥H1  0  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='11)  for a suitable positive constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We can hence apply a standard Galerkin method to deduce the  existence of a unique solution within the function space of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  We come back now to System (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Since uk(t, ·) is divergent free, it can be written in terms of a stream  function Φk = Φk(t, y), which is in C([0, T], H2) with ∂tΦk ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' H1(0, 1)), for any real T ≤ Tσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Furthermore, because ikΦk = vk, the function Φk is identically null in y = 0 (in the sense of trace),  therefore  uk(t, y) = ∂yΦk(t, y)  �  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Φk(t, y) :=  � y  0  uk(t, z)dz  �  ⇒  vk(t, y) = −ikΦk(t, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='12)  The initial data of Φk at any y ∈ (0, 1) are given by  Φin,k(y) :=  � y  0  uin,k(z)dz,  Φt,in,k(y) :=  � y  0  ut,in,k(z)dz,  which ensures Φin,k ∈ H2(0, 1) ∩ H1  0(0, 1) and Φt,in,k ∈ H1(0, 1)(remark that Φin,k(1) = 0, since uin,k is  average free).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='8), we deduce that Φk is solution in (0, Tσ) × (0, 1) of the following system:  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  ∂2  t ∂yΦk+ikUsh(y)∂t∂yΦk−ikU ′  sh(y)∂tΦk+∂t∂yΦk+Ush(y)ik∂yΦk−ikΦkU ′  sh(y) − ∂3  yΦk = 0,  (Φk, ∂tΦk)|t=0 = (Φin,k, Φt,in,k)  (∂yΦk)|y=0,1 = 0,  Φk|y=0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='13)  We next provide some heuristics on the L2-estimates satisfied by Φk and show that, unfortunately, System  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='13) (as it is written) may not prevent the stream function Φk to exponentially growth as exp(a|k|), for  11  some positive a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' To this end, we first isolate the linear operator in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='13) that affects only  the derivative ∂yΦk and transfer the remaining terms in Φk on the right-hand side of the identity:  ((∂t + 1)(∂t + ikUsh) − ∂2  y)∂yΦk = (∂t + 1)ikU ′  shΦk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='14)  We will shortly see that the operator on the left-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='14) is crucial for our next analysis (in  particular, to define a new state variable ψk in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='17)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We first outline, however, that, in the current form,  equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='14) is still ineffective and does not predict the crucial Gevrey-3 regularity of our solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Indeed, we infer that a standard energy approach would provide (at best) an L2-estimate of ∂yΦk = uk  of the form  1  2  d  dt  �  ∥(∂t + 1)∂yΦk∥2  L2 + ∥∂t∂yΦk∥2  L2 + 2∥∂2  yΦk∥2  L2  �  ≤ C|k|  �  ∥(∂t + 1)∂yΦk∥2  L2 + ∥∂t∂yΦk∥2  L2  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='15)  for a suitable positive constant C, which is also calibrated with the following Poincar´e-type inequality of  Φk in the domain y ∈ (0, 1):  ∥Φk(t)∥L2 =  � � 1  0  |Φk(t, y)|2dy  � 1  2  =  � � 1  0  ���  � y  0  ∂yΦk(t, z)dz  ���  2  dy  � 1  2  ≤  � � 1  0  y  � y  0  ���∂yΦk(t, z)  ���  2  dzdy  � 1  2  ≤ 1  2  � � 1  0  ���∂yΦk(t, z)  ���  2  dz  � 1  2  ≤ ∥∂yΦk(t)∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='16)  Hence, roughly speaking, in this regime the L2-norms of ∂yΦk and ∂t∂yΦk would growth exponentially  as ec|k|(∥∂yΦin,k∥L2 + ∥∂t∂yΦin,k∥L2), a setting which is typical of analytic solutions (which are of course  much more regular than any Gevrey-class m, m > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' To achieve the Gevrey regularity, we shall therefore  perform a further development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' To this end, we introduce a new state variable ψk : [0, Tσ) × (0, 1) → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Following the approach used in [11] for the classical Prandtl equation, ψk = ψk(t, y) is chosen in a form  that gets rid of the terms ikU ′  sh∂tΦk and ikU ′  shΦk at the left-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' More precisely, we define  ψk as the unique solution in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' H2) ∩ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' H2) of the following PDE:  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  ((∂t + 1)(∂t + ikUsh) − ∂2  y)ψk = Φk  (0, T) × (0, 1),  (ψk, ∂tψk)|t=0 = (0, 0)  (0, 1),  ψk|y=0,1 = 0  (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='17)  The most compelling reason for this definition is a meaningful cancellation that occurs when coupling  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='17) together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' More precisely, equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='14) implies that ψk satisfies  ((∂t + 1)(∂t + ikUsh) − ∂2  y)∂y((∂t + 1)(∂t + ikUsh) − ∂2  y)ψk = ikU ′  sh(∂t + 1)Φk,  which is  �  (∂t +1)(∂t +ikUsh)−∂2  y  �2∂yψk +  �  (∂t +1)(∂t +ikUsh)−∂2  y  ��  ikU ′  sh(∂t +1)ψk  �  = ikU ′  sh(∂t +1)Φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='18)  The second term at left-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='18) almost coincides with ikU ′  sh(∂t+1)Φk at the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' To  complete the aforementioned cancellation, we first invoke the commutator ik[U ′  sh, ∂2  y]ψk = ikU ′  sh∂2  y(ψk) −  ik∂2  y(U ′  shψk), in order to write the second term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='18) as  �  (∂t +1)(∂t +ikUsh)−∂2  y  ��  ikU ′  sh(∂t +1)ψk  �  = ikU ′  sh  �  (∂t +1)(∂t +ikUsh)−∂2  y  ��  (∂t +1)ψk  �  −ik[U ′  sh, ∂2  y]ψk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Finally, we plug this identity into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='18), to gather  �  (∂t + 1)(∂t + ikUsh) − ∂2  y  �2∂yψk + ikU ′  sh(∂t + 1)  � �  (∂t + 1)(∂t + ikUsh) − ∂2  y  �  ψk  �  ��  �  =Φk  �  +  −[ikU ′  sh(y), ∂2  y]  �  (∂t + 1)ψk  �  = ikU ′  sh(∂t + 1)Φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='19)  12  Recalling that ψk satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='17), we remark that ikU ′  sh(∂t+1)Φk appears both on the left- and right-hand  sides of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We thus obtain the following final form of the ψk-equation:  �  (∂t + 1)(∂t + ikUsh) − ∂2  y  �2∂yψk = ik[U ′  sh, ∂2  y]  �  (∂t + 1)ψk  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='20)  which was indeed claimed at the beginning in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We shall remark that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='20) still presents a forcing  term ik[U ′  sh, ∂2  y]  �  (∂t + 1)ψk  �  , which growths linearly like |k| at high frequencies |k| ≫ 1 (similarly as  ikU ′  sh(∂t +1)Φk in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='14)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Nonetheless, the operator ((∂t +1)(∂t +ikUsh)−∂2  y)2 on ∂yψk has now doubled  in order (in comparison with just ((∂t + 1)(∂t + ikUsh) − ∂2  y) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='14)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' This will unlock more-refined  estimates on ∂yψk (and its derivatives) than the ones in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='15) for ∂yΦk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We formalise these heuristics in  the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3  Reaching Gevrey-class 3  In the forthcoming analysis, we illustrate how the derived equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='20) succeeds in enabling an L2-  estimate of Gevrey-3 type to the new state variable ∂yψk, as well its time derivative ∂t∂yψk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' This estimate  is a consequence of the following statement, that we set as the basis of our development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' It guarantees  that the functions (∂t + 1)∂yψk and ∂2  yψk satisfy an improved Gronwall-type inequality, as described by  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' The following estimate on the functions (∂t + 1)∂yψk and ∂2  yψk holds true, for any  frequency k ∈ Z and at any time t ∈ (0, Tσ):  sup  s∈[0,t]  ���(∂t + 1)∂yψk(s)  ��  L2 +  ��∂2  yψk(s)  ��  L2  �  ≤ gk(t) + λk(t)3  2  � t  0  (t − s)2 sup  τ∈[0,s]  ���(∂t + 1)∂yψk(τ)  ��  L2 +  ��∂2  yψk(τ)  ��  L2  �  ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='21)  The functions gk, λk are increasing in time and depend uniquely on k ∈ Z, the shear flow Ush and the  initial data (uin,k, ut,in,k, Φin,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' More precisely, gk, λk are defined by  gk(t) := 4t  �  |k|  �  ∥U ′  sh∥L∞∥Φin,k∥L2 + ∥Ush∥L∞∥uin,k∥L2  �  + ∥ut,in,k∥L2 + (3 +  √  2)∥uin,k∥L2  �  λk(t) := 2  5  6 |k|  1  3  �  ∥U ′′′  sh∥L∞ + 2∥U ′′  sh∥L∞  � 1  3 e  t  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='22)  for k ∈ Z and t ∈ (0, Tσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Since the proof of this Proposition is rather technical, we postpone it to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5 and we focus the next  paragraphs on the remaining steps to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Let us observe that inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='21) encompasses the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3) of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' This ensures therefore  that the following improved Gronwall’s inequality holds true for any t ∈ [0, Tσ):  ��(∂t + 1)∂yψk(t)  ��  L2 +  ��∂2  yψk(t)  ��  L2 ≤ sup  s∈[0,t]  ��(∂t + 1)∂yψk(s)  ��  L2 ≤ gk(t)  �  1 + (λk(t)t)3  6  �  eλk(t)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  We have essentially achieved the claimed regularity of Gevrey 3, at least for (∂t + 1)∂yψk and ∂2  yψk, since  the definitions of gk and λk in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='22) imply that  sup  s∈[0,t]  ���(∂t+1)∂yψk(s)  ��  L2+  ��∂2  yψk(s)  ��  L2  �  ≤4t  �  |k|  �  ∥U ′  sh∥L∞∥Φin,k∥L2 + ∥Ush∥L∞∥uin,k∥L2  �  +∥ut,in,k∥L2+  +(3+  √  2)∥uin,k∥L2  ��  1+  2  √  2|k|  �  ∥U ′′′  sh∥L∞ + 2∥U ′′  sh∥L∞  �  ett3  3  �  exp  �  |k|  1  32  5  6  �  ∥U ′′′  sh∥L∞+2∥U ′′  sh∥L∞  � 1  3 e  t  3t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='23)  13  In particular, the L2-norm increases exponentially at worst as |k|1/3, which we can counteract (at least  locally in time), by imposing that the initial data exponentially decay with the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' To formalise  this principle, we shall however first transfer all frequencies |k| of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='23) to the exponential function, as  well as transfer these Gevrey-3 estimates also to ψk, ∂yψk and ∂t∂yψk (instead of just (∂t + 1)∂yψk and  ∂2  yψk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We cope with this issue in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Assume that the sequence of initial data (uin,k, ut,in,k, Φin,k)k∈Z satisfies  sup  k∈Z  �  eσ|k|  1  3 �  ∥uin,k∥L2 + ∥ut,in,k∥L2 + ∥Φin,k∥L2  ��  < +∞,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='24)  for a given radius σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Let α : [0, Tσ) → R+ be defined by  α(t) := σ  2 − 2  5  6  �  ∥U ′′′  sh∥L∞ + 2∥U ′′  sh∥L∞  � 1  3 e  t  3 t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='25)  Then the sequence (ψk)k∈N generates a function ψ : [0, Tσ) × T × (0, 1) → R through the Fourier series  ψ(t, x, y) =  �  k∈Z  ψk(t, y)eikx,  (t, x, y) ∈ [0, Tσ) × T × (0, 1)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='26)  such that ψ, ∂yψ and ∂tψ are all in L∞(0, Tδ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' G3  α(t),xH1  0,y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' the following estimate on the  norms holds true at any time t ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Tσ):  ∥ψ(t)∥G3  α(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='xH1  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='y+∥∂yψ(t)∥G3  α(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='xH1  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='y+∥∂2  tyψ(t)∥G3  α(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='xH1  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='y+∥∂2  yψ(t)∥G3  α(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='xH1  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='y=  sup  k∈Z  �  eα(t)|k|  1  3��ψk(t)  ��  L2  �  +sup  k∈Z  �  eα(t)|k|  1  3��∂yψk(t)  ��  L2  �  +sup  k∈Z  �  eα(t)|k|  1  3��∂2  tyψk(t)  ��  L2  �  +sup  k∈Z  �  eα(t)|k|  1  3��∂2  yψk(t)  ��  L2  �  ≤ Cσ(t)  �  1+∥Ush∥L∞ +∥U ′  sh∥L∞ +∥U ′′  sh∥L∞ +∥U ′′′  sh∥L∞  �2  sup  k∈Z  �  eσ|k|  1  3 �  ∥Φin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='k∥L2 +∥uin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='k∥L2+∥ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='in,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='k∥L2  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='27)  where Cσ(t) = 170 · max{1, 12/σ}6t(1 + t)3et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' If the sequence (ψk)k∈Z and its derivatives satisfy the corresponding inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='27), then it  generates trivially a function ψ : [0, Tδ) × T × (0, 1) → R as described in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='26) , since the series converges  strongly in L∞(0, Tσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' L2(T × (0, 1)) and the limit has explicit Fourier coefficients given by (ψk)k∈Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Our  main objective is therefore to prove uniquely the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='27) in the coefficient (ψk)k∈Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  We first show that the function (∂t + 1)∂yψk and ∂2  yψk satisfies a similar inequality, namely  eα(t)|k|  1  3 ���(∂t + 1)∂yψk(t)  ��  L2 +  ��∂2  yψk(t)  ��  L2  �  ≤ Cσ(t)  5  �  1 + ∥Ush∥L∞+  + ∥U ′  sh∥L∞ + ∥U ′′  sh∥L∞ + ∥U ′′′  sh∥L∞  �2  sup  ˜k∈Z  �  eσ|˜k|  1  3 �  ∥Φin,˜k∥L2 +∥uin,˜k∥L2+∥ut,in,˜k∥L2  ��  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='28)  for any k ∈ Z and t ∈ [0, Tσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' When k = 0, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='28) is essentially a direct consequence of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='23), which  implies in particular  ��(∂t + 1)∂yψ0(t)  ��  L2 +  ��∂2  yψ0(t)  ��  L2 ≤ 4t  �  ∥uin,0∥L2 + ∥ut,in,0∥L2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  We turn our attention therefore to |k| ≥ 1, so that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='23) yields  sup  s∈[0,t]  ���(∂t+1)∂yψk(s)  ��  L2+  ��∂2  yψk(s)  ��  L2  �  ≤ 4t(3 +  √  2)  �  1+∥U ′  sh∥L∞ +∥Ush∥L∞  ��  ∥Φin,k∥L2+∥uin,k∥L2+  +∥ut,in,k∥L2  �4  √  2  3 et(1 + t)3�  1 + ∥U ′′′  sh∥L∞ + ∥U ′′  sh∥L∞  �  |k|2 exp  �  2  5  6 |k|  1  3  �  ∥U ′′′  sh∥L∞ + 2∥U ′′  sh∥L∞  � 1  3 e  t  3 t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  14  Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' we collect all terms of the shear flow Ush within a single parenthesis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' we extrapolate eσ|k|1/3 in  front of the initial data and we choose s = t on the supremum at the left-hand side  ��(∂t + 1)∂yψk(t)  ��  L2 +  ��∂2  yψk(t)  ��  L2 ≤ 34t(1 + t)3et�  1 + ∥Ush∥L∞ + ∥U ′  sh∥L∞+  +∥U ′′  sh∥L∞ + ∥U ′′′  sh∥L∞�2  �  sup  ˜k∈Z  �  eσ|˜k|  1  3 �  ∥Φin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='˜k∥L2+∥uin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='˜k∥L2 + ∥ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='in,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='˜k∥L2  ���  ×  × |k|2 exp  �  − σ|k|  1  3 +2  1  3 |k|  1  3  �  ∥U ′′′  sh∥L∞ +2∥U ′′  sh∥L∞  � 1  3e  t  3 t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Since one has a ≤ ea, for any positive real number a > 0, we can bound the term |k|2, by means of  |k|2 =  �  |k|  1  3 �6 =  �12  σ  �6� σ  12|k|  1  3  �6  ≤  �12  σ  �6�  e  σ  12 |k|  1  3 �6  =  �12  σ  �6  e  σ  2 |k|  1  3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Hence, remarking that 34(12/σ)6t(1 + t)3et ≤ Cσ(t)/5, we deduce that  ��(∂t + 1)∂yψk(t)  ��  L2 +  ��∂2  yψk(t)  ��  L2 ≤ Cσ(t)  4  �  1 + ∥Ush∥L∞ + ∥U ′  sh∥L∞ + ∥U ′′  sh∥L∞ + ∥U ′′′  sh∥L∞�2×  ×  �  sup  ˜k∈Z  �  eσ|˜k|  1  3 �  ∥Φin,˜k∥L2 + ∥uin,˜k∥L2 + ∥ut,in,˜k∥L2  ���  exp  �  −  �σ  2 −2  1  3  �  ∥U ′′′  sh∥L∞ +2∥U ′′  sh∥L∞  � 1  3e  t  3 t  �  �  ��  �  =α(t)  |k|  1  3  �  ,  which coincides with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We now transfer the estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='28) directly to the functions ψk, ∂yψk and  ∂t∂yψk in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We begin with by developing ∂yψk through  ∂yψk(t, y) = e−tet∂yψk(t, y) = e−t  � t  0  ∂s(es∂yψk(s, y))ds =  � t  0  es−t�  (∂t + 1)∂yψk  �  (s, y)ds,  for any (t, y) ∈ (0, Tσ) × (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Hence, by taking the L2-norm on both left and right-hand sides and  keeping in mind that α is decreasing in time, we remark that  eα(t)|k|  1  3 ∥∂yψk(t)∥L2 ≤ eα(t)|k|  1  3  � t  0  es−t∥(∂t + 1)∂yψk(s)∥L2ds ≤  � t  0  es−teα(s)|k|  1  3 ∥(∂t + 1)∂yψk(s)∥L2ds  ≤ Cσ(t)  5  �  1+∥Ush∥L∞+∥U ′  sh∥L∞ +∥U ′′  sh∥L∞ +∥U ′′′  sh∥L∞  �2  sup  ˜k∈Z  �  eσ|˜k|  1  3 �  ∥Φin,˜k∥L2 +∥uin,˜k∥L2+∥ut,in,˜k∥L2  ��  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='29)  where we have used  � t  0 es−tds = 1 − e−t < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Invoking the Poincar´e inequality ∥ψk∥L2 ≤ ∥∂yψk∥L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' it is  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='easy at this stage to check that  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='k∈Z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='eα(t)|k|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3��ψk(t)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='eα(t)|k|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='≤ 3 sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='eα(t)|k|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='3 ��∂yψk(t)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='+ sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='eα(t)|k|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='3 ��(∂t + 1)∂yψk(t)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='+ sup  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='k∈Z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='eα(t)|k|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3 ��∂2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='yψk(t)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='L2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  We finally couple the last inequality together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='28) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='29), which finally implies the estimate  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' This concludes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4  Transferring Gevrey 3 to the velocity field  We shall now transfer the Gevrey regularity from the function ψ of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4 to a solution u of the  original hyperbolic Prandtl equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  15  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Assume that uin is in G3  σ,xH1  0,y, while ut,in is in G3  x,σL2  y, for a given σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Let  Tσ > 0 and β, γ : [0, Tσ) → R be as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Then the sequence (uk)k∈N generates a function  u : [0, Tσ) × T × (0, 1) → R through the inverse Fourier transform  u(t, x, y) =  �  k∈Z  uk(t, y)eikx,  (t, x, y) ∈ [0, Tσ) × T × (0, 1),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='30)  such that u ∈ L∞(0, Tσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Gβ(t),xH1  0,y) and ∂tu ∈ L∞(0, Tσ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Gγ(t),xL2  y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In particular, the following estimate  holds true at any time t ∈ [0, Tσ):  ∥u(t)∥G3  β(t),xL2y + ∥∂yu(t)∥G3  β(t),xL2y + ∥∂tu(t)∥G3  γ(t),xL2y  = sup  k∈Z  �  eβ(t)|k|  1  3 ��uk(t)  ��  L2  �  + sup  k∈Z  �  eβ(t)|k|  1  3 ��∂yuk(t)  ��  L2  �  + sup  k∈Z  �  eγ(t)|k|  1  3 ��∂tuk(t)  ��  L2  �  ≤ ˜Cσ(t)  �  1 + ∥Ush∥L∞ + ∥U ′  sh∥L∞ + ∥U ′′  sh∥L∞ + ∥U ′′′  sh∥L∞  �3�  ∥uin∥G3σ,xH1  0,y + ∥ut,in∥G3σ,xL2y  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='31)  where ˜Cσ(t) = 104 max{1, 12/σ}15(1 + t)5et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Similarly as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4, we focus this entire proof to show the estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='31) on the sequence  (uk)k∈Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' The solution u is then automatically determined by the Fourier series (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  We fix momentarily the frequency k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' If the initial data (uin,k, uin,k,t) are identically null, then the  solution uk of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='8) is identically null (since the equation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='8) for a fixed k ∈ Z is linear, hyperbolic  and damped) and the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='31) is automatically satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We shall thus focus this proof to the  case (uin,k, uin,k,t) ̸= (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  We begin with by setting the function fk :=  �  (∂t +1)(∂t +ikUsh)−∂2  y  �  ∂yψk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Thanks to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='17), fk satisfies  fk(t, y) = ikU ′  sh(y)(∂t + 1)ψk(t, y) + ∂yΦk(t, y) = ikU ′  sh(y)(∂t + 1)ψk(t, y) + uk(t, y),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='32)  for any (t, y) ∈ [0, Tσ) × (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Because of the boundary conditions on ψk and uk, the function fk fulfills  homogeneous Dirichlet conditions fk|y=0,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Furthermore its initial data are determined by  fk(0, y) = uin,k(y)  ∂tfk(0, y) = ikU ′  sh(y)∂2  t ψk(0, y) + ut,in,k(y) = ikU ′  sh(y)Φin,k(y) + ut,in,k(y)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='33)  for any y ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Thanks to identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='20), we remark moreover that fk is also solution of  �  (∂t + 1)(∂t + ikUsh) − ∂2  y  �  fk = [ikU ′  sh, ∂2  y]  �  (∂t + 1)ψk  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='34)  Starting from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='34), we aim to develop some suitable estimates on the L2-norm of (∂t + 1)fk and ∂yfk,  which we will eventually transfer to uk, ∂yuk and ∂tuk, making use of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We shall observe that we  have now a complete control on the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='34), because of the uniform estimates given by  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  We multiply (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='34) with the conjugate (∂t + 1)fk, then we integrate along (0, t)×(0, 1) for a time t ∈ [0, Tσ)  and finally we calculate the real part Re of the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' This leads in particular to  1  2  ��(∂t + 1)fk(t)  ��2  L2 + 1  2  ��∂yfk(t)  ��2  L2 +  � t  0  ��∂yfk(s)  ��2  L2ds = 1  2  ��uin,k + ikU ′  shΦin,k + ut,in,k  ��2  L2+  +1  2  ��∂yuin,k  ��2  L2 + Re  �  ik  � t  0  � 1  0  [ikU ′  sh, ∂2  y]  �  (∂t + 1)ψk  �  (∂t + 1)fkdyds  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='35)  where we have used the initial conditions in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='33), as well as  Re  �  ik  � t  0  � 1  0  Ush(y)|(∂t + 1)fk(s, y)|2dyds  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  16  Multiplying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='35) by 2 and applying the Cauchy-Schwarz inequality on the last integral, we obtain  ��(∂t + 1)fk(t)  ��2  L2 +  ��∂yfk(t)  ��2  L2 + 2  � t  0  ��∂yfk(s)  ��2  L2ds ≤  ��uin,k + ikU ′  shΦin,k + ut,in,k  ��2  L2+  +∥∂yuin,k∥2  L2 + 2|k|  � t  0  ��[U ′  sh, ∂2  y]  �  (∂t + 1)ψk(s)  ���  L2∥(∂t + 1)fk(s)∥L2ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='36)  We next address the last integral in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='35) with the commutator [U ′  sh, ∂2  y]((∂t + 1)ψk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' First, we remark  ��[U ′  sh, ∂2  y]  �  (∂t + 1)ψk(s)  ���  L2 =  ��U ′  sh∂2  y  �  (∂t + 1)ψk(s)  �  − ∂2  y  �  U ′  sh(∂t + 1)ψk(s)  ���  L2  =  ��U ′′′  sh(∂t + 1)ψk(s) + 2U ′′  sh(∂t + 1)∂yψk(s)  ��  L2  ≤ ∥U ′′′  sh∥L∞∥(∂t + 1)ψk(s)∥L2 + 2∥U ′′  sh∥L∞∥(∂t + 1)∂yψk(s)∥L2  ≤  �  ∥U ′′′  sh∥L∞ + 2∥U ′′  sh∥L∞�  ∥(∂t + 1)∂yψk(s)∥L2  Hence, we plug this last inequality into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='36), we take the Supermum within the time interval [0, t] and  finally we divide the result by sups∈[0,t](∥(∂t + 1)fk(s)∥2  L2 + ∥(∂t + 1)fk(s)∥2  L2)1/2 (which is not null, since  the initial data are not all zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' This leads to  sup  s∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='t]  �  ∥(∂t + 1)fk(s)∥L2 + ∥∂yfk(s)∥L2  �  ≤  √  2 sup  s∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='t]  �  ∥(∂t + 1)fk(s)∥2  L2 + ∥∂yfk(s)∥2  L2  � 1  2  ≤  √  2  sups∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='t]  �  ∥(∂t + 1)fk(s)∥2  L2 + ∥∂yfk(s)∥2  L2  �  sups∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='t]  �  ∥(∂t + 1)fk(s)∥2  L2 + ∥∂yfk(s)∥2  L2  � 1  2  ≤  √  2  ��uin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='k + ikU ′  shΦin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='k + ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='in,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='k  ��2  L2 + ∥∂yuin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='k∥2  L2  sups∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='t]  �  ∥(∂t + 1)fk(s)∥2  L2 + ∥∂yfk(s)∥2  L2  � 1  2  +  + 2  √  2  �  ∥U ′′′  sh∥L∞ + 2∥U ′′  sh∥L∞�  |k|  � t  0  ∥(∂t + 1)∂yψk(s)∥L2  ∥(∂t + 1)fk(s)∥L2  sups∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='t]  �  ∥(∂t + 1)fk(s)∥2  L2 + ∥∂yfk(s)∥2  L2  � 1  2  ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  At s = 0, the functions (∂t + 1)fk(s) and ∂yfk(s) coincide with uin,k + ikU ′  shΦin,k + ut,in,k and ∂yuin,k,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We deduce hence that  sup  s∈[0,t]  �  ∥(∂t + 1)fk(s)∥L2 + ∥∂yfk(s)∥L2  �  ≤  √  2  ���uin,k + ikU ′  shΦin,k + ut,in,k  ��2  L2 + ∥∂yuin,k∥2  L2  � 1  2 +  +2  √  2  �  ∥U ′′′  sh∥L∞ +2∥U ′′  sh∥L∞�  |k|  � t  0  ∥(∂t+1)∂yψk(s)∥L2ds ≤  √  2  �  ∥uin,k∥L2+|k|∥U ′  sh∥L∞∥Φin,k∥L2+  +∥ut,in,k∥L2 + ∥∂yuin,k∥L2  �  + 2  √  2  �  ∥U ′′′  sh∥L∞ + 2∥U ′′  sh∥L∞�  |k|  � t  0  ∥(∂t + 1)∂yψk(s)∥L2ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='37)  We are now in the condition to reveal the uniform estimates of the Gevrey-three regularity on the sequences  ((∂t + 1)fk)k∈Z and (∂yfk)k∈Z, by establishing the corresponding exponential growth on the modes |k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  To this end, we take advantage of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4, so that the last integral in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='37) enables  � t  0  ∥(∂t + 1)∂yψk(s)∥L2ds ≤  � � t  0  Cσ(s)e−α(s)|k|  1  3 ds  ��  1 + ∥Ush∥L∞ + ∥U ′  sh∥L∞ + ∥U ′′  sh∥L∞+  +∥U ′′′  sh∥L∞  �2  sup  ˜k∈Z  �  eσ|˜k|  1  3 �  ∥Φin,˜k∥L2 +∥uin,˜k∥L2+∥ut,in,˜k∥L2+∥∂yuin,˜k∥L2  ��  ,  17  where we recall that Cσ in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4 is defined as Cσ(t) = 170 max{1, 12/σ}6t(1+t)3et, while the radius  α(t) = σ/2 − 2  5  6(∥U ′′′  sh∥L∞ + 2∥U ′′  sh∥L∞)  1  3et/3t (which satisfies α(t) ≥ 7σ/8 > 0, for any t ∈ [0, Tσ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Since  s ∈ [0, t] → Cσ(s)e−α(s)|k|  1  3 is a non-decreasing function, it can be bound by its value at s = t, so that  � t  0  ∥(∂t + 1)∂yψk(s)∥L2ds ≤ tCσ(t)e−α(t)|k|  1  3 �  1 + ∥Ush∥L∞ + ∥U ′  sh∥L∞ + ∥U ′′  sh∥L∞+  +∥U ′′′  sh∥L∞  �2  sup  ˜k∈Z  �  eσ|˜k|  1  3 �  ∥Φin,˜k∥L2+∥uin,˜k∥L2+∥ut,in,˜k∥L2+∥∂yuin,˜k∥L2  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  We thus couple this last inequality together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='37), which guarantees  ∥(∂t + 1)fk(t)∥L2 + ∥∂yfk(t)∥L2 ≤  √  2  �  ∥uin,k∥L2 + |k|∥U ′  sh∥L∞∥Φin,k∥L2 + ∥ut,in,k∥L2 + ∥∂yuin,k∥L2  �  +  +4  √  2  �  ∥U ′′′  sh∥L∞ + ∥U ′′  sh∥L∞�  |k|tCσ(t)e−α(t)|k|  1  3 �  1 + ∥Ush∥L∞ + ∥U ′  sh∥L∞+  +∥U ′′  sh∥L∞ + ∥U ′′′  sh∥L∞  �2  sup  ˜k∈Z  �  eσ|˜k|  1  3 �  ∥Φin,˜k∥L2+∥uin,˜k∥L2+∥ut,in,˜k∥L2 +∥∂yuin,k∥L2  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Finally, to obtain our uniform estimates, we multiply both left and right-hand sides with eβ(t)|k|1/3 (where  β(t) = α(t) − σ/4 is defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='8)), to obtain  eβ(t)|k|  1  3 �  ∥(∂t + 1)fk(t)∥L2 + ∥∂yfk(t)∥L2  �  ≤  √  2eβ(t)|k|  1  3 �  ∥uin,k∥L2 + |k|∥U ′  sh∥L∞∥Φin,k∥L2 + ∥ut,in,k∥L2+  +∥∂yuin,k∥L2  �  + 4  √  2e− σ  4 |k|  1  3 �  ∥U ′′′  sh∥L∞ + ∥U ′′  sh∥L∞�  |k|tCσ(t)  �  1 + ∥Ush∥L∞ + ∥U ′  sh∥L∞+  +∥U ′′  sh∥L∞ + ∥U ′′′  sh∥L∞  �2  sup  ˜k∈Z  �  eσ|˜k|  1  3 �  ∥Φin,˜k∥L2+∥uin,˜k∥L2+∥ut,in,˜k∥L2 +∥∂yuin,k∥L2  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  The left-hand side already reveals the Gevrey-three norm on ((∂t + 1)fk)k∈Z and (∂yfk)k∈Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We shall  however provide a uniform estimate of the right-hand side, with respect to the frequencies k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' To this  end, we first observe that  |k| =  � 4  σ  �3�σ  4 |k|  1  3  �3  ≤  � 4  σ  �3  e  3σ  4 |k|  1  3 ,  |k| =  �12  σ  �3� σ  12|k|  1  3  �3  ≤  �12  σ  �3  e  σ  4 |k|  1  3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='38)  Therefore, since 1 ≤ max{1, 4/σ}3e3σ|k|1/3/4, we gather that  eβ(t)|k|  1  3 �  ∥(∂t+1)fk(t)∥L2 +∥∂yfk(t)∥L2  �  ≤  √  2e  �  β(t)+ 3σ  4  �  |k|  1  3 max  �  1, 4  σ  �3�  ∥uin,k∥L2+∥U ′  sh∥L∞∥Φin,k∥L2+  +∥ut,in,k∥L2 + ∥∂yuin,k∥L2  �  + 4  √  2e− σ  4 |k|  1  3 �  ∥U ′′′  sh∥L∞ + ∥U ′′  sh∥L∞��12  σ  �3  e  σ  4 |k|  1  3 tCσ(t)  �  1 + ∥Ush∥L∞+  +∥U ′  sh∥L∞ + ∥U ′′  sh∥L∞ + ∥U ′′′  sh∥L∞  �2  sup  ˜k∈Z  �  eσ|˜k|  1  3 �  ∥Φin,˜k∥L2+∥uin,˜k∥L2 +∥ut,in,˜k∥L2+∥∂yuin,˜k∥L2  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  This provides the required uniform estimate in k ∈ Z, since 3σ/4 + β(t) = α(t) + σ/2 ≤ σ for any  t ∈ [0, Tσ), hence  eβ(t)|k|  1  3 �  ∥(∂t + 1)fk(t)∥L2 + ∥∂yfk(t)∥L2  �  ≤ C2,σ(t)  �  1 + ∥Ush∥L∞ + ∥U ′  sh∥L∞ + ∥U ′′  sh∥L∞+  +∥U ′′′  sh∥L∞  �3  sup  ˜k∈Z  �  eσ|˜k|  1  3 �  ∥Φin,˜k∥L2 + ∥uin,˜k∥L2 + ∥ut,in,˜k∥L2 + ∥∂yuin,˜k∥L2  ��  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='39)  18  where the function t ∈ (0, Tσ) → C2,σ(t) is now defined by  C2,σ(t) :=  √  2 max  �  1, 4  σ  �3  + 4  √  2  �12  σ  �3  tCσ(t)  =  √  2 max  �  1, 4  σ  �3  + 4  √  2  �12  σ  �3  170 · max  �  1, 12  σ  �6  t2(1 + t)3et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='40)  We now take advantage of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='39), in order to transfer the corresponding Gevrey-three estimates to the  sequences (uk)k∈Z, (∂yuk)k∈Z and (∂tuk)k∈Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We begin with ∂yuk and we invoke identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='32), which  ensures that ∂yuk = ∂yfk − ikU ′′  sh(∂t + 1)ψk − ikU ′  sh(∂t + 1)∂yψk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Hence  eβ(t)|k|  1  3 ∥∂yuk(t)∥L2 ≤ eβ(t)|k|  1  3 �  ∥∂yfk∥L2 + |k|∥U ′  sh∥L∞∥(∂t + 1)∂yψk∥L2 + |k|∥U ′′  sh∥L∞∥(∂t + 1)ψk∥L2  �  ,  which we couple together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='38), the Poincar´e inequality ∥(∂t + 1)ψk∥L2 ≤ ∥(∂t + 1)∂yψk∥L2 and the  relation α(t) = β(t) + σ/4, to obtain  eβ(t)|k|  1  3 ∥∂yuk(t)∥L2 ≤ eβ(t)|k|  1  3  �  ∥∂yfk∥L2 +  �12  σ  �3  e  σ  4 |k|  1  3 �  ∥U ′  sh∥L∞ + ∥U ′′  sh∥L∞�  ∥(∂t + 1)∂yψk∥L2  �  ≤ eβ(t)|k|  1  3 ∥∂yfk∥L2 +  �12  σ  �3  eα(t)|k|  1  3 �  ∥U ′  sh∥L∞ + ∥U ′′  sh∥L∞�  ∥(∂t + 1)∂yψk∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Thanks to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='39) and the Poincar´e inequality ∥uk∥L2 ≤ ∥∂yuk∥L2, we establish therefore the uniform  estimate  eβ(t)|k|  1  3 �  ∥uk(t)∥L2 + ∥∂yuk(t)∥L2  �  ≤  �  C2,σ(t) +  �12  σ  �3  Cσ(t)  ��  1 + ∥Ush∥L∞ + ∥U ′  sh∥L∞+  +∥U ′′  sh∥L∞ + ∥U ′′′  sh∥L∞  �3  sup  ˜k∈Z  �  eσ|˜k|  1  3 �  ∥Φin,˜k∥L2 + ∥uin,˜k∥L2 + ∥ut,in,˜k∥L2 + ∥∂yuin,˜k∥L2  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='41)  Next, we aim to address the sequence of the time derivative (∂tuk)k∈Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We invoke once more the relation  uk = fk − ikU ′  sh(∂t + 1)ψk in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='32) and we decompose ∂tuk as follows:  ∂tuk = ∂tfk − ikU ′  sh(∂t + 1)∂tψk  = (∂t + 1)fk − fk − ikU ′  sh  �  (∂t + ikUsh)(∂t + 1) − ∂2  y  �  ψk − k2U ′  shUsh(∂t + 1)ψk − ikU ′  sh∂2  yψk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Recalling that ((∂t + ikUsh)(∂t + 1) − ∂2  y)ψk = Φk from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='17), we gather that  ∂tuk = (∂t + 1)fk − fk − ikU ′  shΦk − k2U ′  shUsh(∂t + 1)ψk − ikU ′  sh∂2  yψk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  A straightforward calculation leads hence to the estimate  eγ(t)|k|  1  3 ∥∂tuk(t)∥L2 ≤ eγ(t)|k|  1  3  �  ∥(∂t + 1)fk(t)∥L2 + ∥fk(t)∥L2 + |k|∥U ′  sh∥L∞∥Φk(t)∥L2+  + k2∥U ′  sh∥L∞∥Ush∥L∞∥(∂t + 1)ψk(t)∥L2 + |k|∥U ′  sh∥L∞∥∂2  yψk(t)∥L2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  We now remark that γ(t) ≤ β(t), for any t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Thus, making use of the Poincar´e inequalities given by  ∥fk(t)∥L2 ≤ ∥∂yfk(t)∥L2 and ∥Φk(t)∥L2 ≤ ∥∂yΦk(t)∥L2 = ∥uk∥L2, as well as recalling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='38) together with  k2 =  �24  σ  �6� σ  24|k|  1  3  �6  ≤  �24  σ  �6�  e  σ  24 |k|  1  3 �6  = 64  �12  σ  �6  e  σ  4 |k|  1  3 ,  19  we finally obtain  eγ(t)|k|  1  3 ∥∂tuk(t)∥L2 ≤ eβ(t)|k|  1  3 �  ∥(∂t + 1)fk(t)∥L2 + ∥∂yfk(t)∥L2  �  +  �12  σ  �3  ∥U ′  sh∥L∞eβ(t)|k|  1  3 ∥uk(t)∥L2+  + 64  �12  σ  �6  ∥U ′  sh∥L∞∥Ush∥L∞eα(t)|k|  1  3 ∥(∂t + 1)∂yψk(t)∥L2 +  �12  σ  �3  ∥U ′  sh∥L∞eα(t)|k|  1  3 ∥∂2  yψk(t)∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  We hence plug (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='28), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='39) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='41) into this last relation, to gather  eγ(t)|k|  1  3 ∥∂tuk(t)∥L2 ≤  �  C2,σ(t)  � �� �  from all fk  +  �12  σ  �3 �  C2,σ(t) +  �12  σ  �3  Cσ(t)  �  �  ��  �  from uk  +64  �12  σ  �6 Cσ(t)  5  � �� �  from (∂t+1)∂yψk  +  �12  σ  �3 Cσ(t)  5  � �� �  from ∂2yψk  ��  1+  +∥Ush∥L∞ + ∥U ′  sh∥L∞ + ∥U ′′  sh∥L∞ + ∥U ′′′  sh∥L∞  �4  sup  ˜k∈Z  �  eσ|˜k|  1  3 �  ∥uin,˜k∥L2 + ∥ut,in,˜k∥L2 + ∥∂yuin,˜k∥L2  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  To simplify the summation of the terms depending on Cσ and C2,σ, we make use of their definitions in  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='40), so that  C2,σ(t) +  �12  σ  �3�  C2,σ(t) +  �12  σ  �3  Cσ(t)  �  + 64  �12  σ  �6 Cσ(t)  5  +  �12  σ  �3 Cσ(t)  5  ≤ max  �  1, 12  σ  �6�  2C2,σ(t) + 14Cσ(t)  �  ≤ 5000 max  �  1, 12  σ  �15  (1 + t)5et =  ˜Cσ(t)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  This provides indeed the following estimate on ∂tuk  eγ(t)|k|  1  3 ∥∂tuk(t)∥L2 ≤  ˜Cσ(t)  2  �  1 + ∥Ush∥L∞ + ∥U ′  sh∥L∞+  + ∥U ′′  sh∥L∞ + ∥U ′′′  sh∥L∞  �4  sup  ˜k∈Z  �  eσ|˜k|  1  3 �  ∥uin,˜k∥L2 + ∥ut,in,˜k∥L2 + ∥∂yuin,˜k∥L2  ��  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='42)  which together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='41) (and the fact that C2,σ(t) + (12/σ)3Cσ(t) ≤ ˜C(t)/2) imply finally the claimed  inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' This concludes the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5  A suitable test function  In order to conclude the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4, we need to establish Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3 about the uniform  estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='21) on (∂t + 1)∂yψk and ∂2  yψk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In the present section we establish a suitable test function  ωτ,k for equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='20), that reveals some advantageous estimates, when analysing the L2-inner product  between ωτ,k and equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' For a given positive time τ ∈ (0, Tσ), we consider ωτ,k as the unique  solution of the following backward-in-time linear problem:  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  ((∂t − 1)(∂t + ikUsh) − ∂2  y)ωτ,k = (∂t + 1)∂yψk  (0, τ) × (0, 1),  (ωτ,k, ∂tωτ,k)|t=τ = (0, 0)  (0, 1),  ωτ,k|y=0,1 = 0  (0, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='43)  With the next lemma, we determine the relations between certain meaningful norms of ωτ,k and the ones  of (∂t + 1)∂yψk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' The solution ωτ,k of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='43) satisfies at any time t ∈ [0, τ]  sup  s∈(t,τ)  ∥(∂t − 1)ωτ,k(s)∥L2 ≤ 2  � τ  t  ∥(∂t + 1)∂yψk(s)∥L2ds,  sup  s∈(t,τ)  ∥∂yωτ,k(s)∥L2 ≤  √  2  � τ  t  ∥(∂t + 1)∂yψk(s)∥L2ds,  sup  s∈(t,τ)  ∥ωτ,k(s)∥L2 ≤ 2eτ−t  � τ  t  (s − t)∥(∂t + 1)∂yψk(s)∥L2ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='44)  20  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We multiply the first equation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='43) with the complex conjugate (∂t − 1)ωτ,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Hence, for a given  time ˜t ∈ (0, τ), we integrate the achieved identity within the domain (˜t, τ) × (0, 1) and we extrapolate the  corresponding real part:  −1  2∥(∂t − 1)ωτ,k(˜t)∥2  L2 − 1  2∥∂yωτ,k(˜t)∥2  L2 −  � τ  ˜t  ∥∂yωτ,k(s)∥2  L2ds  =  � τ  ˜t  � 1  0  Re  �  (∂t + 1)∂yψk · (∂t − 1)ωτ,k  �  (s, y)dyds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='45)  We multiply (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='45) by −1 and we take the Supremum of within ˜t ∈ (t, T), for a fixed t ∈ (0, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Thanks  to Cauchy-Schwarz, we hence establish that  sup  ˜t∈(t,τ)  ∥(∂t − 1)ωτ,k(˜t)∥2  L2 ≤ 2  � τ  t  ∥(∂t + 1)∂yψk(s)∥L2∥(∂t − 1)ωτ,k(s)∥L2ds  ≤ 2  � τ  t  ∥(∂t + 1)∂yψk(s)∥L2ds sup  s∈(t,τ)  ∥(∂t − 1)ωτ,k(s)∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  This corresponds to the first inequality of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Next, we deal with the norm ∥∂yωτ,k∥L2 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' By  invoking (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='45), we have first  sup  s∈(t,T)  ∥∂yωτ,k(s)∥L2 ≤  � � τ  t  ∥(∂t + 1)∂yψk(s)∥L2ds sup  s∈(t,T)  ∥(∂t − 1)ωτ,k(s)∥L2  � 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  The result is thus obtained by invoking the first inequality of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' To conclude the proof, we deal now  with the last inequality of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Since ωτ,k is null at t = τ, we have  ∥ωτ,k(t)∥L2 =  ��� −  � τ  t  ∂tωτ,k(s)ds  ���  L2 ≤  � τ  t  ∥∂tωτ,k(s)∥L2ds  ≤  � τ  t  ∥(∂t − 1)ωτ,k(s)∥L2ds +  � τ  t  ∥ωτ,k(s)∥L2ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Furthermore, the first estimate in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='44) guarantees that  ∥ωτ,k(t)∥L2 ≤ 2  � τ  t  � τ  s  ∥(∂t + 1)∂yψk(z)∥L2dzds +  � τ  t  ∥ωτ,k(s)∥L2ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  The result is then achieved by applying the Gronwall’s lemma:  ∥ωτ,k(t)∥L2 ≤ 2eτ−t  � τ  t  � τ  s  ∥(∂t + 1)∂yψk(z)∥L2ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  This concludes the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='6  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3  This section is devoted to the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3, which is based on the specific test function ωτ,k,  introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We begin with, by recalling system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='20) for the evolution of ∂yψk:  �  (∂t + 1)(∂t + ikUsh) − ∂2  y  �2∂yψk = [ikU ′  sh, ∂2  y]  �  (∂t + 1)ψk  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='46)  with initial data ∂tψk|t=0 = ψk|t=0 = 0 and boundary conditions ψk|y=0,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Next, we fix a general time  τ ∈ (0, Tσ) and we multiply the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='46) with the conjugate ωτ,k of the test function defined in  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' By integrating the result along (0, τ) × (0, 1), we obtain the following identity:  � τ  0  � 1  0  ��  (∂t + 1)(∂t + ikUsh(y)) − ∂2  y  �2∂yψk  �  (t, y)ωτ,k(t, y)dydt  =  � τ  0  � 1  0  [ikU ′  sh(y), ∂2  y]  �  (∂t + 1)ψk(t, y)  �  ωτ,k(t, y)dydt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='47)  21  We aim therefore to integrate by parts the integral at the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' To this end, we first develop the  operator (∂t + 1)(∂t + ikUsh(y)) − ∂2  y into ∂2  t + (1 + ikUsh(y))∂t + ikUsh(y) − ∂2  y, which localises the order  of each derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Hence, by considering momentarily the derivative ∂2  t of second order, we gather  � τ  0  � 1  0  �  ∂2  t  �  (∂t + 1)(∂t + ikUsh(y)) − ∂2  y  �  ∂yψk  �  (t, y)ωτ,k(t, y)dydt  =  � τ  0  � 1  0  ��  (∂t + 1)(∂t + ikUsh(y)) − ∂2  y  �  ∂yψk  �  (t, y)∂2  t ωτ,k(t, y)dydt+  +  � 1  0  �  ∂t  �  (∂t + 1)(∂t + ikUsh(y)) − ∂2  y  �  ∂yψk  �  (0, y)ωτ,k(0, y)dy+  −  � 1  0  ��  (∂t + 1)(∂t + ikUsh(y)) − ∂2  y  �  ∂yψk  �  (0, y)∂tωτ,k(0, y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='48)  The last two integrals of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='48) are set at t = 0 and can hence be recasted in terms of the initial data of  the velocity field uin,k, ∂tuin,k and of the stream function Φin,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Indeed, recalling that ψk is also solution  of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='17), we remark that the second integrand at the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='48) satisfies  �  ∂t((∂t + 1)(∂t + ikUsh) − ∂2  y)∂yψk  �  (0, y) = ikU ′  sh(y)  �  ∂2  t ψk(0, y) + ∂tψk(0, y)  �  + ∂t∂yΦk(0, y),  for any y ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' This can be simplified further, since ∂t∂yΦk(0, y) = ∂yΦt,in,k = ut,in,k, ∂tψk|t=0 = 0 and  equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='17) implies that ∂2  t ψk|t=0 = Φin,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Thus  �  ∂t((∂t + 1)(∂t + ikUsh) − ∂2  y)∂yψk  �  (0, y) = ikU ′  sh(y)Φin,k(y) + ut,in,k(y),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='49)  for any y ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' An analogous approach leads moreover to the following identity for the third integrand  at the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='48):  �  ((∂t + 1)(∂t + ikUsh) − ∂2  y)∂yψk  �  (0, y) = ∂yΦk(0, y) = uin,k(y),  y ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='50)  Therefore, thanks to the relations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='49) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='50), we can reformulate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='48) as follows:  � τ  0  � 1  0  �  ∂2  t  �  (∂t + 1)(∂t + ikUsh(y)) − ∂2  y  �  ∂yψk  �  (t, y)ωτ,k(t, y)dydt  =  � τ  0  � 1  0  ��  (∂t + 1)(∂t + ikUsh(y)) − ∂2  y  �  ∂yψk  �  (t, y)∂2  t ωτ,k(t, y)dydt+  +  � 1  0  �  ikU ′  sh(y)Φin,k(y) + ut,in,k(y)  �  ωτ,k(0, y)dy −  � 1  0  uin,k(y)∂tωτ,k(0, y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='51)  We now come back to our original identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='47) and we shall now integrate by parts the operator  (1 + ikUsh)∂t, with a a first order derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' As for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='51), our aim is once more to recast the resulting  integrals at t = 0 in terms of the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' A direct calculation guarantees that  � τ  0  � 1  0  �  (1 + ikUsh)∂t  �  (∂t + 1)(∂t + ikUsh) − ∂2  y  �  ∂yψk  �  (t, y)ωτ,k(t, y)dydt  =  � τ  0  � 1  0  ��  (∂t + 1)(∂t + ikUsh) − ∂2  y  �  ∂yψk  �  (t, y)(1 − ikUsh(y))∂tωτ,k(t, y)dydt+  +  � 1  0  �  (1 + ikUsh)  �  (∂t + 1)(∂t + ikUsh) − ∂2  y  �  ∂yψk  �  (0, y)ωτ,k(0, y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='52)  22  Hence, recalling from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='50) that [((1 + ikUsh)((∂t + 1)(∂t + ikUsh) − ∂2  y)∂yψk](0, y) = uin,k(y), we obtain  � τ  0  � 1  0  �  (1 + ikUsh)∂t  �  (∂t + 1)(∂t + ikUsh)−∂2  y  �  ∂yψk  �  (t, y)ωτ,k(t, y)dydt  =  � τ  0  � 1  0  ��  (∂t + 1)(∂t + ikUsh(y)) − ∂2  y  �  ∂yψk  �  (t, y)(1 − ikUsh(y))∂tωτ,k(t, y)dydt+  +  � 1  0  (1 + ikUsh(y))uin,k(y)ωτ,k(0, y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='53)  To conclude the integration by parts related to the operator ((∂t + 1)(∂t + ikUsh(y)) − ∂2  y) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='47), we  shall now treat −∂2  y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Making use of the homogeneous conditions ωk|y=0,1 = 0 on the test function, we  have that  −  � τ  0  � 1  0  �  ∂2  y  �  (∂t + 1)(∂t + ikUsh(y)) − ∂2  y  �  ∂yψk  �  (t, y)ωτ,k(t, y)dydt  =  � τ  0  � 1  0  �  ∂y  �  (∂t + 1)(∂t + ikUsh(y)) − ∂2  y  �  ∂yψk  �  (t, y)∂yωτ,k(t, y)dydt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Now, recalling from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='32) that ((∂t + 1)(∂t + ikUsh(y)) − ∂2  y)∂yψk = fk = ikU ′  sh(y)(∂t + 1)ψk + uk is null  in y = 0, 1, we obtain  −  � τ  0  � 1  0  �  ∂2  y  �  (∂t + 1)(∂t + ikUsh(y)) − ∂2  y  �  ∂yψk  �  (t, y)ωτ,k(t, y)dydt  = −  � τ  0  � 1  0  ��  (∂t + 1)(∂t + ikUsh(y)) − ∂2  y  �  ∂yψk  �  (t, y)∂2yωτ,k(t, y)dydt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='54)  As final result, we couple the identities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='51),(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='53) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='54), so that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='47) can be recasted as  � τ  0  � 1  0  �  (∂t + 1)(∂t + ikUsh(y)) − ∂2  y  �  ∂yψk(t, y)  �  (∂t − 1)(∂t + ikUsh(y)) − ∂2y  �  ωτ,k(t, y)dydt+  +  � 1  0  �  ikU ′  sh(y)Φin,k(y) + ut,in,k(y) + (1 + ikUsh(y))uin,k(y)  �  ωτ,k(0, y)dy−  −  � 1  0  uin,k(y)∂tωτ,k(0, y)dy =  � τ  0  � 1  0  [ikU ′  sh(y), ∂2  y]  �  (∂t + 1)ψk(t, y)  �  ωτ,k(t, y)dydt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='55)  Next, we make use of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='55), in order to derive suitable estimates on the L2-norms of (∂t + 1)∂yψk and  ∂2  yψk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' These estimates shall not depend upon ωτ,k, hence we aim to get rid of this test function making  use of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='6 and system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' By extrapolating the real part of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='55),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' the first integral becomes  Re  � τ  0  � 1  0  �  (∂t + 1)(∂t + ikUsh(y)) − ∂2  y  �  ∂yψk(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' y)(∂t + 1)∂yψk(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' y)dydt  =  � τ  0  � 1  0  Re  �  ∂t(∂t + 1)∂yψk(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' y)(∂t + 1)∂yψk(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' y)  �  dydt+  +  � τ  0  � 1  0  Re  �  ikUsh(y)∥(∂t + 1)∂yψk(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' y)∥2  L2  �  �  ��  �  =0  dydt +  � τ  0  ��∂2  yψk(s)  ��2  L2ds  = 1  2  ��(∂t + 1)∂yψk(τ)  ��2  L2 + 1  2  ��∂yψk(τ)  ��2  L2 +  � τ  0  ��∂2  yψk(s)  ��2  L2ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  where we have used in the last identity the fact that ∂yψk and ∂t∂yψk are identically null at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Furthermore, when dealing with the real part of the second and third integrals in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='55), we obtain  ����Re  � 1  0  �  ikU ′  sh(y)Φin,k(y) + ut,in,k(y) + (1 + ikUsh(y))uin,k(y)  �  ωτ,k(0, y)dy  ����  ≤  �  |k|∥U ′  sh∥L∞∥Φin,k∥L2 + ∥ut,in,k∥L2 +  �  1 + |k|∥Ush∥L∞�  ∥uin,k∥L2  �  ∥ωτ,k(0)∥L2,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='56)  23  as well as  ����Re  � 1  0  uin,k(y)∂tωτ,k(0, y)dy  ���� ≤ ∥uin,k∥L2∥∂tωτ,k(0)∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='57)  Finally, the real part of the right-hand side in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='55) fulfills  ����Re  � τ  0  � 1  0  [ikU ′  sh(y), ∂2  y]  �  (∂t + 1)ψk(t, y)  �  ωτ,k(t, y)dydt  ����  ≤ |k|  � τ  0  ∥[U ′  sh, ∂2  y](∂t + 1)ψk(t)∥L2∥ωτ,k(t)∥L2dt  ≤ |k|  � τ  0  ∥[U ′  sh, ∂2  y](∂t + 1)ψk(t)∥L2∥ωτ,k(t)∥L2dt  ≤ |k|  � τ  0  ∥U ′′′  sh(∂t + 1)ψk(t) + 2U ′′  sh(∂t + 1)∂yψk(t)∥L2∥ωτ,k(t)∥L2dt  ≤ |k|  � τ  0  �  ∥U ′′′  sh∥L∞ + 2∥U ′′  sh∥L∞  �  ∥(∂t + 1)∂yψk(t)∥L2∥ωτ,k(t)∥L2dt,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='58)  where we have also made use of the Poincar´e inequality in y ∈ (0, 1): ∥(∂t+1)ψk∥L2 ≤ ∥(∂t+1)∂yψk∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  We can summarise hence our last estimates, by coupling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='55) together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='56), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='57) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  This guarantees that for any τ ∈ (0, Tσ)  1  2  ��(∂t + 1)∂yψk(τ)  ��2  L2 + 1  2  ��∂2  yψk(τ)  ��2  L2 +  � τ  0  ��∂2  yψk(t)  ��2  L2dt  ≤  �  |k|∥U ′  sh∥L∞∥Φin,k∥L2 + ∥ut,in,k∥L2 +  �  1 + |k|∥Ush∥L∞�  ∥uin,k∥L2  �  ∥ωτ,k(0)∥L2  + ∥uin,k∥L2∥∂tωτ,k(0)∥L2 + |k|  �  ∥U ′′′  sh∥L∞ + 2∥U ′′  sh∥L∞  � � τ  0  ∥(∂t + 1)∂yψk(t)∥L2∥ωτ,k(t)∥L2dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='59)  The right-hand side still depends upon the test function ωk,τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' We are however in the condition to get rid  of that, by applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' This implies in particular (together with Poincar´e) that  ∥ωk,τ(0)∥L2 ≤ ∥∂yωk,τ(0)∥L2 ≤  √  2  � τ  0  ∥(∂t + 1)∂yψk(s)∥L2ds  ≤  √  2 τ sup  s∈[0,τ]  ∥(∂t + 1)∂yψk(s)∥L2,  ∥∂tωk,τ(0)∥L2 ≤ ∥(∂t − 1)ωk,τ(0)∥L2 + ∥ωk,τ(0)∥L2 ≤ (2 +  √  2)  � τ  0  ∥(∂t + 1)∂yψk(s)∥L2ds  ≤  √  2(1 +  √  2)τ sup  s∈[0,τ]  ∥(∂t + 1)∂yψk(s)∥L2,  ∥ωk(t)∥L2 ≤ eτ−t  � τ  t  (s − t)∥(∂t + 1)∂yψk(s)∥L2ds  ≤ eτ  � τ  t  (s − t)ds sup  s∈[0,τ]  ∥(∂t + 1)∂yψk(s)∥L2 = eτ(τ − t)2  2  sup  s∈[0,τ]  ∥(∂t + 1)∂yψk(s)∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Thus, by taking now the supremum in τ ∈ (0, t) for a general t ∈ (0, Tσ) (and re-denoting the variables  of integration), we can recast (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='59) uniquely in terms of the stream function ψk as follows:  1  2 sup  s∈(0,t)  ���(∂t + 1)∂yψk(s)  ��2  L2 +  ��∂2  yψk(s)  ��2  L2  �  +  � t  0  ��∂2  yψk(s)  ��2  L2ds ≤  sup  s∈(0,t)  ∥(∂t+1)∂yψk(s)∥L2  �√  2  �  |k|∥U ′  sh∥L∞∥Φin,k∥L2 + ∥ut,in,k∥L2 +  �  1 + |k|∥Ush∥L∞�  ∥uin,k∥L2  �  t+  + 2(1 +  √  2)∥uin∥L2τ + |k|  �  ∥U ′′′  sh∥L∞ + 2∥U ′′  sh∥L∞  �  et  � t  0  ∥(∂t + 1)∂yψk(s)∥L2(t − s)2ds  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  24  If sups∈(0,t) ∥(∂t+1)∂yψk(s)∥L2 = 0, then the main estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='21) is automatically satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' On the other  hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' in case this term is not identically null,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' we have that  sup  s∈(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='t)  ���(∂t + 1)∂yψk(s)  ��  L2 +  ��∂2  yψk(s)  ��  L2  �  ≤  √  2  �  sup  s∈(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='t)  ���(∂t + 1)∂yψk(s)  ��2  L2 +  ��∂2  yψk(s)  ��2  L2  �� 1  2  ≤  2  √  2  sups∈(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='t) ∥(∂t+1)∂yψk(s)∥L2  1  2 sup  s∈(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='t)  ���(∂t + 1)∂yψk(s)  ��2  L2 +  ��∂2  yψk(s)  ��2  L2  �  ≤ 2  √  2  �√  2  �  |k|∥U ′  sh∥L∞∥Φin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='k∥L2 + ∥ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='in,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='k∥L2 +  �  1 + |k|∥Ush∥L∞�  ∥uin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='k∥L2  �  t+  + 2(1 +  √  2)∥uin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='k∥L2t + |k|  �  ∥U ′′′  sh∥L∞ + 2∥U ′′  sh∥L∞  �  et  � t  0  ∥(∂t + 1)∂yψk(s)∥L2(t − s)2ds  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  We hence reorganise the last inequality into the following compact form:  sup  s∈[0,t]  ���(∂t + 1)∂yψk(s)  ��  L2 +  ��∂2  yψk(s)  ��  L2  �  ≤ gk(t) + λk(t)3  2  � t  0  (t − s)2 sup  τ∈[0,s]  ∥(∂t + 1)∂yψk(τ)∥L2ds  where the functions gk(t) and λk(t) are defined by means of  gk(t) := 4t  �  |k|  �  ∥U ′  sh∥L∞∥Φin,k∥L2 + ∥Ush∥L∞∥uin,k∥L2  �  + ∥ut,in,k∥L2 + (3 +  √  2)∥uin,k∥L2  �  λk(t) := 2  5  6 |k|  1  3  �  ∥U ′′′  sh∥L∞ + 2∥U ′′  sh∥L∞  � 1  3 e  t  3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  This last inequality corresponds to our claimed estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' This concludes therefore the proof of  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  3  Conclusion and remarks on the non-linear system  In this section, we investigate why Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4 cannot be proven for the nonlinear counterparts of  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1) without further ado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Clearly, this is a consequence of the nonlinear structure but furthermore, the  hyperbolic regime interferes with the known cancellation properties of the classical Prandtl/Navier-Stokes  equations in anaggravating way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  To begin with, we observe that several candidates exist for which Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4 might hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Some  of the represent simplifications of other formulations but, nevertheless, they contain drawbacks which  cannot be dealt with easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' The simplest form of the nonlinear hyperbolic Prandtl equation consists of  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  τ∂ttu + ∂tu + u∂xu + v∂yu − ∂2  yu =  �  τ∂t + 1  ��  ∂tuE + uE∂xuE�  ,  (0, T) × X × (0, +∞),  ∂xu + ∂yv = 0  (0, T) × X × (0, +∞),  (u, v)|y=0 = 0  lim  y→+∞ u = uE  (0, T) × X,  (u, ut)|t=0 = (uin, ut,in)  X × (0, +∞),  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1)  This system looks promising when trying to implement the strategy of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' However, besides the fact  that the second time derivative produces difficulties (see below), a quick look at the linearization  ∂2  t u + ∂tu + Ush(y)∂xu + v U ′  sh(y) − ∂yyu = 0,  on (0, T) × T × (0, 1)  suffices to realize the Eigenvalues contain a positive real part in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' More precisely, solutions corre-  sponding to a frequency k in x will behave like e  √  |k|t which restricts well-posedness theory to the Gevrey  2 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' For a hyperbolic equation, this is expected and actually proven for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1) in [19] (see also [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  25  In conclusion, it is essential to maintain the convective structure of the hyperbolic Prandtl equations (as  shown in Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' By Cattaneo’s law, it reads  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  �  τ∂t + 1  ��  ∂tu + u∂xu + v∂yu  �  − ∂2  yu =  �  τ∂t + 1  ��  ∂tuE + uE∂xuE�  ,  (0, T) × X × (0, +∞),  ∂xu + ∂yv = 0  (0, T) × X × (0, +∞),  (u, v)|y=0 = 0  lim  y→+∞ u = uE  (0, T) × X,  (u, ut)|t=0 = (uin, ut,in)  X × (0, +∞),  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2)  Unfortunately, the time derivative on the convective term brings several other difficulties with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' At first,  note that one of the terms, ∂tu∂xu, competes with the damping mechanism for large values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Even for the  hyperbolic Navier-Stokes equations, this circumstance poses a fundamental issue (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Secondly and much more inherent to the strategy followed in [11] and Section 2, the additional (second)  time-derivatives produce corresponding terms terms on the right-hand side of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Following  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2, we realize that two commutators need to be evaluated in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='18) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In the classical  Prandtl regime, the solenoidality of (u, v) and cancellation of curl-related terms enters the analysis, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  [∂y, ∂t + u∂x + v∂y − ∂yy]u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  None of these instances persist in the hyperbolic version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Instead, new terms arise like ∂xtv ∼ ∂xxtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Confronted with a 4-order time derivative on the left-hand side, this does not present an improvement  over the standard Gevrey 2 regularity result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' A new, different (perhaps related) cancellation mechanism  seems to be necessary but it is not clear how the system can be closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  At this point, we remark that the independence of the shear flow Ush on (t, x) is exploited heavily in  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' A third variant of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1), substituting the first equation by  (τ∂tt + τu∂x∂t + ηv∂y∂t + ∂t + u∂x + v∂y − ∂yy)u = 0,  bears similar problems, although the competing damping term ∂xu∂tu is not present here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' However, the  potential improvement of the above equation might lie in the transport structure related to the time-  derivatives of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  In sum, we conjecture that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='4 hints stability results around shear flows for the nonlinear  hyperbolic equations in Gevrey 3 class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' On the other hand, a general well-posedness theory for arbitrary  initial data in G3 does not seem to be achievable without major novelties or improvements on the strategy  followed in [11] and this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Acknowledgment  The author would like to thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Paicu for the several helpful advises on various technical issues  examined in this Paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' The first author was partially supported by the Bavarian Funding Programme  for the Initiation of International Projects (F¨orderkennzeichen: BayIntAn UWUE 2022 139).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' The third  author was partially supported by GNAMPA and INDAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Data Availability Statement  Data sharing is not applicable to this article, since no datasets were generated or analysed during the  current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Conflict of interest  The authors declare that they have no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  26  References  [1]  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' “Global existence of solutions for hyperbolic Navier-Stokes equations in three space  dimensions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' “Well-posedness of the Prandtl equation in  Sobolev spaces”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='  [3]  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' “Homogeneous hydrostatic flows with convex velocity profiles”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In: Nonlinearity 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='  [4]  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Brenier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' “Remarks on the derivation of the hydrostatic Euler equations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In: Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='7 (2003), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='  [5]  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Brenier, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Natalini, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Puel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' “On a relaxation approximation of the incompressible Navier-  Stokes equations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='4 (2004), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' 1021–1028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  [6]  M Carrassi and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Morro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' “Modified Navier-Stokes equation and its consequences on sound disper-  sion”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In: Nuovo Cimento della Societa Italiana di Fisica B B 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2 (1972), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='  [7]  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Cattaneo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' “Sulla conduzione del calore”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In: Atti Sem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Fis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Modena 3 (1949), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='  [8]  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Cattaneo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' “Sur une forme de l’´equation de la chaleur ´eliminant le paradoxe d’une propagation  instantan´ee”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In: C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='  [9]  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Coulaud, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Hachicha, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Raugel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' “Hyperbolic quasilinear Navier-Stokes Equations in R2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  In press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  [10]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Dalibard and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Masmoudi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' “Separation for the stationary Prandtl equation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In: Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Hautes ´Etudes Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' 130 (2019), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' 187–297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  [11]  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Dietert and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' G´erard-Varet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' In: Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' PDE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1 (2019), Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' 8, 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  [12]  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Lombardo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Sammartino, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Sciacca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='  Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' 364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Lecture Note Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Press, Cambridge, 2009, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' 81–  120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  [13]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' G´erard-Varet and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Dormy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' In: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='2 (2010), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' 591–609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  [14]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' G´erard-Varet and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Masmoudi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' “Well-posedness for the Prandtl system without analyticity or  monotonicity”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In: Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' ´Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Sup´er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' (4) 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='6 (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' 1273–1325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  [15]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' G´erard-Varet, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Masmoudi, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Vicol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' “Well-posedness of the hydrostatic Navier-Stokes  equations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' PDE 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='  [16]  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Grenier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' “On the derivation of homogeneous hydrostatic equations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In: M2AN Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='5 (1999), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' 965–970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  [17]  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Grenier, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' Nguyen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='  In: Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='16 (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' Grenier, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' Nguyen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' Li and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' In: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3 (2020), 717–775.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Wong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Wong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' 204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1 (2012), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' 231–271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  [23]  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Oleinik and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Samokhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Mathematical Models in Boundary Layer Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Applied Math-  ematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Taylor & Francis, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' Raugel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' “Une Perturbation Hyperbolique des ´Equations de Navier-Stokes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In:  Esaim: Proceedings 21 (2007), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' 65–87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' “Global existence and the decay of solutions to the Prandtl system with  small analytic data”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In: Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' 241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1 (2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' 403–446.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' “Global hydrostatic approximation of the hyperbolic Navier-Stokes system  with small Gevrey class 2 data”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In: Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' China Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='6 (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' Zhang, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' “On the hydrostatic approximation of the Navier-Stokes equa-  tions in a thin strip”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In: Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' Saal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' “Hyperbolic Navier-Stokes equations I: Local well-posedness”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In: Evol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Control Theory 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1 (2012), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' Saal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' “Hyperbolic Navier-Stokes equations II: Global existence of small solutions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  In: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='1 (2012), 217–234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' “Ill-posedness of the hydrostatic Euler and Navier-Stokes equations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In: Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content='3 (2009), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Caflisch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' “Zero viscosity limit for analytic solutions, of the Navier-Stokes  equation on a half-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Existence for Euler and Prandtl equations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In: Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' Riegels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Berlin, Heidelberg: Springer Berlin Heidelberg, 1961,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' Wang, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' “Gevrey stability of hydrostatic approximate for the Navier-  Stokes equations in a thin domain”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In: Nonlinearity 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+page_content=' Wong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' “Blowup of solutions of the hydrostatic Euler equations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='3 (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content=' 1119–1125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
+page_content='  28' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9AyT4oBgHgl3EQfYfeT/content/2301.00205v1.pdf'}
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+Strategic Defense of Feedback-Controlled
+Parallel Queues against Reliability and
+Security Failures
+Qian Xie, Jiayi Wang and Li Jin ∗
+January 30, 2023
+Abstract
+Parallel traffic service systems such as transportation, manufacturing, and computer
+systems typically involve feedback control (e.g., dynamic routing) to ensure stability
+and to improve throughput. Such control relies on connected cyber components for
+computation and communication. These components are susceptible to random mal-
+functions and malicious attacks, which motivates the design of strategic defense that
+are both traffic-stabilizing and cost-efficient under reliability/security failures. In this
+paper, we consider a parallel queuing system with dynamic routing subject to such fail-
+ures. For the reliability setting, we consider an infinite-horizon Markov decision process
+where the system operator strategically activates the protection mechanism upon each
+job arrival based on the traffic state. We use Hamilton-Jacobi-Bellman equation to
+show that the optimal protection strategy is a deterministic threshold policy. For the
+security setting, we extend the model to an infinite-horizon stochastic game where the
+attacker strategically manipulates routing assignment. We show that a Markov per-
+fect equilibrium of this game always exists and that both players follow a threshold
+strategy at each equilibrium. For both settings, we also consider the stability of the
+traffic queues in the face of failures. Finally, we develop approximate dynamic program-
+ming algorithms to compute the optimal/equilibrium policies and present numerical
+examples for validation and illustration.
+Keywords: Queuing systems, cyber-physical security, stochastic games, Markov decision
+processes, HJB equation, Lyapunov function.
+∗This work was in part supported by US NSF Award CMMI-1949710, C2SMART University Transporta-
+tion Center, NYU Tandon School of Engineering, SJTU-UM Joint Institute, J. Wu & J. Sun Endowment
+Fund, and Cornell University McMullen Fellowship. Q. Xie is with the School of Operations Research and
+Information Engineering, Cornell University, USA. J. Wang is with the Department of Electrical and Com-
+puter Engineering, University of California, San Diego, USA. L. Jin is with the UM Joint Institute, and the
+Department of Automation, Shanghai Jiao Tong University, China. Q. Xie and L. Jin were with the Tandon
+School of Engineering, New York University, USA. J. Wang was with the UM Joint Institute, Shanghai Jiao
+Tong University, China. (emails: qx66@cornell.edu, jiw139@ucsd.edu, li.jin@sjtu.edu.cn).
+1
+arXiv:2301.11530v1  [eess.SY]  27 Jan 2023
+
+1
+Introduction
+Parallel queuing system is a typical model for characterizing a service system of multiple
+servers, each with a waiting queue. Real-world instances include packet switching networks
+[35, 21], manufacturing systems [24], transportation facilities [25], etc. Such systems use
+feedback from the state observation to generate routing decisions that ensure stability and
+improve throughput.
+Meanwhile, the feedback control relies heavily on connected cyber
+components for data collection and transmission. Such cyber components are subject to
+persistent threat due to random malfunctions and malicious attacks [8]. For instance, trans-
+portation infrastructure information (e.g., traffic sensors, traffic lights) and vehicle commu-
+nications can be intruded and manipulated [17, 38, 1]. Real-world incidents indicated that a
+simulated traffic jam can cause motorists to deviate from their planned routes [20]. Similar
+security risks also exist in industrial control [3] and communication systems [2, 31, 12]. Such
+real-world systems will not be accepted by the authorities, the industry, and the public,
+unless the security problems are well addressed. However, cyber security risks have not been
+sufficiently studied in conjunction with the physical queuing dynamics. Moreover, it is eco-
+nomically infeasible and technically unnecessary to perfectly avoid cyber failures. Therefore,
+it is crucial to understand the impact due to such threats and to design strategic defense
+mechanisms.
+In response to such concerns, we try to address the following two research questions:
+(i) How to model the security vulnerabilities and quantify the security risks for parallel
+queuing systems?
+(ii) How to design traffic-stabilizing, cost-efficient defense strategies against failures?
+For the first question, we consider two scenarios of failures, viz.
+reliability failures and
+security failures. We quantitatively characterize the security risks in terms of failure-induced
+queuing delays and defending costs. For the second question, we analyze the stability criteria
+of the failure-prone system with defense, and characterize the structure of the cost-efficient
+strategies.
+We also develop algorithms to compute such strategies, and discuss how to
+incorporate the stability condition. Our results are demonstrated via a series of numerical
+examples and simulations.
+This paper is related to two lines of work: queuing control and game theory. On the
+queuing side, the majority of the existing analysis and design are based on perfect observation
+of the states (i.e., queue lengths) and perfect implementation of the control [13, 23, 15, 21, 26].
+Besides, researchers have noted the impact of delayed [28, 32], erroneous [6], or decentralized
+information [36]. Although these results provide hints for our problem, they do not directly
+apply to the security setting with failures such as imperfect sensing (state observation) and
+imperfect control implementation. On the game side, a variety of game-theoretic models have
+been applied to studying cyber-physical security in transportation [43, 29], communication
+[7, 2, 31], and infrastructure systems [46, 31].
+However, to the best of our knowledge,
+security risks of feedback-controlled systems have not been well studied from a combined
+game-theoretic and feedback-control perspective, which is essential for capturing the coupling
+between the queuing dynamics and the attacker-defender interactions.
+Our model includes two parts: the physical model (parallel queuing system) and the
+cyber model (dynamic routing subject to failures). Specifically, we consider a system of
+2
+
+parallel servers and queues with possibly faulty dynamic routing.
+Dynamic routing is a
+classical feedback control strategy that assigns jobs to one of the parallel queues according
+to the current system state (queue length). We assume that the routing instructions can be
+faulty due to the following two threats:
+1. Reliability failures. Due to random malfunctions, a fault may occur to each arrival
+with a constant probability. For instance, the operator may lose the observation of the
+state or fail to deliver the routing instruction temporarily [19], and then the job joins
+a random queue; see Fig. 1a.
+2. Security failures. An adversary may strategically launch attacks on the arrivals for
+selfish or malicious intent. Typical attack modes include (i) denial-of-service that cuts
+off the operator’s observation of the state [45, 44, 1], (ii) falsification that injects falsi-
+fied data or creates fictitious traffic to mislead the operator [1, 38], and (iii) spoofing
+that manipulates the routing instruction (e.g., sending jobs to a busy server) by im-
+personating the operator [17, 38]. The consequence of an attack is that an incoming
+job joins a queue selected by the attacker; see Fig. 1b.
+The system operator can protect/defend1the routing instructions for incoming jobs based on
+the state observation.
+The activation of protection/defense mechanisms induces a cost rate
+on the operator. For the reliability setting, we formulate the operator’s trade-off between
+queuing costs and protecting costs as an infinite-horizon, continuous-time, and discrete-state
+Markov decision process. For the security setting, we formulate the interaction between the
+attacker and the operator as an infinite-horizon stochastic game.
+To study the stability of the queuing system, previous works typically relied on character-
+ization or approximation of the steady-state distribution of the queuing state [18]; however,
+this approach is hard to be synthesized with failure models. In addition, it is hard to study
+the steady-state distribution of queuing systems with state-dependent transition rates. To
+address such challenges, we use a Lyapunov function-based approach to study the stability
+(in this paper we consider mean boundedness [33]) of queuing systems under control and to
+obtain upper bounds for the mean number of jobs in the system [34]. Such an approach has
+been applied to queuing systems in no-failure scenarios [27, 10, 14, 47]. Importantly, we use
+this approach to derive stability criterion for protection under reliability failures and defense
+against the security failures.
+To analyze the cost efficiency of the system operator’s decision, we formulate the opti-
+mization problem in terms of queuing and protecting costs and then derive its Hamiltonian-
+Jacobian-Bellman (HJB) equation [48]. We show that the optimal protecting policy under
+reliability failures is a deterministic threshold policy: the operator either protects or does
+not protect, according to threshold functions in the multi-dimensional state space. Similar
+approaches have been discussed in [5, 22] for two queues, and we generalize the analysis to
+n queues and to failure-prone settings. For the attacker-defender game, we again use HJB
+equation to show the threshold properties of the Markov perfect equilibria.
+The above analysis leads to useful insights for designing strategic protection/defense
+that are both stabilizing and cost-efficient. A key finding is that the system operator has
+1In the rest of the paper, we use the word “protect” for the reliability setting and “defend” for the security
+setting.
+3
+
+𝜆
+𝑋1(𝑡)
+𝑋𝑛(𝑡)
+Server 1
+𝑎
+𝑝1
+𝑝2
+Server 𝑛
+𝜇
+𝜇
+𝛽
+𝑋1(𝑡) = min
+"
+𝑋𝑖(𝑡)
+1 − 𝑎(1 − 𝑏)
+𝑎(1 − 𝑏)
+𝑋2(𝑡)
+Server 2
+𝜇
+𝑝𝑛
+⋮
+System Operator
+Random Fault
+Success
+Failure
+(a) Scenario of reliability failures.
+𝜆
+𝑋1(𝑡)
+𝑋𝑛(𝑡)
+Server 1
+𝛼
+Server 𝑛
+𝜇
+𝜇
+𝛽
+1 − 𝑎(1 − 𝑏)
+𝑎(1 − 𝑏)
+𝑋2(𝑡)
+Server 2
+𝜇
+⋮
+𝑋1(𝑡) = min
+!
+𝑋𝑖(𝑡)
+𝑋𝑛(𝑡) = max
+!
+𝑋𝑖(𝑡)
+System Operator
+Malicious Attack
+Success
+Failure
+(b) Scenario of security failures.
+Figure 1: An n-queue system with shortest-queue routing under reliability/security
+failures. See Section 2 for definitions of notations.
+4
+
+OOa higher incentive to protect/defend if the queues are more “unbalanced”.
+In addition,
+our numerical analysis shows that 1) the incentive to protect increases with the failure
+probability, decreases with the technological cost, and increases with the demand-capacity
+ratio; 2) the optimal protecting policy performs better than static policies such as never
+protect and always protect. We also note that the optimal decision is not always stabilizing.
+Considering this, we propose how to compute the stability-constrained optimal policy by
+imposing the stability condition on the HJB equation.
+Our contributions lie in the following three aspects:
+• Modeling: 1) We build a framework for modeling the cyber-physical vulnerabilities of
+queuing systems with feedback control (dynamic routing) subject to reliability/security
+failures.
+2) We propose a formulation of protection under reliability failures as an
+infinite-horizon Markov decision process and defense against security failures as an
+attacker-defender stochastic game.
+• Analysis: 1) We provide stability criteria under failures and control based on Lya-
+punov functions. 2) We show the threshold properties of the optimal protection and
+the game equilibria on multidimensional state space based on HJB equations.
+• Design: Our theoretical results provide insights on the design of traffic-stabilizing
+and cost-efficient protecting policy and defending response. We also propose approx-
+imate dynamic programming algorithms to numerically compute the optimal policy
+and equilibrium strategies.
+The rest of this paper is organized as follows. Section 2 introduces the queuing model
+and the failure models. Section 3 studies protection against reliability failures and Section 4
+analyzes defense against security failures. Section 5 gives a concluding remark.
+2
+Parallel queuing system and failure models
+2.1
+Parallel queuing system
+Consider a queuing system with n identical servers in parallel. Jobs arrive according to a
+Poisson process of rate λ > 0. Each server serves jobs at an exponential rate of µ > 0. We use
+X(t) =
+�
+X1(t)
+X2(t)
+· · ·
+Xn(t)
+�T to denote the number of jobs at time t, either waiting
+or being served, in the n servers, respectively. The state space of the parallel queuing system
+is Zn
+≥0. Specifically, the initial system state (queue length) is X(0) = x = [x1 x2 · · · xn]T.
+We use x + (−)ei to denote adding (subtracting) 1 from xi. Since the queue lengths
+are always non-negative, i.e. xi ≥ 0, we use (x − ei)+ = max(x − ei, 0) to avoid the case
+that subtracting 1 makes the element negative. Let xmin = mini xi and xmax = maxi xi.
+We use x−i to denote variables in x other than xi. We call x a diagonal vector if x1 =
+x2 = · · · = xn and a non-diagonal vector otherwise. Denote the one-norm of the vector x as
+||x||1:= x1 + x2 + · · · + xn. Then ||X(t)||1 means the total number of jobs in the system at
+time t. We use x ≻ 0 to denote that x is not a zero vector, i.e., ||x||1> 0.
+5
+
+Without any failures, any incoming job is allocated to the shortest queue.
+If there
+are multiple shortest queues, then the job is randomly allocated to one of them with (not
+necessarily equal) probabilities.
+2.2
+Reliability failures
+Suppose that when a job arrives at the system, the fault occurs with a constant probability
+a ∈ (0, 1]. The system operator can deploy additional security resources to ensure correct
+routing.
+Let ∆({NP, P}) := {(1 − b, b) : b ∈ [0, 1]} denote the probability distribution
+over the action set {not protect, protect}. The system operator selects a Markovian policy
+β : Zn
+≥0 → ∆({NP, P}) that depends only on the current state but not on the history of
+states and actions. We write b(x) as the state-dependent protecting probability at state
+x ∈ Zn
+≥0. With a slight abuse of notation, when the policy is deterministic, we rewrite the
+mapping as β : Zn
+≥0 → {NP, P}.
+Protecting a job induces a one-time technological cost of cb > 0. If the fault occurs but
+the system operator does not protect, the routing instruction is faulty, and the job joins a
+random queue; the probability of joining the ith queue is pi ∈ [0, 1] where �n
+i=1 pi = 1, and
+we define
+pmax = max(p1, p2, · · · , pn).
+Otherwise, the routing instruction is correct, and the job joins the shortest queue, as shown
+in Fig. 1a.
+The objective of the system operator is to balance the queuing cost and the protecting
+cost. We formulate this problem as an infinite-horizon continuous-time Markov decision
+process. Specifically, the operator aims to minimize the expected cumulative discounted cost
+J(x; β):
+J∗(x) := min
+β
+J(x; β)
+= min
+β
+E
+� � ∞
+0
+e−ρtC(X(t); β)dt
+���X(0) = x
+�
+,
+(1)
+where ρ ∈ (0, 1) is the discounted factor and C : Zn
+≥0 → R is the cost rate defined as
+C(ξ; β) = ||ξ||1+cbb(ξ).
+The optimal protecting policy is essentially the policy that minimizes the expected cumula-
+tive discounted cost.
+Definition 1 (Optimal protecting policy) The optimal protecting policy β∗ against re-
+liability failures is defined as:
+β∗(x) := argmin
+β
+J(x; β),
+∀x ∈ Zn
+≥0.
+6
+
+2.3
+Security failures
+Suppose that when each job arrives, a malicious attacker is able to manipulate the routing
+such that the job is allocated to a non-shortest queue. For the sake of simplicity, we consider
+the attacker’s best action (and thus the operator’s worst case), i.e., the job goes to the longest
+queue, as shown in Fig. 1b. Let ∆({NA, A}) := {(1−a, a) : a ∈ [0, 1]} denote the probability
+distribution over the action set {not attack, attack}. The attacker selects a (possibly mixed)
+Markov strategy α : Zn
+≥0 → ∆({NA, A}). With a slight abuse of notation, we write a(x) as
+the state-dependent attacking probability. Note that here a(x) has a different meaning from
+the constant fault probability a in the reliability failure setting. Attacking a job induces a
+one-time technological cost of ca > 0.
+The system operator’s action is similar to that in the reliability setting. The only differ-
+ence is that in the security setting, the system operator knows there is a strategic attacker
+making decisions simultaneously. We formulate the interaction between the attacker and the
+operator (also called defender) as an infinite-horizon stochastic game with Markov strategies
+that do not depend on the history of states and actions. The attacker aims to maximize the
+expected cumulative discounted reward V (x; α, β) given the operator’s Markov strategy β:
+V ∗
+A(x; β) := max
+α
+V (x; α, β)
+= max
+α
+E
+� � ∞
+0
+e−ρtR(X(t); α, β)dt
+���X(0) = x
+�
+,
+where R : Zn
+≥0 → R is the immediate reward defined as
+R(ξ; α, β) = ||ξ||1+cbb(ξ) − caa(ξ).
+Here the attacker’s reward not only considers the queuing cost and the attacking cost, but
+also minus the defending cost. This is because the attacker has the incentive to increase
+the system’s total operating cost, viz. the sum of queuing and defending costs. Similarly,
+the operator aims to minimize the expected cumulative discounted loss given the attacker’s
+Markov strategy α:
+V ∗
+B(x; α) = min
+β
+V (x; α, β).
+We can also define the Markov perfect equilibrium of such an attacker-defender game:
+Definition 2 (Markov perfect equilibrium) The equilibrium attacking (resp.
+defend-
+ing) strategy α∗ (resp. β∗) satisfies that for each state x ∈ Zn
+≥0,
+α∗(x) = argmax
+α
+V (x; α, β∗) = argmax
+α
+V ∗
+A(x; β∗),
+β∗(x) = argmin
+β
+V (x; α∗, β) = argmin
+β
+V ∗
+B(x; α∗).
+The equilibrium value of the attacker (defender) is V ∗
+A(x; β∗) (resp. V ∗
+B(x; α∗)). In particular,
+(α∗, β∗) is a Markov perfect equilibrium.
+7
+
+3
+Protection against reliability failures
+In this section, we consider the design of the system operator’s state-dependent protecting
+policy from two aspects: stability and optimality.
+It is well known that a parallel n-queue system is stabilizable if and only if the demand
+is less than the total capacity, i.e., λ < nµ. In the following results, we will see that even
+this condition is met, in the absence of defense, reliability failures can still destabilize the
+queuing system, especially when the probability of failures is high and when the random
+faulty routing is highly heterogeneous; the following summarizes the above insights.
+Proposition 1 The unprotected n-queue system with faulty probability a is stable if and only
+if
+λ < nµ,
+(2a)
+apmaxλ < µ.
+(2b)
+Furthermore, when the system is stable, the long-time average number of jobs is upper-
+bounded by
+¯X := 1
+t lim sup
+t→∞
+t
+�
+τ=0
+E[X(τ)] ≤
+λ + nµ
+2
+�
+µ − max(apmax, 1
+n)λ
+�.
+The next result provides a stability criterion for an n-queue system with a given protecting
+policy. The proof of this result is presented in Section 3.1.
+Theorem 1 (Stability under reliability failures) Consider an n-queue system subject
+to reliability failures with probability a > 0.
+Suppose that the system operator selects a
+Markovian protecting policy β characterized by a state-dependent probability b(x) ∈ [0, 1] at
+each state x ∈ Zn
+≥0. Then we have the following:
+(i) The system is stable if for every non-diagonal vector x, the protecting probability b(x)
+satisfies
+b(x) > 1 −
+µ||x||1−λxmin
+aλ
+� n�
+i=1
+pixi − xmin
+�.
+(3)
+(ii) When (2a) holds, there must exist a policy satisfying (3). When (2b) holds, every
+policy satisfies (3).
+(iii) If (3) holds, the long-time average number of jobs in the system is upper-bounded by
+¯X ≤ λ + nµ
+2c
+,
+(4)
+where
+c = min
+x≻0
+�
+µ − λ xmin
+||x||1
+− a(1 − b(x))λ
+�n
+i=1 pixi − xmin
+||x||1
+�
+.
+8
+
+The next result characterizes the structure of the optimal protecting policy for the dynamic
+routing. The key findings are: 1) the optimal protecting policy is deterministic, i.e., the
+system operator either protects or does not protect; 2) the system operator chooses to
+protect when the queue lengths are more “unbalanced”.
+Theorem 2 (Optimal protecting policy) Consider a n-queue system subject to reliabil-
+ity failures. The optimal protecting policy β∗ has the following properties:
+(i) β∗ is deterministic in the sense that either β∗(x) = NP or β∗(x) = P for each x ∈ Zn
+≥0.
+(ii) β∗ is a threshold policy characterized by n threshold functions fm (m = 1, 2, · · · , n) via
+b∗(x) = 1
+�
+n�
+m=1
+(fm(x) > 0)
+�
+,
+where for each m = 1, 2, · · · , n,
+(a) fm : Zn
+≥0 → R separates the polyhedron Xm = {x ∈ Zn
+≥0 | xi ≥ xm, ∀1 ≤ i ≤ n}
+into two subsets: {x ∈ Xm | β∗(x) = NP} and {x ∈ Xm | β∗(x) = P} by means
+of
+b∗(x) = 1{fm(x) > 0},
+∀x ∈ Xm;
+(b) the optimal protecting probability b∗(x) is monotonically non-decreasing (resp.
+non-increasing) in xi (∀i ̸= m) (resp. xm) while other variables x−i (resp. x−m)
+are fixed.
+0
+5
+10
+15
+20
+25
+30
+length of queue 1 x1
+0
+5
+10
+15
+20
+25
+30
+length of queue 2 x2
+not protect
+protect
+1
+2
+Figure 2: The characterization of the optimal protecting policy β∗ for a two-queue system
+(p1 = 0.1, p2 = 0.9, ρ = 1, a = 0.9).
+Here the threshold functions qualitatively characterize the degree of “unbalancedness”.
+They separate the state space into n + 1 subsets of states: one containing “balanced” states
+such that the action “not protect” is optimal, and the other n containing “unbalanced” states
+such that the action “protect” is optimal. See Fig. 2. The concept “threshold function” has
+appeared in previous works [5, 22, 41]. The rest of this section is devoted to the proofs and
+discussions of Theorem 1-2.
+9
+
+3.1
+Stability under reliability failures
+In this subsection, we provide a proof of the stability condition under the protected case
+(Theorem 1) and leave the proof of the stability condition under the unprotected case (Propo-
+sition 1) to Appendix A.1. Both proofs are based on Foster-Lyapunov drift condition [34].
+Proof of Theorem 1. (i) and (iii) Consider the quadratic Lyapunov function
+W(x) = 1
+2
+n
+�
+i=1
+x2
+i .
+(5)
+By applying infinitesimal generator [11, 4] under protecting policy β (denoted as Lβ) we
+have
+LβW(x) =a(1 − b(x))λ
+2
+n
+�
+i=1
+pi
+�
+(xi + 1)2 − x2
+i
+�
++ (1 − a(1 − b(x))) λ
+2
+�
+(xmin + 1)2 − x2
+min
+�
++ µ
+2
+n
+�
+i=1
+1xi>0
+�
+(xi − 1)2 − x2
+i
+�
+= a(1 − b(x))λ
+n
+�
+i=1
+pixi + (1 − a(1 − b(x))) λxmin
+− µ
+n
+�
+i=1
+xi + 1
+2λ + 1
+2
+n
+�
+i=1
+1{xi > 0}µ
+≤ a(1 − b(x))λ
+� n
+�
+i=1
+pixi − xmin
+�
++ (λxmin
+− µ||x||1) + 1
+2(λ + nµ).
+By (3) there exists constants c = min
+x≻0{µ − λ xmin
+||x||1 − a(1 − b(x))λ
+�n
+i=1 pixi−xmin
+||x||1}
+> 0 and d =
+1
+2(λ + nµ) such that
+LβW(x) ≤ −c||x||1+d,
+∀x ∈ Zn
+≥0.
+(6)
+By [34, Theorem 4.3], this drift condition implies the upper bound (4) and thus the stability.
+□
+(ii) When λ < nµ, for every non-diagonal vector x, we have µ||x||1−λxmin > µ||x||1−λ ||x||1
+n
+=
+�
+µ − λ
+n
+�
+||x||1> 0 and �n
+i=1 pixi − xmin > 0, then
+1 −
+µ||x||1−λxmin
+aλ
+� n�
+i=1
+pixi − xmin
+� < 1.
+10
+
+Thus, b(x) ≡ 1 satisfies the stability condition (3) and β(x) ≡ P is a stabilizing policy that
+exists.
+When max(apmax, 1/n)λ < µ, for every non-diagonal vector x, we have aλ �n
+i=1 pixi +
+(1 − a)λxmin ≤ max(apmax, 1/n)λ||x||1< µ||x||1, and then
+1 −
+µ||x||1−λxmin
+aλ
+� n�
+i=1
+pixi − xmin
+� < 0.
+Thus, every policy satisfies the stability criterion (3).
+□
+Theorem 1 provides a stability criterion for any state-dependent protecting probability.
+This implies that the operator needs to protect, i.e., choose some positive protecting proba-
+bility to stabilize the system at certain states (queue lengths). We will use such stabilizing
+threshold probabilities to obtain a stability-constrained optimal policy. See Section 3.3 and
+Appendix A.5.
+3.2
+Optimal protecting policy
+A standard way to solve the discounted infinite-horizon minimization problem (1) is to write
+down its HJB equation for optimality [9, Chapter 4]:
+0 = min
+β {||x||1+cbb(x) − ρJ∗(x) + LβJ∗(x)}.
+(7)
+We can rewrite it as the following recurrence form:
+(ρ + λ + nµ)J∗(x) = min
+β
+�
+||x||1+cbb(x)+
+µ
+n
+�
+i=1
+J∗((x − ei)+) + λ min
+1≤j≤n J∗(x + ej) + (1 − b(x))aλ
+�
+n
+�
+i=1
+piJ∗(x + ei) − min
+1≤j≤n J∗(x + ej)
+��
+.
+(8)
+The optimal protecting policy β∗ is essentially the solution of (7) and (8). A direct observa-
+tion is that when there is no failure, i.e., a = 0, the system operator never needs to protect
+(i.e., ∀x, β∗(x) = NP); when all queue lengths are equal, i.e., x1 = x2 = · · · = xn, the opera-
+tor deterministically chooses not to protect at this state x (i.e., β∗(x) = NP). Furthermore,
+Theorem 2(i) indicates that β∗ is deterministic (not probabilistic) at each state, i.e., either
+protects or does not protect.
+Proof of Theorem 2(i). The expression to be minimized in the right-hand side of the HJB
+equation (8) is linear in b(x), so the minimum is reached at the endpoints, that is, b(x) = 0
+or b(x) = 1.
+□
+11
+
+Now the HJB equation (8) turns into
+(ρ + λ + nµ)J∗(x) = min
+b∈{0,1}
+�
+||x||1+cbb
++µ
+n
+�
+i=1
+J∗((x − ei)+) + λ min
+1≤j≤n J∗(x + ej)
++(1 − b)aλ
+�
+n
+�
+i=1
+piJ∗(x + ei) − min
+1≤j≤n J∗(x + ej)
+��
+:= min
+b∈{0,1}
+�
+c(x, b) +
+�
+x′
+q(x′|x, b)J∗(x′)
+�
+,
+where q(x′|x, b) are the transition rates from state x to state x′ given protecting probability
+b. For example, q((x − ei)+|x, b) = µ for b = 0, 1.
+Using the uniformization trick [30, 37], we have
+J∗(x) := min
+b
+�
+˜c(x, b) + γ
+�
+x′
+p(x′|x, b)J∗(x′)
+�
+,
+(9)
+where Λ = λ + nµ, γ = Λ/(ρ + Λ), ˜c(x, b) = c(x, b)/(ρ + Λ) and p(x′|x, b) = q(x′|x, b)/Λ.
+Without loss of generality, we assume ρ + Λ = 1 in the following.
+Before proving Theorem 2, we first present properties of the optimal cost function J∗
+since they are useful for showing the threshold property of optimal policy β∗.
+Proposition 2 The optimal cost function J∗ : Zn
+≥0 → R has the following properties:
+(i) (symmetry) J∗ is symmetric, i.e., J∗(x) = J∗(σx) where σx is a permutation of x.
+(ii) (Schur convexity) J∗ is Schur convex, i.e., J∗(x + ei) ≥ J∗(x + ej) if xi ≥ xj.
+(iii) (monotonicity) J∗ is non-decreasing, i.e., J∗(x) ≥ J∗(y) if xi ≥ yi for all i (1 ≤ i ≤ n).
+(iv) (convexity) J∗ is convex in each variable, i.e., J∗(x+ei)−J∗(x) ≤ J∗(x+2ei)−J∗(x+
+ei).
+(v) (supermodularity) J∗ is supermodular, i.e., J∗(x+ei+ej)+J∗(x) ≥ J∗(x+ei)+J∗(x+
+ej).
+The proof of Proposition 2 is left to Appendix A.3 and we will particularly use property
+(i)-(ii), i.e., symmetry and Schur convexity to prove Theorem 2.
+Proof of Theorem 2(ii). Let m = argmini xi. Similar to the idea in [5, Chapter 4.6], we
+can demonstrate (a) the existence of the threshold functions by showing (b) the monotonicity
+of the optimal protecting probability:
+b∗(x + ei) ≥ b∗(x),
+∀i ̸= m
+b∗(x + em) ≤ b∗(x).
+(10)
+12
+
+Now we prove (10). Because of Schur convexity, J∗(x + ei) ≥ J∗(x + em) (∀i ̸= m). We can
+rewrite (9) as
+J∗(x) = min
+b∈{0,1}
+�
+||x||1+cbb + µ
+n
+�
+i=1
+J∗((x − ei)+) + λ
+J∗(x + em) + (1 − b)aλ
+�
+n
+�
+i=1
+piJ∗(x + ei) − J∗(x + em)
+��
+.
+Let ∆∗(x) =
+n�
+i=1
+piJ∗(x + ei) − J∗(x + em). Note that by Definition 1 and Theorem 2(i),
+b∗(x) = 1 if ∆∗(x) > cb and b∗(x) = 0 if ∆∗(x) < cb.
+Then the monotonicity of b∗ is
+essentially the monotonicity of ∆∗. Thus, (10) is equivalent to
+∆∗(x + ei) ≥ ∆∗(x),
+∀i ̸= m
+∆∗(x + em) ≤ ∆∗(x).
+(11)
+We defer the proof of (11) to Appendix A.2. The high-level idea is to use induction based
+on value iteration.
+□
+To obtain an estimated optimal policy, we propose an algorithm called truncated policy
+iteration (TPI). See Algorithm 1 in Appendix A.5. It is adapted from the classic policy
+iteration algorithm [42] and based on the following value iteration form of the HJB equation
+(9):
+Jk+1(x) = min
+b
+�
+˜c(x, b) + γ
+�
+x′
+p(x′|x, b)Jk(x′)
+�
+.
+(12)
+Now we use the estimated optimal policy to conduct numerical analysis on 1) the relationship
+between the incentive to protect and the system parameters; 2) the comparison between the
+optimal policy and two naive static policies: always protect and never protects.
+We first analyze the tipping points when the system operator starts to protect “riskier”
+states under the optimal policy β∗, i.e., ∃x s.t. β∗(x) = P, as the failure probability a and
+technological cost cb change. It can be seen from Fig. 3 that the incentive to protect is
+non-decreasing in the failure probability a, non-increasing in the technological cost cb and
+non-decreasing in the demand-capacity ratio (a.k.a. utilization ratio) ρ = λ/µ. That is, the
+system operator has higher incentive to protect when 1) the failure probability is higher; 2)
+the technological cost is lower; 3) the utilization ratio is higher.
+In the following simulation, we will see that the optimal policy β∗ can significantly reduce
+the security risk, compared to the static policies: β(x) ≡ P (always protects) and β(x) ≡ NP
+(never protects). See Fig. 4 where the yellow curves are below the red curves and the green
+curves. The Monte Carlo simulation result is based on the cumulative discounted cost within
+50000s. Here the cumulative discounted cost is calculated as the sum of the total queuing
+cost and total technological cost in the episode, and we normalized it to be a value between
+0 and 1. Note that under the static policy β(x) ≡ P (i.e., ∀x, b(x) = 1), the job always joins
+the shortest queue regardless of the failure probability, so the cumulative discounted cost is
+a constant (red curve).
+13
+
+0.00
+0.25
+0.50
+0.75
+1.00
+failure probability a
+0
+1
+2
+3
+4
+tech cost cb
+x, β*(x) = NP
+x, β*(x) = P
+(a) ρ = 1.6
+0.00
+0.25
+0.50
+0.75
+1.00
+failure probability a
+0
+1
+2
+3
+4
+tech cost cb
+x, β*(x) = NP
+x, β*(x) = P
+(b) ρ = 0.4
+Figure 3: The tipping points when the system operator starts to protect “riskier” states
+under the optimal policy as the failure probability and the technological cost change.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+failure probability a
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+normalized 
+ cumulative cost
+β(x) ≡ NP
+β ∗
+β(x) ≡ P
+(a) cb = 0.05
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+failure probability a
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+normalized 
+ cumulative cost
+β(x) ≡ NP
+β ∗
+β(x) ≡ P
+(b) cb = 0.5
+Figure 4: Comparison of the normalized cumulative discounted costs between the optimal
+policy and the static policies (p1 = 0.1, p2 = 0.9, ρ = 1.6).
+14
+
+3.3
+Stability-constrained optimal policy
+The optimal policy may not always be stabilizing. For example, the optimal policy under
+the system parameters p1 = 0.1, p2 = 0.9, ρ = 1, a = 0.9 does not satisfy the stability
+condition (3). In this case, we can choose the optimal policy from the subset of policies
+satisfying the stability criterion, called stability-constrained optimal policy. The operator
+needs to randomize over {P, NP} at some states, see Fig. 5. We denote such policy as ˆβ∗
+with state-dependent protecting probability ˆb∗(x). Specifically, we can consider the stability-
+constrained MDP by imposing (3) on the HJB equation. Appendix A.5 gives a corresponding
+modification of the TPI algorithm.
+0
+5
+10
+15
+20
+25
+30
+length of queue 1 x1
+0
+5
+10
+15
+20
+25
+30
+length of queue 2 x2
+not protect
+protect
+mixed
+1
+2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+protecting probability ˆb*
+Figure 5: The characterization of the stability-constrained optimal policy ˆβ∗ for a
+two-queue system (p1 = 0.1, p2 = 0.9, ρ = 1, a = 0.9).
+Besides, we can have a quick check of the existence of a stabilizing policy and stabiliz-
+ability of the optimal policy using stability conditions (2a)-(2b) as follows:
+• When (2a)-(2b) hold, i.e., max(apmax, 1/n)λ < µ, the optimal protecting policy is also
+stabilizing.
+• When only (2a) holds, i.e., λ/n < µ ≤ apmaxλ, the optimal protecting policy may not
+be stabilizing.
+• When (2a) does not hold, i.e., λ ≥ µn, no stabilizing protecting policy exists.
+4
+Defense against security failures
+In this section, we analyze the attacker’s attacking strategy and system operator’s defending
+strategy from two aspects: stability and game equilibrium.
+The following criterion can be used for checking the stability of the n-queue system under
+any state-dependent attacking and defending strategies:
+15
+
+Theorem 3 (Stability under security failures) Consider an n-queue system subject to
+security failures. Suppose that at each state x ∈ Zn
+≥0, the attacker (resp. system operator)
+attacks (resp. defends) each job following Markov strategy α (resp. β) characterized by a
+state-dependent probability a(x) ∈ [0, 1] (resp. b(x) ∈ [0, 1]). Then we have the following:
+(i) The system is stable if for every non-diagonal vector x, the attacking and defending
+probabilities satisfy
+a(x) (1 − b(x)) < µ||x||1−λxmin
+λ(xmax − xmin).
+(13)
+(ii) When λ < nµ, there must exist a strategy β with defending probability b(x) satisfying
+(13).
+(iii) Furthermore, if (13) holds, then the long-time average number of jobs is upper-bounded
+by
+¯X ≤ λ + nµ
+2c
+,
+(14)
+where
+c = min
+x≻0
+�
+µ − λ xmin
+||x||1
+− a(x)(1 − b(x))λxmax − xmin
+||x||1
+�
+.
+The next result characterizes the structure of the strategy of the Markov perfect equilibria
+of the stochastic game: the equilibrium defending probability is higher when the queue
+lengths are more “unbalanced”.
+Theorem 4 (Markov perfect equilibrium) For each Markov perfect equilibrium (α∗, β∗)
+of the attacker-defender game, we have the following:
+(i) (α∗, β∗) is qualitatively different over the following three subsets of the state space Zn
+≥0:
+(a) S1 = {x ∈ Zn
+≥0 | (α∗(x), β∗(x)) = (NA, NP)}; (“low risk”)
+(b) S2 = {x ∈ Zn
+≥0 | (α∗(x), β∗(x)) = (A, NP)}; (“medium risk”)
+(c) S3 = {x ∈ Zn
+≥0 | (α∗(x), β∗(x)) is mixed}. (“high risk”)
+(ii) The boundaries between S1 and S2, as well as those between S2 and S3 are characterized
+by threshold functions gij, hij (1 ≤ i ̸= j ≤ n) as follows:
+S1 =
+�
+x ∈ Zn
+≥0 |
+�
+1≤i̸=j≤n
+(gij(x) < 0)
+�
+,
+S2 =
+�
+x ∈ Zn
+≥0 |
+�
+1≤i̸=j≤n
+(gij(x) > 0 ∨ hij(x) < 0)
+�
+,
+S3 =
+�
+x ∈ Zn
+≥0 |
+�
+1≤i̸=j≤n
+(hij(x) > 0)
+�
+.
+where for each i, j = 1, 2, · · · , n (i ̸= j),
+16
+
+(a) gij, hij : Zn
+≥0 → R separate the polyhedron Xij = {x ∈ Zn
+≥0 | xi = xmax, xj = xmin}
+into three subsets: S1 ∩ Xij, S2 ∩ Xij and S3 ∩ Xij;
+(b) state x has a lower (resp. higher) or equal security level than state x + ei (resp.
+x + ej).
+Here the threshold functions also characterize the degree of “unbalancedness”.
+Intu-
+itively, S1–S3 correspond to various security risk levels, and thus correspond to the incentive
+of the operator to defend: when the queues are more “unbalanced” and the risk is higher,
+the operator has a higher incentive to defend. Fig. 6 visualizes the equilibria for a two-queue
+system, which help understand Theorem 4. For a detailed argument about the relation-
+ship between the security levels and the system parameters (e.g., technological costs and
+utilization ratio), see Section 4.2.
+0
+5
+10
+15
+20
+length of queue 1
+0
+5
+10
+15
+20
+length of queue 2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+attacking probability a*
+0
+5
+10
+15
+20
+length of queue 1
+0
+5
+10
+15
+20
+length of queue 2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+defending probability b*
+Figure 6: The equilibrium attacking and defending strategies for a two-queue system
+(ρ = 1, ca = 0.1, cb = 0.2).
+We also find that the security game has four equilibrium regimes under different combi-
+nations of attacking cost ca and defending cost cb; see Fig. 7. Each regime is labeled with
+corresponding subsets of Markov perfect equilibria and security levels.
+The rest of this section is devoted to the proofs of Theorem 3-4, as well as an additional
+discussion on Markov perfect equilibrium.
+17
+
+Figure 7: Equilibrium regimes of the security game (ρ = 1.6).
+4.1
+Stability under security failures
+Proof of Theorem 3. By applying infinitesimal generator Lα,β under the attacking strategy
+α and the defending strategy β to the same Lyapunov function (5), we have
+Lα,βW(x) =a(x)(1 − b(x))λxmax + (1 − a(x)(1 − b(x)))
+λxmin −
+n
+�
+i=1
+µxi + 1
+2λ + 1
+2
+n
+�
+i=1
+1{xi > 0}µ
+≤a(x)(1 − b(x))λ(xmax − xmin)
++ λxmin − µ||x||1+1
+2(λ + nµ).
+Hence, by (13) there exists c = min
+x≻0{µ − λ xmin
+||x||1 − a(x)(1 − b(x))λ xmax−xmin
+||x||1
+)} > 0 and d =
+1
+2(λ + nµ) such that
+Lα,βW(x) ≤ −c||x||1+d,
+∀x ∈ Zn
+≥0.
+By [34, Theorem 4.3], this drift condition implies the upper bound (14) and thus the stability.
+Besides, when λ < nµ, for every non-diagonal vector x, we have µ||x||1≥ nµxmin > λxmin,
+and xmax > xmin, implying
+µ||x||1−λxmin
+λ(xmax−xmin) > 0. Thus, no matter what strategy the attacker
+chooses, the defending strategy with b(x) ≡ 1 satisfies the stability condition (13).
+□
+4.2
+Markov perfect equilibrium
+For the stochastic attacker-defender game, we first show the existence of Markov perfect
+equilibrium.
+18
+
+20
+20
+2
+queue
+15
+15
+10
+low risk
+3.0
+10
+low risk
+length
+5
+2.5
+5
+medium risk
+0
+cost
+2.0
+0
+0
+510
+15
+20
+0
+5101520
+length of queue 1
+defending
+1.5
+V
+lengthofqueue1
+20
+20
+2
+1.0
+2
+queue
+15
+15
+0.5
+low risk
+low risk
+III
+10
+of
+10
+0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+length 
+5
+attacking cost Ca
+5
+medium risk
+high risk
+0
+05101520
+0
+5101520
+lengthofgueue1
+lengthofgueue1Proposition 3 Markov perfect equilibrium (α∗, β∗) of the stochastic attacker-defender game
+always exists.
+Proof. Note that the state space Zn
+≥0 is countable and the action space {0, 1} is finite (and
+thus compact). By [16, Theorem 1], Markov perfect equilibrium (also called discounted equi-
+librium point of policies) exists.
+□
+Next, we discuss the derivation of Markov perfect equilibria. According to Shapley’s
+extension on minimax theorem for stochastic game [39], the attacker and the defender have
+the same equilibrium (minimax) value:
+V ∗
+B(x; α∗) = V ∗
+A(x; β∗) = V ∗(x).
+Thus, we only need to compute the minimax value V ∗ of the stochastic game. Similar to the
+derivation of (9), we obtain the following HJB equation of the minimax problem (assuming
+ρ + λ + nµ = 1):
+V ∗(x) = max
+α
+min
+β
+�
+||x||1+cbb(x) − caa(x)+
+µ
+�
+i
+V ∗((x − ei)+) + λ min
+j
+V ∗(x + ej) + a(x)(1 − b(x))
+λ
+�
+max
+j
+V ∗(x + ej) − min
+j
+V ∗(x + ej)
+��
+.
+(15)
+For each state x ∈ Zn
+≥0, let δ∗(x) = λ(maxj V (x+ej)−minj V (x+ej)) and build an auxiliary
+matrix game
+M(x, V ∗) =
+�
+||x||1+µ
+n
+�
+i=1
+V ∗((x − ei)+) + λ min
+j
+V ∗(x + ej)
+�
+�1
+1
+1
+1
+�
++
+�
+0
+cb
+−ca + δ∗(x)
+−ca + cb
+�
+.
+(16)
+Then given δ∗(x), the equilibrium strategies (α∗(x), β∗(x)) can be obtained by Shapley-Snow
+method [40], a convenient algorithm for finding the minimax value and equilibrium strategies
+of any two-player zero-sum game.
+Proof of Theorem 4(i).
+Consider the matrix game M(x, V ∗) defined as (16) where the
+attacker and the system operator are the row player and the column player.
+Based on
+Shapley-Snow method, the equilibrium strategies (α∗(x), β∗(x)) are in the following three
+cases depending on the relationship between δ∗(x) and the technological costs ca, cb > 0:
+(a) When δ∗(x) ≤ ca, it is obvious that α∗(x) = NA (i.e., a∗(x) = 0) is a dominant
+strategy. Then cb > 0 implies β∗(x) = NP (i.e., b∗(x) = 0). That is, the attacker has
+no incentive to attack, and thus the defender does not need to defend. At this pure
+strategy equilibrium, the security risk is low.
+19
+
+(b) When the defending cost cb is higher then the attacking cost ca, and ca < δ∗(x) ≤ cb, it
+is obvious that β∗(x) = NP (i.e., b∗(x) = 0) is a dominant strategy. Then cb > −ca +cb
+implies α∗(x) = A (i.e., a∗(x) = 1). That is, the defender has no incentive to defend
+and consequently the attacker prefers to attack. At this pure strategy equilibrium, the
+security risk is higher than the first case but tolerable.
+(c) When δ∗(x) > max{ca, cb} > 0, no saddle point exists. Then both the attacker and the
+system operator consider mixed strategies such that a∗(x) =
+cb
+δ∗(x), b∗(x) = 1 −
+ca
+δ∗(x).
+Particularly, the operator needs to select positive protecting probability, and now the
+security risk is high.
+The above three cases correspond to the three subsets of states. Note that the subset S2 is
+empty when ca > cb.
+□
+From the above proof, we observe that for fixed technological costs ca and cb, the security
+risk level is only higher when δ∗ is larger. Then as in the proof of Theorem 2, we use the
+fact that the monotonicity of the security risk level is equivalent to the monotonicity of δ∗
+to show the threshold property of the equilibrium. Besides, since δ∗ is also related to the
+equilibrium value function V ∗, we also need the following properties of V ∗.
+Proposition 4 The equilibrium value function V ∗ : Zn
+≥0 → R has the following properties:
+(i) V ∗ is symmetric, i.e., V ∗(x) = V ∗(σx) where σx is a permutation of x.
+(ii) V ∗ is Schur convex, i.e., V ∗(x + ei) ≥ V ∗(x + ej) if xi ≥ xj.
+(iii) V ∗ is non-decreasing, i.e., V ∗(x) ≥ V ∗(y) if xi ≥ yi for all i (1 ≤ i ≤ n).
+(iv) V ∗ is convex in each variable, i.e., V ∗(x + ei) − V ∗(x) ≤ V ∗(x + 2ei) − V ∗(x + ei).
+(v) V ∗ is supermodular, i.e., V ∗(x + ei + ej) + V ∗(x) ≥ V ∗(x + ei) + V ∗(x + ej).
+Since the proof of Proposition 4 is analogous to the proof of Proposition 2, we omit it here.
+Now we are ready to present the proof of Theorem 4(ii) which uses symmetry (property (i))
+and Schur Convexity (property (ii)).
+Proof of Theorem 4(ii). By symmetry, without loss of generality, we only need to consider
+the case when x1 = xmax, xn = xmin. Besides, Schur convexity implies that V (x + e1) =
+maxj V (x + ej), V (x + en) = minj V (x + ej). We can rewrite the HJB equation (15) as
+V ∗(x) = max
+α
+min
+β
+�
+||x||1+cbb(x) − caa(x) + λV ∗(x + en)
++µ
+�
+i
+V ∗((x − ei)+) + a(x)(1 − b(x))δ∗(x)
+�
+,
+(17)
+where δ∗(x) = λ(V ∗(x + e1) − V ∗(x + en)).
+Since the monotonicity of the security risk
+level of the states is equivalent to the monotonicity of δ∗, and implies the existence of the
+threshold functions, it is sufficient to show that δ∗ is monotonically non-decreasing (resp.
+20
+
+non-increasing) in the largest variable x1 (resp. the smallest variable xn) when other variables
+are fixed; that is,
+δ∗(x + e1) ≥ δ∗(x),
+δ∗(x + en) ≤ δ∗(x).
+(18)
+The proof of (18) also uses induction based on value iteration and can be found in Ap-
+pendix A.4.
+□
+Last, we discuss the numerical computation of the minimax value V ∗(x) and the equi-
+librium strategies (α∗(x), β∗(x)) for each state x. Based on the value iteration form of HJB
+equation (15), we develop an algorithm adapted from Shapley’s algorithm [39, 2, 44]. See
+Algorithm 2 in Appendix 2.
+The algorithm proceeds as follows. In each iteration and for each state x, let δ(x) =
+λ(maxj V (x + ej) − minj V (x + ej)) and build an auxiliary matrix game M(x, V ) similar to
+(16); then update V (x) with the minimax value val(M) given by Shapley-Snow method:
+• when δ(x) ≤ ca, val(M) = ||x||1+µ
+n�
+i=1
+V ((x − ei)+) + λ minj V (x + ej);
+• when ca < δ(x) ≤ cb, val(M) = ||x||1−ca + µ
+n�
+i=1
+V ((x − ei)+) + λ maxj V (x + ej);
+• when δ(x) > max{ca, cb}, val(M) = ||x||1+cb + µ
+n�
+i=1
+V ((x − ei)+) + λ minj V (x + ej) −
+cacb
+δ(x).
+When V (x) converges to V ∗(x), we again use Shapley-Snow method to solve the matrix
+game M(x, V ∗) and obtain the estimation of the equilibrium (α∗(x), β∗(x)).
+4.3
+Equilibrium regimes
+In the last subsection, we have seen that no medium risk states when ca > cb. Then in this
+subsection, we will discuss the existence of different security levels under different combina-
+tions of ca and cb.
+In Fig. 7, various regimes correspond to particular combinations of security levels. Under
+large attacking cost, the attacker has no incentive to attack, then only the low risk states exist
+(see regime IV). When the attacking cost goes smaller but still greater than the defending
+cost (ca > cb), not only the low risk states but also the high risk states exist (see regime III)
+since the attacker has less incentive to attack. As the defending cost increases to be greater
+than the attacking cost (cb > ca), the defender has less incentive to defend, and now all risk
+levels including the medium risk exist (see regime II).
+5
+Concluding Remarks
+In this work, we analyze the reliability and security risks of feedback-controlled parallel queu-
+ing systems and propose advice for strategic protection and defense. Our stability analysis
+21
+
+shows that without secure feedback control, the queuing system is vulnerable to random
+faults and malicious attacks, and our stability conditions can be used to check whether
+the feedback-controlled queuing system is stable under such failures. We also propose cost-
+efficient strategic defense advise for the system operator based on queue length, utilization
+ratio, and technical costs.
+Our theoretical results also provide practical insights for the
+design of fault-tolerant feedback control such as vehicle navigation, signal-free intersection
+control, flight dispatch, and data packet routing. Interesting future directions include 1) the
+extension to general queuing networks, and 2) the design of efficient (in both time and space)
+algorithms for estimating the optimal policy and equilibrium strategies.
+Acknowledgement
+The authors appreciate the discussions with Manxi Wu, Siddhartha Banerjee, Zhengyuan
+Zhou, Yu Tang, Haoran Su, Xi Xiong, and Nairen Cao. Undergraduate student Dorothy Ng
+also contributed to this project.
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+A
+Appendices
+A.1
+Proof of Proposition 1
+Here we provide the proof of the stability condition under the unprotected space. The proof
+uses the standard results on the subdivision of a Poisson process and stability condition of
+the join the shortest queue system.
+25
+
+Proof of proposition 1. The unprotected n-queue system has n + 1 classes of jobs. The
+i-th class enters server i as a Poisson process of rate apiλ (1 ≤ i ≤ n). The (n + 1)-th
+class enters the n-queue system as a Poisson process of rate (1 − a)λ; when a job of this
+class arrives, the job joins the shorter queue. By [18, Theorem 1], the (n + 1)-class, n-queue
+system is stable if and only if
+max (apiλ/µ, λ/(nµ)) < 1,
+which is equivalent to (2a)-(2b).
+Consider the same quadratic Lyapunov function (5) and apply infinitesimal generator,
+we have
+LW(x) =aλ1
+2
+n
+�
+i=1
+pi
+�
+(xi + 1)2 − x2
+i
+�
++ (1 − a)λ1
+2
+�
+(xmin + 1)2 − x2
+min
+�
++ µ1
+2
+n
+�
+i=1
+Ixi>0
+�
+(xi − 1)2 − x2
+i
+�
+=aλ
+n
+�
+i=1
+pixi + (1 − a)λxmin − µ
+n
+�
+i=1
+xi + 1
+2λ
++ 1
+2
+n
+�
+i=1
+Ixi>0µ
+≤ (max(apmax, 1/n)λ − µ) ||x||1+1
+2(λ + nµ).
+Hence, by (2a)–(2b) there exists a constant c = µ−max(apmax, 1/n)λ > 0 and d = 1
+2(λ+nµ)
+such that
+LW(x) ≤ −c||x||1+d,
+∀x ∈ Zn
+≥0.
+By [34, Theorem 4.3], this drift condition implies the upper bound and thus the stability. □
+A.2
+Induction part of Theorem 2
+In this subsection, we continue the proof of Theorem 2 with the induction proving (11).
+Let ∆k(x) =
+n�
+i=1
+piJk(x + ei) − Jk(x + em), it is sufficient to show for all k ∈ N,
+∆k(x + ei) ≥ ∆k(x),
+∀i ̸= m
+∆k(x + em) ≤ ∆k(x).
+One can verify that the above hold for k = 0, 1, 2. Here we consider multiple base cases to
+avoid reaching trivial conclusions, say all inequalities are just equalities.
+26
+
+Now we show the inductive step. According to the value iteration (12), we have ∀j ̸= m,
+∆k+1(x + ej) − ∆k+1(x)
+=µ
+n
+�
+i=1
+[∆k((x + ej − ei)+) − ∆k((x − ei)+)]
++ λ[∆k(x + ej + em) − ∆k(x + em)]
++ f k(x + ej) − f k(x),
+∆k+1(x + em) − ∆k+1(x)
+=µ
+n
+�
+i=1
+[∆k((x + em − ei)+) − ∆k((x − ei)+)]
++ λ[∆k(x + 2em) − ∆k(x + em)]
++ f k(x + em) − f k(x),
+where f k(x) = �n
+i=1 pi min
+�
+cb, aλ∆k(x + ei)
+�
+− min
+�
+cb, aλ∆k(x + em)
+�
+.
+Note that based on the induction hypothesis, we have ∀j ̸= m,
+∆k((x + ej − ei)+) ≥ ∆k((x − ei)+) ≥ ∆k((x + em − ei)+),
+∆k(x + ej + ei) ≥ ∆k(x + ei) ≥ ∆k(x + em + ei),
+∆k(x + ej + em) ≥ ∆k(x + em) ≥ ∆k(x + 2em).
+Then we can conclude that
+∆k+1(x + ej) ≥ ∆k+1(x)
+(∀j ̸= m)
+∆k+1(x) ≤ ∆k+1(x + em).
+A.3
+Proof of Proposition 2
+We present the proofs of property (i) and (ii) below, and omit the proofs of (iii)-(v), as we
+only need symmetry and Schur covexity for proving Theorem 2 while the proofs of other
+properties are tedious and analogous.
+Proof of Proposition 2. (i) Note that for any x ∈ Zn
+≥0,
+• ||σx||1= ||x||1,
+• {σ((x − e1)+), · · · , σ((x − en)+)} is a permutation of {(x − e1)+, · · · , (x − en)+},
+27
+
+• {σ(x + e1), · · · , σ(x + en)} is a permutation of {x + e1, · · · , x + en},
+then by (9) we can conclude that J∗(x) = J∗(σx).
+(ii) We will use induction to prove xi ≥ xj implies Jk(x+ei) ≥ Jk(x+ej) for any x ∈ Zn
+≥0
+and k ∈ N.
+For the base cases, it is easy to verify that J0 = 0, J1(x) = ||x||1 and J2(x) = (1 +
+Λ)||x||1+λ − µ �
+i Ixi>0. Then we have xi ≥ xj ⇒ J2(x + ei) ≥ J2(x + ej) for any x. Note
+that the inequality is strict for some x, say (1, 0, · · · , 0). Now we show the inductive step.
+According to the value iteration (12), we have
+Jk+1(x + ei) − Jk+1(x + ej)
+=µ
+n
+�
+l=1
+[Jk((x + ei − el)+) − Jk((x + ej − el)+)]
++λ
+�
+min
+l
+Jk(x + ei + el) − min
+l
+Jk(x + ej + el)
+�
++ min
+�
+cb, aλ
+�
+n
+�
+l=1
+plJk(x + ei + el) − min
+l
+Jk(x + ei + el)
+��
+− min
+�
+cb, aλ
+�
+n
+�
+l=1
+plJk(x + ej + el) − min
+l
+Jk(x + ej + el)
+��
+.
+Note that based on the induction hypothesis, when xi ≥ xj, ∀l (1 ≤ l ≤ n) we have
+Jk((x − el)+ + ei) ≥ Jk((x − el)+ + ej),
+Jk(x + ei + el) ≥ Jk(x + ej + el),
+and thus
+Jk((x + ei − el)+) ≥ Jk((x + ej − el)+),
+min
+l
+Jk(x + ei + el) ≥ min
+l
+Jk(x + ej + el).
+Then we can conclude that
+Jk+1(x + ei) ≥ Jk+1(x + ej).
+Therefore, the Schur convexity xi ≥ xj implies J∗(x + ei) ≥ J∗(x + ej) always holds.
+□
+A.4
+Induction part of Theorem 4
+In this subsection, we continue the proof of Theorem 4 with the induction proving (18).
+Let δk(x) = λ(V k(x + e1) − V k(x + en)), it is sufficient to show for all k ∈ N,
+δk(x + e1) ≥ δk(x),
+δk(x + en) ≤ δk(x).
+28
+
+For the base cases, one can verify that the above inequalities hold for k = 0, 1, 2. Now we
+show the inductive step. According to the value iteration form of formula (17),
+δk+1(x + e1) − δk+1(x)
+=µ[δk(x) − δk((x − e1)+)]
++µ[δk((x + e1 − en)+) − δk((x − en)+)]
++λ[δk(x + e1 + en) − δk(x + en)]
++gk(x + 2e1) − gk(x + e1 + en) − gk(x + e1) + gk(x + en),
+where
+gk(x) = max
+�
+0, min
+�
+δk(x) − ca, cb − cacb
+δk(x)
+��
+.
+Note that based on the induction hypothesis, we have
+δk((x + e1 − en)+) ≥ δk((x − en)+) ≥ δk(x) ≥ δk((x − e1)+),
+δk(x + 2e1) ≥ δk(x + e1) ≥ δk(x + e1 + en) ≥ δk(x + en).
+Then we can conclude that δk+1(x + e1) ≥ δk+1(x) and prove δk+1(x) ≤ δk+1(x + en) in a
+similar way.
+A.5
+Truncated policy iteration
+In this subsection, we present the truncated policy iteration algorithm for estimating stability-
+constrained optimal policy. This algorithm is adapted from the classic policy iteration algo-
+rithm [42] by combining the stability condition (3). Since the original state space is countably
+infinite, here we set a boundary to make the state space finite so that the algorithm can ter-
+minate in finite steps.
+29
+
+Algorithm 1 Truncated policy iteration for estimating optimal policy β ≈ β∗ (continuing)
+Algorithm parameter: small ϵ > 0
+Input: arrival rate λ, service rate µ, number of queues n, one-time protecting cost cb, the
+probability of joining queues p1, p2, · · · , pn
+Initialize arrays: J(x) ∈ R and b(x) ∈ {0, 1} arbitrarily (e.g., J(x) = 0, b(x) = 0) for all
+x ∈ X = {0, 1, · · · , B}n
+# B is the queue length upper bound 1
+repeat
+repeat
+∆ ← 0
+# Policy evaluation
+foreach x ∈ X do
+v ← J(x)
+c ← |x|+cbb(x)
+J(x) ← c + �
+x′ p(x′|x, b)J(x′)
+∆ ← max(∆, |v − J(x, b)|)
+end
+until ∆ < ϵ;
+stable ← True
+# Policy improvement
+foreach x ∈ X do
+old-action← b(x)
+if aλ
+��
+i
+piJ∗(x + ei) − minj J∗(x + ej)
+�
+≤ cb then
+b(x) = 0 2
+end
+else
+b(x) = 1
+end
+if old-action̸= π(x) then
+stable ← False
+end
+end
+until stable = True;
+Output β = (1 − b)NP + bP ≈ β∗
+A.6
+Adapted Shapley’s algorithm
+Here we present the adapted Shpaley’s algorithm for computing the minimax value and
+equilibrium strategies of the attacker-defender stochastic game. In each iteration, it builds
+an auxiliary matrix game and obtains the minimax value using Shapley-Snow method [40].
+1Since the estimation errors are relative large at the boundary, we can set B to be larger than the real
+upper bound.
+2For computing the stability-constrained optimal policy, we can modify the algorithm as follows. In the
+initialization, for every x ∈ X, if x1 = · · · = xn then continue, else set θ(x) = max
+�
+1−
+µ||x||1−λxmin
+aλ
+�
+n
+�
+i=1
+pixi−xmin
+�, 0
+�
+.
+Then in the policy improvement, replace b(x) = 0 with b(x) = θ(x).
+30
+
+Algorithm 2 Adapted Shapley’s algorithm for estimating equilibrium strategies β ≈ β∗,
+α ≈ α∗ (continuing)
+Set V (x) = 0 for all x ∈ X = {0, 1, 2, · · · , B}n 1
+repeat
+∆ ← 0
+foreach x ∈ X do
+v ← V (x)
+Build auxiliary matrix game M(x, V )
+Compute the minimax value val(M) by using Shapley-Snow method
+V (x) ← val(M)
+∆ ← |v − V (x)|
+end
+until ∆ < ϵ;
+foreach x ∈ X do
+Build auxiliary matrix game M(x, V )
+Compute α and β from M(x, V ) by using Shapley-Snow method
+end
+31
+
diff --git a/ldFJT4oBgHgl3EQfZSxF/content/tmp_files/load_file.txt b/ldFJT4oBgHgl3EQfZSxF/content/tmp_files/load_file.txt
new file mode 100644
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+++ b/ldFJT4oBgHgl3EQfZSxF/content/tmp_files/load_file.txt
@@ -0,0 +1,794 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf,len=793
+page_content='Strategic Defense of Feedback-Controlled  Parallel Queues against Reliability and  Security Failures  Qian Xie, Jiayi Wang and Li Jin ∗  January 30, 2023  Abstract  Parallel traffic service systems such as transportation, manufacturing, and computer  systems typically involve feedback control (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', dynamic routing) to ensure stability  and to improve throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Such control relies on connected cyber components for  computation and communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' These components are susceptible to random mal-  functions and malicious attacks, which motivates the design of strategic defense that  are both traffic-stabilizing and cost-efficient under reliability/security failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' In this  paper, we consider a parallel queuing system with dynamic routing subject to such fail-  ures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' For the reliability setting, we consider an infinite-horizon Markov decision process  where the system operator strategically activates the protection mechanism upon each  job arrival based on the traffic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' We use Hamilton-Jacobi-Bellman equation to  show that the optimal protection strategy is a deterministic threshold policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' For the  security setting, we extend the model to an infinite-horizon stochastic game where the  attacker strategically manipulates routing assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' We show that a Markov per-  fect equilibrium of this game always exists and that both players follow a threshold  strategy at each equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' For both settings, we also consider the stability of the  traffic queues in the face of failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Finally, we develop approximate dynamic program-  ming algorithms to compute the optimal/equilibrium policies and present numerical  examples for validation and illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Keywords: Queuing systems, cyber-physical security, stochastic games, Markov decision  processes, HJB equation, Lyapunov function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  ∗This work was in part supported by US NSF Award CMMI-1949710, C2SMART University Transporta-  tion Center, NYU Tandon School of Engineering, SJTU-UM Joint Institute, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Wu & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Sun Endowment  Fund, and Cornell University McMullen Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Xie is with the School of Operations Research and  Information Engineering, Cornell University, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Wang is with the Department of Electrical and Com-  puter Engineering, University of California, San Diego, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Jin is with the UM Joint Institute, and the  Department of Automation, Shanghai Jiao Tong University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Xie and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Jin were with the Tandon  School of Engineering, New York University, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Wang was with the UM Joint Institute, Shanghai Jiao  Tong University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' (emails: qx66@cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='edu, jiw139@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='edu, li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='jin@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='11530v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='SY]  27 Jan 2023  1  Introduction  Parallel queuing system is a typical model for characterizing a service system of multiple  servers, each with a waiting queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Real-world instances include packet switching networks  [35, 21], manufacturing systems [24], transportation facilities [25], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Such systems use  feedback from the state observation to generate routing decisions that ensure stability and  improve throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Meanwhile, the feedback control relies heavily on connected cyber  components for data collection and transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Such cyber components are subject to  persistent threat due to random malfunctions and malicious attacks [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' For instance, trans-  portation infrastructure information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', traffic sensors, traffic lights) and vehicle commu-  nications can be intruded and manipulated [17, 38, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Real-world incidents indicated that a  simulated traffic jam can cause motorists to deviate from their planned routes [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Similar  security risks also exist in industrial control [3] and communication systems [2, 31, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Such  real-world systems will not be accepted by the authorities, the industry, and the public,  unless the security problems are well addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' However, cyber security risks have not been  sufficiently studied in conjunction with the physical queuing dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Moreover, it is eco-  nomically infeasible and technically unnecessary to perfectly avoid cyber failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Therefore,  it is crucial to understand the impact due to such threats and to design strategic defense  mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  In response to such concerns, we try to address the following two research questions:  (i) How to model the security vulnerabilities and quantify the security risks for parallel  queuing systems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (ii) How to design traffic-stabilizing, cost-efficient defense strategies against failures?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  For the first question, we consider two scenarios of failures, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  reliability failures and  security failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' We quantitatively characterize the security risks in terms of failure-induced  queuing delays and defending costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' For the second question, we analyze the stability criteria  of the failure-prone system with defense, and characterize the structure of the cost-efficient  strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  We also develop algorithms to compute such strategies, and discuss how to  incorporate the stability condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Our results are demonstrated via a series of numerical  examples and simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  This paper is related to two lines of work: queuing control and game theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' On the  queuing side, the majority of the existing analysis and design are based on perfect observation  of the states (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', queue lengths) and perfect implementation of the control [13, 23, 15, 21, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Besides, researchers have noted the impact of delayed [28, 32], erroneous [6], or decentralized  information [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Although these results provide hints for our problem, they do not directly  apply to the security setting with failures such as imperfect sensing (state observation) and  imperfect control implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' On the game side, a variety of game-theoretic models have  been applied to studying cyber-physical security in transportation [43, 29], communication  [7, 2, 31], and infrastructure systems [46, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  However, to the best of our knowledge,  security risks of feedback-controlled systems have not been well studied from a combined  game-theoretic and feedback-control perspective, which is essential for capturing the coupling  between the queuing dynamics and the attacker-defender interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Our model includes two parts: the physical model (parallel queuing system) and the  cyber model (dynamic routing subject to failures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Specifically, we consider a system of  2  parallel servers and queues with possibly faulty dynamic routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Dynamic routing is a  classical feedback control strategy that assigns jobs to one of the parallel queues according  to the current system state (queue length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' We assume that the routing instructions can be  faulty due to the following two threats:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Reliability failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Due to random malfunctions, a fault may occur to each arrival  with a constant probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' For instance, the operator may lose the observation of the  state or fail to deliver the routing instruction temporarily [19], and then the job joins  a random queue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Security failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' An adversary may strategically launch attacks on the arrivals for  selfish or malicious intent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Typical attack modes include (i) denial-of-service that cuts  off the operator’s observation of the state [45, 44, 1], (ii) falsification that injects falsi-  fied data or creates fictitious traffic to mislead the operator [1, 38], and (iii) spoofing  that manipulates the routing instruction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', sending jobs to a busy server) by im-  personating the operator [17, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' The consequence of an attack is that an incoming  job joins a queue selected by the attacker;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  The system operator can protect/defend1the routing instructions for incoming jobs based on  the state observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  The activation of protection/defense mechanisms induces a cost rate  on the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' For the reliability setting, we formulate the operator’s trade-off between  queuing costs and protecting costs as an infinite-horizon, continuous-time, and discrete-state  Markov decision process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' For the security setting, we formulate the interaction between the  attacker and the operator as an infinite-horizon stochastic game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  To study the stability of the queuing system, previous works typically relied on character-  ization or approximation of the steady-state distribution of the queuing state [18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' however,  this approach is hard to be synthesized with failure models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' In addition, it is hard to study  the steady-state distribution of queuing systems with state-dependent transition rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' To  address such challenges, we use a Lyapunov function-based approach to study the stability  (in this paper we consider mean boundedness [33]) of queuing systems under control and to  obtain upper bounds for the mean number of jobs in the system [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Such an approach has  been applied to queuing systems in no-failure scenarios [27, 10, 14, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Importantly, we use  this approach to derive stability criterion for protection under reliability failures and defense  against the security failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  To analyze the cost efficiency of the system operator’s decision, we formulate the opti-  mization problem in terms of queuing and protecting costs and then derive its Hamiltonian-  Jacobian-Bellman (HJB) equation [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' We show that the optimal protecting policy under  reliability failures is a deterministic threshold policy: the operator either protects or does  not protect, according to threshold functions in the multi-dimensional state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Similar  approaches have been discussed in [5, 22] for two queues, and we generalize the analysis to  n queues and to failure-prone settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' For the attacker-defender game, we again use HJB  equation to show the threshold properties of the Markov perfect equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  The above analysis leads to useful insights for designing strategic protection/defense  that are both stabilizing and cost-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' A key finding is that the system operator has  1In the rest of the paper, we use the word “protect” for the reliability setting and “defend” for the security  setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  3  𝜆  𝑋1(𝑡)  𝑋𝑛(𝑡)  Server 1  𝑎  𝑝1  𝑝2  Server 𝑛  𝜇  𝜇  𝛽  𝑋1(𝑡) = min  "  𝑋𝑖(𝑡)  1 − 𝑎(1 − 𝑏)  𝑎(1 − 𝑏)  𝑋2(𝑡)  Server 2  𝜇  𝑝𝑛  ⋮  System Operator  Random Fault  Success  Failure  (a) Scenario of reliability failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  𝜆  𝑋1(𝑡)  𝑋𝑛(𝑡)  Server 1  𝛼  Server 𝑛  𝜇  𝜇  𝛽  1 − 𝑎(1 − 𝑏)  𝑎(1 − 𝑏)  𝑋2(𝑡)  Server 2  𝜇  ⋮  𝑋1(𝑡) = min  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  𝑋𝑖(𝑡)  𝑋𝑛(𝑡) = max  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  𝑋𝑖(𝑡)  System Operator  Malicious Attack  Success  Failure  (b) Scenario of security failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Figure 1: An n-queue system with shortest-queue routing under reliability/security  failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' See Section 2 for definitions of notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  4  OOa higher incentive to protect/defend if the queues are more “unbalanced”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  In addition,  our numerical analysis shows that 1) the incentive to protect increases with the failure  probability, decreases with the technological cost, and increases with the demand-capacity  ratio;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' 2) the optimal protecting policy performs better than static policies such as never  protect and always protect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' We also note that the optimal decision is not always stabilizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Considering this, we propose how to compute the stability-constrained optimal policy by  imposing the stability condition on the HJB equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Our contributions lie in the following three aspects:  Modeling: 1) We build a framework for modeling the cyber-physical vulnerabilities of  queuing systems with feedback control (dynamic routing) subject to reliability/security  failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  2) We propose a formulation of protection under reliability failures as an  infinite-horizon Markov decision process and defense against security failures as an  attacker-defender stochastic game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Analysis: 1) We provide stability criteria under failures and control based on Lya-  punov functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' 2) We show the threshold properties of the optimal protection and  the game equilibria on multidimensional state space based on HJB equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Design: Our theoretical results provide insights on the design of traffic-stabilizing  and cost-efficient protecting policy and defending response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' We also propose approx-  imate dynamic programming algorithms to numerically compute the optimal policy  and equilibrium strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Section 2 introduces the queuing model  and the failure models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Section 3 studies protection against reliability failures and Section 4  analyzes defense against security failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Section 5 gives a concluding remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  2  Parallel queuing system and failure models  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='1  Parallel queuing system  Consider a queuing system with n identical servers in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Jobs arrive according to a  Poisson process of rate λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Each server serves jobs at an exponential rate of µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' We use  X(t) =  �  X1(t)  X2(t)  · ·  Xn(t)  �T to denote the number of jobs at time t, either waiting  or being served, in the n servers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' The state space of the parallel queuing system  is Zn  ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Specifically, the initial system state (queue length) is X(0) = x = [x1 x2 · · · xn]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  We use x + (−)ei to denote adding (subtracting) 1 from xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Since the queue lengths  are always non-negative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' xi ≥ 0, we use (x − ei)+ = max(x − ei, 0) to avoid the case  that subtracting 1 makes the element negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Let xmin = mini xi and xmax = maxi xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  We use x−i to denote variables in x other than xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' We call x a diagonal vector if x1 =  x2 = · · · = xn and a non-diagonal vector otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Denote the one-norm of the vector x as  ||x||1:= x1 + x2 + · · · + xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Then ||X(t)||1 means the total number of jobs in the system at  time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' We use x ≻ 0 to denote that x is not a zero vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', ||x||1> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  5  Without any failures, any incoming job is allocated to the shortest queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  If there  are multiple shortest queues, then the job is randomly allocated to one of them with (not  necessarily equal) probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='2  Reliability failures  Suppose that when a job arrives at the system, the fault occurs with a constant probability  a ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' The system operator can deploy additional security resources to ensure correct  routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Let ∆({NP, P}) := {(1 − b, b) : b ∈ [0, 1]} denote the probability distribution  over the action set {not protect, protect}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' The system operator selects a Markovian policy  β : Zn  ≥0 → ∆({NP, P}) that depends only on the current state but not on the history of  states and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' We write b(x) as the state-dependent protecting probability at state  x ∈ Zn  ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' With a slight abuse of notation, when the policy is deterministic, we rewrite the  mapping as β : Zn  ≥0 → {NP, P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Protecting a job induces a one-time technological cost of cb > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' If the fault occurs but  the system operator does not protect, the routing instruction is faulty, and the job joins a  random queue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' the probability of joining the ith queue is pi ∈ [0, 1] where �n  i=1 pi = 1, and  we define  pmax = max(p1, p2, · · · , pn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Otherwise, the routing instruction is correct, and the job joins the shortest queue, as shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  The objective of the system operator is to balance the queuing cost and the protecting  cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' We formulate this problem as an infinite-horizon continuous-time Markov decision  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Specifically, the operator aims to minimize the expected cumulative discounted cost  J(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' β):  J∗(x) := min  β  J(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' β)  = min  β  E  � � ∞  0  e−ρtC(X(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' β)dt  ���X(0) = x  �  ,  (1)  where ρ ∈ (0, 1) is the discounted factor and C : Zn  ≥0 → R is the cost rate defined as  C(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' β) = ||ξ||1+cbb(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  The optimal protecting policy is essentially the policy that minimizes the expected cumula-  tive discounted cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Definition 1 (Optimal protecting policy) The optimal protecting policy β∗ against re-  liability failures is defined as:  β∗(x) := argmin  β  J(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' β),  ∀x ∈ Zn  ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='3  Security failures  Suppose that when each job arrives, a malicious attacker is able to manipulate the routing  such that the job is allocated to a non-shortest queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' For the sake of simplicity, we consider  the attacker’s best action (and thus the operator’s worst case), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', the job goes to the longest  queue, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Let ∆({NA, A}) := {(1−a, a) : a ∈ [0, 1]} denote the probability  distribution over the action set {not attack, attack}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' The attacker selects a (possibly mixed)  Markov strategy α : Zn  ≥0 → ∆({NA, A}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' With a slight abuse of notation, we write a(x) as  the state-dependent attacking probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Note that here a(x) has a different meaning from  the constant fault probability a in the reliability failure setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Attacking a job induces a  one-time technological cost of ca > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  The system operator’s action is similar to that in the reliability setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' The only differ-  ence is that in the security setting, the system operator knows there is a strategic attacker  making decisions simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' We formulate the interaction between the attacker and the  operator (also called defender) as an infinite-horizon stochastic game with Markov strategies  that do not depend on the history of states and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' The attacker aims to maximize the  expected cumulative discounted reward V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' α, β) given the operator’s Markov strategy β:  V ∗  A(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' β) := max  α  V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' α, β)  = max  α  E  � � ∞  0  e−ρtR(X(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' α, β)dt  ���X(0) = x  �  ,  where R : Zn  ≥0 → R is the immediate reward defined as  R(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' α, β) = ||ξ||1+cbb(ξ) − caa(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Here the attacker’s reward not only considers the queuing cost and the attacking cost, but  also minus the defending cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' This is because the attacker has the incentive to increase  the system’s total operating cost, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' the sum of queuing and defending costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Similarly,  the operator aims to minimize the expected cumulative discounted loss given the attacker’s  Markov strategy α:  V ∗  B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' α) = min  β  V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  We can also define the Markov perfect equilibrium of such an attacker-defender game:  Definition 2 (Markov perfect equilibrium) The equilibrium attacking (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  defend-  ing) strategy α∗ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' β∗) satisfies that for each state x ∈ Zn  ≥0,  α∗(x) = argmax  α  V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' α, β∗) = argmax  α  V ∗  A(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' β∗),  β∗(x) = argmin  β  V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' α∗, β) = argmin  β  V ∗  B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' α∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  The equilibrium value of the attacker (defender) is V ∗  A(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' β∗) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' V ∗  B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' α∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' In particular,  (α∗, β∗) is a Markov perfect equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  7  3  Protection against reliability failures  In this section, we consider the design of the system operator’s state-dependent protecting  policy from two aspects: stability and optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  It is well known that a parallel n-queue system is stabilizable if and only if the demand  is less than the total capacity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', λ < nµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' In the following results, we will see that even  this condition is met, in the absence of defense, reliability failures can still destabilize the  queuing system, especially when the probability of failures is high and when the random  faulty routing is highly heterogeneous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' the following summarizes the above insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Proposition 1 The unprotected n-queue system with faulty probability a is stable if and only  if  λ < nµ,  (2a)  apmaxλ < µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (2b)  Furthermore, when the system is stable, the long-time average number of jobs is upper-  bounded by  ¯X := 1  t lim sup  t→∞  t  �  τ=0  E[X(τ)] ≤  λ + nµ  2  �  µ − max(apmax, 1  n)λ  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  The next result provides a stability criterion for an n-queue system with a given protecting  policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' The proof of this result is presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Theorem 1 (Stability under reliability failures) Consider an n-queue system subject  to reliability failures with probability a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Suppose that the system operator selects a  Markovian protecting policy β characterized by a state-dependent probability b(x) ∈ [0, 1] at  each state x ∈ Zn  ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Then we have the following:  (i) The system is stable if for every non-diagonal vector x, the protecting probability b(x)  satisfies  b(x) > 1 −  µ||x||1−λxmin  aλ  � n�  i=1  pixi − xmin  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (3)  (ii) When (2a) holds, there must exist a policy satisfying (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' When (2b) holds, every  policy satisfies (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (iii) If (3) holds, the long-time average number of jobs in the system is upper-bounded by  ¯X ≤ λ + nµ  2c  ,  (4)  where  c = min  x≻0  �  µ − λ xmin  ||x||1  − a(1 − b(x))λ  �n  i=1 pixi − xmin  ||x||1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  8  The next result characterizes the structure of the optimal protecting policy for the dynamic  routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' The key findings are: 1) the optimal protecting policy is deterministic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', the  system operator either protects or does not protect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' 2) the system operator chooses to  protect when the queue lengths are more “unbalanced”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Theorem 2 (Optimal protecting policy) Consider a n-queue system subject to reliabil-  ity failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' The optimal protecting policy β∗ has the following properties:  (i) β∗ is deterministic in the sense that either β∗(x) = NP or β∗(x) = P for each x ∈ Zn  ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (ii) β∗ is a threshold policy characterized by n threshold functions fm (m = 1, 2, · · · , n) via  b∗(x) = 1  �  n�  m=1  (fm(x) > 0)  �  ,  where for each m = 1, 2, · · · , n,  (a) fm : Zn  ≥0 → R separates the polyhedron Xm = {x ∈ Zn  ≥0 | xi ≥ xm, ∀1 ≤ i ≤ n}  into two subsets: {x ∈ Xm | β∗(x) = NP} and {x ∈ Xm | β∗(x) = P} by means  of  b∗(x) = 1{fm(x) > 0},  ∀x ∈ Xm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (b) the optimal protecting probability b∗(x) is monotonically non-decreasing (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  non-increasing) in xi (∀i ̸= m) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' xm) while other variables x−i (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' x−m)  are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  0  5  10  15  20  25  30  length of queue 1 x1  0  5  10  15  20  25  30  length of queue 2 x2  not protect  protect  1  2  Figure 2: The characterization of the optimal protecting policy β∗ for a two-queue system  (p1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='1, p2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='9, ρ = 1, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Here the threshold functions qualitatively characterize the degree of “unbalancedness”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  They separate the state space into n + 1 subsets of states: one containing “balanced” states  such that the action “not protect” is optimal, and the other n containing “unbalanced” states  such that the action “protect” is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' The concept “threshold function” has  appeared in previous works [5, 22, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' The rest of this section is devoted to the proofs and  discussions of Theorem 1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='1  Stability under reliability failures  In this subsection, we provide a proof of the stability condition under the protected case  (Theorem 1) and leave the proof of the stability condition under the unprotected case (Propo-  sition 1) to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Both proofs are based on Foster-Lyapunov drift condition [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' (i) and (iii) Consider the quadratic Lyapunov function  W(x) = 1  2  n  �  i=1  x2  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (5)  By applying infinitesimal generator [11, 4] under protecting policy β (denoted as Lβ) we  have  LβW(x) =a(1 − b(x))λ  2  n  �  i=1  pi  �  (xi + 1)2 − x2  i  �  + (1 − a(1 − b(x))) λ  2  �  (xmin + 1)2 − x2  min  �  + µ  2  n  �  i=1  1xi>0  �  (xi − 1)2 − x2  i  �  = a(1 − b(x))λ  n  �  i=1  pixi + (1 − a(1 − b(x))) λxmin  − µ  n  �  i=1  xi + 1  2λ + 1  2  n  �  i=1  1{xi > 0}µ  ≤ a(1 − b(x))λ  � n  �  i=1  pixi − xmin  �  + (λxmin  − µ||x||1) + 1  2(λ + nµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  By (3) there exists constants c = min  x≻0{µ − λ xmin  ||x||1 − a(1 − b(x))λ  �n  i=1 pixi−xmin  ||x||1}  > 0 and d =  1  2(λ + nµ) such that  LβW(x) ≤ −c||x||1+d,  ∀x ∈ Zn  ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (6)  By [34, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='3], this drift condition implies the upper bound (4) and thus the stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  □  (ii) When λ < nµ, for every non-diagonal vector x, we have µ||x||1−λxmin > µ||x||1−λ ||x||1  n  =  �  µ − λ  n  �  ||x||1> 0 and �n  i=1 pixi − xmin > 0, then  1 −  µ||x||1−λxmin  aλ  � n�  i=1  pixi − xmin  � < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  10  Thus, b(x) ≡ 1 satisfies the stability condition (3) and β(x) ≡ P is a stabilizing policy that  exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  When max(apmax, 1/n)λ < µ, for every non-diagonal vector x, we have aλ �n  i=1 pixi +  (1 − a)λxmin ≤ max(apmax, 1/n)λ||x||1< µ||x||1, and then  1 −  µ||x||1−λxmin  aλ  � n�  i=1  pixi − xmin  � < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Thus, every policy satisfies the stability criterion (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  □  Theorem 1 provides a stability criterion for any state-dependent protecting probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  This implies that the operator needs to protect, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', choose some positive protecting proba-  bility to stabilize the system at certain states (queue lengths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' We will use such stabilizing  threshold probabilities to obtain a stability-constrained optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' See Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='3 and  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='2  Optimal protecting policy  A standard way to solve the discounted infinite-horizon minimization problem (1) is to write  down its HJB equation for optimality [9, Chapter 4]:  0 = min  β {||x||1+cbb(x) − ρJ∗(x) + LβJ∗(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (7)  We can rewrite it as the following recurrence form:  (ρ + λ + nµ)J∗(x) = min  β  �  ||x||1+cbb(x)+  µ  n  �  i=1  J∗((x − ei)+) + λ min  1≤j≤n J∗(x + ej) + (1 − b(x))aλ  �  n  �  i=1  piJ∗(x + ei) − min  1≤j≤n J∗(x + ej)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (8)  The optimal protecting policy β∗ is essentially the solution of (7) and (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' A direct observa-  tion is that when there is no failure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', a = 0, the system operator never needs to protect  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', ∀x, β∗(x) = NP);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' when all queue lengths are equal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', x1 = x2 = · · · = xn, the opera-  tor deterministically chooses not to protect at this state x (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', β∗(x) = NP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Furthermore,  Theorem 2(i) indicates that β∗ is deterministic (not probabilistic) at each state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', either  protects or does not protect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Proof of Theorem 2(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' The expression to be minimized in the right-hand side of the HJB  equation (8) is linear in b(x), so the minimum is reached at the endpoints, that is, b(x) = 0  or b(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  □  11  Now the HJB equation (8) turns into  (ρ + λ + nµ)J∗(x) = min  b∈{0,1}  �  ||x||1+cbb  +µ  n  �  i=1  J∗((x − ei)+) + λ min  1≤j≤n J∗(x + ej)  +(1 − b)aλ  �  n  �  i=1  piJ∗(x + ei) − min  1≤j≤n J∗(x + ej)  ��  := min  b∈{0,1}  �  c(x, b) +  �  x′  q(x′|x, b)J∗(x′)  �  ,  where q(x′|x, b) are the transition rates from state x to state x′ given protecting probability  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' For example, q((x − ei)+|x, b) = µ for b = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Using the uniformization trick [30, 37], we have  J∗(x) := min  b  �  ˜c(x, b) + γ  �  x′  p(x′|x, b)J∗(x′)  �  ,  (9)  where Λ = λ + nµ, γ = Λ/(ρ + Λ), ˜c(x, b) = c(x, b)/(ρ + Λ) and p(x′|x, b) = q(x′|x, b)/Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Without loss of generality, we assume ρ + Λ = 1 in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Before proving Theorem 2, we first present properties of the optimal cost function J∗  since they are useful for showing the threshold property of optimal policy β∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Proposition 2 The optimal cost function J∗ : Zn  ≥0 → R has the following properties:  (i) (symmetry) J∗ is symmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', J∗(x) = J∗(σx) where σx is a permutation of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (ii) (Schur convexity) J∗ is Schur convex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', J∗(x + ei) ≥ J∗(x + ej) if xi ≥ xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (iii) (monotonicity) J∗ is non-decreasing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', J∗(x) ≥ J∗(y) if xi ≥ yi for all i (1 ≤ i ≤ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (iv) (convexity) J∗ is convex in each variable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', J∗(x+ei)−J∗(x) ≤ J∗(x+2ei)−J∗(x+  ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (v) (supermodularity) J∗ is supermodular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', J∗(x+ei+ej)+J∗(x) ≥ J∗(x+ei)+J∗(x+  ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  The proof of Proposition 2 is left to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='3 and we will particularly use property  (i)-(ii), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', symmetry and Schur convexity to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Proof of Theorem 2(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Let m = argmini xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Similar to the idea in [5, Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='6], we  can demonstrate (a) the existence of the threshold functions by showing (b) the monotonicity  of the optimal protecting probability:  b∗(x + ei) ≥ b∗(x),  ∀i ̸= m  b∗(x + em) ≤ b∗(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (10)  12  Now we prove (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Because of Schur convexity, J∗(x + ei) ≥ J∗(x + em) (∀i ̸= m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' We can  rewrite (9) as  J∗(x) = min  b∈{0,1}  �  ||x||1+cbb + µ  n  �  i=1  J∗((x − ei)+) + λ  J∗(x + em) + (1 − b)aλ  �  n  �  i=1  piJ∗(x + ei) − J∗(x + em)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Let ∆∗(x) =  n�  i=1  piJ∗(x + ei) − J∗(x + em).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Note that by Definition 1 and Theorem 2(i),  b∗(x) = 1 if ∆∗(x) > cb and b∗(x) = 0 if ∆∗(x) < cb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Then the monotonicity of b∗ is  essentially the monotonicity of ∆∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Thus, (10) is equivalent to  ∆∗(x + ei) ≥ ∆∗(x),  ∀i ̸= m  ∆∗(x + em) ≤ ∆∗(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (11)  We defer the proof of (11) to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' The high-level idea is to use induction based  on value iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  □  To obtain an estimated optimal policy, we propose an algorithm called truncated policy  iteration (TPI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' See Algorithm 1 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' It is adapted from the classic policy  iteration algorithm [42] and based on the following value iteration form of the HJB equation  (9):  Jk+1(x) = min  b  �  ˜c(x, b) + γ  �  x′  p(x′|x, b)Jk(x′)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (12)  Now we use the estimated optimal policy to conduct numerical analysis on 1) the relationship  between the incentive to protect and the system parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' 2) the comparison between the  optimal policy and two naive static policies: always protect and never protects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  We first analyze the tipping points when the system operator starts to protect “riskier”  states under the optimal policy β∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', ∃x s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' β∗(x) = P, as the failure probability a and  technological cost cb change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' It can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' 3 that the incentive to protect is  non-decreasing in the failure probability a, non-increasing in the technological cost cb and  non-decreasing in the demand-capacity ratio (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' utilization ratio) ρ = λ/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' That is, the  system operator has higher incentive to protect when 1) the failure probability is higher;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' 2)  the technological cost is lower;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' 3) the utilization ratio is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  In the following simulation, we will see that the optimal policy β∗ can significantly reduce  the security risk, compared to the static policies: β(x) ≡ P (always protects) and β(x) ≡ NP  (never protects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' 4 where the yellow curves are below the red curves and the green  curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' The Monte Carlo simulation result is based on the cumulative discounted cost within  50000s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Here the cumulative discounted cost is calculated as the sum of the total queuing  cost and total technological cost in the episode, and we normalized it to be a value between  0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Note that under the static policy β(x) ≡ P (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', ∀x, b(x) = 1), the job always joins  the shortest queue regardless of the failure probability, so the cumulative discounted cost is  a constant (red curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='00  failure probability a  0  1  2  3  4  tech cost cb  x, β*(x) = NP  x, β*(x) = P  (a) ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='00  failure probability a  0  1  2  3  4  tech cost cb  x, β*(x) = NP  x, β*(x) = P  (b) ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='4  Figure 3: The tipping points when the system operator starts to protect “riskier” states  under the optimal policy as the failure probability and the technological cost change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='0  failure probability a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='0  normalized  cumulative cost  β(x) ≡ NP  β ∗  β(x) ≡ P  (a) cb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='0  failure probability a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='0  normalized  cumulative cost  β(x) ≡ NP  β ∗  β(x) ≡ P  (b) cb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='5  Figure 4: Comparison of the normalized cumulative discounted costs between the optimal  policy and the static policies (p1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='1, p2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='9, ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  14  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='3  Stability-constrained optimal policy  The optimal policy may not always be stabilizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' For example, the optimal policy under  the system parameters p1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='1, p2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='9, ρ = 1, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='9 does not satisfy the stability  condition (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' In this case, we can choose the optimal policy from the subset of policies  satisfying the stability criterion, called stability-constrained optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' The operator  needs to randomize over {P, NP} at some states, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' We denote such policy as ˆβ∗  with state-dependent protecting probability ˆb∗(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Specifically, we can consider the stability-  constrained MDP by imposing (3) on the HJB equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='5 gives a corresponding  modification of the TPI algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  0  5  10  15  20  25  30  length of queue 1 x1  0  5  10  15  20  25  30  length of queue 2 x2  not protect  protect  mixed  1  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='0  protecting probability ˆb*  Figure 5: The characterization of the stability-constrained optimal policy ˆβ∗ for a  two-queue system (p1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='1, p2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='9, ρ = 1, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Besides, we can have a quick check of the existence of a stabilizing policy and stabiliz-  ability of the optimal policy using stability conditions (2a)-(2b) as follows:  When (2a)-(2b) hold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', max(apmax, 1/n)λ < µ, the optimal protecting policy is also  stabilizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  When only (2a) holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', λ/n < µ ≤ apmaxλ, the optimal protecting policy may not  be stabilizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  When (2a) does not hold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', λ ≥ µn, no stabilizing protecting policy exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  4  Defense against security failures  In this section, we analyze the attacker’s attacking strategy and system operator’s defending  strategy from two aspects: stability and game equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  The following criterion can be used for checking the stability of the n-queue system under  any state-dependent attacking and defending strategies:  15  Theorem 3 (Stability under security failures) Consider an n-queue system subject to  security failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Suppose that at each state x ∈ Zn  ≥0, the attacker (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' system operator)  attacks (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' defends) each job following Markov strategy α (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' β) characterized by a  state-dependent probability a(x) ∈ [0, 1] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' b(x) ∈ [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Then we have the following:  (i) The system is stable if for every non-diagonal vector x, the attacking and defending  probabilities satisfy  a(x) (1 − b(x)) < µ||x||1−λxmin  λ(xmax − xmin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (13)  (ii) When λ < nµ, there must exist a strategy β with defending probability b(x) satisfying  (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (iii) Furthermore, if (13) holds, then the long-time average number of jobs is upper-bounded  by  ¯X ≤ λ + nµ  2c  ,  (14)  where  c = min  x≻0  �  µ − λ xmin  ||x||1  − a(x)(1 − b(x))λxmax − xmin  ||x||1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  The next result characterizes the structure of the strategy of the Markov perfect equilibria  of the stochastic game: the equilibrium defending probability is higher when the queue  lengths are more “unbalanced”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Theorem 4 (Markov perfect equilibrium) For each Markov perfect equilibrium (α∗, β∗)  of the attacker-defender game, we have the following:  (i) (α∗, β∗) is qualitatively different over the following three subsets of the state space Zn  ≥0:  (a) S1 = {x ∈ Zn  ≥0 | (α∗(x), β∗(x)) = (NA, NP)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' (“low risk”)  (b) S2 = {x ∈ Zn  ≥0 | (α∗(x), β∗(x)) = (A, NP)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' (“medium risk”)  (c) S3 = {x ∈ Zn  ≥0 | (α∗(x), β∗(x)) is mixed}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' (“high risk”)  (ii) The boundaries between S1 and S2, as well as those between S2 and S3 are characterized  by threshold functions gij, hij (1 ≤ i ̸= j ≤ n) as follows:  S1 =  �  x ∈ Zn  ≥0 |  �  1≤i̸=j≤n  (gij(x) < 0)  �  ,  S2 =  �  x ∈ Zn  ≥0 |  �  1≤i̸=j≤n  (gij(x) > 0 ∨ hij(x) < 0)  �  ,  S3 =  �  x ∈ Zn  ≥0 |  �  1≤i̸=j≤n  (hij(x) > 0)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  where for each i, j = 1, 2, · · · , n (i ̸= j),  16  (a) gij, hij : Zn  ≥0 → R separate the polyhedron Xij = {x ∈ Zn  ≥0 | xi = xmax, xj = xmin}  into three subsets: S1 ∩ Xij, S2 ∩ Xij and S3 ∩ Xij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (b) state x has a lower (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' higher) or equal security level than state x + ei (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  x + ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Here the threshold functions also characterize the degree of “unbalancedness”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Intu-  itively, S1–S3 correspond to various security risk levels, and thus correspond to the incentive  of the operator to defend: when the queues are more “unbalanced” and the risk is higher,  the operator has a higher incentive to defend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' 6 visualizes the equilibria for a two-queue  system, which help understand Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' For a detailed argument about the relation-  ship between the security levels and the system parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', technological costs and  utilization ratio), see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  0  5  10  15  20  length of queue 1  0  5  10  15  20  length of queue 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='0  attacking probability a*  0  5  10  15  20  length of queue 1  0  5  10  15  20  length of queue 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='0  defending probability b*  Figure 6: The equilibrium attacking and defending strategies for a two-queue system  (ρ = 1, ca = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='1, cb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  We also find that the security game has four equilibrium regimes under different combi-  nations of attacking cost ca and defending cost cb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Each regime is labeled with  corresponding subsets of Markov perfect equilibria and security levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  The rest of this section is devoted to the proofs of Theorem 3-4, as well as an additional  discussion on Markov perfect equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  17  Figure 7: Equilibrium regimes of the security game (ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='1  Stability under security failures  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' By applying infinitesimal generator Lα,β under the attacking strategy  α and the defending strategy β to the same Lyapunov function (5), we have  Lα,βW(x) =a(x)(1 − b(x))λxmax + (1 − a(x)(1 − b(x)))  λxmin −  n  �  i=1  µxi + 1  2λ + 1  2  n  �  i=1  1{xi > 0}µ  ≤a(x)(1 − b(x))λ(xmax − xmin)  + λxmin − µ||x||1+1  2(λ + nµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Hence, by (13) there exists c = min  x≻0{µ − λ xmin  ||x||1 − a(x)(1 − b(x))λ xmax−xmin  ||x||1  )} > 0 and d =  1  2(λ + nµ) such that  Lα,βW(x) ≤ −c||x||1+d,  ∀x ∈ Zn  ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  By [34, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='3], this drift condition implies the upper bound (14) and thus the stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Besides, when λ < nµ, for every non-diagonal vector x, we have µ||x||1≥ nµxmin > λxmin,  and xmax > xmin, implying  µ||x||1−λxmin  λ(xmax−xmin) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Thus, no matter what strategy the attacker  chooses, the defending strategy with b(x) ≡ 1 satisfies the stability condition (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='2  Markov perfect equilibrium  For the stochastic attacker-defender game, we first show the existence of Markov perfect  equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  18  20  20  2  queue  15  15  10  low risk  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='0  10  low risk  length  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='5  5  medium risk  0  cost  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='0  0  0  510  15  20  0  5101520  length of queue 1  defending  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='5  V  lengthofqueue1  20  20  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='0  2  queue  15  15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='5  low risk  low risk  III  10  of  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='0  length  5  attacking cost Ca  5  medium risk  high risk  0  05101520  0  5101520  lengthofgueue1  lengthofgueue1Proposition 3 Markov perfect equilibrium (α∗, β∗) of the stochastic attacker-defender game  always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Note that the state space Zn  ≥0 is countable and the action space {0, 1} is finite (and  thus compact).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' By [16, Theorem 1], Markov perfect equilibrium (also called discounted equi-  librium point of policies) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  □  Next, we discuss the derivation of Markov perfect equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' According to Shapley’s  extension on minimax theorem for stochastic game [39], the attacker and the defender have  the same equilibrium (minimax) value:  V ∗  B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' α∗) = V ∗  A(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' β∗) = V ∗(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Thus, we only need to compute the minimax value V ∗ of the stochastic game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Similar to the  derivation of (9), we obtain the following HJB equation of the minimax problem (assuming  ρ + λ + nµ = 1):  V ∗(x) = max  α  min  β  �  ||x||1+cbb(x) − caa(x)+  µ  �  i  V ∗((x − ei)+) + λ min  j  V ∗(x + ej) + a(x)(1 − b(x))  λ  �  max  j  V ∗(x + ej) − min  j  V ∗(x + ej)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (15)  For each state x ∈ Zn  ≥0, let δ∗(x) = λ(maxj V (x+ej)−minj V (x+ej)) and build an auxiliary  matrix game  M(x, V ∗) =  �  ||x||1+µ  n  �  i=1  V ∗((x − ei)+) + λ min  j  V ∗(x + ej)  �  �1  1  1  1  �  +  �  0  cb  −ca + δ∗(x)  −ca + cb  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (16)  Then given δ∗(x), the equilibrium strategies (α∗(x), β∗(x)) can be obtained by Shapley-Snow  method [40], a convenient algorithm for finding the minimax value and equilibrium strategies  of any two-player zero-sum game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Proof of Theorem 4(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Consider the matrix game M(x, V ∗) defined as (16) where the  attacker and the system operator are the row player and the column player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Based on  Shapley-Snow method, the equilibrium strategies (α∗(x), β∗(x)) are in the following three  cases depending on the relationship between δ∗(x) and the technological costs ca, cb > 0:  (a) When δ∗(x) ≤ ca, it is obvious that α∗(x) = NA (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', a∗(x) = 0) is a dominant  strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Then cb > 0 implies β∗(x) = NP (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', b∗(x) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' That is, the attacker has  no incentive to attack, and thus the defender does not need to defend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' At this pure  strategy equilibrium, the security risk is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  19  (b) When the defending cost cb is higher then the attacking cost ca, and ca < δ∗(x) ≤ cb, it  is obvious that β∗(x) = NP (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', b∗(x) = 0) is a dominant strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Then cb > −ca +cb  implies α∗(x) = A (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', a∗(x) = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' That is, the defender has no incentive to defend  and consequently the attacker prefers to attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' At this pure strategy equilibrium, the  security risk is higher than the first case but tolerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (c) When δ∗(x) > max{ca, cb} > 0, no saddle point exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Then both the attacker and the  system operator consider mixed strategies such that a∗(x) =  cb  δ∗(x), b∗(x) = 1 −  ca  δ∗(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Particularly, the operator needs to select positive protecting probability, and now the  security risk is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  The above three cases correspond to the three subsets of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Note that the subset S2 is  empty when ca > cb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  □  From the above proof, we observe that for fixed technological costs ca and cb, the security  risk level is only higher when δ∗ is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Then as in the proof of Theorem 2, we use the  fact that the monotonicity of the security risk level is equivalent to the monotonicity of δ∗  to show the threshold property of the equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Besides, since δ∗ is also related to the  equilibrium value function V ∗, we also need the following properties of V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Proposition 4 The equilibrium value function V ∗ : Zn  ≥0 → R has the following properties:  (i) V ∗ is symmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', V ∗(x) = V ∗(σx) where σx is a permutation of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (ii) V ∗ is Schur convex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', V ∗(x + ei) ≥ V ∗(x + ej) if xi ≥ xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (iii) V ∗ is non-decreasing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', V ∗(x) ≥ V ∗(y) if xi ≥ yi for all i (1 ≤ i ≤ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (iv) V ∗ is convex in each variable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', V ∗(x + ei) − V ∗(x) ≤ V ∗(x + 2ei) − V ∗(x + ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (v) V ∗ is supermodular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', V ∗(x + ei + ej) + V ∗(x) ≥ V ∗(x + ei) + V ∗(x + ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Since the proof of Proposition 4 is analogous to the proof of Proposition 2, we omit it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Now we are ready to present the proof of Theorem 4(ii) which uses symmetry (property (i))  and Schur Convexity (property (ii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Proof of Theorem 4(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' By symmetry, without loss of generality, we only need to consider  the case when x1 = xmax, xn = xmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Besides, Schur convexity implies that V (x + e1) =  maxj V (x + ej), V (x + en) = minj V (x + ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' We can rewrite the HJB equation (15) as  V ∗(x) = max  α  min  β  �  ||x||1+cbb(x) − caa(x) + λV ∗(x + en)  +µ  �  i  V ∗((x − ei)+) + a(x)(1 − b(x))δ∗(x)  �  ,  (17)  where δ∗(x) = λ(V ∗(x + e1) − V ∗(x + en)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Since the monotonicity of the security risk  level of the states is equivalent to the monotonicity of δ∗, and implies the existence of the  threshold functions, it is sufficient to show that δ∗ is monotonically non-decreasing (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  20  non-increasing) in the largest variable x1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' the smallest variable xn) when other variables  are fixed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' that is,  δ∗(x + e1) ≥ δ∗(x),  δ∗(x + en) ≤ δ∗(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (18)  The proof of (18) also uses induction based on value iteration and can be found in Ap-  pendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  □  Last, we discuss the numerical computation of the minimax value V ∗(x) and the equi-  librium strategies (α∗(x), β∗(x)) for each state x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Based on the value iteration form of HJB  equation (15), we develop an algorithm adapted from Shapley’s algorithm [39, 2, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' See  Algorithm 2 in Appendix 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  The algorithm proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' In each iteration and for each state x, let δ(x) =  λ(maxj V (x + ej) − minj V (x + ej)) and build an auxiliary matrix game M(x, V ) similar to  (16);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' then update V (x) with the minimax value val(M) given by Shapley-Snow method:  when δ(x) ≤ ca, val(M) = ||x||1+µ  n�  i=1  V ((x − ei)+) + λ minj V (x + ej);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  when ca < δ(x) ≤ cb, val(M) = ||x||1−ca + µ  n�  i=1  V ((x − ei)+) + λ maxj V (x + ej);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  when δ(x) > max{ca, cb}, val(M) = ||x||1+cb + µ  n�  i=1  V ((x − ei)+) + λ minj V (x + ej) −  cacb  δ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  When V (x) converges to V ∗(x), we again use Shapley-Snow method to solve the matrix  game M(x, V ∗) and obtain the estimation of the equilibrium (α∗(x), β∗(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='3  Equilibrium regimes  In the last subsection, we have seen that no medium risk states when ca > cb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Then in this  subsection, we will discuss the existence of different security levels under different combina-  tions of ca and cb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' 7, various regimes correspond to particular combinations of security levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Under  large attacking cost, the attacker has no incentive to attack, then only the low risk states exist  (see regime IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' When the attacking cost goes smaller but still greater than the defending  cost (ca > cb), not only the low risk states but also the high risk states exist (see regime III)  since the attacker has less incentive to attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' As the defending cost increases to be greater  than the attacking cost (cb > ca), the defender has less incentive to defend, and now all risk  levels including the medium risk exist (see regime II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  5  Concluding Remarks  In this work, we analyze the reliability and security risks of feedback-controlled parallel queu-  ing systems and propose advice for strategic protection and defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Our stability analysis  21  shows that without secure feedback control, the queuing system is vulnerable to random  faults and malicious attacks, and our stability conditions can be used to check whether  the feedback-controlled queuing system is stable under such failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' We also propose cost-  efficient strategic defense advise for the system operator based on queue length, utilization  ratio, and technical costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Our theoretical results also provide practical insights for the  design of fault-tolerant feedback control such as vehicle navigation, signal-free intersection  control, flight dispatch, and data packet routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Interesting future directions include 1) the  extension to general queuing networks, and 2) the design of efficient (in both time and space)  algorithms for estimating the optimal policy and equilibrium strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Acknowledgement  The authors appreciate the discussions with Manxi Wu, Siddhartha Banerjee, Zhengyuan  Zhou, Yu Tang, Haoran Su, Xi Xiong, and Nairen Cao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Undergraduate student Dorothy Ng  also contributed to this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
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+page_content=' Securing infrastructure facilities: When does proactive  defense help?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
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+page_content=' Stabilizing queuing networks with model data-independent control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  IEEE Transactions on Control of Network Systems, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  [48] Jiongmin Yong and Xun Yu Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Stochastic controls: Hamiltonian systems and HJB  equations, volume 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Springer Science & Business Media, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  A  Appendices  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='1  Proof of Proposition 1  Here we provide the proof of the stability condition under the unprotected space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' The proof  uses the standard results on the subdivision of a Poisson process and stability condition of  the join the shortest queue system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  25  Proof of proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' The unprotected n-queue system has n + 1 classes of jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' The  i-th class enters server i as a Poisson process of rate apiλ (1 ≤ i ≤ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' The (n + 1)-th  class enters the n-queue system as a Poisson process of rate (1 − a)λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' when a job of this  class arrives, the job joins the shorter queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' By [18, Theorem 1], the (n + 1)-class, n-queue  system is stable if and only if  max (apiλ/µ, λ/(nµ)) < 1,  which is equivalent to (2a)-(2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Consider the same quadratic Lyapunov function (5) and apply infinitesimal generator,  we have  LW(x) =aλ1  2  n  �  i=1  pi  �  (xi + 1)2 − x2  i  �  + (1 − a)λ1  2  �  (xmin + 1)2 − x2  min  �  + µ1  2  n  �  i=1  Ixi>0  �  (xi − 1)2 − x2  i  �  =aλ  n  �  i=1  pixi + (1 − a)λxmin − µ  n  �  i=1  xi + 1  2λ  + 1  2  n  �  i=1  Ixi>0µ  ≤ (max(apmax, 1/n)λ − µ) ||x||1+1  2(λ + nµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Hence, by (2a)–(2b) there exists a constant c = µ−max(apmax, 1/n)λ > 0 and d = 1  2(λ+nµ)  such that  LW(x) ≤ −c||x||1+d,  ∀x ∈ Zn  ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  By [34, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='3], this drift condition implies the upper bound and thus the stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' □  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='2  Induction part of Theorem 2  In this subsection, we continue the proof of Theorem 2 with the induction proving (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Let ∆k(x) =  n�  i=1  piJk(x + ei) − Jk(x + em), it is sufficient to show for all k ∈ N,  ∆k(x + ei) ≥ ∆k(x),  ∀i ̸= m  ∆k(x + em) ≤ ∆k(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  One can verify that the above hold for k = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Here we consider multiple base cases to  avoid reaching trivial conclusions, say all inequalities are just equalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  26  Now we show the inductive step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' According to the value iteration (12), we have ∀j ̸= m,  ∆k+1(x + ej) − ∆k+1(x)  =µ  n  �  i=1  [∆k((x + ej − ei)+) − ∆k((x − ei)+)]  + λ[∆k(x + ej + em) − ∆k(x + em)]  + f k(x + ej) − f k(x),  ∆k+1(x + em) − ∆k+1(x)  =µ  n  �  i=1  [∆k((x + em − ei)+) − ∆k((x − ei)+)]  + λ[∆k(x + 2em) − ∆k(x + em)]  + f k(x + em) − f k(x),  where f k(x) = �n  i=1 pi min  �  cb, aλ∆k(x + ei)  �  − min  �  cb, aλ∆k(x + em)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Note that based on the induction hypothesis, we have ∀j ̸= m,  ∆k((x + ej − ei)+) ≥ ∆k((x − ei)+) ≥ ∆k((x + em − ei)+),  ∆k(x + ej + ei) ≥ ∆k(x + ei) ≥ ∆k(x + em + ei),  ∆k(x + ej + em) ≥ ∆k(x + em) ≥ ∆k(x + 2em).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Then we can conclude that  ∆k+1(x + ej) ≥ ∆k+1(x)  (∀j ̸= m)  ∆k+1(x) ≤ ∆k+1(x + em).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='3  Proof of Proposition 2  We present the proofs of property (i) and (ii) below, and omit the proofs of (iii)-(v), as we  only need symmetry and Schur covexity for proving Theorem 2 while the proofs of other  properties are tedious and analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' (i) Note that for any x ∈ Zn  ≥0,  ||σx||1= ||x||1,  {σ((x − e1)+), · · · , σ((x − en)+)} is a permutation of {(x − e1)+, · · · , (x − en)+},  27  {σ(x + e1), · · · , σ(x + en)} is a permutation of {x + e1, · · · , x + en},  then by (9) we can conclude that J∗(x) = J∗(σx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  (ii) We will use induction to prove xi ≥ xj implies Jk(x+ei) ≥ Jk(x+ej) for any x ∈ Zn  ≥0  and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  For the base cases, it is easy to verify that J0 = 0, J1(x) = ||x||1 and J2(x) = (1 +  Λ)||x||1+λ − µ �  i Ixi>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Then we have xi ≥ xj ⇒ J2(x + ei) ≥ J2(x + ej) for any x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Note  that the inequality is strict for some x, say (1, 0, · · · , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Now we show the inductive step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  According to the value iteration (12), we have  Jk+1(x + ei) − Jk+1(x + ej)  =µ  n  �  l=1  [Jk((x + ei − el)+) − Jk((x + ej − el)+)]  +λ  �  min  l  Jk(x + ei + el) − min  l  Jk(x + ej + el)  �  + min  �  cb, aλ  �  n  �  l=1  plJk(x + ei + el) − min  l  Jk(x + ei + el)  ��  − min  �  cb, aλ  �  n  �  l=1  plJk(x + ej + el) − min  l  Jk(x + ej + el)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Note that based on the induction hypothesis, when xi ≥ xj, ∀l (1 ≤ l ≤ n) we have  Jk((x − el)+ + ei) ≥ Jk((x − el)+ + ej),  Jk(x + ei + el) ≥ Jk(x + ej + el),  and thus  Jk((x + ei − el)+) ≥ Jk((x + ej − el)+),  min  l  Jk(x + ei + el) ≥ min  l  Jk(x + ej + el).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Then we can conclude that  Jk+1(x + ei) ≥ Jk+1(x + ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Therefore, the Schur convexity xi ≥ xj implies J∗(x + ei) ≥ J∗(x + ej) always holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  □  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='4  Induction part of Theorem 4  In this subsection, we continue the proof of Theorem 4 with the induction proving (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Let δk(x) = λ(V k(x + e1) − V k(x + en)), it is sufficient to show for all k ∈ N,  δk(x + e1) ≥ δk(x),  δk(x + en) ≤ δk(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  28  For the base cases, one can verify that the above inequalities hold for k = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Now we  show the inductive step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' According to the value iteration form of formula (17),  δk+1(x + e1) − δk+1(x)  =µ[δk(x) − δk((x − e1)+)]  +µ[δk((x + e1 − en)+) − δk((x − en)+)]  +λ[δk(x + e1 + en) − δk(x + en)]  +gk(x + 2e1) − gk(x + e1 + en) − gk(x + e1) + gk(x + en),  where  gk(x) = max  �  0, min  �  δk(x) − ca, cb − cacb  δk(x)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Note that based on the induction hypothesis, we have  δk((x + e1 − en)+) ≥ δk((x − en)+) ≥ δk(x) ≥ δk((x − e1)+),  δk(x + 2e1) ≥ δk(x + e1) ≥ δk(x + e1 + en) ≥ δk(x + en).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Then we can conclude that δk+1(x + e1) ≥ δk+1(x) and prove δk+1(x) ≤ δk+1(x + en) in a  similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='5  Truncated policy iteration  In this subsection, we present the truncated policy iteration algorithm for estimating stability-  constrained optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' This algorithm is adapted from the classic policy iteration algo-  rithm [42] by combining the stability condition (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' Since the original state space is countably  infinite, here we set a boundary to make the state space finite so that the algorithm can ter-  minate in finite steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  29  Algorithm 1 Truncated policy iteration for estimating optimal policy β ≈ β∗ (continuing)  Algorithm parameter: small ϵ > 0  Input: arrival rate λ, service rate µ, number of queues n, one-time protecting cost cb, the  probability of joining queues p1, p2, · · · , pn  Initialize arrays: J(x) ∈ R and b(x) ∈ {0, 1} arbitrarily (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=', J(x) = 0, b(x) = 0) for all  x ∈ X = {0, 1, · · · , B}n  # B is the queue length upper bound 1  repeat  repeat  ∆ ← 0  # Policy evaluation  foreach x ∈ X do  v ← J(x)  c ← |x|+cbb(x)  J(x) ← c + �  x′ p(x′|x, b)J(x′)  ∆ ← max(∆, |v − J(x, b)|)  end  until ∆ < ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  stable ← True  # Policy improvement  foreach x ∈ X do  old-action← b(x)  if aλ  ��  i  piJ∗(x + ei) − minj J∗(x + ej)  �  ≤ cb then  b(x) = 0 2  end  else  b(x) = 1  end  if old-action̸= π(x) then  stable ← False  end  end  until stable = True;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Output β = (1 − b)NP + bP ≈ β∗  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='6  Adapted Shapley’s algorithm  Here we present the adapted Shpaley’s algorithm for computing the minimax value and  equilibrium strategies of the attacker-defender stochastic game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' In each iteration, it builds  an auxiliary matrix game and obtains the minimax value using Shapley-Snow method [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  1Since the estimation errors are relative large at the boundary, we can set B to be larger than the real  upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  2For computing the stability-constrained optimal policy, we can modify the algorithm as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content=' In the  initialization, for every x ∈ X, if x1 = · · · = xn then continue, else set θ(x) = max  �  1−  µ||x||1−λxmin  aλ  �  n  �  i=1  pixi−xmin  �, 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  Then in the policy improvement, replace b(x) = 0 with b(x) = θ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  30  Algorithm 2 Adapted Shapley’s algorithm for estimating equilibrium strategies β ≈ β∗,  α ≈ α∗ (continuing)  Set V (x) = 0 for all x ∈ X = {0, 1, 2, · · · , B}n 1  repeat  ∆ ← 0  foreach x ∈ X do  v ← V (x)  Build auxiliary matrix game M(x, V )  Compute the minimax value val(M) by using Shapley-Snow method  V (x) ← val(M)  ∆ ← |v − V (x)|  end  until ∆ < ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
+page_content='  foreach x ∈ X do  Build auxiliary matrix game M(x, V )  Compute α and β from M(x, V ) by using Shapley-Snow method  end  31' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFJT4oBgHgl3EQfZSxF/content/2301.11530v1.pdf'}
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+ 
+Developing a Framework for Heterotopias as Discursive Playgrounds:  
+A Comparative Analysis of Non-Immersive and Immersive Technologies 
+Elif Hilal Korkut1  and Elif Surer1, *  
+1 Department of Modeling and Simulation, Graduate School of Informatics, Middle East Technical 
+University, 06800 Ankara, Turkey 
+ 
+*Corresponding Author: 
+Assoc. Prof. Elif Surer 
+Department of Modeling and Simulation 
+Graduate School of Informatics 
+Middle East Technical University 
+06800 Ankara, Turkey 
+elifs@metu.edu.tr 
++90 545 2107888 
+ 
+Elif Hilal KORKUT (elif.korkut@metu.edu.tr) is a Master’s student in the Multimedia Informatics program at Middle 
+East Technical University with a Bachelor’s degree in Architecture. She is interested in generative algorithms, 
+computational design, exhibition and display design, vision and visuality, and game technologies. 
+ 
+Elif SURER (elifs@metu.edu.tr) is currently working as an Associate Professor with the Graduate School of 
+Informatics’ Multimedia Informatics program at Middle East Technical University. Her research interests include 
+serious games, virtual/augmented reality, and reinforcement learning. 
+ 
+
+ 
+Developing a Framework for Heterotopias as Discursive Playgrounds: A 
+Comparative Analysis of Non-Immersive and Immersive Technologies 
+ 
+Abstract 
+The discursive space represents the reordering of knowledge gained through accumulation. 
+In the digital age, multimedia has become the language of information, and the space for 
+archival practices is provided by non-immersive technologies, resulting in the disappearance 
+of several layers from discursive activities. Heterotopias are unique, multilayered epistemic 
+contexts that connect other systems through the exchange of information. This paper 
+describes a process to create a framework for Virtual Reality, Mixed Reality, and personal 
+computer environments based on heterotopias to provide absent layers. This study provides 
+virtual museum space as an informational terrain that contains a “world within worlds” and 
+presents place production as a layer of heterotopia and the subject of discourse. Automation 
+for the individual multimedia content is provided via various sorting and grouping 
+algorithms, and procedural content generation algorithms such as Binary Space Partitioning, 
+Cellular Automata, Growth Algorithm, and Procedural Room Generation. Versions of the 
+framework were comparatively evaluated through a user study involving 30 participants, 
+considering factors such as usability, technology acceptance, and presence. The results of 
+the study show that the framework can serve diverse contexts to construct multilayered 
+digital habitats and is flexible for integration into professional and daily life practices. 
+Keywords: Virtual Museums, Virtual Reality, Arts and Media, Design Tools, Intelligent UI, Culture, 
+Design Research, HCI Design Theory, Multimodal UI, User Interface. 
+
+ 
+2 
+ 
+ 
+ 
+1. Introduction 
+Every culture, field, and individual creates different ways of communicating and constructing knowledge 
+by selecting certain elements, excluding others, and rearranging them in a particular order. The formation 
+and constraints can vary, but the meaning and knowledge production occur through narration and 
+interpretations of these units (Maciag, 2018). The increasing availability and accessibility of digital 
+information, made possible through processes like digitalization and digitization, has changed discursive 
+practices. Actors have been narrating emerged “formless chaos of knowledge” composed of multimedia via 
+technological systems to produce, consume, and preserve information (Manovich, 2001). While technology 
+provided accessibility and usability to users, extensive use of systems facilitated by the widespread adoption 
+of technology and the increasing reliance on personal computers and mobile phones led to a transformation, 
+elimination, and dissociation of several layers in daily life and professional practices. Odom et al. (2014) 
+describe this situation as “placelessness” and “formlessness.” In the absence of certain layers, the user 
+constantly maps the physical interactions and customizations to “virtual possessions” in the digital 
+environment. 
+Being a multilayered discursive space, the missing layers of this new territory can be explored with 
+recourse to Foucault’s heterotopias, “areas of resistance that intensify knowledge” (Foucault, 2008). In this 
+study, we investigated the layers essential for comprehensive production, representation, and interpretation 
+processes for professional and personal “archival practices” through the concept of heterotopia. Foucault 
+provides several strategies to construct discourses: “operating table,” “surfaces of emergence,” “authorities 
+of delimitation,” and “grids of specification” (Foucault, 2019). Based on layers of heterotopias and their 
+potential to integrate strategies of discourse, we provided a framework that can support users in creating 
+their digital habitats via immersive technologies incorporating individual archives. 
+Resistance of heterotopias occurs through adaptations and changes in function resulting from 
+dynamic power relations, which is an essential layer of heterotopias. The individual is regarded as a “focal 
+point of resistance” in modernity, with the ability to shape and manipulate experiences and products in 
+conjunction with the systems to which they are linked (Thompson, 2003). Together with the system, they 
+become “authorities of delimitation.” The developed framework offers the dynamism of heterotopias that 
+can be altered to fit the needs of users, with the ability to import various formats to compose a content layer 
+and modular structure based on object-oriented programming. It empowers individuals to create 
+experiences based on their content in an immersive environment where they become the authority. 
+Heterotopias provide alternative viewpoints of the world and opportunities for people to explore 
+different ways of thinking by juxtaposing and combining many spaces into one site; they problematize 
+received knowledge from the “surfaces of emergence.” The framework functions as an “operating table,” 
+it expands the space of heterotopias and enables users to interact with the elements of multimedia language 
+in changeable settings. Heterotopias are dynamic physical or virtual spaces that exist outside of the normal 
+ordering of time and space. They offer spatial layers to reorder information (Foucault, 2008). The presented 
+framework offers “grids of specification,” forming “liquid architectures” where users can create places 
+according to their needs (Novak, 1992). It includes procedural generation algorithms (PCG) that provide 
+
+ 
+3 
+ 
+ 
+ 
+different levels of autonomy while offering several architectural elements. This way, the framework 
+provides another layer that can be interpreted as the meaning of the space that contains complex relations. 
+This layer includes how the space is narrated, represented, or understood by different groups of people and 
+the contexts that give it meaning (Lehtinen, 2022). 
+Through embodiment, mixed and virtual reality technologies have the potential to create a spatial 
+layer of heterotopias by creating immersive, interactive digital environments that can mimic the feeling of 
+being in a physical place or blend the physical space with the virtual and provide a sense of place and 
+connection to their surroundings (Schultze, 2010). Virtual Reality (VR) and Mixed Reality (MR) 
+technologies are extensively studied to facilitate places and knowledge production by allowing users to 
+engage in activities and simulations of cultural heritage (Cecotti, 2022; Gonizzi Barsanti et al., 2015), 
+construction management (Hepp and Hasebrink, 2018; Safikhani et al., 2022), and architecture (Akin et al., 
+2020; Prabhakaran et al., 2022). While these studies provide the existence of spatial layers, they were 
+developed to target specific fields. Therefore, they do not offer comprehensiveness in terms of multimedia 
+and provide an operating table for only targeted fields which eliminates the possibility of using them in 
+other contexts. 
+In this study, virtual museum space was specifically chosen as an overarching term that can contain 
+various types of entities holding both place and media representation and reproduction layers. Exhibitions 
+inside a virtual museum create a “world within worlds” through narration constructed with places. Foucault 
+(2008) already defined museums as heterotopia; however, not all museums are necessarily discursive 
+heterotopias. The current applications show that large corpora consisting of virtual museums provide 
+abstract or exact copies of a real museum with isolated interactions based on predefined paths (Cecotti, 
+2022; Kersten et al., 2017). There is an increase in the number of studies improved by individualization 
+being incorporated in a static, designed, built environment (Foo et al., 2009; Komianos and Oikonomou, 
+2018). Virtual museum applications offer a wide range of multimedia elements; however, they provide 
+static functions, and personal content has been limited to images. A system that provides the technical 
+capability of integrating diverse personal media such as text, audio, and 3D graphics is not available. Also, 
+the elements are provided in digital “emplacements.” They do not offer the adaptable place production 
+required for reordering individual content. Therefore, they mostly do not provide operating table functions, 
+including the resistance presented by individuals. 
+Building on these insights, we developed a flexible framework that can provide the construction of 
+heterotopias for the broader audience and disciplines, which makes the framework usable for further 
+research and diverse practices. The spatial layers of heterotopia creation are provided via architectural 
+elements open for manipulation and texture, lighting, and scale options and supported via integration of 
+PCG algorithms, especially for novice users. Additionally, by injecting user preferences, we provided 
+different automation levels. Content layers and archival practices are provided via several algorithms that 
+can import, store, and exhibit various data formats and allow analysis and grouping of personal content. In 
+conjunction with the ability to hold various layers, the developed framework can provide a playground 
+based on heterotopias. Throughout the design process, we followed meta-design principles, which helped 
+us to create a flexible and modular framework that can shape via needs of users to serve diverse contexts. 
+
+ 
+4 
+ 
+ 
+ 
+This study compares experiences across three platforms (PC, VR, MR) to understand the effects of different 
+technologies, which can serve as further insights to increase the integration of immersive technologies. We 
+conducted a comparative user study with the participation of 30 respondents. After exposure to the system, 
+participants completed standardized presence, technology acceptance, usability questionnaires, and system-
+specific surveys. Later, we conducted semi-constructed interviews to gain better insights and interpret the 
+objective and subjective data gathered in the previous steps. Results indicate that we managed to provide 
+standard levels of usability and acceptance across technologies, and the framework is suitable for diverse 
+contexts. 
+While VR and MR technologies can offer absent layers of 2D interfaces, from the meta-design 
+perspective (Fischer et al., 2004; Lee et al., 2020), when compared to PC-based tools, immersive 
+technologies are still waiting for more pervasive use, which requires new human-computer interaction 
+(HCI) approaches to integrate those technologies into professional and personal life on a daily basis, to 
+create a comprehensive dialogue between system, user, content, and place. 
+To derive design features that can increase the integration of immersive technologies into different contexts, 
+this study aims to provide answers to the following questions: 
+ 
+R1. In the creation process of heterotopias, which medium provides better experiences in terms of 
+usability, presence, and technology acceptance? 
+ 
+R2. According to which dimensions do technology acceptance, usability, and presence of versions 
+present differences? 
+ 
+R3. Which medium is more applicable for the interaction with different asset types? 
+ 
+R4. How do different mediums change the design approach of users? 
+ 
+R5. How do scale, point of view, lighting, and material aspects affect users’ experiences and tasks? 
+ 
+R6. What are the effects of the autonomy level of procedural generation algorithms on users’ 
+preferences? 
+In what follows, we review works related to immersive technologies, virtual museums, and 
+procedural generation techniques, and to facilitate readability and to preface our user analysis, we briefly 
+outline the conceptual terms that emerged through technology. Then, we describe each aspect of the 
+proposed application and the rationale behind the design decisions. In the next part, we explain our 
+evaluation methods and procedures. Then, we provide the results of the user tests about related concepts 
+from various fields. We discuss the results while reflecting on the extended concepts and terms that emerged 
+through user tests. Finally, to conclude, we summarize our study and add insights into how further details 
+might be searched. 
+2. Literature Review 
+2.1. Discourse and Heterotopias 
+Michel Foucault was a French philosopher who studied the ways in which power and knowledge shape 
+society. He identified certain places, called heterotopias, that exist within a culture and serve as a sort of 
+
+ 
+5 
+ 
+ 
+ 
+“realized utopia” where other places within the culture are represented, disputed, and reversed. Heterotopias 
+are characterized by their ability to juxtapose and compare various spaces and concepts and can manifest 
+in various architectural forms, functions, and sizes (Foucault, 2008). 
+According to Michel Foucault, discourse, or systems of thought and representation, plays an 
+important role in shaping our understanding of the world. He identified several strategies for constructing 
+discourses: “operating table,” “surfaces of emergence,” “authorities of delimitation,” and “grids of 
+specification.” The operating table represents the space in which knowledge is constituted serving as a 
+means of classifying and ordering things, allowing us to make sense of them. Surfaces of emergence are 
+the foundation for the development of new discourses and the creation of new subjects, while an authority 
+of delimitation is a set of rules that determines how objects can become subjects within a discourse. The 
+grids of specification are a taxonomy of concepts used to order objects into a hierarchy within a discourse. 
+By understanding these strategies, Foucault explained how discourses shape our understanding of the world 
+and our actions within it (Foucault, 2005). 
+Heterotopias are distinctive spaces that can contain the processes of discourse. Michel Foucault 
+identified various characteristics of these spaces and provided examples to help illustrate and understand 
+these features. According to Michel Foucault, all cultures produce heterotopias of deviation and crisis such 
+as boarding schools and prisons. Depending on the culture and time period, these spaces may serve a variety 
+of purposes within a society, and their features and functions evolve over time, which provides resistance. 
+He argued that resistance is not separate from power relationships but rather is an integral part of them. 
+Power is not a fixed entity but is constantly negotiated and contested through interactions, leading to 
+changes in the functions of heterotopias (Lehtinen and Brunila,  2021). Heterotopias are spaces that can 
+bring together multiple incompatible spaces and elements in a single location, disrupting the way we 
+understand the world. Foucault also investigated the “order of discourse” as the organization of the rules 
+and systems that govern knowledge production (Foucault, 2005). Heterotopias offer alternative ways of 
+organizing information and the layout of a space can affect the way knowledge is shared and understood, 
+providing “grids of specification.” The characteristics and purpose of these spaces influence the meanings 
+and perceptions that can be experienced through embodiment. 
+Heterotopias are spaces that are connected to certain periods in time. They can either permanently 
+store time, like archives or libraries, or they can provide a brief window into a specific time, like exhibitions 
+or festivals. In modern society, the individual becomes the “focal point of resistance,” and archives, 
+libraries, and museums often represent personal choice and “self-formation,” with the individual acting as 
+a point of resistance to the shaping power of the larger system they exist within. These individuals become 
+“authorities of delimitation,” shaping their own products within the constraints of the system (Foucault, 
+1972). 
+Heterotopias are places where access is controlled, meaning that they are not freely accessible like 
+public spaces. This control can be achieved through technological means, such as the use of devices that 
+have opening and closing systems or personal accounts. Heterotopias are sites that have a particular 
+connection to the environment around them. They may either expose the essence of other real spaces or 
+
+ 
+6 
+ 
+ 
+ 
+create an organized space that is distinct from the disorder of other places. In the context of discourse, this 
+relationship creates “surfaces of emergence,” where new ideas, arguments, or perspectives arise or come to 
+the forefront of the discourse (Foucault, 1972). 
+2.2. Heterotopias and Digital Playgrounds 
+Rousseaux and Thouvenin (2009) explored Informed Virtual Sites (IVS) through Michel Foucault’s 
+heterotopias by superimposing digital and physical spaces. They also named these heterogeneous places 
+heterovirtopias as an extension of Foucault’s terminology. The method of loci, also known as the mind 
+palace, is a memory technique that allows people to recall information by associating the space they know 
+well with the information they want to remember through the connection between object and subject. This 
+memory technique primarily employs spatial memory to remember information efficiently. As another 
+example of informational topologies, Yamada et al. (2017) developed a system called HoloMoL using the 
+mind palace method to help users to memorize by combining information as mixed reality content with 
+physical places. 
+The museum’s potential is not solely derived from its exhibits, but rather from the interactions 
+between the objects on display and the visitors. These interactions, along with the stories and communities 
+that are formed within the museum, contribute to the vitality of both the collections and the visitors. The 
+viewers’ engagement with the objects endows them with meaning and with a sense of symbolic immortality. 
+The museum also possesses a transformative power and an allure due to the mysteries contained within its 
+unseen artifacts and untold stories. Malraux (1967) demonstrates how formal museums compromise the 
+essence of works of art by clustering them together, diminishing the importance of individuals. He proposed 
+a fictitious museum composed of photographs of works of art. Our society is evolving toward a more 
+mediated culture. 
+Following the heterotopia concept, a paradigm shift in terms of contemporary senses has occurred 
+within the museum structure. Online platform projects, such as Google Art and WikiArt, provide high-
+resolution images of selected artworks available to the public all over the world. Although there is a 
+noteworthy difference between “Museum without Walls” and today’s virtual museums on the Web, they 
+have a similar conceptual purpose regarding transforming information and knowledge into forms that are 
+available regardless of distance from it. The concept of the Digital Museum emerged in conjunction with 
+the museum’s expansion beyond physical places through the increased use of new media technologies. 
+Various factors, including media and digital resources, influence the organization of virtual museums. 
+According to Schweibenz (2019), the primary distinction between virtual museums and traditional physical 
+museums is their level of accessibility. 
+With the adoption of digital media and a narrative focus, known as the “narrative turn,” “the 
+memory institutions” galleries, libraries, archives, and museums sector has undergone a transformation. 
+Using technologies like virtual and augmented reality, 360-degree photography, and 3D reconstructions, 
+they have transformed and created places and increased their use of digital spaces including websites 
+(Basaraba, 2021). Immersion and interactivity in immersive environments aim to increase the sense of 
+
+ 
+7 
+ 
+ 
+ 
+presence in a virtual environment; nevertheless, this does not mean that the digital environment is entirely 
+composed of fictional elements. Reconstruction of physical artifacts into digital media uses 3D data 
+acquisition methods, such as photogrammetry and laser scanning. These methods are frequently employed 
+in virtual museums, virtual exploration, and cultural heritage contexts (Gonizzi Barsanti et al., 2015; Haydar 
+et al., 2011; Pietroni et al., 2013). Hayashi et al. (2016) developed a virtual museum capable of displaying 
+planar artifacts using web scraping to extract necessary information. The system allows users to choose the 
+museum’s content that is designed by authors. MR systems in museums can enhance the typical visitor 
+experience by combining historical interactive visualizations with related physical artifacts and displays. 
+MuseumEye application (Hammady, 2018) focuses on different guidance techniques to improve visitors’ 
+experiences. Providing adaptable exhibition spaces, Komianos et al. (2018) provided automated virtual 
+exhibition construction based on adaptive exhibition topologies. They state that the facilitation of visitors’ 
+navigation can enhance users’ visiting experiences. 
+Virtual museums and immersive cultural heritage studies can be interpreted as heterotopias of 
+deviation and crisis. The risk of extinction due to natural or unnatural factors has created a need for the 
+reconstruction of artifacts in a digital medium. The deviation has been occurring due to technological 
+advances that have redefined almost every aspect of various practices. Additionally, COVID-19, as a 
+worldwide crisis, created boundaries between people and places, which accelerated virtual museum studies 
+to provide accessibility. 
+Place-making activities have been provided by games for decades, where players can shape and 
+influence the virtual spaces within the game. These activities can range from building and constructing 
+structures and landscapes to customizing and decorating the appearance of these spaces (Basabara, 2021). 
+For individuals, they provide an opportunity to express creativity and personalize their virtual spaces and 
+foster a sense of ownership and attachment to the game world, as players feel a sense of pride and 
+accomplishment in creating and shaping their own spaces. Boldi et al. (2022) explore the technologies of 
+crisis through video games. According to their study, the COVID-19 pandemic caused routines to be 
+confined to people’s homes, leading some to feel disconnected from their usual places of habitation and to 
+turn to video games as a way to escape. Some people used games to explore virtual worlds that were 
+different from their everyday lives, while others used games to recreate or substitute for meaningful places 
+that were no longer available to them. Researchers suggest that games or online communities could be 
+designed to allow people to explore and customize virtual spaces to strengthen their attachment to them and 
+recreate lost places to help people find new opportunities and meanings in their everyday environments. 
+2.3. Immersive Technologies  
+Virtual, augmented, and mixed realities have been widely studied and have seen a range of applications in 
+various fields. While virtual reality has a well-established definition, mixed reality has been described using 
+a variety of terms, leading to ambiguous definitions. Mixed reality combines elements of both augmented 
+and virtual reality, with the degree of combination determining where it falls on the reality-virtuality 
+continuum (Milgram and Kishino, 1994). Virtual reality is a technology that uses real-world visual 
+perception in artificial computer-generated environments through stereoscopic vision, providing an 
+
+ 
+8 
+ 
+ 
+ 
+immersive experience with motion capture. Paul Milgram and Kishino propose the terms “augmented 
+reality” and “augmented virtuality” to describe environments closer to the center of this spectrum. 
+Augmented reality maps virtual elements onto physical space to create a hybrid of the real and virtual, often 
+through smartphones. Mixed reality technology uses head-mounted displays to combine AR and VR 
+capabilities, such as Microsoft’s HoloLens. 
+2.3.1. Interface and Interaction 
+The user interface (UI) is the means of communication between a user and a computer system or application. 
+Most personal computers use a graphical user interface (GUI) which uses visual elements such as icons, 
+buttons, and menus and input and output devices like a mouse and keyboard to display a screen. Different 
+types of interfaces include form filling, which involves the user entering data into a pre-defined form or 
+template, and direct manipulation (Shneiderman, 1983), where the user interacts with on-screen objects in 
+a way that directly affects their behavior. Command languages are sets of instructions or commands that a 
+user can enter to interact with a computer or software program, often used in text-based interfaces such as 
+in a terminal. Natural language processing (NLP) allows users to communicate with a computer or software 
+program using natural language rather than specific commands, often used in voice assistants and chatbots 
+(Gilbert, 2019). 
+Immersive environments may require more interactive interfaces with specialized input devices 
+such as motion controllers or haptic feedback devices. These interfaces can include head tracking, hand 
+movement tracking, voice control, eye tracking, body movement tracking, and virtual hand manipulation. 
+To effectively manipulate objects in virtual reality environments, interaction techniques should allow for 
+object selection, positioning, and rotation. The design of these techniques, often involving hand 
+manipulation, is important for the overall user experience of a virtual reality environment (Mann et al., 
+2022). 
+2.3.2. Presence, Embodiment, and Experiencing Architecture 
+The human body serves as a system for acquiring, processing, and displaying information from the physical 
+world through the senses. Virtual worlds are digital environments that users can interact with using 
+technology that differ from the physical world. They require technology to experience and interact with 
+objects, spaces, and people. Presence refers to the sensation of being physically present in a given space 
+and time, often experienced in virtual reality environments (Schultze, 2010). Factors that can affect the 
+level of presence include the realism of the sensory stimuli in the environment, the believability of the 
+environment as a real space, the user’s level of engagement with the environment, and the user’s prior 
+knowledge and expectations. Virtual worlds offer a chance to examine the role of the physical body in the 
+communication and the effects of communicating without a physical body (Spence, 2020). Merleau-Ponty’s 
+philosophy (Lehtinen, 2022) emphasizes the concept of embodiment, or how the body shapes and is shaped 
+by our perception and experience of the world. He argues that the body is not simply a means of engaging 
+
+ 
+9 
+ 
+ 
+ 
+with the outside world but also plays a role in processes of perception, understanding, and expression. The 
+body is therefore central to our subjectivity and understanding of the world. 
+In “Experiencing Architecture,” Rasmussen (1964) argues that the sensory experience of a building 
+or space is crucial to our understanding and interaction with the built environment. He believes that 
+architecture should be evaluated based on how it is experienced by the user, rather than just its aesthetic or 
+functional qualities. Rasmussen explores how various sensory experiences, such as sight, sound, touch, and 
+even smell and taste, contribute to our perception of a building or space. He explains that the way we 
+experience architecture is based on the perception of solids and cavities, figure-ground relations, rhythm, 
+scale, proportion, sound, lighting, color, and texture. He divides the process of creating and interpreting 
+architectural forms into two categories: solid-minded and cavity-minded. Solid-minded architecture 
+involves creating forms by combining solids or shells of voids, while cavity-minded architecture involves 
+carving out components from a large solid to create spaces. Rasmussen describes architecture as “the art of 
+playing with solids and cavities.” 
+According to Luck (2014), the design process often involves behaviors that may seem incongruous 
+but serve a deeper purpose. These movements, which pertain to the design of a structure, symbolize 
+something and provide a fleeting visual representation of architectural concepts and the anticipated 
+sensation of movement within the architectural form. This process is referred to as “aesthetic becoming,” 
+and reflects the creative process behind the emergence of these behaviors. The relationship between a 
+building’s physical form, how it is experienced, and the reactions it elicits are subjective, intuitive, and 
+complex. 
+2.3.3. Usability, Acceptance, Trust 
+Technology acceptance refers to the willingness of an individual or organization to use a particular 
+technology. By understanding the factors that impact technology acceptance, designers and developers can 
+create more effective and appealing technologies (Rheingold, 1991). Usability is a measure of how easily 
+an individual can use a system or product to achieve their goals. It affects the performance, efficiency, and 
+satisfaction of the user, as well as their overall experience. Factors that contribute to the usability of a 
+system include the cognitive abilities and limitations of the user, and the design of the system’s interface, 
+organization, and structure (Salanitri et al., 2015). 
+The concept of trust in technology has been studied extensively in the field of human-computer 
+interaction (HCI). Previous research has shown that trust in technology is influenced by factors such as 
+usability, technology acceptance, and presence and that these factors also affect each other (Lippert and 
+Swiercz, 2005). Trust in technology is a multi-dimensional concept, and researchers have identified several 
+dimensions that contribute to trust between users and technology. These dimensions include predictability 
+(the ability of technology to adhere to previously established performance standards), reliability (the 
+perceived dependability of technology in certain situations), and utility (the perceived usefulness of 
+technology) (Mcknight et al., 2011). 
+
+ 
+10 
+ 
+ 
+ 
+2.4. Procedural Content Generation 
+The automatic creation of digital assets through algorithmic means and patterns with little to no user input 
+is known as procedural content generation (PCG). In the gaming industry, algorithms are employed to create 
+complex items like road networks, buildings, living things, as well as landscapes and plants. PCG is being 
+studied in fields other than computer science, interdisciplinary approach in the scientific sciences, including 
+biology, architecture, urban studies, and psychology, is triggering rising interest in other communities, and 
+it has become increasingly prominent (Prusinkiewicz and Lindenmayer, 2012). As the importance of PCG 
+for production increases, researchers are exploring new ways to produce high-quality assets, either with or 
+without human input; therefore, new paradigms have also risen. Various techniques have been produced 
+through machine learning (PCGML) and deep learning, such as neural networks, auto-encoders, and deep 
+convolutional networks; Markov models, n-grams, and multi-dimensional Markov chains; clustering; and 
+matrix factorization (Summerville et al., 2018). 
+Designers are increasingly utilizing autonomous tools to complete complex tasks faster and in novel 
+ways. However, the deterministic nature of these methods can produce similar results repeatedly and may 
+not provide the same level of creative user experience as hand-crafted and designed content. Most recently 
+developed techniques aim to create fully autonomous approaches. Many researchers have proposed 
+methods for the automatic generation of floor plans and buildings, but only a minority allows for 
+customization of the end product of the algorithm. Products that are not open to customization have been 
+criticized in the field of architecture (Porter and Hanna, 2006). In this study, we aim to offer greater control 
+over the system on demand by manipulating the autonomy level of the algorithms and providing a “design 
+space” for the user to customize the end product. Therefore, we sought methods that offer various autonomy 
+levels and design space to the user. 
+Previously, large-scale procedural modeling of virtual worlds typically only resulted in empty 
+structures devoid of internal divisions and interconnections. Different approaches have been proposed to 
+solve this problem, including tile placement, room enlargement, inside-outward extension, and subdivision. 
+Using tile placement strategies, the domain is partitioned, and the resulting grid is filled with tiles 
+representing the rooms. Without relying on the limits of a building as a constraint, inside-out algorithms 
+distribute rooms in accordance with the requirements for room connection. Growth-based algorithms 
+disperse room seeds throughout a building, and the rooms gradually fill the interior space and grow to their 
+full size (Camozzato, 2015). A constrained growth-based method for floor plan generation was presented 
+by Lopes et al. (2010). In their method, the sizes and positions of the rooms are defined by the user. The 
+rooms are then expanded until they become square in shape, and the empty space is then filled with further 
+irregular shapes by moving the rooms. Given that the floor plan can be divided into subdivisions, it is 
+possible to connect adjacent rooms, and subdivisions with doors and windows are also generated for each 
+room. They indicated that minimal complexity and effective data structures are the main reasons for the 
+success of their methods. Graph Approach to Design Generation (GADG), a method for automatically 
+creating rectangular floor plans based on an existing graph extracted from floor plans, has been proposed 
+by Wang et al. (2018). They developed a mechanism to manipulate the rooms based on two transformation 
+
+ 
+11 
+ 
+ 
+ 
+principles, addition, and subtraction, by mapping the floor layouts to connectivity graphs. Users can specify 
+the maximum width-to-height ratio for each created room using the algorithm. 
+Building boundaries are used by subdivision algorithms to divide the interior space into rooms. A 
+technique called binary space partition (BSP) uses hyperplanes to recursively divide a space into convex 
+sets. This subdivision becomes a representation of the scene as a tree data structure known as the BSP tree. 
+Baron (2017) examined many methods for creating procedural material that may be applied to both 2D and 
+3D projects. They created pairings by using five-room and corridor creation algorithms (Random Room 
+Placement, BSP Room Placement, Random Point Connect, Drunkard’s Walk, and BSP Corridors). For 
+indoor contexts, Yang et al. (2022) presented a reconstruction strategy for room layouts. To create the 
+optimum in-door polygonal models, they integrate voxel-based room segmentation and space partitioning. 
+The technique involves room semantic data to divide subspaces. 
+Cellular automata (CA) is another technique used for architectural generation. Cellular automata 
+apply a set of rules to cells, and each cell executes those sets of rules with respect to its neighbors. Cells 
+can acquire a finite number of cell states. Herr and Kvan (2007) show how a variety of cell shapes and sizes 
+used in one CA model may support architectural form findings. 
+Additionally, they proposed a theoretical framework for the integration of CA into the design 
+process. Araghi and Stouffs (2015) explore CA systems for high-density residential building forms. Based 
+on the capability of CA to create complex rule definitions, they also address solutions for architectural 
+problems such as density, accessibility, and natural light via rule definitions. Cruz et al. (2016) discuss 
+variations of classic CA cell shapes to derive a variety of architecturally feasible forms capable of 
+generating aggregate spatial units to match specific spatial configurations. 
+3. Design Rationale and Structure of the Framework 
+Based on the literature regarding heterotopias and discourse, we generated a set of rules to provide 
+consistency in methodology and a comprehensive framework structure. The system should (1) have 
+dynamic power relations and be open to changing its functions to create resistance; (2) provide 
+juxtapositions of multimedia elements on an “operating table” that allow certain interactions; (3) provide a 
+ground for “self-formation” and a “focal point of resistance” through individuals who construct “authorities 
+of delimitation” together with the system; and (4) provide a relationship with other places by either creating 
+an illusionary space or an orderly space composing “grids of specification”; (5) able to create “liquid 
+architectures” according to needs of the users that can convey spatial qualities that are perceived via 
+embodiment. 
+Throughout the methodology, we emphasized end-user development and meta-design principles to 
+empower users, create a design space, and encourage informal use cases for novice and experienced users. 
+Therefore, to simplify the mechanisms and the features that we integrated to transform users into curators 
+of their digital environments, they were carefully selected following the previous studies on multimedia, 
+
+ 
+12 
+ 
+ 
+ 
+architecture, and immersive technologies. In the subsequent sections, implementation details for VR, MR, 
+and PC technologies are given in accordance with the flow and the layers provided. 
+3.1. Virtual Museum as an Operating Table 
+Different form the previous studies, the virtual museum concept in this study is not referred to as a 
+static “emplacement” in digital space. On the contrary, it becomes a dynamic structure, an operating table 
+where the user operates on elements by including, excluding, and transforming them. Designing a virtual 
+museum can be viewed as an infinite process that begins with documentation and continues with 
+imagination and experience in the virtual environment, forming a multilayered discursive space. When 
+personal data is categorized and stored in a digital space, it can be seen as a digital archive. If it is displayed 
+or played, it is considered an individual exhibit; if multiple files are displayed within a context, it is 
+considered a digital exhibition. As it provides a categorization of documentation and displays archives in 
+the form of exhibitions via the built environment, virtual museum space can transform individual archives 
+into personal exhibitions creating “grids of specification.” 
+Through the experience, we used the terminology museum spaces to increase usability and decrease 
+the complexity of the framework. We defined two different modes for the application, namely curator and 
+visitor modes. Curator mode defines the active creation of heterotopias, while visitor mode excludes the 
+control layers. The framework provides a system for users to store and display digital content, including 
+text, images, videos, 3D models, and audio, in a way that allows for easy access and manipulation. To be 
+able to categorize items, we named the assets according to museum conventions, such as painting, sculpture, 
+etc., and added labels to provide more data. We used asset style, artist, location, and time data to provide 
+users with automation in reordering systems of heterotopias. On the other hand, the system is open to 
+manipulation and interpretation. For example, the four variables that are used for sorting and grouping can 
+be systematically employed by the user for different variables. 
+3.2. Dialogue Between the User and the System 
+The hardware provided, the general structure of the system, the interface design, interaction, and 
+navigational capabilities all directly affect the experience. For example, hardware defines the physical 
+comfort level, interaction modalities, and technical borders. The dialogue between the interface and the 
+user affects multiple features of the heterotopia created by the user and shapes the user experience. Through 
+the interface, the user grasps the level of autonomy. Interface, interaction, and navigation provide the 
+narration for the discursive space and sense-making processes. 
+3.2.1. Programming Language and Development Environment 
+To eliminate the results stemming from hardware differences in VR and MR, an Oculus Quest 2 HMD 
+system was used to stream the audial and visual content. With the feature called Passthrough, using the 
+sensors on the headset allows seeing a real-time view of the surroundings together with the virtual content. 
+
+ 
+13 
+ 
+ 
+ 
+The main reason that we used the broader term mixed reality instead of augmented reality in this study is 
+the interaction and anchoring capabilities of the device, which combines AR and VR. The Oculus Insight 
+tracking system is developed by Facebook Inc. and made available on the Oculus Quest 2. Oculus Insight 
+tracking relies on three streams of sensor data. Continuous streams of data provide hand gesture recognition, 
+physical context awareness, and the position of controllers and hands. The Oculus Quest 2 Controllers as 
+interactivity devices and hand tracking were used to allow the users to navigate through the virtual 
+environment (VE). The buttons on the controllers were visible in the virtual environment, including the 
+hand position. The Oculus Quest 2 tracks the head movement to present the correct virtual world image to 
+the eyes (LaValle et al., 2014), and it analyzes the user’s head movement in real-time to control the view, 
+which enables natural interactions, leading to high levels of presence and immersion (Desai et al., 2014). 
+The HMD includes an adjustable head strap that sets the hands free for the controllers. The combined weight 
+of the HMD (503 grams) and controllers (126 grams/per controller) is 755 grams, which facilitates 
+comfortable use for extended periods. 
+All versions of the framework were developed using Unity. Unity game development platform was 
+selected because of its prominent features among the other opponents in the industry, and it is one of the 
+natural development platforms for Oculus Quest 2. The Unity game engine is compatible with various 
+platforms including desktop, mobile, console, web, and more. Unity supports C# programming language 
+which is designed for Common Language Infrastructure by Microsoft. It is a general-purpose, multi-
+paradigm, and object-oriented language that allows developers to build applications that run on the .NET 
+Framework. A programming paradigm called “object-oriented programming” (OOP) is based on the idea 
+that “objects” can hold both data and code that manipulates that data. An object in OOP is a self-contained 
+unit composed of both information and programming code. The code that operates on objects is created to 
+reflect the behavior of real-world objects, and objects are used to represent real-world concepts or objects. 
+OOP is characterized by the use of encapsulation or the bundling of data and the code that manipulates it 
+into a single unit or object. Using encapsulation, the framework becomes open for transformation and 
+additional features. This changeable structure gives the framework adaptability of the heterotopias. Besides 
+Unity, the Microsoft-driven project Mixed Reality Toolkit (MRTK) provides a set of components and 
+features used for the VR and MR parts of the application. The 3D models were created in RhinoCeros and 
+Blender and enhanced by 3D modeling distribution web environments which provide copy-left models such 
+as CG trader, Sketchfab, and TurboSquid. 
+3.2.2. Interface and Interaction 
+The interface is designed to guide the user through the experience and construction of heterotopias layer by 
+layer. Different types of layers are provided via different planes and sequentially presented. For the interface 
+design, we preferred techniques especially suitable for novice users. For example, although the command 
+line is very flexible since the learnability of command languages is generally very poor and not suitable for 
+non-expert users, we did not include any command language interaction. The form fill-in interaction style 
+was aimed at a different set of users than the command language, namely non-expert users. We used a form 
+fill-in type of design for the tasks that require several data entries from the user. For example, the size 
+
+ 
+14 
+ 
+ 
+ 
+properties of generation algorithms (Figure 1) and URL for connection are given in the style of form. To 
+guide the user via the predefined rules and simplify the data entry, instead of using empty input fields, we 
+preferred to use sliders to define and show the range directly to the user when it is applicable. 
+Menus are collections of options that are displayed on the screen, and, upon selection and execution, 
+affect the system’s state. The user chooses a command from a predetermined list of commands listed in 
+menus using a system based on menu selection. Although it can slow down the frequent users, we prefer to 
+use menu interactions in most cases since it is ideal for novice or intermittent users. To increase 
+memorization and eliminate language boundaries instead of labels, we used icons for the command/menu 
+items. To decrease the information overload and visual clutter, the items were clustered into sub-menus 
+according to their functions (Figure 1). In the VR and MR versions, if the feature is not toggled off, menus 
+follow the users as they move. In case of losing the main menu, a hand menu comprising the restart, quit, 
+and call main menu options was included, which can be called via a hand gesture and a button on the 
+controller (Figure 2). The 3D menus designed for VR and MR also allow users to interact via touch. 
+ 
+Figure 1. Example Interface options of the PC version. 
+We employed the direct manipulation principle for all objects that are placeable to the grid and 3D interface 
+elements. Selected objects are highlighted and can be directly manipulated via hands, controllers, or mouse 
+and keyboard according to versions. Direct manipulation is especially important for the embodiment aspects 
+since it requires movement of the body. With this division, we aim to prevent unintended changes in models 
+and remove unnecessary UI elements from the vision of the user, which, otherwise can disturb the users’ 
+experiences. The user experience for all versions starts with a welcome screen and continues with a panel 
+that is used for importing the digital archive. The archive that is analyzed and environmental elements to 
+construct places are provided to the user with a series of panels to construct the discursive habitat. While 
+the PC version uses the keyboard and mouse as input, for VR and MR versions, we offer controllers and 
+hand interaction. The gestures are components of human-computer interaction that have become the subject 
+of attention in multimodal systems. In this study, we used six hand gestures, namely point, select, release, 
+teleport, call hand menu, and call main menu (Oculus). Hand interaction is provided with far and near 
+interaction options. For all versions, the user can manipulate the elements including 3D UI elements, via 
+scale, rotate or translate operations. 
+
+Q8
+B
+800 
+15 
+ 
+ 
+ 
+ 
+Figure 2. Hand Menu and Interfaces in VR environment. 
+3.2.3. Navigation 
+In authoring personal museum experiences, the users should be able to fully control their navigation 
+in the museum and they should be able to freely explore and manipulate the objects to create their own 
+virtual experiences in an interactive and flexible way. In the curation process, the user is not just an observer 
+but can interact with the exhibits via constructive dialogue. The overall process and navigation of the user 
+in the virtual museum enhances the understanding and keeps the interest alive by enriching the aesthetic 
+sensitivities. 
+During the design process, we tried to make sure that all versions were as similar to each other as 
+possible. However, inevitably, we used different methods in terms of navigation and interaction between 
+the PC version and other versions. The PC version includes four types of cameras: first-person, top-down, 
+isometric, and isometric bird view. All cameras can be controlled by W, A, S, and D buttons, and in the 
+top-down view, zoom-in/out options are available with a mouse scroll. Also, there is a transport toggle in 
+the control menu which allows the user to teleport all cameras in the scene with the left mouse click. The 
+gestures are components of human-computer interaction that have become the subject of attention in 
+multimodal systems. Gestures are captured through cameras on Oculus Quest 2 and can recognize various 
+gesture types. Inside the VE, users can navigate, walk or teleport themselves using controllers or hand 
+gestures. The tags are used to define colliders that allow teleportation. Objects with floor and ground tags 
+allow teleportation. To eliminate motion sickness, we did not add movement via controllers. 
+3.3. Content Layer as an act of Self-Formation and Archival Practices 
+Being a force of resistance, individuals should determine the elements of the “surfaces of emergence” as an 
+act of self-formation. Archival practices as a method of analyzing the discourse and its elements comprise 
+the actions of self-formation on an operating table. To provide automation for self-formation activities, we 
+constructed a pipeline that includes importing individual archives and transforming them into operable 
+artifacts. We also provided an analysis of the archives based on grids of specification of museums. 
+
+Can
+Dat
+1961StyfeMagicRealiem
+Genve:poevait 
+16 
+ 
+ 
+ 
+3.3.1. Importing Files and Authority of Delimitation of the Framework 
+The first layer of delimitation of the elements will be represented on the operation table defined by the 
+framework in terms of media types. As the elements of multimedia language, five types of assets and their 
+museum equivalents have been identified. Texts in .txt, images in .jpeg and .png format, 3D models in .obj 
+and .fbx format, audios in .mp3 format, and videos in .mp4 format are acceptable for generating the artifact 
+displays. Within these constraints, being the focal point of resistance, the users are enabled to create a 
+personal collection of 3D digital exhibits according to their interests and preferences. For the PC version, a 
+custom file browser is written, which allows the user to choose the folder consisting of the files that will be 
+uploaded to the application. For the VR and MR versions, instead of using a file browser due to the security 
+properties of the device, the application pulls the data from Google Drive. To provide this feature, we 
+include the virtual keyboard provided by Oculus and an input field where the user can write only the file 
+ID of the data file. Using the ID, a direct link for Google Drive is constructed for a data file in .JSON 
+format. The data file needs to include the file IDs of the objects to be downloaded into the database of the 
+framework. 
+3.3.2. Artifact Classes 
+In addition to assets, the user can provide a data file in .CSS or .JSON formats which include certain features 
+of the artifacts. Data columns are defined as the name, artist, style, location, time, size, and description. 
+Five different classes were constructed inheriting from the base class according to acceptable asset types, 
+which are paintings, sculptures, videos, sounds, and text. Each art object has common properties, such as 
+name, description, artist, and style. According to the class, different properties are attained. For example, 
+paintings and sculptures have location and size properties. According to file extensions, the tool copies the 
+files into the data folder and groups them into artifact groups. If the data provided by the user is inside the 
+folder, data is parsed into columns and for each line, a new art object instance is constructed according to 
+its class. Algorithm searches for matching names comparing the data and files and constructs dictionaries 
+and separate lists for the art objects. If no data file is provided, instantiated objects use the filename as a 
+name and are placed without a description panel and with a generic scale. After importing processes 
+according to extensions of the files, buttons are constructed for each file under the corresponding panel to 
+spawn the artifacts in virtual environments. For the paintings, sprites of the buttons use the file directly, for 
+other types of icons indicating the type are used. The name of the button is generated according to the name 
+of the files. The placement of all objects will be explained in the grid system. 
+3.3.3. Artifact Holders 
+Following the end-user development, the system should provide automation to transform the assets of the 
+individuals into artifacts that are open to manipulation so that users can benefit by making changes, plans, 
+and experiments with the artifacts and finding the best position for them both in the virtual and the real 
+environment. Based on features of OOP, for each artifact type, an artifact holder is designed which contains 
+
+ 
+17 
+ 
+ 
+ 
+several scripts and elements that are necessary for the interaction. Holders are higher entities in the 
+hierarchy, placed under the manipulation of the object’s position, rotation, and scale. Each holder consists 
+of a label panel where the short descriptions are placed, the Game object that will contain the assets, the 
+Details button to open the ArtCanvas panel, a collider to manipulate, and a grid object collection script to 
+fix the space between artifact and label. If the dimensions of a painting or sculpture asset are provided, the 
+container object of the asset is scaled according to data, and the space between the artifact display and label 
+remains stable. Additionally, each artifact holder has asset-specific features. For example, holders of video 
+and sound have a play and pause button (Figure 3). 
+ 
+Figure 3. Artifact Holders in VR Environment. 
+For each class, artifact holders are designed as prefabs. When an artifact button is pressed, an 
+artifact holder is constructed according to the related prefab. If data is provided, the name, artist, style, time, 
+and location columns are given on the label. Also, we provide an ArtCanvas panel where the information 
+in the description column is given with the artifact. The video, text, and painting artifacts are visually 
+represented on the left side of the panel. For the sound and sculptures, icons are placed. On the right side 
+of the canvas, the details section is provided. Users are also able to see a list of all assets they uploaded as 
+a list in the artifact analysis panel where the information is provided based on the categorization of artifacts. 
+3.3.4. Analysis of Artifacts and Grids of Specification of the Framework 
+The first classification of the assets is performed according to formats to provide an accessible archive. If 
+the data is provided, it is parsed into columns and for each class, artifacts are grouped according to four 
+specifications. The framework offers three strings and one integer for archival indexes which are named in 
+accordance with the terminology of museums and named as artist, style, location, and time. The constructed 
+lists are provided through an artifact analysis panel. These groups are also prepared for the PCG algorithms 
+to calculate the dimensions of the rooms according to grouping options. On the other hand, based on the 
+data types, users can generate different types of specifications. 
+
+目 
+18 
+ 
+ 
+ 
+3.4. Spatial Layers as an Act of Self-Formation and Architectural Elements 
+The primary purpose of using architectural elements in a virtual museum context is to reorder the 
+multimedia elements constructing spatial grids of specification. In this study, the idea of a “virtual museum” 
+does not necessarily indicate a traditional built environment. On the contrary, it extends the space and forms 
+a “liquid architecture” around the user. Every digital object that is placed, architectural or not, becomes 
+information. Beyond being a partition for artifacts we offered place-production practices to provide 
+“aesthetic becoming” experiences as an act of self-formation. 
+The layers of space in a heterotopia demonstrate how complex and multifaceted these spaces are, 
+with different meanings and functions. These layers can be thought of as different levels of reality that are 
+provided by various technologies. One layer of spatial meaning in a heterotopia is its actual physical 
+location in the world and its meaning within context. PC, VR, and MR technologies provide different sets 
+of relations with places and place production. The relationship with the physical location is provided via 
+mixed reality technologies. According to the context, the user can transform the physical space with digital 
+elements or create different places that are completely digital. Heterotopias have layers of spatial meaning 
+that relate to the way they are experienced by individuals. Through embodiment and telepresence, 
+technology provides users with endless possibilities. In this study, we provided architectural practices to 
+increase the transformative effects of heterotopias, and through embodiment and presence, we aimed to 
+present an operating table for those practices. To be able to provide an easy-to-use generic framework, we 
+examined the architectural studies to identify essential features. According to Rasmussen’s research in 
+“Experiencing Architecture,” various factors such as the perception of solid and cavity forms, the 
+distinction between foreground and background elements, the size and dimensions of architectural features, 
+the appropriate use of proportion and scale, the acoustics of the space, the lighting, the textures, and the 
+rhythm all contribute to the overall sensory experience of a building or space. Therefore, we aimed to 
+provide those elements in a technological landscape. 
+3.4.1. Construction Elements 
+For the construction of architectural aspects, we identify the main construction elements as walls, floors, 
+roofs, windows, stairs, and doors. Additionally, we add two different classes: landscape elements and 
+furniture. All classes are provided in different panels. The main elements are available as default objects, 
+and they are editable in terms of position, scale, and rotation. 
+3.4.2. Textures, Materials, and Colors 
+Inside the Unity engine, 2D textures are tiled to provide materials. For the wall, floor, roof, door, and 
+window objects, different textures are provided. For the consistency of place and ease of use, generic objects 
+in the same layer share the same material properties. We added sliders that control the RGB values of 
+materials that are used by the construction objects. Additionally, we provided different skybox materials 
+for VR and PC versions. 
+
+ 
+19 
+ 
+ 
+ 
+3.4.3. Scale 
+Although all objects are scalable, we add different scale options to increase usability. The same wall, floor, 
+and roof tiles are also provided in different scales, such as 1x1, 2x2, and 4x4. At the beginning of the 
+experience, the framework provides two different scale modes for VR and MR environments; human scale 
+(1:1) and model scale (1:20) (Figure 4). All models are automatically scaled according to the selection. 
+Additionally, vertical construction components are scaled according to the selected grid height. 
+Figure 4. Human-Scale and Model-Scale Interaction. 
+3.4.4. Lighting Settings 
+Three different light sources are used in the project. Sunlight, as directional light, ceiling lights as light 
+strips, and art objects have their lighting as a spotlight. In the lighting settings panel, each lighting type has 
+a toggle that can be turned on and off (Figure 5). A slider is added to control the temperature of the light 
+with values in Kelvin which are converted to the color of light. 
+ 
+
+香
+目
+x
+目
+8 
+20 
+ 
+ 
+ 
+Figure 5. Artifacts under neutral and warm temperatures. 
+3.4.5. Placement and Grid System 
+The grid system was constructed to provide placement and snapping to correct positions to avoid 
+overlapping. Users can set grid level and grid height properties. Grid level defines the number of floor 
+levels, and grid height defines the height of the walls and the distance between two grid levels. For all 
+placeable objects, including artifacts, three different Scriptable Objects (SO) were created. The SOs are 
+types of assets in the Unity engine that allow saving data in an asset file rather than as a mono behavior 
+attached to a game object. This allows data to be stored in a way that is independent of any specific game 
+object and allows data to be easily reused and shared between different parts of the framework. The first 
+type is floor objects which can only be placed on the layers identified as ground. The grid system is used 
+for snapping floor objects to correct positions and holding the position data of floor objects. The second 
+type includes vertical elements that are placed edges of the floor tiles such as windows, walls, and doors. 
+Each floor object includes four colliders that are placed on the corners. This way, during the placement of 
+the vertical objects on floors, the grid system snaps and rotates the vertical elements according to the edges 
+of the floor objects on collision. 
+The third type includes other elements and artifacts which are placeable in every location. All SO 
+types include two variations of one prefab, which are ghost objects and placed objects. The initial code on 
+the scriptable objects adds the additional scripts that are necessary for interaction and hierarchy according 
+to types. When the instantiation button is pressed, first, a ghost object is spawned which follows the pointer 
+in MR and VR, the mouse position in the PC version. To place the object, the left click of a mouse, the 
+select gesture with the right hand or right trigger is used according to the version. To cancel the selection, 
+the right mouse click, and the select gesture with the left hand or left trigger button is used. For the delete 
+operation, a toggle is added which allows users to delete the selected objects when it is on. If the model 
+scale is selected, the grid system is placed on a table and every spawned object becomes a child object of 
+the table. This provides a conversion of the scale and movement of the objects with the table. 
+
+208
+凉明 
+21 
+ 
+ 
+ 
+3.5. Architecture and Archives Combined: Automated Generation of the Built Environment 
+The algorithms provide reordering mechanisms in 3D dimensional spaces constructing grids of 
+specification based on archival data while offering fast prototyping to architectural practices. Based on 
+discussions in architecture, heterotopias, and meta-design principles regarding power, autonomy, and 
+control, we adapted the algorithms which can offer design space and control to the users (Figure 6). 
+According to the requirements of this study, algorithms are able to generate content based on user data, and 
+generated content is open for customization. 
+ 
+Figure 6. Transformation of Binary Space Partitioning Algorithm’s output. 
+Rasmussen divides the production and interpretation processes of architectural forms into two: 
+solid-minded and cavity-minded. One can start with a simple framework and add to it, or one can carve out 
+certain components from a large solid to start. Drawing from this separation, we defined two scale options 
+namely, room-scale and building scale, and employed one algorithm that can produce content based on 
+dimension, window, and door requirements for room generation where the user can identify the dimensions 
+of the solid and the number of cavities. We have provided two different generation techniques for building 
+scale production. We created a solid-minded generation combining Binary Space Partitioning (BSP) and 
+Cellular Automata (CA) algorithms and a constrained growth algorithm to provide cavity-minded 
+production. While these algorithms are mostly used with random number generators to create different 
+game-level designs, we injected user preferences and user data to create levels of automation. Each 
+algorithm provides three levels of automation. Users can prefer to generate a 3D environment based on data 
+provided, with random numbers, or they can define the parameters through the interface. This approach 
+offers creative control over mechanisms that presents different levels of granularity. In all three cases, we 
+made architectural choices that would ensure coherence in the environment produced to create a consistent 
+user experience. Users can re-design the produced models of a building by using the same operation that 
+they can perform on other assets. Without the scale operation of the user, the height of all vertical 
+
+GridLevel 
+22 
+ 
+ 
+ 
+components is defined by the parameter given by the user at the beginning of the experience. To this end, 
+as an input to algorithms, we used the same assets that we provided as main construction objects. 
+3.5.1. Calculation Strategies Based on Data 
+To be able to calculate the number of rooms and minimum dimensions, the data provided by the user is 
+parsed into columns (Algorithm 1). According to style, artist, and location data, artifacts are grouped and 
+provided via an artifact analysis panel together with artifact types. The number of rooms is defined 
+according to the categorization selection of the user. For example, if the style is selected, the number of 
+styles also defines the number of rooms. In this way, each artifact is assigned to a certain room. Since the 
+dimensions of audio, video, and text artifact holders are already defined, their dimensions in the X-axis are 
+directly added to the sum. Dimensions of the paintings and sculptures are extracted from data and if they 
+are not available, the default dimension in the X-axis is defined as one meter. 
+ 
+Algorithm 1: Room Number and Size Calculation 
+1 
+Parse user data into columns 
+2 
+Group artifacts by style, artist, and location 
+3 
+Define the number of rooms based on user categorization selection 
+4 
+Assign each artifact to a room 
+5 
+foreach Room do 
+6 
+ 
+foreach Artifact in a room do 
+7 
+ 
+ 
+if the artifact is audio, video, or text then 
+8 
+ 
+ 
+ 
+Add dimension in X-axis to sum 
+9 
+ 
+ 
+ 
+else 
+10 
+ 
+ 
+ 
+if Dimensions are available then 
+11 
+ 
+ 
+ 
+ 
+Extract the X-axis dimension from the data and add it to the sum 
+12 
+ 
+ 
+ 
+ 
+else 
+13 
+ 
+ 
+ 
+ 
+Set the default dimension as 1 meter and add it to the sum 
+14 
+ 
+ 
+ 
+end if 
+15 
+ 
+ 
+end if 
+16 
+ 
+ 
+Add constant label and space sizes to sum 
+17 
+ 
+ 
+Divide the X-axis dimension by four and add to the sum as spacing 
+18 
+ 
+end for 
+19 
+ 
+Add minimum entrance size for two parallel walls (2-meters x 2) to sum 
+20 
+ 
+Divide the sum by 2 to get the sum of two perpendicular walls 
+21 
+end for 
+ 
+ 
+ 
+
+ 
+23 
+ 
+ 
+ 
+Additionally, the sizes of the labels and the space between displayed objects are static which is added to 
+the total sum for each artifact. To create lists, artifacts are queued according to the date aspect. The artifacts 
+that are not provided with a date are added according to alphabetical order. From the constructed list, space 
+between each artifact is calculated based on their sizes and added to the sum together with the minimum 
+entrance size for two parallel walls. In the final step, the calculated length is divided into two which gives 
+the sum of two walls that are perpendicular to each other. Calculated lengths are provided to other 
+algorithms as a parameter that will define the dimensions of the rooms. 
+3.5.2. Constrained Growth Algorithm 
+A procedural method’s capacity to define its limits depends on space, allowing it to precisely use the region 
+it generated as input before continuing to generate within it (Algorithm 2). Using a building outline for a 
+floor plan generation allows for determining the borders of the procedural method. To provide consistent 
+footprints for users, we searched for floor plans of existing museums. Using the websites such as Archdaily 
+and Divisare, we obtained 20 different floor plans that have different complexity levels in terms of space. 
+Using OpenCV, we extracted the outlines of the floor plans with filling operation, and we transformed them 
+into footprints. We used footprints as the solids that will hold the cavities that will grow inside them. After 
+several experiments, we defined the sizes of .png files that contain footprints as 128x128 pixels due to 
+computation limitations. The prepared footprints according to pixel and color requirements are provided as 
+options to users. To operate the algorithm, the user first selects the footprint to work on. 
+Depending on the preference, the user can define the starting growth points by selecting the pixel 
+form image of the footprint or if data is available, the user can choose the option from sorting categories 
+(style, artist, location) that defines the number of rooms, and the algorithm selects the random points 
+according to optimal distances. If there is no data available, the user can also prefer the algorithm to select 
+the number and location of the points randomly. The starting points form a square with a 1-pixel void and 
+8 pixels around as boundaries (Figure 7). After the selection of starting points based on pixel-based search, 
+starting points begin to grow until they reach the borders of the footprint or borders of each other forming 
+cavities inside the footprint. When the growing process is completed, corners of the containers are found 
+by continuously adding and comparing the positions of the pixels which are owned by a certain room. 
+According to corners, center positions, and length of the walls are defined to place 3D wall tiles. For each 
+room, the 3D floor and their mirrored version of roof tiles are placed according to empty pixels and walls. 
+
+ 
+24 
+ 
+ 
+ 
+ 
+Figure 7. Example Outputs of Constrained Growth Algorithm. 
+3.5.3. Room Generation 
+In the room generation algorithm, a cubic room is generated using the generic wall, roof, and floor assets 
+as a solid. For the data-based version of the algorithm, the user can select the sorting options for the assets 
+and each room according to the number and sizes of the artifacts that the room will contain, the width and 
+depth dimensions of the room are calculated, and the initial room is scaled according to calculations. For 
+each room, a button is produced to spawn the room in the virtual environment. In the version based on user 
+preferences, initially, the user places a generic room that holds the generation algorithm. The user can play 
+with the cavities and solids by defining the dimensions of the room, the number of windows, and the number 
+of doors. The assets that are used to construct the initial solid together and window and door assets are 
+given to the algorithm. When the user presses generate button; first, floor and wall tiles are replaced 
+according to the given dimensions. Then, the algorithm changes wall tiles to window and door tiles 
+according to the given numbers by the user (Figure 8). 
+ 
+
+ 
+25 
+ 
+ 
+ 
+ 
+Algorithm 2: Constrained Growth Algorithm 
+1 
+Initialize new list cornerList 
+2 
+From (minX, minY) search four directions for a boundary pixel 
+3 
+if it is found then 
+4 
+ 
+Add to the cornerList 
+5  
+else 
+6 
+ 
+Move 
+7 
+end if 
+8 
+Sort corners clockwise 
+9 
+if a room is placed, then 
+10 
+ 
+Check Region 
+11 
+ 
+if the region is clean, then 
+12 
+ 
+ 
+Draw room boundaries 
+13 
+ 
+ 
+Add a room to the room list  
+14 
+ 
+ 
+else 
+15 
+ 
+ 
+Terminate the Room 
+16 
+ 
+end if 
+17 
+end if 
+18 
+if it is not paused, then 
+19  
+Iterate through the list of rooms and find candidate walls 
+20  
+while the number of growable walls > 0 do 
+21  
+ 
+Grow room 
+22  
+ 
+Update walls 
+23  
+ 
+Update texture  
+24  
+end while 
+25 end if 
+
+ 
+26 
+ 
+ 
+ 
+Figure 8. Example Outputs of Room Generation Algorithm. 
+3.5.4. Binary Space Partitioning and Cellular Automata 
+Cellular automata (CA) and binary space partitioning (BSP) are two different algorithms that can be used 
+separately or in combination to generate layouts. BSP is a technique used in computer graphics to divide a 
+2D or 3D space into smaller subspaces, or nodes. In general, the number of subspaces of BSP is defined 
+with a random number from a range with a seed provided for variation. It is mostly used for generating 
+complex 2D dungeon patterns. This study used BSP to produce interconnected rooms and corridors; 
+however, different from the most common techniques, instead of using random numbers we constrain the 
+algorithm via user preferences. 
+A type of mathematical system known as a cellular automaton is made up of a grid of cells, each 
+of which has a limited number of possible states. According to a set of rules, each cell’s state is determined 
+by the states of its neighbors. This means that by interacting with one another in cellular automata, simple 
+rules can be used to create complex patterns and structures. In video games, CA is typically used to create 
+more natural patterns like caves and forests. In this study, we created a 3D model of the generated grid data 
+based on the cell state using CA. The basic idea of BSP is that any plane can divide space into two spaces. 
+Therefore, it creates different areas by recursively dividing the level space. Any half-space that we continue 
+to define a plane in will be further split into two smaller subspaces. Subspaces are created and a spatial 
+binary tree is formed through continued division. The divisions may be placed at random points so that not 
+all areas are of the same size and shape. Once the areas are a suitable size, each area is converted to a room, 
+and connections are added between adjacent regions. In the data-based version of the BSP, the number of 
+rooms and their dimensions are provided by a data analysis algorithm that calculates parameters according 
+to the sorting option selected by the user. Other variables are randomly produced by the algorithm and 
+
+ 
+27 
+ 
+ 
+ 
+altered via seed. In the absence of data, with the defined seed, the algorithm randomizes all parameters used 
+for the generation. For the version based on user preferences, the user can define maximum and minimum 
+values for dimensions of corridors and rooms, dimensions of footprint, and the number of rooms so that the 
+user can give the parameters of the solids. 
+We generated a layout using BSP to start by defining the initial boundary which is the footprint 
+that has dimensions depending on the user’s preferences (Algorithm 3). Given sizes for the footprint can 
+change the overall placement of the layout. For example, if the user prefers to create longitudinal side-by-
+side blocks or more compact structures, it can be done by playing with the width and depth ratio. The next 
+step is to divide this footprint into smaller spaces in accordance with the number of rooms and dimensions 
+defined by the user (Figure 9). In binary space partitioning, a seed is a starting value that is used to generate 
+a random sequence of values. The random sequence is then used to determine the layout. We used the seed 
+to randomly select the location and orientation of the division to create variety. The seed value is used as 
+the input to a random number generator, which produces a sequence of random values based on the seed. 
+By using the same seed value, it is possible to generate the same layout multiple times. The divisions are 
+created based on the room and corridor dimensions and seed value which continues until the number of 
+rooms defined by the user is reached. 
+Each node in the constructed tree structure represents a subspace within the layout. To create the 
+actual layout of the tree, the tree is then traversed, and at each node, if the node can create a room, it is 
+added to the layout and the algorithm tries to place a corridor in all four cardinal directions. If there is 
+enough space, the algorithm places a corridor and moves on to the next node. If there is not enough space, 
+the algorithm terminates the process and restarts to division. The BSP algorithm operates on cells, which 
+makes it suitable to combine with the CA for a rule-based 3D generation. We defined several states for the 
+cells to construct the 3D model of the generated data structure. Before the division, all the cells are in the 
+state of empty when all the rooms take place in the spatial binary tree cells change their states to the wall, 
+corner wall floor, door, and window. While wall and roof tiles do not need any rotation, for wall, corner 
+wall, and door tiles there are different states which apply the rotational operations. Applying the rules, 3D 
+assets take their places. For the roof, instead of creating a new state, we injected the responsibility to the 
+cells in the floor state, which can create roof tiles by duplication according to the defined height. 
+
+ 
+28 
+ 
+ 
+ 
+ 
+Figure 9. Example Outputs of Binary Space Partitioning and Cellular Automata. 
+4. Evaluation 
+4.1. Ethics 
+Ethical Approval of Research was approved by the Middle East Technical University Human Subjects 
+Ethics Committee with the approval protocol code of 0244-ODTUİAEAK-2022 in April 2022. Before the 
+start of the study, every participant signed an informed consent form. The form includes a brief description 
+of the study’s aims and objectives, the importance of participants’ input, data collection methods, 
+possession of participants’ personal information, the intended use of their data, and what is expected of 
+them. Participants had been warned about the potential negative effects of VR platforms, such as motion 
+sickness and nausea, and it was explained that participants had the right to leave at any time if they feel 
+uncomfortable or disturbed during the experiences. All the information collected for this study is 
+anonymized and figures that may include personal data were excluded. No compensation was provided to 
+participants. 
+ 
+
+ 
+29 
+ 
+ 
+ 
+ 
+Algorithm 3: Binary Space Partitioning and Cellular Automata 
+1 
+Initialize the layout with the given footprint dimensions 
+2 
+Use the seed value as the input to a random number generator 
+3 
+while layout.num_nodes < user.num.rooms do 
+4 
+ 
+Select a random location and orientation for the division 
+5 
+ 
+Divide the layout into smaller subspaces  
+6 
+end while 
+7 
+Traverse the tree structure of the layout 
+8 
+Recursively visit each node in the tree structure of the layout 
+9 
+if the node can create a room, then 
+10 
+ 
+Add it to the layout 
+11 
+ 
+Try to place corridors in all four cardinal directions 
+12 
+ 
+else return 
+13 
+end if 
+14 
+if layout.num_nodes = user.num.rooms, then 
+15 
+ 
+Iterate over the cells in the layout and place 3D models according to the 
+states 
+16 
+ 
+else return 
+17 
+end if 
+4.2. Participants 
+A total of 30 participants were recruited. Recruited respondents were between the ages of 18–35, with a 
+mean age of 22.52 ± 4.13 years. Each participant had a normal or corrected-to-normal vision (self-reported). 
+The scope and focus of the research topic, the methods utilized to collect the data, and the amount of 
+information that was acquired from each participant are all factors that can affect the sample size (Braun 
+and Clarke, 2019). Enough information should be gathered to adequately address the research questions. 
+Given the numerous methods of data collection and the depth of information gathered from the participants, 
+30 participants were an adequate number, and the analysis from which produced important insights. 
+4.3. Procedure 
+We conducted our evaluation using a quantitative survey in the form of questionnaires. Additionally, semi-
+structured interviews were undertaken as an exploratory method to provide more detailed and in-depth 
+insights about the use of the application and preferences. Before the version test, participants were asked to 
+fill out the Gamer Motivation Profile (Yee, 2016), Immersive Tendencies (ITQ), and Tool Competence 
+questionnaires. The users were provided with identical data and museum collections to avoid 
+inconsistencies resulting from the data and media. Following the procedure, participants were given access 
+to their collections. Users were first asked to freely use the application, then test the specific features, such 
+as lighting settings and procedural generation algorithms. Once participants had used all three versions, the 
+participants were given the questionnaires intended to assess several aspects of the versions. For the 
+qualitative evaluation of the framework, a widely used System Usability Scale (SUS) (Brooke, 1996), 
+Presence Questionnaire (PQ) (Witmer and Singer, 1998), and Technology Acceptance Model (TAM) 
+
+ 
+30 
+ 
+ 
+ 
+(Venkatesh & Davis, 2000) questionnaires were adopted and distributed to the participants. In addition to 
+those, we composed three questionnaires for lighting, procedural generation algorithm, and interaction 
+preferences. Later, a short semi-structured interview was conducted. Each session lasted about one hour. 
+4.4. Data Collection Methods 
+4.4.1. Semi-structured Interview 
+After all the platforms had been used by the participants, the semi-structured interview was held after the 
+session to gather further information about general impressions of using the versions. The guide consisted 
+of 11 questions that focused on participants’ interpretation and perception of the framework, experiences 
+of using the different versions how they could be improved, and which versions and options they preferred 
+and why. Additionally, informal conversations were made to identify possible factors that had been 
+overseen, such as mistakes in applying the method and software errors. In this interview, they were also 
+asked to describe their experience using the application in their own words. 
+4.4.2. Questionnaires 
+The System Usability Scale is a 10-statement self-report scale that is used to analyze and study a system’s 
+usability for general evaluations. It is assessed on a Likert scale of 1 to 5, where 1 is for strongly disagreeing 
+and 5 is for strongly agreeing. The comprised 10 questions consist of five positive and five negative 
+statements. The SUS score is calculated using the collected data, and the result falls between 0 and 100. 
+SUS is widely used in the literature to compare and evaluate the success of outputs. 
+The sense of presence experienced via exposure to an immersive environment has frequently been 
+connected to the effectiveness of the environment. Presence is frequently described as a “feeling of being 
+there” when a person feels as though they are in a different place. The Presence Questionnaire was first 
+developed by Witmer and Singer (1998). It comprises several subsections, but only a few of them are 
+incorporated in this study. Presence was rated on a 7-point Likert scale ranging from 1 = strongly disagree 
+to 7 = strongly agree. 
+A methodology called the Technology Acceptance Model (TAM) seeks to determine whether new 
+technology will be accepted, rejected, or usable. The approach was first presented by Davis (1987), and it 
+is widely used in the literature. TAM consists of different subsections and different adoptions. A revised 
+form of the TAM questionnaire (Venkatesh & Davis, 2000) is used and it consists of 10 questions that cover 
+two subsections: perceived usefulness (PU) and perceived ease of use (PEU). The first six questions are 
+used to measure PU, and the other four are used to calculate PEU. TAM was rated on a 7-point Likert scale 
+ranging from 1 = strongly disagree to 7 = strongly agree. 
+The results of the effectiveness of the immersive systems, however, may be mediated by individual 
+differences, aspects of the technology, and negative side effects. Measuring these mediating effects is of 
+great importance in understanding the relationship between and among them, and how to maximize the 
+
+ 
+31 
+ 
+ 
+ 
+effectiveness of the immersive environments. To measure several individual differences, the Immersive 
+Tendency Questionnaire (Witmer and Singer, 1998) was introduced. The immersive tendency is an 
+individual’s proclivity to become immersed in the simulation. ITQ consists of 18 items and is composed of 
+four subsections that include involvement (five items), focus (five items), emotions (five items), and 
+propensity to play video games (three items). 
+To measure the tool competence level of participants regarding VR and MR, they were asked to 
+rate their experience level on a 7-point Likert scale ranging from 1 = I have no experience to 7 = I am an 
+expert. In the PC version, we utilized the most common navigation and interaction methods (W, A, S, and 
+D buttons for navigation, and mouse interaction for camera movements), which are not required for MR 
+and VR. Therefore, instead of asking about experience level in PC, we employed a survey called the Quantic 
+Foundry Gamer Motivation Profile to measure participants’ degrees of video game experience in the study. 
+The questionnaire comprises six questions that focus on respondents’ favorite games and game-playing 
+habits and were created utilizing factor analysis and historical investigation. It assesses experience based 
+on four levels, ranging from non-gamer to hard-core gamer. 
+Four additional open-ended question sets were answered by the users for this study, namely, 
+Framework Competence, Interaction Preference, Procedural Generation Algorithm Selection, and Lighting 
+Preferences. Tool Competence Questionnaire includes two questions to assess experience level regarding 
+VR and MR. There is one question for each version (VR and MR) with two options (hand 
+interaction/controllers) for interaction mode preferences. In the Lighting Preferences Questionnaire, there 
+are two questions for different roles (curator/visitor) per version (VR and PC) with three temperature 
+options (cold/neutral/warm). Procedural Generation Algorithm Selection consists of four questions for each 
+version (based on data, based on user preferences) for three procedural generation algorithms to identify 
+ease of use, perceived control, understandability of the algorithm, and the quality of the design outcomes. 
+It is rated on a 5-point Likert scale ranging from 1 = none to 5 = completely. 
+5. Results 
+The comparative results presented in this chapter are based on the responses given by the 30 participants 
+for the PC version with MacBook M1 Pro and VR and MR version with Oculus Quest 2. Responses to 
+open-ended questions were analyzed using the analysis tool MAXQDA (2022). Statistical analyses were 
+conducted using the software program JASP (2022 V.0.16.4) to compute and validate the results. For each 
+questionnaire, reliability tests were run to determine whether the mean values’ internal consistency was 
+satisfactory. We compute Cronbach’s alpha values, which is a method often used to validate surveys (Table 
+1). 
+ 
+ 
+ 
+
+ 
+32 
+ 
+ 
+ 
+Table 1. Interpretation of Cronbach alpha (α) values (Tavakol et al., 2001). 
+Cronbach’s  
+Cronbach’s α Comments 
+α ≥ 0.9 
+Outstanding 
+0.7 ≤ α < 0.9 
+Good 
+0.6 ≤ α < 0.7 
+Acceptable 
+0.5 ≤ α < 0.6 
+Weak 
+α ≤ 0.4 
+Unacceptable 
+5.1. Semi-structured Interview 
+The first and second interview questions require users to explain their general impressions regarding the 
+framework and their interpretation of the usage area. Figures 10, 11, and 12 present the frequency mapping 
+of words that are commonly used to describe different versions of the framework and potential use cases. 
+ 
+Figure 10. Frequency Mapping of Most Used Words by Participants to Describe the VR version of the 
+framework. Produced with MAXQDA. 
+
+inciusive
+exmerimenta
+design
+education
+productive
+complex
+fun
+quick
+exnansion
+inspire
+dimensional unlimited
+understande
+arcnitecture
+hard
+mind
+medum
+Vr
+display
+interest
+study
+Drototyne
+build
+teach
+space
+creative
+composition
+generative
+art
+difficult
+effective
+visualization
+creation
+brainstorm
+exhibition
+interaction
+functional 
+33 
+ 
+ 
+ 
+ 
+Figure 11. Frequency Mapping of Most Used Words by Participants to Describe the MR version of the 
+framework. Produced with MAXQDA. 
+ 
+Figure 12. Frequency Mapping of Most Used Words by Participants to Describe the PC version of the 
+framework. Produced with MAXQDA. 
+5.2. Interaction Preferences 
+Users were asked to select the medium that they preferred according to identified asset types. Figure 13 
+shows the users’ preferences for interaction mediums with different asset types. 
+ 
+
+brainstorm
+design
+transformation
+dimensional composition
+inspire
+experimental art study
+understand
+difficult presentation teach
+showcaselearn
+functional
+exhibition
+mr
+hard
+productive
+generative
+fun
+display
+complex quick
+architecture
+configuration
+creation
+space
+build production interest
+visualization
+Drototvne inclusive
+interior-design
+effective
+creative
+interactioncreation
+composition
+mse
+understand
+fun
+production
+archive
+vision
+quick
+design
+simul ation
+gallery
+artistic
+pc
+conttiguration
+easy
+brainstorm
+nterest
+game
+art
+effective
+architecture
+jrototype
+functional
+display
+experimental
+generative
+interact
+inclusive
+creative 
+34 
+ 
+ 
+ 
+ 
+Figure 13. Interaction preferences of participants with different asset types. 
+In the VR and MR versions, all users tested the hand interaction and controllers. Figure 14 shows 
+the users’ preferences for interaction with different versions of the application. 
+ 
+Figure 14. Interaction preferences of participants for VR and MR Version. 
+5.3. Lighting Temperature Preferences 
+Three parts of the temperature scale were given as a preference for the respondents for different modes of 
+the framework. Since the MR version does not utilize the virtual lighting system, it was not included in the 
+survey. The results are demonstrated in Figure 15. 
+ 
+
+Video
+Text
+Image
+PC
+MR
+PC
+VRArtCanvas
+3.3%
+13.3%
+26.7%
+16.7%
+VR ArtCanvas
+VR
+PC ArtCanvas
+VR
+3.3%
+36.7%
+20.0%
+50.0%
+MR
+VR
+MR
+36.7%
+23.3%
+PC ArtCanvas
+23.3%
+46.7%
+3D Model
+Audio
+PC
+16.7%
+VR
+36.7%
+MR
+40.0%
+VR
+50.0%
+MR
+PC
+33.3%
+23.3%MR Version
+VR Version
+Hand Interaction
+33.3%
+Hand Interaction
+46.7%
+Controllers
+53.3%
+Controllers
+66.7% 
+35 
+ 
+ 
+ 
+ 
+Figure 15. Lighting preferences of participants for different modes and immersive environments. 
+5.4. Procedural Generation Algorithms 
+Four different criteria were identified to assess the success of the implemented algorithms. The results are 
+divided into two—generation based on data and generation based on user preferences. To test the 
+correlation between the number of options provided to users and other eligibility indicators, they are 
+presented together in Table 2. 
+ 
+Table 2. Mean and standard deviation results of the questionnaire for procedural generation based on user 
+preferences. Answers to the questionnaires are on a 1 to 5 scale; a higher score indicates a more positive 
+attitude. 
+ 
+Growth 
+Algorithm 
+Binary Space Partitioning and Cellular 
+Automata  
+Room 
+Generation 
+Number of 
+Options 
+3 
+7 
+4 
+Perceived Control  
+3.56 ± 1.04 
+3.76 ± 0.97 
+4.03 ± 0.80 
+Ease of Use  
+3.90 ± 0.75 
+3.7 ± 0.98 
+4.00 ± 0.78 
+Comprehensibility  
+4.00 ± 0.81 
+3.66 ± 0.92 
+4.00 ± 0.87 
+Design Outcome  
+3.54 ± 0.97 
+3.43 ± 0.89 
+4.03 ± 0.92 
+
+VisitorMode(PC)
+Visitor Mode (VR)
+Cold
+13.3%
+Neutral
+Neutral
+Warm
+40.0%
+43.3%
+46.7%
+Warm
+43.3%
+Cold
+13.3%
+Curator Mode (PC)
+CuratorMode(VR)
+Warm
+16.7%
+Cold
+Neutral
+36.7%
+36.7%
+Neutral
+Cold
+56.7%
+26.7%
+Warm
+26.7% 
+36 
+ 
+ 
+ 
+Since perceived control, the number of options, and ease of use are not relevant to generation based 
+on data, only design outcome and comprehensibility of the algorithms were investigated. The results are 
+given in Table 3. 
+Table 3. Mean and standard deviation results of the questionnaire for procedural generation based on data. 
+Answers to the questionnaires are on a 1 to 5 scale; a higher score indicates a more positive attitude. 
+ 
+Growth 
+Algorithm 
+Binary Space Partitioning and 
+Cellular Automata 
+Room 
+Generation 
+Design Outcome  
+3.23 ± 0.89 
+3.90 ± 0.75 
+3.40 ± 0.85 
+Comprehensibility  
+3.66 ± 0.71 
+2.86 ± 0.84 
+3.30 ± 0.83 
+ 
+The results show that although the comprehensibility of the combination of BSP and Cellular 
+Automata algorithm is low in comparison to other algorithms, it has the highest score in terms of design 
+outcome. 
+5.5. System Usability Scale 
+Although SUS does not have subscales, the answers should be computed in accordance with SUS scoring 
+algorithms to interpret the results (Brooke, 1996). To create consistent scoring as required, the scoring 
+formulas include different calculation methods for the questionnaire’s negative and positive questions. 
+Table 4 shows Cronbach’s alpha values, while Table 5 and Figure 16 display the findings of SUS’ statistical 
+study. 
+Table 4. Cronbach’s Alpha Values of System Usability Scale. 
+Version 
+PC 
+VR 
+MR 
+Cronbach’s  
+0.653 
+0.874 
+0.670 
+Table 5. Mean and Standard Deviation Results of System Usability Scale. 
+Version 
+PC 
+VR 
+MR 
+Mean 
+79.91 ± 10.89 
+70.00 ± 14.03 
+77.51 ± 12.99 
+ 
+The results of the SUS were found reliable since the calculated Cronbach’s alpha (α) value was 
+higher than 0.5 for all versions. (PC (0.653 > 0.5), VR (0.874 > 0.5), MR (0.670 > 0.5)). The overall System 
+Usability Scores (SUS) for 30 participants indicated high rates of system usability, with an average score 
+of 79.91% for PC which is defined as “good” and is graded as “B” in the system usability assessment. 
+Comparing the standard deviations reveals greater consistency in the participants’ PC scores. On the other 
+
+ 
+37 
+ 
+ 
+ 
+hand, the usability score of the MR version (77.51%) is higher than the VR version (70.00%), but both are 
+in the range of “B” and are defined as “good” as well. 
+ 
+ 
+Figure 16. Comparative Results of SUS Scores in Boxplot Format. 
+5.6. Technology Acceptance Model 
+TAM questionnaire does not have a structured question set nor a scoring sheet to compare the results. 
+Therefore, the results were examined only numerically using descriptive statistics. For the user study, the 
+most related two subsections of the TAM model were selected, which are perceived ease of use and 
+perceived usefulness. The questionnaire was analyzed with its subsection. The results are given in Table 6, 
+Table 7, and Figure 17 comparatively for each version. 
+Table 6. Cronbach’s Alpha Values of Technology Acceptance Model. 
+Version 
+PC 
+VR 
+MR 
+Cronbach’s  
+0.640 
+0.857 
+0.848 
+ 
+The collected data in total were found reliable since the calculated Cronbach’s alpha (α) values 
+were higher than 0.5 (PC (0.640 > 0.5), VR (0.857 > 0.5), MR (0.848 > 0.5)). 
+ 
+ 
+
+System Usability Scale
+PC
+VRMR
+100
+90
+80
+X
+X
+70
+60
+Data
+50
+40
+30
+20
+10
+0
+Versions 
+38 
+ 
+ 
+ 
+Table 7. Mean and standard deviation results of the Technology Acceptance Model. Answers to the 
+questionnaires are on a 1 to 7 scale; a higher score indicates a more positive attitude. 
+Version 
+PC 
+VR 
+MR 
+Perceived Usefulness  
+5.63 ± 1.05 
+5.47 ± 1.22 
+5.5 ± 1.24 
+Ease of Use  
+5.54 ± 1.05 
+4.88 ±1.13 
+5.15 ± 1.41 
+Total Mean  
+5.60 ± 1.05 
+5.23 ± 1.22 
+5.29 ± 1.35 
+ 
+According to the results, PC’s mean value, PU, and PUE scores were found to be higher than the 
+VR and MR versions. While the MR version’s score is slightly higher than the VR version, the most notable 
+score is the PEU score of the VR version, which is lower than other versions. 
+ 
+ 
+Figure 17. Comparative Results of Technology Acceptance Model Questionnaire in Boxplot Format. 
+5.7. Presence Questionnaire 
+The Presence Questionnaire (PQ) has subsections that represent diverse characteristics of the framework. 
+By focusing on the subsections of the questionnaire, such as realism, possibility to act, quality of the 
+interface, possibility to investigate, and self-evaluation of performance, a total of 19 questions were posed 
+to assess versions’ affordance in presence. Each subsection’s overall system performance is calculated 
+independently, and the results are then combined to get a total presence score. The scoring sheet provided 
+by Laboratoire de Cyberpsychologie de l’UQO (L’UQO, 2002), which shows the minimum scores for a 
+successful outcome, was used as a guideline for comprehending the presence capabilities of the successful 
+versions. The results are given in Table 8, Table 9, and Figure 18 comparatively for each version. 
+
+Technology Acceptance Model
+PC
+VRMR
+7
+6
+X
+5
+Data
+4
+3
+2
+1
+0
+Versions 
+39 
+ 
+ 
+ 
+Table 8. Cronbach’s Alpha Values of Presence Questionnaire. 
+Version 
+PC 
+VR 
+MR 
+Cronbach’s  
+0.717 
+0.822 
+0.685 
+Table 9. Mean and standard deviation results of PQ. Answers to the questionnaires are on a 1 to 7 scale; a 
+higher score indicates a more positive attitude. 
+Version 
+PC 
+VR 
+MR 
+Realism  
+5.26 ± 1.20 
+5.02 ±1.31 
+5.02 ±1.45 
+Possibility to Act  
+5.40 ± 1.15 
+5.25 ± 1.46 
+4.93 ± 1.37 
+Quality of Interface  
+4.37 ± 1.52 
+4.33 ± 1.56  
+4.93 ± 1.49 
+Self-evaluation of Performance  
+5.53 ± 0.92 
+5.38 ± 1.09 
+4.78 ± 1.43 
+Total Mean  
+5.14 ± 1.27 
+4.97 ± 1.46 
+4.93 ± 1.44 
+ 
+ 
+Figure 18. Comparative results of Presence Questionnaire in Boxplot Format. 
+According to the results of reliability tests, the results of the questionnaire were found eligible. 
+While the total mean scores of versions are similar to each other, several outliers can be seen in Figure 18 
+for the PC version. 
+6. Discussion 
+Heterotopias are unique spaces that can connect different systems through the exchange of information and 
+are characterized by the presence of multiple layers. Based on the features of discursive heterotopias, the 
+framework described in this study is designed to be flexible and modular, which can be used in a variety of 
+
+Presence Questionnaire
+PC
+VRMR
+7
+6
+5
+Data
+4
+3
+2
+1
+0
+Versions 
+40 
+ 
+ 
+ 
+contexts to create digital habitats that can provide a space for discourse to occur. Because of the 
+complexities of heterotopias, a variety of evaluation approaches are required to test the convenience of the 
+methods regarding the objectives. 
+To test the framework with individual archives, while some participants prepared a collection of 
+assets right before the sessions, some preferred to use existing materials in their online or offline databases. 
+While the pre-made dataset provided us with essential insights into more technical aspects, the potential of 
+the framework to construct unique heterotopias revealed itself with personal experiences and archives, 
+proving the framework’s applicability to diverse contexts. 
+The colocation of multimedia elements through the “operating table” approach has created several 
+possibilities for participants. For example, participants preferred to use their photographs, videos, notes, 
+and screenshots, similar to the accumulative aspect of heterotopias in terms of time and elements. They 
+generated different rooms constructing “grids of specification” according to their categorization, which 
+created multilayered experiences directly related to self-reflection and self-formation. Individual archives 
+and spatial layers provided by the framework allowed users to create a “world within worlds.” 
+[P.17] “I considered the experience as walking inside my brain and organizing it, so it could work. Also, 
+the framework can be extended via deep learning algorithms for further customization and guidance.” 
+ 
+Participants who used the assets related to their professional life have created relations and 
+experiences based on “surfaces of emergence” that cannot be provided without place and embodiment. For 
+example, participants from backgrounds related to the design studies used the models they produced 
+together with the sketches, notes, and videos of production processes and presented narrative exhibitions. 
+They created in-situ demonstrations and performed architectural design activities experiencing the creative 
+process of “aesthetic becoming.” 
+[P.22] “I think most of the design tools that we used are like trucks; this application was like a sports car.” 
+ 
+[P.04] “I can easily use this tool for my interior design practices, and I especially found the mixed reality 
+version very useful.” 
+ 
+Participants with medical backgrounds placed anatomical models and texts, transforming the 
+environment into a puzzle game where they could manipulate the parts of the model with hand interaction. 
+Additionally, two participants from computer sciences added new features with dynamic models via scripts 
+embedded in 3D models based on OOP using the adaptability of the framework. Throughout the semi-
+structured interview, we investigated the reasons behind the results of the questionnaires, participants’ 
+personal opinions about the versions of the framework, and their perception of the framework for possible 
+use cases. Through the open-ended questions, we explored the interpretation of the framework from the 
+users’ perspective. Open-ended questions were used to gather data on how users perceive and interpret the 
+framework. An analysis of the responses to these questions, using frequency mapping, showed that the 
+different technologies used in the framework resulted in various interpretations. For example, words such 
+as “game,” “creative,” “fun,” and “art” were commonly used to describe the PC version, while the VR 
+
+ 
+41 
+ 
+ 
+ 
+version was more frequently described using words such as “experimental,” “design,” “fun,” “architecture,” 
+“visualization,” and “inclusive.” On the other hand, the MR version was often described using words such 
+as “interior-design,” “prototype,” “interaction,” and “transformation.” A comparison of the frequency 
+mappings suggests that MR and VR technologies may provide more comprehensive and multilayered 
+experiences that involve embodied cognition. Also, most participants provided individual cases based on 
+their experiences and needs. 
+[P.02] “I would prefer to use this tool when I teach my students geography and history.” 
+ 
+[P.25] “We can use this tool for designing, displaying, and teaching the systems in our mechanical 
+engineering studies.” 
+ 
+The richness of the cases provided by the participants implies that the framework is applicable to 
+various contexts. The dynamic autonomy layers and understanding of self-formation based on heterotopias 
+provided design space for the users. They became the “focal points of resistance,” producing informal use 
+cases which were not structured by the framework but by the users. 
+The concept of heterotopia can be interpreted as a complex set of relationships which in this case, 
+also involves the user experience which is affected by various elements from hardware to navigation. 
+According to statements of respondents, different technologies and interaction modalities that are provided 
+serve better for certain discourses. Also, according to the activities, the users change their preferences in 
+environmental settings such as lighting and scale. 
+The multimedia elements of the discursive space were provided with different representation 
+methods. Users’ environment preferences differ according to media type. For the 3D models, users 
+preferred to interact with them in VR and MR environments. Participants stated that MR provides better 
+scale perception in relation to surroundings, and both other versions were more impressive. Most users 
+prefer to display videos and images on a human scale. However, due to distractions created by physical 
+objects in the MR version, the majority of users preferred the VR version for videos since it required a long-
+term focus. For the audio, most of the users preferred the VR environment, finding the combination of a 
+completely artificial environment with audio more engaging. For the text interactions, the majority of the 
+users tend to read in the ArtCanvas view, which is the most traditional option. Therefore, the dominant 
+media format of discourse can affect the preferences of users in terms of technology. 
+When creating a multimedia system, it is important to consider the nature of the interaction and the 
+anticipated results in terms of learning, sensory and emotional engagement, and satisfaction. The hand 
+interaction and controllers were essential for conducting activities through embodiment. The hand 
+interaction has similarities with the command language, and the users need to remember the gesture to 
+complete the operation. Users mostly preferred to use controllers rather than hand tracking for interactions. 
+Using buttons on controllers was more accessible and easier than hand gestures, especially for novice users. 
+The results showed that there was an increase in the number of participants who preferred controllers in the 
+
+ 
+42 
+ 
+ 
+ 
+MR version. This result suggests that being able to receive information from physical locations increases 
+the request for physical body engagements. 
+It is possible to consistently stimulate human emotions and actions in virtual settings, which adds 
+to the overall effectiveness of a heterotopia. The stimuli’s visual integrity results in a high level of 
+immersion. In this study, we looked into the lighting preferences of participants in the curator and visitor 
+modes/roles. Results showed that there are differences in lighting preferences performing two different 
+interactions. While the majority of the users preferred natural and cold light during the curation process, 
+warm lighting settings were preferred during the visiting mode. In the VR environment, there is an increase 
+in the selection of cold temperatures. The relationship between physiological input and an abstract 
+conception in the curation process emphasizes that spatial perception is ultimately experiential and lived. 
+Users’ perceptions of and integration with virtual elements may change as a result of spatial perception 
+blended with an embodiment. Commonly, users indicated that they were able to focus on details under cold 
+and neutral lighting, and warmer temperatures made them feel more comfortable during observation. 
+Control and power mechanisms are important elements for both heterotopias and architectural 
+practices. Heterotopias redefine those relations and provide an “order of discourse” with different syntaxes, 
+where the syntax is constructed via places. The integration of technology changes the dynamics and defines 
+the borders of control and power for the user. An increase in the autonomy of technology has created 
+discussions in architectural practices. Architectural design practices cannot be distinctively separated from 
+their tools. As technology advanced, design and art practices have changed their systems from physical to 
+digital space. Architectural drafting, representation, building, and most critically, architectural design, have 
+all been impacted by modern technologies, which have changed the production and final output. The general 
+tendency of the tools points out the resistance of technology instead of users. 
+By providing a flexible framework and various autonomy levels for automated generation 
+algorithms, we aimed to understand the dynamics between the technology, the user, and the transformation 
+or construction of spatial layers of heterotopias provided via digital mediums. With the level of manual 
+design provided to users, the overall outcome depends on how procedural generation is guided, constrained, 
+and changed by the user. Moreover, since procedural generation is inherently deterministic, the integration 
+of manual design steps is essential for users to inject variety and creativity. We aim to understand the impact 
+of the level of autonomy on the outcome and user preferences. Here, we are particularly interested in the 
+extent to which the method led to the intended results and higher levels of control, considering the tension 
+between usability and comprehensibility associated with the design process of the users. Therefore, it is 
+worthwhile to explore how the different outcomes are perceived by users. 
+According to the results, the growth algorithm provides the highest comprehensibility among the 
+data-based versions. During the process, users were able to see the initial area that they or the algorithm 
+would work on, which increased their understanding of how the algorithm works. In the user-based version, 
+users can select the starting point for growth, but the size and shape of the rooms are determined by the 
+algorithm, which the user cannot control. In the BSP version, the size of the rooms is chosen by the user, 
+but their placement is controlled by the algorithm. The room generation algorithm allows the user to control 
+
+ 
+43 
+ 
+ 
+ 
+both the placement and size of the final product. Overall, the growth algorithm provides a higher level of 
+perceived control than BSP, but a lower level than the room generation algorithm. According to survey 
+results, BSP combined with CA produces the best design outcomes but has a lower level of 
+comprehensibility compared to the other algorithms. When participants generated the content based on their 
+preferences, options that were given to users increased comprehensibility; however, this approach 
+decreased the design outcome produced by the algorithm. This implies that the data-based version meets 
+the space needs of the users better than the user-based version. With the higher number of options presented, 
+BSP provided a less perceived sense of control and ease of use. The data-based version of the room 
+generation algorithm has lower scores for design outcome than BSP, but higher scores for perceived control, 
+ease of use, and design outcome in the user-based version. 
+Most of the participants exhibit different strategies according to versions depending on their 
+embodiment level, point of view, and scale. The spatial experiences in virtual environments provide an 
+active interpretation process based on the awareness of surroundings. Therefore, the design approaches of 
+participants differ according to versions. Comparing produced scales of the algorithms, BSP has the 
+capability of producing large-scale designs while room generation can produce only one space, and the 
+growth algorithm is able to produce within the borders of the footprint. Therefore, users tend to use the BSP 
+algorithm in the PC version where they can observe the outcome with a completely top-down view. Room 
+generation was mostly preferred in the MR version where users can change the surroundings with their 
+virtual versions. The growth algorithm was found more suitable for the VR version since the users can 
+produce multiple rooms, but still be able to navigate themselves without losing their tracks. Results also 
+indicate that the comprehensibility of the method increases the perceived level of control. 
+Another objective of the study was to develop a framework for immersive technology based on 
+heterotopias that is accessible and easy to use for a general audience, including both novice and experienced 
+users where they can conduct discursive practices. The level of usability and acceptance can change an 
+individual’s control over the system by providing more resistance to the technology. Therefore, the 
+framework was tested using the Technology Acceptance Model (TAM) and the System Usability Scale 
+(SUS). The results showed that the different versions of the framework were convenient for users with 
+different backgrounds and levels of competence, pointing out that the layers of heterotopia in a digital 
+medium can provide an “operation table” for various cases and individuals. The SUS was used to assess 
+the complexity, consistency, and ease of use of the system, and all three versions scored in a similar range. 
+However, some participants preferred the VR version although it had lower usability scores, indicating that 
+engagement may be a factor in the success of immersive technologies. 
+[P.17] “I can use the VR version for hours when I want to get away from reality.” 
+ 
+[P.04] “MR version was pretty interesting, the PC version was easier to use, but VR was more fun.” 
+ 
+According to user feedback analyzed through frequency analysis, the MR and VR versions of the 
+framework were perceived as having both positive and negative qualities. While they were described as 
+“experimental”, “useful”, “quick”, “functional”, and “interesting”, they were also considered “difficult”, 
+
+ 
+44 
+ 
+ 
+ 
+“hard”, and “complex” to use. In contrast, the PC version was described as “easy “to use and more “game”-
+like. The high volume of information available within the heterotopias created an experimental and creative 
+space for users but also resulted in interfaces that may be overwhelming for those who are new to the 
+system. Some users noted that the numerous features hindered usability and increased cognitive load, 
+disrupting their ability to fully immerse themselves in the environment. The intensification of information 
+in heterotopias both produced “experimental” and “creative” ground for the users and also expressed itself 
+via interfaces, which created exhaustion for novice users. Some participants indicated that the number of 
+features decreases usability and increases the cognitive load, which disturbs the feeling of presence. 
+[P.25] “I think the number of features should be less. I felt there were so many features to use, which put 
+me under stress.” 
+ 
+To interpret the results of the TAM questionnaire in more detail, we examined the subsections. 
+Even though the PC version had the highest scores, the results of the versions in terms of PU were similar 
+to each other. The virtual reality (VR) version had the lowest PEU score, while the mixed reality (MR) 
+version was perceived as being more understandable and requiring less effort. This difference may be due 
+to the nature of VR, which can disconnect users from the real world. Some users found the MR version 
+safer than the VR version, while the VR version was considered more enjoyable to use. The HMD used in 
+the study creates a virtual boundary around the user and displays the real world when in close proximity to 
+a physical object. However, some novice users who were new to being fully immersed in a virtual world 
+may have had lower trust levels towards the technology and preferred to use the teleportation option for 
+navigation, reducing their interaction with the virtual environment. In terms of performance, all three 
+versions were produced based on the same system; therefore, they can sustain the same predictability level. 
+However, other factors, reliability, and utility affect each other. In the case of VR, lower reliability to 
+technology produces lower perceived usefulness results. 
+[P.03] “Since I was able to see the physical environment, I felt safer in MR than VR.” 
+ 
+[P.19] “I became disoriented in the VR, and at one point, I felt constrained by the headset and wanted to 
+remove it. The MR version was not as impressive as the VR version, but I would prefer to use the MR 
+version.” 
+ 
+The results showed that PC had higher presence scores than VR and MR, and this difference was 
+attributed to the different levels of competence and familiarity that users had with these different 
+technologies. Analysis of various subcategories, such as realism, quality of the interface, and the possibility 
+of examination, found that each technology had its own strengths and weaknesses in these areas. For 
+example, the PC version had higher scores in terms of naturalness of interaction and the ability to survey 
+the environment, since the keyboard and mouse interaction are more natural for most of the users, and the 
+PC version provides several camera options to the survey environment. On the other hand, the VR version 
+had higher scores in terms of visual aspects and consistency with real-world experiences. The MR version 
+
+ 
+45 
+ 
+ 
+ 
+had lower scores due to problems that occurred with hand interaction and the distractions caused by the 
+colocation of physical and virtual content.  
+The results suggest that different technologies can have different impacts on cognitive load and 
+users’ focus, and it is important to consider these factors when choosing a technology for a particular task. 
+Overall, total scores indicate that a multi-layer approach to constructing heterotopias with the given features 
+provided high scores in presence, usability, and acceptance for users with different experience levels. 
+7. Conclusion 
+The discursive space refers to the understanding and insights gained through the study and examination of 
+specialized knowledge about the world. With the increasing use of digital technologies, multimedia has 
+become the primary means of communication in the digital age and the language of discursive practices. 
+Heterotopias are special, multi-faceted contexts that facilitate the exchange of information and link to other 
+systems with “surfaces of emergence.” They are unique in their ability to connect and integrate various 
+perspectives and knowledge bases. In this study, we presented a method for creating a framework for virtual 
+and mixed reality environments, as well as personal computers, that allows for the exchange of knowledge 
+and information through the creation of multilayered digital spaces based on heterotopias. The framework 
+aims to fill in the gaps left by the disappearance of certain layers in archival practices, particularly spatial 
+layers, due to the increasing reliance on technology that excludes embodiment. We provided the concept of 
+a virtual museum as an “operating table” for discursive methodologies, and instead of static settings, we 
+offered dynamic, “liquid architectures” which find their forms around the individual archives composed of 
+elements of multimedia. 
+The framework was tested with 30 participants and was found to be useful, easy to use, and 
+adaptable to various professional and personal settings. The effectiveness of the framework and 
+transformative impact of heterotopias are presented via the results of the qualitative and quantitative 
+approaches. We provided various sorting and grouping algorithms, and procedural content generation 
+algorithms such as Binary Space Partitioning, Cellular Automata, Growth Algorithm, and Procedural Room 
+Generation which offer different levels of autonomy to the automated generation algorithms, giving users 
+an opportunity for reflection, modification, and control over the design. Results indicate that place-
+production, transformation, and archival practices have a mutual relationship that is essential for “self-
+formation” activities. Integration of the layers and features discovered within heterotopias with digital 
+technologies, which offer a spatial “aesthetic becoming” at several levels of reality, was found productive 
+for providing discursive practices within digital habitats. Blending the meta-design and end-user 
+development approaches with the layers of heterotopias we provided sites where the users become “focal 
+points of resistance.” 
+To provide design features and outcomes that can direct future work in the HCI community, we 
+analyzed the various factors that can influence user experiences and preferences. The results of the 
+questionnaires and interviews demonstrate that the three evaluated versions are different, each having its 
+own characteristics, strengths, and weaknesses. Through a comparative study, we demonstrate how 
+
+ 
+46 
+ 
+ 
+ 
+different reality levels might augment different abilities of users by designing and curating virtual museums 
+of their activity. The level of the hybridity of the physical and digital spaces, interactions, the meaning and 
+perception of the space, and the elements that are contained by the space can be re-interpreted by different 
+users in VR, MR, and PC environments. 
+8. Future Directions 
+This framework targeted essential features to provide a system for diverse contexts. Therefore, we did not 
+add detailed technical requirements for a specific practice. Based on the framework structure, modules can 
+be added to serve different professions. According to the literature review, most architects and designers 
+prefer to work on several types of representations with multiple screens. With the additional features, this 
+framework can solve colocation problems of the design processes and provide more detailed production. 
+To reorder the content, we used four variables based on museology which can be reinterpreted by users. 
+However, for more complex structures, features for different reordering methodologies can be generated. 
+One of the factors that can affect the environment’s realism is audio assets that are not provided in 
+accordance with physical distance. To enhance the realism of the environment and navigate through these 
+environments in a more realistic and immersive way, sound localization can be provided. According to 
+participants’ statements, the number of provided features can become exhausting for some novice users. 
+Therefore, features can be provided separately for different experience levels. 
+In this study, we examined individual experiences separately for each technology to compare and 
+understand specific preferences and impacts. As a future study, providing a social layer and more 
+transformative systems between PC and immersive technologies can increase the richness and relations 
+provided via heterotopias created by the users and increase the usability of the framework. Future studies 
+can include asymmetric and symmetric collaborative approaches including the social relations to practices. 
+Disclosure statement 
+No potential conflict of interest was reported by the author(s). 
+Data availability statement 
+The data that support the findings of this study are available from the corresponding author, [E.S.], upon 
+reasonable request. 
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+
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+page_content='Developing a Framework for Heterotopias as Discursive Playgrounds: A Comparative Analysis of Non-Immersive and Immersive Technologies Elif Hilal Korkut1  and Elif Surer1, * 1 Department of Modeling and Simulation, Graduate School of Informatics, Middle East Technical University, 06800 Ankara, Turkey  Corresponding Author: Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Elif Surer Department of Modeling and Simulation Graduate School of Informatics Middle East Technical University 06800 Ankara, Turkey elifs@metu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='tr +90 545 2107888  Elif Hilal KORKUT (elif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='korkut@metu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='tr) is a Master’s student in the Multimedia Informatics program at Middle East Technical University with a Bachelor’s degree in Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' She is interested in generative algorithms, computational design, exhibition and display design, vision and visuality, and game technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Elif SURER (elifs@metu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='tr) is currently working as an Associate Professor with the Graduate School of Informatics’ Multimedia Informatics program at Middle East Technical University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Her research interests include serious games, virtual/augmented reality, and reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Developing a Framework for Heterotopias as Discursive Playgrounds: A Comparative Analysis of Non-Immersive and Immersive Technologies  Abstract The discursive space represents the reordering of knowledge gained through accumulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In the digital age, multimedia has become the language of information, and the space for archival practices is provided by non-immersive technologies, resulting in the disappearance of several layers from discursive activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Heterotopias are unique, multilayered epistemic contexts that connect other systems through the exchange of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' This paper describes a process to create a framework for Virtual Reality, Mixed Reality, and personal computer environments based on heterotopias to provide absent layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' This study provides virtual museum space as an informational terrain that contains a “world within worlds” and presents place production as a layer of heterotopia and the subject of discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Automation for the individual multimedia content is provided via various sorting and grouping algorithms, and procedural content generation algorithms such as Binary Space Partitioning, Cellular Automata, Growth Algorithm, and Procedural Room Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Versions of the framework were comparatively evaluated through a user study involving 30 participants, considering factors such as usability, technology acceptance, and presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The results of the study show that the framework can serve diverse contexts to construct multilayered digital habitats and is flexible for integration into professional and daily life practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Keywords: Virtual Museums, Virtual Reality, Arts and Media, Design Tools, Intelligent UI, Culture, Design Research, HCI Design Theory, Multimodal UI, User Interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Introduction Every culture, field, and individual creates different ways of communicating and constructing knowledge by selecting certain elements, excluding others, and rearranging them in a particular order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The formation and constraints can vary, but the meaning and knowledge production occur through narration and interpretations of these units (Maciag, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The increasing availability and accessibility of digital information, made possible through processes like digitalization and digitization, has changed discursive practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Actors have been narrating emerged “formless chaos of knowledge” composed of multimedia via technological systems to produce, consume, and preserve information (Manovich, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' While technology provided accessibility and usability to users, extensive use of systems facilitated by the widespread adoption of technology and the increasing reliance on personal computers and mobile phones led to a transformation, elimination, and dissociation of several layers in daily life and professional practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Odom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2014) describe this situation as “placelessness” and “formlessness.” In the absence of certain layers, the user constantly maps the physical interactions and customizations to “virtual possessions” in the digital environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Being a multilayered discursive space, the missing layers of this new territory can be explored with recourse to Foucault’s heterotopias, “areas of resistance that intensify knowledge” (Foucault, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  In this study, we investigated the layers essential for comprehensive production, representation, and interpretation processes for professional and personal “archival practices” through the concept of heterotopia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Foucault provides several strategies to construct discourses: “operating table,” “surfaces of emergence,” “authorities of delimitation,” and “grids of specification” (Foucault, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Based on layers of heterotopias and their potential to integrate strategies of discourse, we provided a framework that can support users in creating their digital habitats via immersive technologies incorporating individual archives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Resistance of heterotopias occurs through adaptations and changes in function resulting from dynamic power relations, which is an essential layer of heterotopias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The individual is regarded as a “focal point of resistance” in modernity, with the ability to shape and manipulate experiences and products in conjunction with the systems to which they are linked (Thompson, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Together with the system, they become “authorities of delimitation.” The developed framework offers the dynamism of heterotopias that can be altered to fit the needs of users, with the ability to import various formats to compose a content layer and modular structure based on object-oriented programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' It empowers individuals to create experiences based on their content in an immersive environment where they become the authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Heterotopias provide alternative viewpoints of the world and opportunities for people to explore different ways of thinking by juxtaposing and combining many spaces into one site;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' they problematize received knowledge from the “surfaces of emergence.” The framework functions as an “operating table,” it expands the space of heterotopias and enables users to interact with the elements of multimedia language in changeable settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Heterotopias are dynamic physical or virtual spaces that exist outside of the normal ordering of time and space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' They offer spatial layers to reorder information (Foucault, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The presented framework offers “grids of specification,” forming “liquid architectures” where users can create places according to their needs (Novak, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' It includes procedural generation algorithms (PCG) that provide  3  different levels of autonomy while offering several architectural elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' This way, the framework provides another layer that can be interpreted as the meaning of the space that contains complex relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' This layer includes how the space is narrated, represented, or understood by different groups of people and the contexts that give it meaning (Lehtinen, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Through embodiment, mixed and virtual reality technologies have the potential to create a spatial layer of heterotopias by creating immersive, interactive digital environments that can mimic the feeling of being in a physical place or blend the physical space with the virtual and provide a sense of place and connection to their surroundings (Schultze, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Virtual Reality (VR) and Mixed Reality (MR) technologies are extensively studied to facilitate places and knowledge production by allowing users to engage in activities and simulations of cultural heritage (Cecotti, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Gonizzi Barsanti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', 2015), construction management (Hepp and Hasebrink, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Safikhani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', 2022), and architecture (Akin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Prabhakaran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' While these studies provide the existence of spatial layers, they were developed to target specific fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Therefore, they do not offer comprehensiveness in terms of multimedia and provide an operating table for only targeted fields which eliminates the possibility of using them in other contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  In this study, virtual museum space was specifically chosen as an overarching term that can contain various types of entities holding both place and media representation and reproduction layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Exhibitions inside a virtual museum create a “world within worlds” through narration constructed with places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Foucault (2008) already defined museums as heterotopia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' however, not all museums are necessarily discursive heterotopias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The current applications show that large corpora consisting of virtual museums provide abstract or exact copies of a real museum with isolated interactions based on predefined paths (Cecotti, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Kersten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' There is an increase in the number of studies improved by individualization being incorporated in a static, designed, built environment (Foo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Komianos and Oikonomou, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Virtual museum applications offer a wide range of multimedia elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' however, they provide static functions, and personal content has been limited to images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' A system that provides the technical capability of integrating diverse personal media such as text, audio, and 3D graphics is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Also, the elements are provided in digital “emplacements.” They do not offer the adaptable place production required for reordering individual content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Therefore, they mostly do not provide operating table functions, including the resistance presented by individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Building on these insights, we developed a flexible framework that can provide the construction of heterotopias for the broader audience and disciplines, which makes the framework usable for further research and diverse practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The spatial layers of heterotopia creation are provided via architectural elements open for manipulation and texture, lighting, and scale options and supported via integration of PCG algorithms, especially for novice users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Additionally, by injecting user preferences, we provided different automation levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Content layers and archival practices are provided via several algorithms that can import, store, and exhibit various data formats and allow analysis and grouping of personal content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In conjunction with the ability to hold various layers, the developed framework can provide a playground based on heterotopias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Throughout the design process, we followed meta-design principles, which helped us to create a flexible and modular framework that can shape via needs of users to serve diverse contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  4  This study compares experiences across three platforms (PC, VR, MR) to understand the effects of different technologies, which can serve as further insights to increase the integration of immersive technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' We conducted a comparative user study with the participation of 30 respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' After exposure to the system, participants completed standardized presence, technology acceptance, usability questionnaires, and system- specific surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Later, we conducted semi-constructed interviews to gain better insights and interpret the objective and subjective data gathered in the previous steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Results indicate that we managed to provide standard levels of usability and acceptance across technologies, and the framework is suitable for diverse contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' While VR and MR technologies can offer absent layers of 2D interfaces, from the meta-design perspective (Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', 2020), when compared to PC-based tools, immersive technologies are still waiting for more pervasive use, which requires new human-computer interaction (HCI) approaches to integrate those technologies into professional and personal life on a daily basis, to create a comprehensive dialogue between system, user, content, and place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To derive design features that can increase the integration of immersive technologies into different contexts, this study aims to provide answers to the following questions:  R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In the creation process of heterotopias, which medium provides better experiences in terms of usability, presence, and technology acceptance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' According to which dimensions do technology acceptance, usability, and presence of versions present differences?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Which medium is more applicable for the interaction with different asset types?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' How do different mediums change the design approach of users?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  R5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' How do scale, point of view, lighting, and material aspects affect users’ experiences and tasks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  R6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' What are the effects of the autonomy level of procedural generation algorithms on users’ preferences?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In what follows, we review works related to immersive technologies, virtual museums, and procedural generation techniques, and to facilitate readability and to preface our user analysis, we briefly outline the conceptual terms that emerged through technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Then, we describe each aspect of the proposed application and the rationale behind the design decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In the next part, we explain our evaluation methods and procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Then, we provide the results of the user tests about related concepts from various fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' We discuss the results while reflecting on the extended concepts and terms that emerged through user tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Finally, to conclude, we summarize our study and add insights into how further details might be searched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Literature Review 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Discourse and Heterotopias Michel Foucault was a French philosopher who studied the ways in which power and knowledge shape society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' He identified certain places, called heterotopias, that exist within a culture and serve as a sort of  5  “realized utopia” where other places within the culture are represented, disputed, and reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Heterotopias are characterized by their ability to juxtapose and compare various spaces and concepts and can manifest in various architectural forms, functions, and sizes (Foucault, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' According to Michel Foucault, discourse, or systems of thought and representation, plays an important role in shaping our understanding of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' He identified several strategies for constructing discourses: “operating table,” “surfaces of emergence,” “authorities of delimitation,” and “grids of specification.” The operating table represents the space in which knowledge is constituted serving as a means of classifying and ordering things, allowing us to make sense of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Surfaces of emergence are the foundation for the development of new discourses and the creation of new subjects, while an authority of delimitation is a set of rules that determines how objects can become subjects within a discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The grids of specification are a taxonomy of concepts used to order objects into a hierarchy within a discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' By understanding these strategies, Foucault explained how discourses shape our understanding of the world and our actions within it (Foucault, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Heterotopias are distinctive spaces that can contain the processes of discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Michel Foucault identified various characteristics of these spaces and provided examples to help illustrate and understand these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  According to Michel Foucault, all cultures produce heterotopias of deviation and crisis such as boarding schools and prisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Depending on the culture and time period, these spaces may serve a variety of purposes within a society, and their features and functions evolve over time, which provides resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' He argued that resistance is not separate from power relationships but rather is an integral part of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Power is not a fixed entity but is constantly negotiated and contested through interactions, leading to changes in the functions of heterotopias (Lehtinen and Brunila,  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Heterotopias are spaces that can bring together multiple incompatible spaces and elements in a single location, disrupting the way we understand the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Foucault also investigated the “order of discourse” as the organization of the rules and systems that govern knowledge production (Foucault, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Heterotopias offer alternative ways of organizing information and the layout of a space can affect the way knowledge is shared and understood, providing “grids of specification.” The characteristics and purpose of these spaces influence the meanings and perceptions that can be experienced through embodiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Heterotopias are spaces that are connected to certain periods in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' They can either permanently store time, like archives or libraries, or they can provide a brief window into a specific time, like exhibitions or festivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  In modern society, the individual becomes the “focal point of resistance,” and archives, libraries, and museums often represent personal choice and “self-formation,” with the individual acting as a point of resistance to the shaping power of the larger system they exist within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' These individuals become “authorities of delimitation,” shaping their own products within the constraints of the system (Foucault, 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Heterotopias are places where access is controlled, meaning that they are not freely accessible like public spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' This control can be achieved through technological means, such as the use of devices that have opening and closing systems or personal accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Heterotopias are sites that have a particular connection to the environment around them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' They may either expose the essence of other real spaces or  6  create an organized space that is distinct from the disorder of other places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In the context of discourse, this relationship creates “surfaces of emergence,” where new ideas, arguments, or perspectives arise or come to the forefront of the discourse (Foucault, 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Heterotopias and Digital Playgrounds Rousseaux and Thouvenin (2009) explored Informed Virtual Sites (IVS) through Michel Foucault’s heterotopias by superimposing digital and physical spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' They also named these heterogeneous places heterovirtopias as an extension of Foucault’s terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The method of loci, also known as the mind palace, is a memory technique that allows people to recall information by associating the space they know well with the information they want to remember through the connection between object and subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' This memory technique primarily employs spatial memory to remember information efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' As another example of informational topologies, Yamada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2017) developed a system called HoloMoL using the mind palace method to help users to memorize by combining information as mixed reality content with physical places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The museum’s potential is not solely derived from its exhibits, but rather from the interactions between the objects on display and the visitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' These interactions, along with the stories and communities that are formed within the museum, contribute to the vitality of both the collections and the visitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  The viewers’ engagement with the objects endows them with meaning and with a sense of symbolic immortality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The museum also possesses a transformative power and an allure due to the mysteries contained within its unseen artifacts and untold stories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Malraux (1967) demonstrates how formal museums compromise the essence of works of art by clustering them together, diminishing the importance of individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' He proposed a fictitious museum composed of photographs of works of art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Our society is evolving toward a more mediated culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Following the heterotopia concept, a paradigm shift in terms of contemporary senses has occurred within the museum structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Online platform projects, such as Google Art and WikiArt, provide high- resolution images of selected artworks available to the public all over the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Although there is a noteworthy difference between “Museum without Walls” and today’s virtual museums on the Web, they have a similar conceptual purpose regarding transforming information and knowledge into forms that are available regardless of distance from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The concept of the Digital Museum emerged in conjunction with the museum’s expansion beyond physical places through the increased use of new media technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Various factors, including media and digital resources, influence the organization of virtual museums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' According to Schweibenz (2019), the primary distinction between virtual museums and traditional physical museums is their level of accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  With the adoption of digital media and a narrative focus, known as the “narrative turn,” “the memory institutions” galleries, libraries, archives, and museums sector has undergone a transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Using technologies like virtual and augmented reality, 360-degree photography, and 3D reconstructions, they have transformed and created places and increased their use of digital spaces including websites (Basaraba, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Immersion and interactivity in immersive environments aim to increase the sense of  7  presence in a virtual environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' nevertheless, this does not mean that the digital environment is entirely composed of fictional elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Reconstruction of physical artifacts into digital media uses 3D data acquisition methods, such as photogrammetry and laser scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' These methods are frequently employed in virtual museums, virtual exploration, and cultural heritage contexts (Gonizzi Barsanti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Haydar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Pietroni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Hayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2016) developed a virtual museum capable of displaying planar artifacts using web scraping to extract necessary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The system allows users to choose the museum’s content that is designed by authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' MR systems in museums can enhance the typical visitor experience by combining historical interactive visualizations with related physical artifacts and displays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' MuseumEye application (Hammady, 2018) focuses on different guidance techniques to improve visitors’ experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Providing adaptable exhibition spaces, Komianos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2018) provided automated virtual exhibition construction based on adaptive exhibition topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' They state that the facilitation of visitors’ navigation can enhance users’ visiting experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Virtual museums and immersive cultural heritage studies can be interpreted as heterotopias of deviation and crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The risk of extinction due to natural or unnatural factors has created a need for the reconstruction of artifacts in a digital medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  The deviation has been occurring due to technological advances that have redefined almost every aspect of various practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Additionally, COVID-19, as a worldwide crisis, created boundaries between people and places, which accelerated virtual museum studies to provide accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Place-making activities have been provided by games for decades, where players can shape and influence the virtual spaces within the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' These activities can range from building and constructing structures and landscapes to customizing and decorating the appearance of these spaces (Basabara, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For individuals, they provide an opportunity to express creativity and personalize their virtual spaces and foster a sense of ownership and attachment to the game world, as players feel a sense of pride and accomplishment in creating and shaping their own spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Boldi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2022) explore the technologies of crisis through video games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' According to their study, the COVID-19 pandemic caused routines to be confined to people’s homes, leading some to feel disconnected from their usual places of habitation and to turn to video games as a way to escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Some people used games to explore virtual worlds that were different from their everyday lives, while others used games to recreate or substitute for meaningful places that were no longer available to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Researchers suggest that games or online communities could be designed to allow people to explore and customize virtual spaces to strengthen their attachment to them and recreate lost places to help people find new opportunities and meanings in their everyday environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Immersive Technologies Virtual, augmented, and mixed realities have been widely studied and have seen a range of applications in various fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' While virtual reality has a well-established definition, mixed reality has been described using a variety of terms, leading to ambiguous definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Mixed reality combines elements of both augmented and virtual reality, with the degree of combination determining where it falls on the reality-virtuality continuum (Milgram and Kishino, 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Virtual reality is a technology that uses real-world visual perception in artificial computer-generated environments through stereoscopic vision, providing an  8  immersive experience with motion capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Paul Milgram and Kishino propose the terms “augmented reality” and “augmented virtuality” to describe environments closer to the center of this spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Augmented reality maps virtual elements onto physical space to create a hybrid of the real and virtual, often through smartphones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Mixed reality technology uses head-mounted displays to combine AR and VR capabilities, such as Microsoft’s HoloLens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Interface and Interaction The user interface (UI) is the means of communication between a user and a computer system or application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Most personal computers use a graphical user interface (GUI) which uses visual elements such as icons, buttons, and menus and input and output devices like a mouse and keyboard to display a screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Different types of interfaces include form filling, which involves the user entering data into a pre-defined form or template, and direct manipulation (Shneiderman, 1983), where the user interacts with on-screen objects in a way that directly affects their behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Command languages are sets of instructions or commands that a user can enter to interact with a computer or software program, often used in text-based interfaces such as in a terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Natural language processing (NLP) allows users to communicate with a computer or software program using natural language rather than specific commands, often used in voice assistants and chatbots (Gilbert, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Immersive environments may require more interactive interfaces with specialized input devices such as motion controllers or haptic feedback devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' These interfaces can include head tracking, hand movement tracking, voice control, eye tracking, body movement tracking, and virtual hand manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To effectively manipulate objects in virtual reality environments, interaction techniques should allow for object selection, positioning, and rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The design of these techniques, often involving hand manipulation, is important for the overall user experience of a virtual reality environment (Mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Presence, Embodiment, and Experiencing Architecture The human body serves as a system for acquiring, processing, and displaying information from the physical world through the senses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Virtual worlds are digital environments that users can interact with using technology that differ from the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' They require technology to experience and interact with objects, spaces, and people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Presence refers to the sensation of being physically present in a given space and time, often experienced in virtual reality environments (Schultze, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Factors that can affect the level of presence include the realism of the sensory stimuli in the environment, the believability of the environment as a real space, the user’s level of engagement with the environment, and the user’s prior knowledge and expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Virtual worlds offer a chance to examine the role of the physical body in the communication and the effects of communicating without a physical body (Spence, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Merleau-Ponty’s philosophy (Lehtinen, 2022) emphasizes the concept of embodiment, or how the body shapes and is shaped by our perception and experience of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' He argues that the body is not simply a means of engaging  9  with the outside world but also plays a role in processes of perception, understanding, and expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The body is therefore central to our subjectivity and understanding of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In “Experiencing Architecture,” Rasmussen (1964) argues that the sensory experience of a building or space is crucial to our understanding and interaction with the built environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' He believes that architecture should be evaluated based on how it is experienced by the user, rather than just its aesthetic or functional qualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Rasmussen explores how various sensory experiences, such as sight, sound, touch, and even smell and taste, contribute to our perception of a building or space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' He explains that the way we experience architecture is based on the perception of solids and cavities, figure-ground relations, rhythm, scale, proportion, sound, lighting, color, and texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' He divides the process of creating and interpreting architectural forms into two categories: solid-minded and cavity-minded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Solid-minded architecture involves creating forms by combining solids or shells of voids, while cavity-minded architecture involves carving out components from a large solid to create spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Rasmussen describes architecture as “the art of playing with solids and cavities.” According to Luck (2014), the design process often involves behaviors that may seem incongruous but serve a deeper purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  These movements, which pertain to the design of a structure, symbolize something and provide a fleeting visual representation of architectural concepts and the anticipated sensation of movement within the architectural form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' This process is referred to as “aesthetic becoming,” and reflects the creative process behind the emergence of these behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The relationship between a building’s physical form, how it is experienced, and the reactions it elicits are subjective, intuitive, and complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Usability, Acceptance, Trust Technology acceptance refers to the willingness of an individual or organization to use a particular technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' By understanding the factors that impact technology acceptance, designers and developers can create more effective and appealing technologies (Rheingold, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Usability is a measure of how easily an individual can use a system or product to achieve their goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' It affects the performance, efficiency, and satisfaction of the user, as well as their overall experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Factors that contribute to the usability of a system include the cognitive abilities and limitations of the user, and the design of the system’s interface, organization, and structure (Salanitri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The concept of trust in technology has been studied extensively in the field of human-computer interaction (HCI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Previous research has shown that trust in technology is influenced by factors such as usability, technology acceptance, and presence and that these factors also affect each other (Lippert and Swiercz, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Trust in technology is a multi-dimensional concept, and researchers have identified several dimensions that contribute to trust between users and technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' These dimensions include predictability (the ability of technology to adhere to previously established performance standards), reliability (the perceived dependability of technology in certain situations), and utility (the perceived usefulness of technology) (Mcknight et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Procedural Content Generation The automatic creation of digital assets through algorithmic means and patterns with little to no user input is known as procedural content generation (PCG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In the gaming industry, algorithms are employed to create complex items like road networks, buildings, living things, as well as landscapes and plants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' PCG is being studied in fields other than computer science, interdisciplinary approach in the scientific sciences, including biology, architecture, urban studies, and psychology, is triggering rising interest in other communities, and it has become increasingly prominent (Prusinkiewicz and Lindenmayer, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' As the importance of PCG for production increases, researchers are exploring new ways to produce high-quality assets, either with or without human input;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' therefore, new paradigms have also risen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Various techniques have been produced through machine learning (PCGML) and deep learning, such as neural networks, auto-encoders, and deep convolutional networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Markov models, n-grams, and multi-dimensional Markov chains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' clustering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' and matrix factorization (Summerville et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Designers are increasingly utilizing autonomous tools to complete complex tasks faster and in novel ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' However, the deterministic nature of these methods can produce similar results repeatedly and may not provide the same level of creative user experience as hand-crafted and designed content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Most recently developed techniques aim to create fully autonomous approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Many researchers have proposed methods for the automatic generation of floor plans and buildings, but only a minority allows for customization of the end product of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Products that are not open to customization have been criticized in the field of architecture (Porter and Hanna, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In this study, we aim to offer greater control over the system on demand by manipulating the autonomy level of the algorithms and providing a “design space” for the user to customize the end product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Therefore, we sought methods that offer various autonomy levels and design space to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Previously, large-scale procedural modeling of virtual worlds typically only resulted in empty structures devoid of internal divisions and interconnections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Different approaches have been proposed to solve this problem, including tile placement, room enlargement, inside-outward extension, and subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Using tile placement strategies, the domain is partitioned, and the resulting grid is filled with tiles representing the rooms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Without relying on the limits of a building as a constraint, inside-out algorithms distribute rooms in accordance with the requirements for room connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Growth-based algorithms disperse room seeds throughout a building, and the rooms gradually fill the interior space and grow to their full size (Camozzato, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  A constrained growth-based method for floor plan generation was presented by Lopes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In their method, the sizes and positions of the rooms are defined by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The rooms are then expanded until they become square in shape, and the empty space is then filled with further irregular shapes by moving the rooms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Given that the floor plan can be divided into subdivisions, it is possible to connect adjacent rooms, and subdivisions with doors and windows are also generated for each room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' They indicated that minimal complexity and effective data structures are the main reasons for the success of their methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Graph Approach to Design Generation (GADG), a method for automatically creating rectangular floor plans based on an existing graph extracted from floor plans, has been proposed by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' They developed a mechanism to manipulate the rooms based on two transformation  11  principles, addition, and subtraction, by mapping the floor layouts to connectivity graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Users can specify the maximum width-to-height ratio for each created room using the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Building boundaries are used by subdivision algorithms to divide the interior space into rooms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' A technique called binary space partition (BSP) uses hyperplanes to recursively divide a space into convex sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' This subdivision becomes a representation of the scene as a tree data structure known as the BSP tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Baron (2017) examined many methods for creating procedural material that may be applied to both 2D and 3D projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' They created pairings by using five-room and corridor creation algorithms (Random Room Placement, BSP Room Placement, Random Point Connect, Drunkard’s Walk, and BSP Corridors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For indoor contexts, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2022) presented a reconstruction strategy for room layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To create the optimum in-door polygonal models, they integrate voxel-based room segmentation and space partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The technique involves room semantic data to divide subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Cellular automata (CA) is another technique used for architectural generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Cellular automata apply a set of rules to cells, and each cell executes those sets of rules with respect to its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Cells can acquire a finite number of cell states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Herr and Kvan (2007) show how a variety of cell shapes and sizes used in one CA model may support architectural form findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Additionally, they proposed a theoretical framework for the integration of CA into the design process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Araghi and Stouffs (2015) explore CA systems for high-density residential building forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Based on the capability of CA to create complex rule definitions, they also address solutions for architectural problems such as density, accessibility, and natural light via rule definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Cruz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2016) discuss variations of classic CA cell shapes to derive a variety of architecturally feasible forms capable of generating aggregate spatial units to match specific spatial configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Design Rationale and Structure of the Framework Based on the literature regarding heterotopias and discourse, we generated a set of rules to provide consistency in methodology and a comprehensive framework structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The system should (1) have dynamic power relations and be open to changing its functions to create resistance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2) provide juxtapositions of multimedia elements on an “operating table” that allow certain interactions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (3) provide a ground for “self-formation” and a “focal point of resistance” through individuals who construct “authorities of delimitation” together with the system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' and (4) provide a relationship with other places by either creating an illusionary space or an orderly space composing “grids of specification”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (5) able to create “liquid architectures” according to needs of the users that can convey spatial qualities that are perceived via embodiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Throughout the methodology, we emphasized end-user development and meta-design principles to empower users, create a design space, and encourage informal use cases for novice and experienced users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Therefore, to simplify the mechanisms and the features that we integrated to transform users into curators of their digital environments, they were carefully selected following the previous studies on multimedia,  12  architecture, and immersive technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In the subsequent sections, implementation details for VR, MR, and PC technologies are given in accordance with the flow and the layers provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Virtual Museum as an Operating Table Different form the previous studies, the virtual museum concept in this study is not referred to as a static “emplacement” in digital space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' On the contrary, it becomes a dynamic structure, an operating table where the user operates on elements by including, excluding, and transforming them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Designing a virtual museum can be viewed as an infinite process that begins with documentation and continues with imagination and experience in the virtual environment, forming a multilayered discursive space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' When personal data is categorized and stored in a digital space, it can be seen as a digital archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' If it is displayed or played, it is considered an individual exhibit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' if multiple files are displayed within a context, it is considered a digital exhibition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' As it provides a categorization of documentation and displays archives in the form of exhibitions via the built environment, virtual museum space can transform individual archives into personal exhibitions creating “grids of specification.” Through the experience, we used the terminology museum spaces to increase usability and decrease the complexity of the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' We defined two different modes for the application, namely curator and visitor modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Curator mode defines the active creation of heterotopias, while visitor mode excludes the control layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The framework provides a system for users to store and display digital content, including text, images, videos, 3D models, and audio, in a way that allows for easy access and manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To be able to categorize items, we named the assets according to museum conventions, such as painting, sculpture, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', and added labels to provide more data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' We used asset style, artist, location, and time data to provide users with automation in reordering systems of heterotopias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' On the other hand, the system is open to manipulation and interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For example, the four variables that are used for sorting and grouping can be systematically employed by the user for different variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Dialogue Between the User and the System The hardware provided, the general structure of the system, the interface design, interaction, and navigational capabilities all directly affect the experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For example, hardware defines the physical comfort level, interaction modalities, and technical borders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The dialogue between the interface and the user affects multiple features of the heterotopia created by the user and shapes the user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Through the interface, the user grasps the level of autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Interface, interaction, and navigation provide the narration for the discursive space and sense-making processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Programming Language and Development Environment To eliminate the results stemming from hardware differences in VR and MR, an Oculus Quest 2 HMD system was used to stream the audial and visual content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' With the feature called Passthrough, using the sensors on the headset allows seeing a real-time view of the surroundings together with the virtual content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  13  The main reason that we used the broader term mixed reality instead of augmented reality in this study is the interaction and anchoring capabilities of the device, which combines AR and VR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The Oculus Insight tracking system is developed by Facebook Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' and made available on the Oculus Quest 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Oculus Insight tracking relies on three streams of sensor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Continuous streams of data provide hand gesture recognition, physical context awareness, and the position of controllers and hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The Oculus Quest 2 Controllers as interactivity devices and hand tracking were used to allow the users to navigate through the virtual environment (VE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The buttons on the controllers were visible in the virtual environment, including the hand position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The Oculus Quest 2 tracks the head movement to present the correct virtual world image to the eyes (LaValle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', 2014), and it analyzes the user’s head movement in real-time to control the view, which enables natural interactions, leading to high levels of presence and immersion (Desai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The HMD includes an adjustable head strap that sets the hands free for the controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The combined weight of the HMD (503 grams) and controllers (126 grams/per controller) is 755 grams, which facilitates comfortable use for extended periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' All versions of the framework were developed using Unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Unity game development platform was selected because of its prominent features among the other opponents in the industry, and it is one of the natural development platforms for Oculus Quest 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The Unity game engine is compatible with various platforms including desktop, mobile, console, web, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Unity supports C# programming language which is designed for Common Language Infrastructure by Microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' It is a general-purpose, multi- paradigm, and object-oriented language that allows developers to build applications that run on the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='NET Framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' A programming paradigm called “object-oriented programming” (OOP) is based on the idea that “objects” can hold both data and code that manipulates that data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' An object in OOP is a self-contained unit composed of both information and programming code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The code that operates on objects is created to reflect the behavior of real-world objects, and objects are used to represent real-world concepts or objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' OOP is characterized by the use of encapsulation or the bundling of data and the code that manipulates it into a single unit or object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Using encapsulation, the framework becomes open for transformation and additional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' This changeable structure gives the framework adaptability of the heterotopias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Besides Unity, the Microsoft-driven project Mixed Reality Toolkit (MRTK) provides a set of components and features used for the VR and MR parts of the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  The 3D models were created in RhinoCeros and Blender and enhanced by 3D modeling distribution web environments which provide copy-left models such as CG trader, Sketchfab, and TurboSquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Interface and Interaction The interface is designed to guide the user through the experience and construction of heterotopias layer by layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Different types of layers are provided via different planes and sequentially presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For the interface design, we preferred techniques especially suitable for novice users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For example, although the command line is very flexible since the learnability of command languages is generally very poor and not suitable for non-expert users, we did not include any command language interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The form fill-in interaction style was aimed at a different set of users than the command language, namely non-expert users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' We used a form fill-in type of design for the tasks that require several data entries from the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For example, the size  14  properties of generation algorithms (Figure 1) and URL for connection are given in the style of form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To guide the user via the predefined rules and simplify the data entry, instead of using empty input fields, we preferred to use sliders to define and show the range directly to the user when it is applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Menus are collections of options that are displayed on the screen, and, upon selection and execution, affect the system’s state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The user chooses a command from a predetermined list of commands listed in menus using a system based on menu selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Although it can slow down the frequent users, we prefer to use menu interactions in most cases since it is ideal for novice or intermittent users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To increase memorization and eliminate language boundaries instead of labels, we used icons for the command/menu items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To decrease the information overload and visual clutter, the items were clustered into sub-menus according to their functions (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In the VR and MR versions, if the feature is not toggled off, menus follow the users as they move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In case of losing the main menu, a hand menu comprising the restart, quit, and call main menu options was included, which can be called via a hand gesture and a button on the controller (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The 3D menus designed for VR and MR also allow users to interact via touch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Example Interface options of the PC version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' We employed the direct manipulation principle for all objects that are placeable to the grid and 3D interface elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Selected objects are highlighted and can be directly manipulated via hands, controllers, or mouse and keyboard according to versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Direct manipulation is especially important for the embodiment aspects since it requires movement of the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' With this division, we aim to prevent unintended changes in models and remove unnecessary UI elements from the vision of the user, which, otherwise can disturb the users’ experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The user experience for all versions starts with a welcome screen and continues with a panel that is used for importing the digital archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The archive that is analyzed and environmental elements to construct places are provided to the user with a series of panels to construct the discursive habitat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' While the PC version uses the keyboard and mouse as input, for VR and MR versions, we offer controllers and hand interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The gestures are components of human-computer interaction that have become the subject of attention in multimodal systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In this study, we used six hand gestures, namely point, select, release, teleport, call hand menu, and call main menu (Oculus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Hand interaction is provided with far and near interaction options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For all versions, the user can manipulate the elements including 3D UI elements, via scale, rotate or translate operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Q8  B  800  15  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Hand Menu and Interfaces in VR environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Navigation In authoring personal museum experiences, the users should be able to fully control their navigation in the museum and they should be able to freely explore and manipulate the objects to create their own virtual experiences in an interactive and flexible way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In the curation process, the user is not just an observer but can interact with the exhibits via constructive dialogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The overall process and navigation of the user in the virtual museum enhances the understanding and keeps the interest alive by enriching the aesthetic sensitivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' During the design process, we tried to make sure that all versions were as similar to each other as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' However, inevitably, we used different methods in terms of navigation and interaction between the PC version and other versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The PC version includes four types of cameras: first-person, top-down, isometric, and isometric bird view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' All cameras can be controlled by W, A, S, and D buttons, and in the top-down view, zoom-in/out options are available with a mouse scroll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Also, there is a transport toggle in the control menu which allows the user to teleport all cameras in the scene with the left mouse click.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The gestures are components of human-computer interaction that have become the subject of attention in multimodal systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Gestures are captured through cameras on Oculus Quest 2 and can recognize various gesture types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Inside the VE, users can navigate, walk or teleport themselves using controllers or hand gestures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The tags are used to define colliders that allow teleportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Objects with floor and ground tags allow teleportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To eliminate motion sickness, we did not add movement via controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Content Layer as an act of Self-Formation and Archival Practices Being a force of resistance, individuals should determine the elements of the “surfaces of emergence” as an act of self-formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Archival practices as a method of analyzing the discourse and its elements comprise the actions of self-formation on an operating table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To provide automation for self-formation activities, we constructed a pipeline that includes importing individual archives and transforming them into operable artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' We also provided an analysis of the archives based on grids of specification of museums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Can  Dat  1961StyfeMagicRealiem  Genve:poevait  16  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Importing Files and Authority of Delimitation of the Framework The first layer of delimitation of the elements will be represented on the operation table defined by the framework in terms of media types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' As the elements of multimedia language, five types of assets and their museum equivalents have been identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Texts in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='txt, images in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='jpeg and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='png format, 3D models in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='obj and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='fbx format, audios in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='mp3 format, and videos in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='mp4 format are acceptable for generating the artifact displays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Within these constraints, being the focal point of resistance, the users are enabled to create a personal collection of 3D digital exhibits according to their interests and preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For the PC version, a custom file browser is written, which allows the user to choose the folder consisting of the files that will be uploaded to the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For the VR and MR versions, instead of using a file browser due to the security properties of the device, the application pulls the data from Google Drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To provide this feature, we include the virtual keyboard provided by Oculus and an input field where the user can write only the file ID of the data file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Using the ID, a direct link for Google Drive is constructed for a data file in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='JSON format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The data file needs to include the file IDs of the objects to be downloaded into the database of the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Artifact Classes In addition to assets, the user can provide a data file in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='CSS or .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='JSON formats which include certain features of the artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Data columns are defined as the name, artist, style, location, time, size, and description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Five different classes were constructed inheriting from the base class according to acceptable asset types, which are paintings, sculptures, videos, sounds, and text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Each art object has common properties, such as name, description, artist, and style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' According to the class, different properties are attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For example, paintings and sculptures have location and size properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' According to file extensions, the tool copies the files into the data folder and groups them into artifact groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' If the data provided by the user is inside the folder, data is parsed into columns and for each line, a new art object instance is constructed according to its class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Algorithm searches for matching names comparing the data and files and constructs dictionaries and separate lists for the art objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' If no data file is provided, instantiated objects use the filename as a name and are placed without a description panel and with a generic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' After importing processes according to extensions of the files, buttons are constructed for each file under the corresponding panel to spawn the artifacts in virtual environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For the paintings, sprites of the buttons use the file directly, for other types of icons indicating the type are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  The name of the button is generated according to the name of the files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The placement of all objects will be explained in the grid system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Artifact Holders Following the end-user development, the system should provide automation to transform the assets of the individuals into artifacts that are open to manipulation so that users can benefit by making changes, plans, and experiments with the artifacts and finding the best position for them both in the virtual and the real environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Based on features of OOP, for each artifact type, an artifact holder is designed which contains  17  several scripts and elements that are necessary for the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Holders are higher entities in the hierarchy, placed under the manipulation of the object’s position, rotation, and scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Each holder consists of a label panel where the short descriptions are placed, the Game object that will contain the assets, the Details button to open the ArtCanvas panel, a collider to manipulate, and a grid object collection script to fix the space between artifact and label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' If the dimensions of a painting or sculpture asset are provided, the container object of the asset is scaled according to data, and the space between the artifact display and label remains stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Additionally, each artifact holder has asset-specific features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For example, holders of video and sound have a play and pause button (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Artifact Holders in VR Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For each class, artifact holders are designed as prefabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' When an artifact button is pressed, an artifact holder is constructed according to the related prefab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' If data is provided, the name, artist, style, time, and location columns are given on the label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Also, we provide an ArtCanvas panel where the information in the description column is given with the artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The video, text, and painting artifacts are visually represented on the left side of the panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For the sound and sculptures, icons are placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' On the right side of the canvas, the details section is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Users are also able to see a list of all assets they uploaded as a list in the artifact analysis panel where the information is provided based on the categorization of artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Analysis of Artifacts and Grids of Specification of the Framework The first classification of the assets is performed according to formats to provide an accessible archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' If the data is provided, it is parsed into columns and for each class, artifacts are grouped according to four specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The framework offers three strings and one integer for archival indexes which are named in accordance with the terminology of museums and named as artist, style, location, and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The constructed lists are provided through an artifact analysis panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' These groups are also prepared for the PCG algorithms to calculate the dimensions of the rooms according to grouping options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  On the other hand, based on the data types, users can generate different types of specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  目  18  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Spatial Layers as an Act of Self-Formation and Architectural Elements The primary purpose of using architectural elements in a virtual museum context is to reorder the multimedia elements constructing spatial grids of specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In this study, the idea of a “virtual museum” does not necessarily indicate a traditional built environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' On the contrary, it extends the space and forms a “liquid architecture” around the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Every digital object that is placed, architectural or not, becomes information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Beyond being a partition for artifacts we offered place-production practices to provide “aesthetic becoming” experiences as an act of self-formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The layers of space in a heterotopia demonstrate how complex and multifaceted these spaces are, with different meanings and functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' These layers can be thought of as different levels of reality that are provided by various technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' One layer of spatial meaning in a heterotopia is its actual physical location in the world and its meaning within context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' PC, VR, and MR technologies provide different sets of relations with places and place production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The relationship with the physical location is provided via mixed reality technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' According to the context, the user can transform the physical space with digital elements or create different places that are completely digital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Heterotopias have layers of spatial meaning that relate to the way they are experienced by individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Through embodiment and telepresence, technology provides users with endless possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In this study, we provided architectural practices to increase the transformative effects of heterotopias, and through embodiment and presence, we aimed to present an operating table for those practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To be able to provide an easy-to-use generic framework, we examined the architectural studies to identify essential features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' According to Rasmussen’s research in “Experiencing Architecture,” various factors such as the perception of solid and cavity forms, the distinction between foreground and background elements, the size and dimensions of architectural features, the appropriate use of proportion and scale, the acoustics of the space, the lighting, the textures, and the rhythm all contribute to the overall sensory experience of a building or space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Therefore, we aimed to provide those elements in a technological landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Construction Elements For the construction of architectural aspects, we identify the main construction elements as walls, floors, roofs, windows, stairs, and doors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Additionally, we add two different classes: landscape elements and furniture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' All classes are provided in different panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The main elements are available as default objects, and they are editable in terms of position, scale, and rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Textures, Materials, and Colors Inside the Unity engine, 2D textures are tiled to provide materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  For the wall, floor, roof, door, and window objects, different textures are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For the consistency of place and ease of use, generic objects in the same layer share the same material properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' We added sliders that control the RGB values of materials that are used by the construction objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Additionally, we provided different skybox materials for VR and PC versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  19  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Scale Although all objects are scalable, we add different scale options to increase usability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The same wall, floor, and roof tiles are also provided in different scales, such as 1x1, 2x2, and 4x4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' At the beginning of the experience, the framework provides two different scale modes for VR and MR environments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' human scale (1:1) and model scale (1:20) (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' All models are automatically scaled according to the selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Additionally, vertical construction components are scaled according to the selected grid height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Human-Scale and Model-Scale Interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Lighting Settings Three different light sources are used in the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Sunlight, as directional light, ceiling lights as light strips, and art objects have their lighting as a spotlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In the lighting settings panel, each lighting type has a toggle that can be turned on and off (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' A slider is added to control the temperature of the light with values in Kelvin which are converted to the color of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  香  目  x  目  8  20  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Artifacts under neutral and warm temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Placement and Grid System The grid system was constructed to provide placement and snapping to correct positions to avoid overlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Users can set grid level and grid height properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Grid level defines the number of floor levels, and grid height defines the height of the walls and the distance between two grid levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For all placeable objects, including artifacts, three different Scriptable Objects (SO) were created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The SOs are types of assets in the Unity engine that allow saving data in an asset file rather than as a mono behavior attached to a game object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' This allows data to be stored in a way that is independent of any specific game object and allows data to be easily reused and shared between different parts of the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The first type is floor objects which can only be placed on the layers identified as ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The grid system is used for snapping floor objects to correct positions and holding the position data of floor objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The second type includes vertical elements that are placed edges of the floor tiles such as windows, walls, and doors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Each floor object includes four colliders that are placed on the corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' This way, during the placement of the vertical objects on floors, the grid system snaps and rotates the vertical elements according to the edges of the floor objects on collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The third type includes other elements and artifacts which are placeable in every location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  All SO types include two variations of one prefab, which are ghost objects and placed objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The initial code on the scriptable objects adds the additional scripts that are necessary for interaction and hierarchy according to types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' When the instantiation button is pressed, first, a ghost object is spawned which follows the pointer in MR and VR, the mouse position in the PC version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To place the object, the left click of a mouse, the select gesture with the right hand or right trigger is used according to the version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To cancel the selection, the right mouse click, and the select gesture with the left hand or left trigger button is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For the delete operation, a toggle is added which allows users to delete the selected objects when it is on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' If the model scale is selected, the grid system is placed on a table and every spawned object becomes a child object of the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' This provides a conversion of the scale and movement of the objects with the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  208  凉明  21  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Architecture and Archives Combined: Automated Generation of the Built Environment The algorithms provide reordering mechanisms in 3D dimensional spaces constructing grids of specification based on archival data while offering fast prototyping to architectural practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Based on discussions in architecture, heterotopias, and meta-design principles regarding power, autonomy, and control, we adapted the algorithms which can offer design space and control to the users (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' According to the requirements of this study, algorithms are able to generate content based on user data, and generated content is open for customization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Transformation of Binary Space Partitioning Algorithm’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Rasmussen divides the production and interpretation processes of architectural forms into two: solid-minded and cavity-minded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' One can start with a simple framework and add to it, or one can carve out certain components from a large solid to start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Drawing from this separation, we defined two scale options namely, room-scale and building scale, and employed one algorithm that can produce content based on dimension, window, and door requirements for room generation where the user can identify the dimensions of the solid and the number of cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' We have provided two different generation techniques for building scale production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' We created a solid-minded generation combining Binary Space Partitioning (BSP) and Cellular Automata (CA) algorithms and a constrained growth algorithm to provide cavity-minded production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' While these algorithms are mostly used with random number generators to create different game-level designs, we injected user preferences and user data to create levels of automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Each algorithm provides three levels of automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Users can prefer to generate a 3D environment based on data provided, with random numbers, or they can define the parameters through the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' This approach offers creative control over mechanisms that presents different levels of granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  In all three cases, we made architectural choices that would ensure coherence in the environment produced to create a consistent user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Users can re-design the produced models of a building by using the same operation that they can perform on other assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Without the scale operation of the user, the height of all vertical  GridLevel  22  components is defined by the parameter given by the user at the beginning of the experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To this end, as an input to algorithms, we used the same assets that we provided as main construction objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Calculation Strategies Based on Data To be able to calculate the number of rooms and minimum dimensions, the data provided by the user is parsed into columns (Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' According to style, artist, and location data, artifacts are grouped and provided via an artifact analysis panel together with artifact types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The number of rooms is defined according to the categorization selection of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For example, if the style is selected, the number of styles also defines the number of rooms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In this way, each artifact is assigned to a certain room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Since the dimensions of audio, video, and text artifact holders are already defined, their dimensions in the X-axis are directly added to the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Dimensions of the paintings and sculptures are extracted from data and if they are not available, the default dimension in the X-axis is defined as one meter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Algorithm 1: Room Number and Size Calculation 1 Parse user data into columns 2 Group artifacts by style,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' artist,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' and location 3 Define the number of rooms based on user categorization selection 4 Assign each artifact to a room 5 foreach Room do 6  foreach Artifact in a room do 7  if the artifact is audio,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' video,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' or text then 8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='Add dimension in X axis to sum  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='else  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='if Dimensions are available then 11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='Extract the X-axis dimension from the data and add it to the sum 12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='else  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='Set the default dimension as 1 meter and add it to the sum 14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='end if  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='end if  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='Add constant label and space sizes to sum 17  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='Divide the X-axis dimension by four and add to the sum as spacing 18  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='end for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='Add minimum entrance size for two parallel walls (2-meters x 2) to sum 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='Divide the sum by 2 to get the sum of two perpendicular walls 21 end for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='Additionally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' the sizes of the labels and the space between displayed objects are static which is added to the total sum for each artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To create lists, artifacts are queued according to the date aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The artifacts that are not provided with a date are added according to alphabetical order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' From the constructed list, space between each artifact is calculated based on their sizes and added to the sum together with the minimum entrance size for two parallel walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In the final step, the calculated length is divided into two which gives the sum of two walls that are perpendicular to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Calculated lengths are provided to other algorithms as a parameter that will define the dimensions of the rooms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Constrained Growth Algorithm A procedural method’s capacity to define its limits depends on space, allowing it to precisely use the region it generated as input before continuing to generate within it (Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Using a building outline for a floor plan generation allows for determining the borders of the procedural method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To provide consistent footprints for users, we searched for floor plans of existing museums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Using the websites such as Archdaily and Divisare, we obtained 20 different floor plans that have different complexity levels in terms of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Using OpenCV, we extracted the outlines of the floor plans with filling operation, and we transformed them into footprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  We used footprints as the solids that will hold the cavities that will grow inside them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' After several experiments, we defined the sizes of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='png files that contain footprints as 128x128 pixels due to computation limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The prepared footprints according to pixel and color requirements are provided as options to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To operate the algorithm, the user first selects the footprint to work on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Depending on the preference, the user can define the starting growth points by selecting the pixel form image of the footprint or if data is available, the user can choose the option from sorting categories (style, artist, location) that defines the number of rooms, and the algorithm selects the random points according to optimal distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' If there is no data available, the user can also prefer the algorithm to select the number and location of the points randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The starting points form a square with a 1-pixel void and 8 pixels around as boundaries (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' After the selection of starting points based on pixel-based search, starting points begin to grow until they reach the borders of the footprint or borders of each other forming cavities inside the footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' When the growing process is completed, corners of the containers are found by continuously adding and comparing the positions of the pixels which are owned by a certain room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' According to corners, center positions, and length of the walls are defined to place 3D wall tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  For each room, the 3D floor and their mirrored version of roof tiles are placed according to empty pixels and walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  24  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Example Outputs of Constrained Growth Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Room Generation In the room generation algorithm, a cubic room is generated using the generic wall, roof, and floor assets as a solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For the data-based version of the algorithm, the user can select the sorting options for the assets and each room according to the number and sizes of the artifacts that the room will contain, the width and depth dimensions of the room are calculated, and the initial room is scaled according to calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For each room, a button is produced to spawn the room in the virtual environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In the version based on user preferences, initially, the user places a generic room that holds the generation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The user can play with the cavities and solids by defining the dimensions of the room, the number of windows, and the number of doors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The assets that are used to construct the initial solid together and window and door assets are given to the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' When the user presses generate button;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' first, floor and wall tiles are replaced according to the given dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Then, the algorithm changes wall tiles to window and door tiles according to the given numbers by the user (Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  25  Algorithm 2: Constrained Growth Algorithm 1 Initialize new list cornerList 2 From (minX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' minY) search four directions for a boundary pixel 3 if it is found then 4  Add to the cornerList  5  else  6  Move 7 end if 8 Sort corners clockwise 9 if a room is placed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' then 10  Check Region  11  if the region is clean,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' then 12  Draw room boundaries  13  Add a room to the room list 14  else  15  Terminate the Room  16  end if 17 end if 18 if it is not paused,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' then 19 Iterate through the list of rooms and find candidate walls 20 while the number of growable walls > 0 do 21  Grow room  22  Update walls  23  Update texture  24  end while  25 end if  26  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Example Outputs of Room Generation Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Binary Space Partitioning and Cellular Automata Cellular automata (CA) and binary space partitioning (BSP) are two different algorithms that can be used separately or in combination to generate layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' BSP is a technique used in computer graphics to divide a 2D or 3D space into smaller subspaces, or nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In general, the number of subspaces of BSP is defined with a random number from a range with a seed provided for variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' It is mostly used for generating complex 2D dungeon patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' This study used BSP to produce interconnected rooms and corridors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' however, different from the most common techniques, instead of using random numbers we constrain the algorithm via user preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' A type of mathematical system known as a cellular automaton is made up of a grid of cells, each of which has a limited number of possible states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' According to a set of rules, each cell’s state is determined by the states of its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' This means that by interacting with one another in cellular automata, simple rules can be used to create complex patterns and structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In video games, CA is typically used to create more natural patterns like caves and forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In this study, we created a 3D model of the generated grid data based on the cell state using CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The basic idea of BSP is that any plane can divide space into two spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Therefore, it creates different areas by recursively dividing the level space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Any half-space that we continue to define a plane in will be further split into two smaller subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Subspaces are created and a spatial binary tree is formed through continued division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The divisions may be placed at random points so that not all areas are of the same size and shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Once the areas are a suitable size, each area is converted to a room, and connections are added between adjacent regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In the data-based version of the BSP, the number of rooms and their dimensions are provided by a data analysis algorithm that calculates parameters according to the sorting option selected by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Other variables are randomly produced by the algorithm and  27  altered via seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In the absence of data, with the defined seed, the algorithm randomizes all parameters used for the generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For the version based on user preferences, the user can define maximum and minimum values for dimensions of corridors and rooms, dimensions of footprint, and the number of rooms so that the user can give the parameters of the solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' We generated a layout using BSP to start by defining the initial boundary which is the footprint that has dimensions depending on the user’s preferences (Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Given sizes for the footprint can change the overall placement of the layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For example, if the user prefers to create longitudinal side-by- side blocks or more compact structures, it can be done by playing with the width and depth ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The next step is to divide this footprint into smaller spaces in accordance with the number of rooms and dimensions defined by the user (Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In binary space partitioning, a seed is a starting value that is used to generate a random sequence of values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The random sequence is then used to determine the layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' We used the seed to randomly select the location and orientation of the division to create variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The seed value is used as the input to a random number generator, which produces a sequence of random values based on the seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' By using the same seed value, it is possible to generate the same layout multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  The divisions are created based on the room and corridor dimensions and seed value which continues until the number of rooms defined by the user is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Each node in the constructed tree structure represents a subspace within the layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To create the actual layout of the tree, the tree is then traversed, and at each node, if the node can create a room, it is added to the layout and the algorithm tries to place a corridor in all four cardinal directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' If there is enough space, the algorithm places a corridor and moves on to the next node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' If there is not enough space, the algorithm terminates the process and restarts to division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The BSP algorithm operates on cells, which makes it suitable to combine with the CA for a rule-based 3D generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' We defined several states for the cells to construct the 3D model of the generated data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Before the division, all the cells are in the state of empty when all the rooms take place in the spatial binary tree cells change their states to the wall, corner wall floor, door, and window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' While wall and roof tiles do not need any rotation, for wall, corner wall, and door tiles there are different states which apply the rotational operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Applying the rules, 3D assets take their places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For the roof, instead of creating a new state, we injected the responsibility to the cells in the floor state, which can create roof tiles by duplication according to the defined height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  28  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Example Outputs of Binary Space Partitioning and Cellular Automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Evaluation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Ethics Ethical Approval of Research was approved by the Middle East Technical University Human Subjects Ethics Committee with the approval protocol code of 0244-ODTUİAEAK-2022 in April 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Before the start of the study, every participant signed an informed consent form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The form includes a brief description of the study’s aims and objectives, the importance of participants’ input, data collection methods, possession of participants’ personal information, the intended use of their data, and what is expected of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Participants had been warned about the potential negative effects of VR platforms, such as motion sickness and nausea, and it was explained that participants had the right to leave at any time if they feel uncomfortable or disturbed during the experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' All the information collected for this study is anonymized and figures that may include personal data were excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' No compensation was provided to participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  29  Algorithm 3: Binary Space Partitioning and Cellular Automata 1 Initialize the layout with the given footprint dimensions 2 Use the seed value as the input to a random number generator 3 while layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='num_nodes < user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='rooms do 4  Select a random location and orientation for the division 5  Divide the layout into smaller subspaces 6 end while 7 Traverse the tree structure of the layout 8 Recursively visit each node in the tree structure of the layout 9 if the node can create a room, then 10  Add it to the layout 11  Try to place corridors in all four cardinal directions 12  else return 13 end if 14 if layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='num_nodes = user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='rooms, then 15  Iterate over the cells in the layout and place 3D models according to the states 16  else return 17 end if 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Participants A total of 30 participants were recruited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Recruited respondents were between the ages of 18–35, with a mean age of 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='52 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='13 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Each participant had a normal or corrected-to-normal vision (self-reported).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The scope and focus of the research topic, the methods utilized to collect the data, and the amount of information that was acquired from each participant are all factors that can affect the sample size (Braun and Clarke, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Enough information should be gathered to adequately address the research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Given the numerous methods of data collection and the depth of information gathered from the participants, 30 participants were an adequate number, and the analysis from which produced important insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Procedure We conducted our evaluation using a quantitative survey in the form of questionnaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Additionally, semi- structured interviews were undertaken as an exploratory method to provide more detailed and in-depth insights about the use of the application and preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Before the version test, participants were asked to fill out the Gamer Motivation Profile (Yee, 2016), Immersive Tendencies (ITQ), and Tool Competence questionnaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The users were provided with identical data and museum collections to avoid inconsistencies resulting from the data and media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Following the procedure, participants were given access to their collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Users were first asked to freely use the application, then test the specific features, such as lighting settings and procedural generation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Once participants had used all three versions, the participants were given the questionnaires intended to assess several aspects of the versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For the qualitative evaluation of the framework, a widely used System Usability Scale (SUS) (Brooke, 1996), Presence Questionnaire (PQ) (Witmer and Singer, 1998), and Technology Acceptance Model (TAM)  30  (Venkatesh & Davis, 2000) questionnaires were adopted and distributed to the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In addition to those, we composed three questionnaires for lighting, procedural generation algorithm, and interaction preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Later, a short semi-structured interview was conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Each session lasted about one hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Data Collection Methods 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Semi-structured Interview After all the platforms had been used by the participants, the semi-structured interview was held after the session to gather further information about general impressions of using the versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The guide consisted of 11 questions that focused on participants’ interpretation and perception of the framework, experiences of using the different versions how they could be improved, and which versions and options they preferred and why.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Additionally, informal conversations were made to identify possible factors that had been overseen, such as mistakes in applying the method and software errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In this interview, they were also asked to describe their experience using the application in their own words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Questionnaires The System Usability Scale is a 10-statement self-report scale that is used to analyze and study a system’s usability for general evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' It is assessed on a Likert scale of 1 to 5, where 1 is for strongly disagreeing and 5 is for strongly agreeing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The comprised 10 questions consist of five positive and five negative statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  The SUS score is calculated using the collected data, and the result falls between 0 and 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' SUS is widely used in the literature to compare and evaluate the success of outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The sense of presence experienced via exposure to an immersive environment has frequently been connected to the effectiveness of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Presence is frequently described as a “feeling of being there” when a person feels as though they are in a different place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The Presence Questionnaire was first developed by Witmer and Singer (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' It comprises several subsections, but only a few of them are incorporated in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Presence was rated on a 7-point Likert scale ranging from 1 = strongly disagree to 7 = strongly agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' A methodology called the Technology Acceptance Model (TAM) seeks to determine whether new technology will be accepted, rejected, or usable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The approach was first presented by Davis (1987), and it is widely used in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' TAM consists of different subsections and different adoptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' A revised form of the TAM questionnaire (Venkatesh & Davis, 2000) is used and it consists of 10 questions that cover two subsections: perceived usefulness (PU) and perceived ease of use (PEU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The first six questions are used to measure PU, and the other four are used to calculate PEU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' TAM was rated on a 7-point Likert scale ranging from 1 = strongly disagree to 7 = strongly agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  The results of the effectiveness of the immersive systems, however, may be mediated by individual differences, aspects of the technology, and negative side effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Measuring these mediating effects is of great importance in understanding the relationship between and among them, and how to maximize the  31  effectiveness of the immersive environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To measure several individual differences, the Immersive Tendency Questionnaire (Witmer and Singer, 1998) was introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The immersive tendency is an individual’s proclivity to become immersed in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' ITQ consists of 18 items and is composed of four subsections that include involvement (five items), focus (five items), emotions (five items), and propensity to play video games (three items).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To measure the tool competence level of participants regarding VR and MR, they were asked to rate their experience level on a 7-point Likert scale ranging from 1 = I have no experience to 7 = I am an expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In the PC version, we utilized the most common navigation and interaction methods (W, A, S, and D buttons for navigation, and mouse interaction for camera movements), which are not required for MR and VR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Therefore, instead of asking about experience level in PC, we employed a survey called the Quantic Foundry Gamer Motivation Profile to measure participants’ degrees of video game experience in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The questionnaire comprises six questions that focus on respondents’ favorite games and game-playing habits and were created utilizing factor analysis and historical investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' It assesses experience based on four levels, ranging from non-gamer to hard-core gamer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Four additional open-ended question sets were answered by the users for this study, namely, Framework Competence, Interaction Preference, Procedural Generation Algorithm Selection, and Lighting Preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Tool Competence Questionnaire includes two questions to assess experience level regarding VR and MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' There is one question for each version (VR and MR) with two options (hand interaction/controllers) for interaction mode preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In the Lighting Preferences Questionnaire, there are two questions for different roles (curator/visitor) per version (VR and PC) with three temperature options (cold/neutral/warm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Procedural Generation Algorithm Selection consists of four questions for each version (based on data, based on user preferences) for three procedural generation algorithms to identify ease of use, perceived control, understandability of the algorithm, and the quality of the design outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' It is rated on a 5-point Likert scale ranging from 1 = none to 5 = completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Results The comparative results presented in this chapter are based on the responses given by the 30 participants for the PC version with MacBook M1 Pro and VR and MR version with Oculus Quest 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Responses to open-ended questions were analyzed using the analysis tool MAXQDA (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Statistical analyses were conducted using the software program JASP (2022 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='4) to compute and validate the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  For each questionnaire, reliability tests were run to determine whether the mean values’ internal consistency was satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' We compute Cronbach’s alpha values, which is a method often used to validate surveys (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  32  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Interpretation of Cronbach alpha (α) values (Tavakol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Cronbach’s \uf061 Cronbach’s α Comments α ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='9 Outstanding 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='7 ≤ α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='9 Good 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='6 ≤ α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='7 Acceptable 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='5 ≤ α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='6 Weak α ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='4 Unacceptable 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Semi-structured Interview The first and second interview questions require users to explain their general impressions regarding the framework and their interpretation of the usage area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Figures 10, 11, and 12 present the frequency mapping of words that are commonly used to describe different versions of the framework and potential use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Frequency Mapping of Most Used Words by Participants to Describe the VR version of the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Produced with MAXQDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  inciusive  exmerimenta  design  education  productive  complex  fun  quick  exnansion  inspire  dimensional unlimited  understande  arcnitecture  hard  mind  medum  Vr  display  interest  study  Drototyne  build  teach  space  creative  composition  generative  art  difficult  effective  visualization  creation  brainstorm  exhibition  interaction  functional  33  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Frequency Mapping of Most Used Words by Participants to Describe the MR version of the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Produced with MAXQDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Frequency Mapping of Most Used Words by Participants to Describe the PC version of the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Produced with MAXQDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Interaction Preferences Users were asked to select the medium that they preferred according to identified asset types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Figure 13 shows the users’ preferences for interaction mediums with different asset types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='brainstorm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='design  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='transformation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='dimensional composition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='inspire  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='experimental art study  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='understand  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='difficult presentation teach  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='showcaselearn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='functional  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='exhibition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='mr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='hard  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='productive  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='generative  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='fun  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='display  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='complex quick  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='architecture  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='configuration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='creation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='space  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='build production interest  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='visualization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='Drototvne inclusive  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='interior design  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='effective  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='creative  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='interactioncreation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='composition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='mse  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='understand  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='fun  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='production  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='archive  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='vision  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='quick  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='design  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='simul ation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='gallery  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='artistic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='pc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='conttiguration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='easy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='brainstorm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content='game  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='art  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='effective  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='architecture  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='jrototype  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='functional  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='display  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='experimental  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='generative  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='interact  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='inclusive  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='creative  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='34  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Interaction preferences of participants with different asset types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In the VR and MR versions, all users tested the hand interaction and controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Figure 14 shows the users’ preferences for interaction with different versions of the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Interaction preferences of participants for VR and MR Version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Lighting Temperature Preferences Three parts of the temperature scale were given as a preference for the respondents for different modes of the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Since the MR version does not utilize the virtual lighting system, it was not included in the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The results are demonstrated in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Video  Text  Image  PC  MR  PC  VRArtCanvas  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3%  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3%  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='7%  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='7%  VR ArtCanvas  VR  PC ArtCanvas  VR  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3%  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='7%  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='0%  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='0%  MR  VR  MR  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='7%  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3%  PC ArtCanvas  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3%  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='7%  3D Model  Audio  PC  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='7%  VR  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='7%  MR  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='0%  VR  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='0%  MR  PC  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3%  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3%MR Version  VR Version  Hand Interaction  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3%  Hand Interaction  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='7%  Controllers  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3%  Controllers  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='7%  35  Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Lighting preferences of participants for different modes and immersive environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Procedural Generation Algorithms Four different criteria were identified to assess the success of the implemented algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The results are divided into two—generation based on data and generation based on user preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To test the correlation between the number of options provided to users and other eligibility indicators, they are presented together in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Mean and standard deviation results of the questionnaire for procedural generation based on user preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Answers to the questionnaires are on a 1 to 5 scale;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' a higher score indicates a more positive attitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Growth Algorithm Binary Space Partitioning and Cellular Automata Room Generation Number of Options 3 7 4 Perceived Control 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='56 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='97 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='80 Ease of Use 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='98 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='78 Comprehensibility 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='81 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='92 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='87 Design Outcome 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='97 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='89 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='92  VisitorMode(PC)  Visitor Mode (VR)  Cold  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3%  Neutral  Neutral  Warm  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='0%  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3%  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='7%  Warm  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3%  Cold  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='3%  Curator Mode (PC)  CuratorMode(VR)  Warm  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='7%  Cold  Neutral  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='7%  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='7%  Neutral  Cold  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='7%  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='7%  Warm  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='7%  36  Since perceived control, the number of options, and ease of use are not relevant to generation based on data, only design outcome and comprehensibility of the algorithms were investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The results are given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Mean and standard deviation results of the questionnaire for procedural generation based on data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Answers to the questionnaires are on a 1 to 5 scale;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' a higher score indicates a more positive attitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Growth  Algorithm  Binary Space Partitioning and  Cellular Automata  Room  Generation  Design Outcome  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='89  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='75  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='85  Comprehensibility  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='71  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='84  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='83  The results show that although the comprehensibility of the combination of BSP and Cellular Automata algorithm is low in comparison to other algorithms, it has the highest score in terms of design outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' System Usability Scale Although SUS does not have subscales, the answers should be computed in accordance with SUS scoring algorithms to interpret the results (Brooke, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To create consistent scoring as required, the scoring formulas include different calculation methods for the questionnaire’s negative and positive questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Table 4 shows Cronbach’s alpha values, while Table 5 and Figure 16 display the findings of SUS’ statistical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Cronbach’s Alpha Values of System Usability Scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Version PC VR MR Cronbach’s \uf061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='653 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='874 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='670 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Mean and Standard Deviation Results of System Usability Scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Version PC VR MR Mean 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='91 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='89 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='00 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='03 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='51 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='99  The results of the SUS were found reliable since the calculated Cronbach’s alpha (α) value was higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='5 for all versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (PC (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='653 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='5), VR (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='874 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='5), MR (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='670 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The overall System Usability Scores (SUS) for 30 participants indicated high rates of system usability, with an average score of 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='91% for PC which is defined as “good” and is graded as “B” in the system usability assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Comparing the standard deviations reveals greater consistency in the participants’ PC scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' On the other  37  hand, the usability score of the MR version (77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='51%) is higher than the VR version (70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='00%), but both are in the range of “B” and are defined as “good” as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Comparative Results of SUS Scores in Boxplot Format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Technology Acceptance Model TAM questionnaire does not have a structured question set nor a scoring sheet to compare the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Therefore, the results were examined only numerically using descriptive statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For the user study, the most related two subsections of the TAM model were selected, which are perceived ease of use and perceived usefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The questionnaire was analyzed with its subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The results are given in Table 6, Table 7, and Figure 17 comparatively for each version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Cronbach’s Alpha Values of Technology Acceptance Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Version PC VR MR Cronbach’s \uf061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='640 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='857 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='848  The collected data in total were found reliable since the calculated Cronbach’s alpha (α) values were higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='5 (PC (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='640 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='5), VR (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='857 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='5), MR (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='848 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  System Usability Scale  PC  VRMR  100  90  80  X  X  70  60  Data  50  40  30  20  10  0  Versions  38  Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Mean and standard deviation results of the Technology Acceptance Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Answers to the questionnaires are on a 1 to 7 scale;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' a higher score indicates a more positive attitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Version PC VR MR Perceived Usefulness 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='63 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='47 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='24 Ease of Use 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='54 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='88 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='15 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='41 Total Mean 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='60 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='23 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='29 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='35  According to the results, PC’s mean value, PU, and PUE scores were found to be higher than the VR and MR versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' While the MR version’s score is slightly higher than the VR version, the most notable score is the PEU score of the VR version, which is lower than other versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Comparative Results of Technology Acceptance Model Questionnaire in Boxplot Format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Presence Questionnaire The Presence Questionnaire (PQ) has subsections that represent diverse characteristics of the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' By focusing on the subsections of the questionnaire, such as realism, possibility to act, quality of the interface, possibility to investigate, and self-evaluation of performance, a total of 19 questions were posed to assess versions’ affordance in presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Each subsection’s overall system performance is calculated independently, and the results are then combined to get a total presence score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The scoring sheet provided by Laboratoire de Cyberpsychologie de l’UQO (L’UQO, 2002), which shows the minimum scores for a successful outcome, was used as a guideline for comprehending the presence capabilities of the successful versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The results are given in Table 8, Table 9, and Figure 18 comparatively for each version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Technology Acceptance Model  PC  VRMR  7  6  X  5  Data  4  3  2  1  0  Versions  39  Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Cronbach’s Alpha Values of Presence Questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Version PC VR MR Cronbach’s \uf061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='717 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='822 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='685 Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Mean and standard deviation results of PQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Answers to the questionnaires are on a 1 to 7 scale;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' a higher score indicates a more positive attitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Version PC VR MR Realism 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='26 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='02 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='31 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='02 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='45 Possibility to Act 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='40 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='25 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='46 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='93 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='37 Quality of Interface 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='37 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='52 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='33 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='56 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='93 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='49 Self-evaluation of Performance 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='92 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='38 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='09 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='78 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='43 Total Mean 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='14 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='97 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='46 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='93 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='44  Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Comparative results of Presence Questionnaire in Boxplot Format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' According to the results of reliability tests, the results of the questionnaire were found eligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' While the total mean scores of versions are similar to each other, several outliers can be seen in Figure 18 for the PC version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Discussion Heterotopias are unique spaces that can connect different systems through the exchange of information and are characterized by the presence of multiple layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Based on the features of discursive heterotopias, the framework described in this study is designed to be flexible and modular, which can be used in a variety of  Presence Questionnaire  PC  VRMR  7  6  5  Data  4  3  2  1  0  Versions  40  contexts to create digital habitats that can provide a space for discourse to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Because of the complexities of heterotopias, a variety of evaluation approaches are required to test the convenience of the methods regarding the objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To test the framework with individual archives, while some participants prepared a collection of assets right before the sessions, some preferred to use existing materials in their online or offline databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' While the pre-made dataset provided us with essential insights into more technical aspects, the potential of the framework to construct unique heterotopias revealed itself with personal experiences and archives, proving the framework’s applicability to diverse contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The colocation of multimedia elements through the “operating table” approach has created several possibilities for participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For example, participants preferred to use their photographs, videos, notes, and screenshots, similar to the accumulative aspect of heterotopias in terms of time and elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' They generated different rooms constructing “grids of specification” according to their categorization, which created multilayered experiences directly related to self-reflection and self-formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Individual archives and spatial layers provided by the framework allowed users to create a “world within worlds.” [P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='17] “I considered the experience as walking inside my brain and organizing it, so it could work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Also, the framework can be extended via deep learning algorithms for further customization and guidance.”  Participants who used the assets related to their professional life have created relations and experiences based on “surfaces of emergence” that cannot be provided without place and embodiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For example, participants from backgrounds related to the design studies used the models they produced together with the sketches, notes, and videos of production processes and presented narrative exhibitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' They created in-situ demonstrations and performed architectural design activities experiencing the creative process of “aesthetic becoming.” [P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='22] “I think most of the design tools that we used are like trucks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' this application was like a sports car.”  [P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='04] “I can easily use this tool for my interior design practices, and I especially found the mixed reality version very useful.”  Participants with medical backgrounds placed anatomical models and texts, transforming the environment into a puzzle game where they could manipulate the parts of the model with hand interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Additionally, two participants from computer sciences added new features with dynamic models via scripts embedded in 3D models based on OOP using the adaptability of the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Throughout the semi- structured interview, we investigated the reasons behind the results of the questionnaires, participants’ personal opinions about the versions of the framework, and their perception of the framework for possible use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Through the open-ended questions, we explored the interpretation of the framework from the users’ perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Open-ended questions were used to gather data on how users perceive and interpret the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' An analysis of the responses to these questions, using frequency mapping, showed that the different technologies used in the framework resulted in various interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' words such as “game,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' “creative,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' “fun,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' and “art” were commonly used to describe the PC version,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' while the VR  41  version was more frequently described using words such as “experimental,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' “design,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' “fun,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' “architecture,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' “visualization,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' and “inclusive.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' the MR version was often described using words such as “interior-design,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' “prototype,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' “interaction,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' and “transformation.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' A comparison of the frequency mappings suggests that MR and VR technologies may provide more comprehensive and multilayered experiences that involve embodied cognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Also, most participants provided individual cases based on their experiences and needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' [P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='02] “I would prefer to use this tool when I teach my students geography and history.”  [P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='25] “We can use this tool for designing, displaying, and teaching the systems in our mechanical engineering studies.”  The richness of the cases provided by the participants implies that the framework is applicable to various contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The dynamic autonomy layers and understanding of self-formation based on heterotopias provided design space for the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' They became the “focal points of resistance,” producing informal use cases which were not structured by the framework but by the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The concept of heterotopia can be interpreted as a complex set of relationships which in this case, also involves the user experience which is affected by various elements from hardware to navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' According to statements of respondents, different technologies and interaction modalities that are provided serve better for certain discourses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Also, according to the activities, the users change their preferences in environmental settings such as lighting and scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The multimedia elements of the discursive space were provided with different representation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Users’ environment preferences differ according to media type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For the 3D models, users preferred to interact with them in VR and MR environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Participants stated that MR provides better scale perception in relation to surroundings, and both other versions were more impressive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Most users prefer to display videos and images on a human scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' However, due to distractions created by physical objects in the MR version, the majority of users preferred the VR version for videos since it required a long- term focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  For the audio, most of the users preferred the VR environment, finding the combination of a completely artificial environment with audio more engaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For the text interactions, the majority of the users tend to read in the ArtCanvas view, which is the most traditional option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Therefore, the dominant media format of discourse can affect the preferences of users in terms of technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' When creating a multimedia system, it is important to consider the nature of the interaction and the anticipated results in terms of learning, sensory and emotional engagement, and satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The hand interaction and controllers were essential for conducting activities through embodiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The hand interaction has similarities with the command language, and the users need to remember the gesture to complete the operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Users mostly preferred to use controllers rather than hand tracking for interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Using buttons on controllers was more accessible and easier than hand gestures, especially for novice users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The results showed that there was an increase in the number of participants who preferred controllers in the  42  MR version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' This result suggests that being able to receive information from physical locations increases the request for physical body engagements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' It is possible to consistently stimulate human emotions and actions in virtual settings, which adds to the overall effectiveness of a heterotopia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The stimuli’s visual integrity results in a high level of immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In this study, we looked into the lighting preferences of participants in the curator and visitor modes/roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Results showed that there are differences in lighting preferences performing two different interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' While the majority of the users preferred natural and cold light during the curation process, warm lighting settings were preferred during the visiting mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In the VR environment, there is an increase in the selection of cold temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The relationship between physiological input and an abstract conception in the curation process emphasizes that spatial perception is ultimately experiential and lived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Users’ perceptions of and integration with virtual elements may change as a result of spatial perception blended with an embodiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Commonly, users indicated that they were able to focus on details under cold and neutral lighting, and warmer temperatures made them feel more comfortable during observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Control and power mechanisms are important elements for both heterotopias and architectural practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Heterotopias redefine those relations and provide an “order of discourse” with different syntaxes, where the syntax is constructed via places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The integration of technology changes the dynamics and defines the borders of control and power for the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' An increase in the autonomy of technology has created discussions in architectural practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Architectural design practices cannot be distinctively separated from their tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' As technology advanced, design and art practices have changed their systems from physical to digital space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Architectural drafting, representation, building, and most critically, architectural design, have all been impacted by modern technologies, which have changed the production and final output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The general tendency of the tools points out the resistance of technology instead of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' By providing a flexible framework and various autonomy levels for automated generation algorithms, we aimed to understand the dynamics between the technology, the user, and the transformation or construction of spatial layers of heterotopias provided via digital mediums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' With the level of manual design provided to users, the overall outcome depends on how procedural generation is guided, constrained, and changed by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Moreover, since procedural generation is inherently deterministic, the integration of manual design steps is essential for users to inject variety and creativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  We aim to understand the impact of the level of autonomy on the outcome and user preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Here, we are particularly interested in the extent to which the method led to the intended results and higher levels of control, considering the tension between usability and comprehensibility associated with the design process of the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Therefore, it is worthwhile to explore how the different outcomes are perceived by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' According to the results, the growth algorithm provides the highest comprehensibility among the data-based versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' During the process, users were able to see the initial area that they or the algorithm would work on, which increased their understanding of how the algorithm works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In the user-based version, users can select the starting point for growth, but the size and shape of the rooms are determined by the algorithm, which the user cannot control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In the BSP version, the size of the rooms is chosen by the user, but their placement is controlled by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The room generation algorithm allows the user to control  43  both the placement and size of the final product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Overall, the growth algorithm provides a higher level of perceived control than BSP, but a lower level than the room generation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' According to survey results, BSP combined with CA produces the best design outcomes but has a lower level of comprehensibility compared to the other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' When participants generated the content based on their preferences, options that were given to users increased comprehensibility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' however, this approach decreased the design outcome produced by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' This implies that the data-based version meets the space needs of the users better than the user-based version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' With the higher number of options presented, BSP provided a less perceived sense of control and ease of use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The data-based version of the room generation algorithm has lower scores for design outcome than BSP, but higher scores for perceived control, ease of use, and design outcome in the user-based version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Most of the participants exhibit different strategies according to versions depending on their embodiment level, point of view, and scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The spatial experiences in virtual environments provide an active interpretation process based on the awareness of surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Therefore, the design approaches of participants differ according to versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Comparing produced scales of the algorithms, BSP has the capability of producing large-scale designs while room generation can produce only one space, and the growth algorithm is able to produce within the borders of the footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Therefore, users tend to use the BSP algorithm in the PC version where they can observe the outcome with a completely top-down view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Room generation was mostly preferred in the MR version where users can change the surroundings with their virtual versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The growth algorithm was found more suitable for the VR version since the users can produce multiple rooms, but still be able to navigate themselves without losing their tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Results also indicate that the comprehensibility of the method increases the perceived level of control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Another objective of the study was to develop a framework for immersive technology based on heterotopias that is accessible and easy to use for a general audience, including both novice and experienced users where they can conduct discursive practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The level of usability and acceptance can change an individual’s control over the system by providing more resistance to the technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Therefore, the framework was tested using the Technology Acceptance Model (TAM) and the System Usability Scale (SUS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  The results showed that the different versions of the framework were convenient for users with different backgrounds and levels of competence, pointing out that the layers of heterotopia in a digital medium can provide an “operation table” for various cases and individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The SUS was used to assess the complexity, consistency, and ease of use of the system, and all three versions scored in a similar range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' However, some participants preferred the VR version although it had lower usability scores, indicating that engagement may be a factor in the success of immersive technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' [P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='17] “I can use the VR version for hours when I want to get away from reality.”  [P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='04] “MR version was pretty interesting, the PC version was easier to use, but VR was more fun.”  According to user feedback analyzed through frequency analysis, the MR and VR versions of the framework were perceived as having both positive and negative qualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' While they were described as “experimental”, “useful”, “quick”, “functional”, and “interesting”, they were also considered “difficult”,  44  “hard”, and “complex” to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In contrast, the PC version was described as “easy “to use and more “game”- like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The high volume of information available within the heterotopias created an experimental and creative space for users but also resulted in interfaces that may be overwhelming for those who are new to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Some users noted that the numerous features hindered usability and increased cognitive load, disrupting their ability to fully immerse themselves in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The intensification of information in heterotopias both produced “experimental” and “creative” ground for the users and also expressed itself via interfaces, which created exhaustion for novice users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Some participants indicated that the number of features decreases usability and increases the cognitive load, which disturbs the feeling of presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' [P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='25] “I think the number of features should be less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' I felt there were so many features to use, which put me under stress.”  To interpret the results of the TAM questionnaire in more detail, we examined the subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Even though the PC version had the highest scores, the results of the versions in terms of PU were similar to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The virtual reality (VR) version had the lowest PEU score, while the mixed reality (MR) version was perceived as being more understandable and requiring less effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' This difference may be due to the nature of VR, which can disconnect users from the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Some users found the MR version safer than the VR version, while the VR version was considered more enjoyable to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The HMD used in the study creates a virtual boundary around the user and displays the real world when in close proximity to a physical object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' However, some novice users who were new to being fully immersed in a virtual world may have had lower trust levels towards the technology and preferred to use the teleportation option for navigation, reducing their interaction with the virtual environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In terms of performance, all three versions were produced based on the same system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' therefore, they can sustain the same predictability level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' However, other factors, reliability, and utility affect each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In the case of VR, lower reliability to technology produces lower perceived usefulness results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' [P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='03] “Since I was able to see the physical environment, I felt safer in MR than VR.”  [P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='19] “I became disoriented in the VR, and at one point, I felt constrained by the headset and wanted to remove it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The MR version was not as impressive as the VR version, but I would prefer to use the MR version.”  The results showed that PC had higher presence scores than VR and MR, and this difference was attributed to the different levels of competence and familiarity that users had with these different technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Analysis of various subcategories, such as realism, quality of the interface, and the possibility of examination, found that each technology had its own strengths and weaknesses in these areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' For example, the PC version had higher scores in terms of naturalness of interaction and the ability to survey the environment, since the keyboard and mouse interaction are more natural for most of the users, and the PC version provides several camera options to the survey environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' On the other hand, the VR version had higher scores in terms of visual aspects and consistency with real-world experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The MR version  45  had lower scores due to problems that occurred with hand interaction and the distractions caused by the colocation of physical and virtual content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The results suggest that different technologies can have different impacts on cognitive load and users’ focus, and it is important to consider these factors when choosing a technology for a particular task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Overall, total scores indicate that a multi-layer approach to constructing heterotopias with the given features provided high scores in presence, usability, and acceptance for users with different experience levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Conclusion The discursive space refers to the understanding and insights gained through the study and examination of specialized knowledge about the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' With the increasing use of digital technologies, multimedia has become the primary means of communication in the digital age and the language of discursive practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Heterotopias are special, multi-faceted contexts that facilitate the exchange of information and link to other systems with “surfaces of emergence.” They are unique in their ability to connect and integrate various perspectives and knowledge bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In this study, we presented a method for creating a framework for virtual and mixed reality environments, as well as personal computers, that allows for the exchange of knowledge and information through the creation of multilayered digital spaces based on heterotopias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  The framework aims to fill in the gaps left by the disappearance of certain layers in archival practices, particularly spatial layers, due to the increasing reliance on technology that excludes embodiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' We provided the concept of a virtual museum as an “operating table” for discursive methodologies, and instead of static settings, we offered dynamic, “liquid architectures” which find their forms around the individual archives composed of elements of multimedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The framework was tested with 30 participants and was found to be useful, easy to use, and adaptable to various professional and personal settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The effectiveness of the framework and transformative impact of heterotopias are presented via the results of the qualitative and quantitative approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' We provided various sorting and grouping algorithms, and procedural content generation algorithms such as Binary Space Partitioning, Cellular Automata, Growth Algorithm, and Procedural Room Generation which offer different levels of autonomy to the automated generation algorithms, giving users an opportunity for reflection, modification, and control over the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Results indicate that place- production, transformation, and archival practices have a mutual relationship that is essential for “self- formation” activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Integration of the layers and features discovered within heterotopias with digital technologies, which offer a spatial “aesthetic becoming” at several levels of reality, was found productive for providing discursive practices within digital habitats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Blending the meta-design and end-user development approaches with the layers of heterotopias we provided sites where the users become “focal points of resistance.” To provide design features and outcomes that can direct future work in the HCI community, we analyzed the various factors that can influence user experiences and preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The results of the questionnaires and interviews demonstrate that the three evaluated versions are different, each having its own characteristics, strengths, and weaknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Through a comparative study, we demonstrate how  46  different reality levels might augment different abilities of users by designing and curating virtual museums of their activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The level of the hybridity of the physical and digital spaces, interactions, the meaning and perception of the space, and the elements that are contained by the space can be re-interpreted by different users in VR, MR, and PC environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Future Directions This framework targeted essential features to provide a system for diverse contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Therefore, we did not add detailed technical requirements for a specific practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Based on the framework structure, modules can be added to serve different professions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' According to the literature review, most architects and designers prefer to work on several types of representations with multiple screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' With the additional features, this framework can solve colocation problems of the design processes and provide more detailed production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To reorder the content, we used four variables based on museology which can be reinterpreted by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' However, for more complex structures, features for different reordering methodologies can be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' One of the factors that can affect the environment’s realism is audio assets that are not provided in accordance with physical distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' To enhance the realism of the environment and navigate through these environments in a more realistic and immersive way, sound localization can be provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  According to participants’ statements, the number of provided features can become exhausting for some novice users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Therefore, features can be provided separately for different experience levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In this study, we examined individual experiences separately for each technology to compare and understand specific preferences and impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' As a future study, providing a social layer and more transformative systems between PC and immersive technologies can increase the richness and relations provided via heterotopias created by the users and increase the usability of the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Future studies can include asymmetric and symmetric collaborative approaches including the social relations to practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Disclosure statement No potential conflict of interest was reported by the author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Data availability statement The data that support the findings of this study are available from the corresponding author, [E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' ], upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' References Akin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' Basaraba, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The emergence of creative and digital place-making: A scoping review across disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' New Media and Society, 14614448211044942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Boldi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', Rapp, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' Playing during a crisis: The impact of commercial video games on the reconfiguration of people’s life during the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Human–Computer Interaction, 1-42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Braun, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' Reflecting on reflexive thematic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=" A method for growth-based procedural floor plan generation (Master's thesis, Pontifícia Universidade Católica do Rio Grande do Sul)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Cecotti, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' Cultural Heritage in Fully Immersive Virtual Reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Virtual Worlds (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' Apress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=', Micoli, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 3D visualization of cultural heritage artefacts with virtual reality devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In 25th International CIPA Symposium 2015 (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' Copernicus Gesellschaft mbH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' (2018, June).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' User experience of markerless augmented reality applications in cultural heritage museums:‘museumeye’as a case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In International Conference on Augmented Reality, Virtual Reality and Computer Graphics (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' Springer, Cham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Automatic generation of personal virtual museum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In 2016 International Conference on Cyberworlds (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=', Roussel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', Maïdi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' Virtual and augmented reality for cultural computing and heritage: a case study of virtual exploration of underwater archaeological sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Virtual Reality, 15(4), 311-327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Hepp, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', & Hasebrink, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Researching transforming communications in times of deep mediatization: A figurational approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Communicative Figurations (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' Palgrave Macmillan, Cham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' Adapting cellular automata to support the architectural design process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Automation in Construction, 16(1), 61-69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Kersten, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=', & Deggim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2017, February).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Development of a virtual museum including a 4D presentation of building history in virtual reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' TC II & CIPA 3D Virtual Reconstruction and Visualization of Complex Architectures, 1–3 March 2017, Nafplio, Greece (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 361-367).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Copernicus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  48  Komianos, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', & Oikonomou, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2018, June).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Adaptive exhibition topologies for personalized virtual museums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In IOP Conference Series: Materials Science and Engineering (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' Placelessness, spacelessness, and formlessness: experiential qualities of virtual possessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' Frontiers in Political Science, 3, 674076.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' Human resource information systems (HRIS) and technology trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Journal of information science, 31(5), 340-353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  Lopes, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=', Smelik, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' (2010, November).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' A constrained growth method for procedural floor plan generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 11th Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Games Simul (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 13-20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Citeseer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Luck, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Seeing architecture in action: Designing, evoking, and depicting space and form in embodied interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' International journal of design creativity and innovation, 2(3), 165-181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Maciag, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Discursive space and its consequences for understanding knowledge and information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Philosophies, 3(4), 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Malraux, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Museum Without Walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Translated From the French by Stuart Gilbert and Francis Price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Mann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Mediated reality with implementations for everyday life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Presence Connect, 1, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Mcknight, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', Carter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', Thatcher, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Trust in a specific technology: An investigation of its components and measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' ACM Transactions on management information systems (TMIS), 2(2), 1-25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Milgram, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' A taxonomy of mixed reality visual displays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' IEICE Transactions on Information and Systems, 77(12), 1321-1329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Novak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' Transmitting architecture: The transphysical city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Digital delirium, 260-271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Pietroni, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', Pagano, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' (2013, October).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' The Etruscanning project: Gesture-based interaction and user experience in the virtual reconstruction of the Regolini-Galassi tomb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In 2013 digital heritage international congress (DigitalHeritage) (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' Methods for investigating architecture: from the physical to the digital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' WIT Transactions on The Built Environment, 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Prabhakaran, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' BIM-based immersive collaborative environment for furniture, fixture and equipment design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Automation in Construction, 142, 104489.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Prusinkiewicz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' The algorithmic beauty of plants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Springer Science & Business Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Rasmussen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' MIT press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Rheingold, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' Virtual reality: exploring the brave new technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Simon & Schuster Adult Publishing Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Rousseaux, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', & Thouvenin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Exploring informed virtual sites through Michel Foucault’s heterotopias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' International Journal of Humanities and Arts Computing, 3(1-2), 175-191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Safikhani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=', & Pirker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Immersive virtual reality for extending the potential of building information modeling in architecture, engineering, and construction sector: systematic review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' International Journal of Digital Earth, 15(1), 503-526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content='  49  Salanitri, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=', Borsci, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', Lawson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', Sharples, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=', & Waterfield, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Relationship between trust and usability in virtual environments: An ongoing study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' In International Conference on Human-Computer Interaction (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+page_content=' Springer, Cham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Schultze, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Embodiment and presence in virtual worlds: a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Journal of Information Technology, 25(4), 434-449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Schultze, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Embodiment and presence in virtual worlds: a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Journal of Information Technology, 25(4), 434-449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' Schweibenz, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FAT4oBgHgl3EQfch31/content/2301.08565v1.pdf'}
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+Incorporating Knowledge into Document Summarization
+— an Application of Prefix-Tuning on GPT-2
+Chen Chen and Wei Emma Zhang
+The University of Adelaide
+South Australia 5005, Australia
+Abstract
+Despite the great development of document
+summarization techniques nowadays, factual
+inconsistencies between the generated sum-
+maries and the original text still occur from
+time to time.
+This paper proposes a prefix-
+tuning-based approach that uses a set of train-
+able continuous prefix prompt together with
+discrete prompts to aid model generation,
+which makes a significant impact on both
+CNN/Daily Mail and XSum summaries gener-
+ated using GPT-2. The improvements on fact
+preservation in the generated summaries indi-
+cates the effectiveness of adopting this prefix-
+tuning-based method in knowledge-enhanced
+document summarization, and also shows a
+great potential on other natural language pro-
+cessing tasks.
+1
+Introduction
+With the explosive growth of information on the
+Internet in recent years, document summarization
+has become an area of increasing interest. There are
+two main approaches for document summarization,
+one is extractive and the other is abstractive (Qu
+et al., 2022). Decades ago, there were more studies
+on extractive abstraction. In recent years, benefits
+from the proposal of Transformer (Vaswani et al.,
+2017) and the emergence of various large-scale
+pre-trained language models such as GPT (Radford
+et al., 2018), BART (Lewis et al., 2020), more and
+more researches now focus on using pre-trained
+language models to do abstractive summarization.
+Abstractive summarization allows us to gener-
+ate natural summaries that are more like human-
+written ones, rather than simply extracting sen-
+tences from the original text, but one problem with
+it is that the summaries a model generates are not
+always factually consistent (Zhu et al., 2021). The
+presence of the model prior makes it likely that the
+model will generate sentences that do not match
+the facts of the original text when we use the pre-
+trained model for document summarization. Al-
+though we can fine tune the model using our own
+data, fine-tuning a model usually still requires a
+large dataset containing tens of thousands of data
+(Brown et al., 2020), which often poses a dilemma
+for the users of the model.
+To solve this problem, one approach is to incor-
+porate knowledge into the model to aid summary
+generation. This reduces the need for training data
+and ensures that the facts we add will always have
+an impact on the model. For example, Liu et al.
+(2021c) proposed KG-BART (knowledge-graph
+BART), which can help BART to generate sum-
+maries by adding KG-encoder and KG-decoder to
+the model, while Zhu et al. (2021) designed a graph
+attention network (GAT) to take in external knowl-
+edge and enhance the factual consistency of the gen-
+erated summaries by passing the output of the GAT
+to the cross attention layer of decoder. However, as
+we can see in the above studies, adding knowledge
+to a model usually requires not only embedding
+knowledge, but also designing new structures to
+receive it, so it is not very practical in many cases,
+still leaving the problem unresolved.
+Stopping to rethink this sequence-to-sequence
+problem of document summarization, we realized
+that, in fact, when generating summaries using pre-
+trained models, we had actually been trying to fig-
+ure out how to generate correct summaries using
+the documents together with the model’s prior. If
+the model’s prior was correct or the input docu-
+ments could effectively influence the summary gen-
+eration, then the generated summaries would be
+correct, and conversely, the summaries would have
+inconsistencies with the original text (Xie et al.,
+2021). Thus, the problem of adding knowledge to a
+model to steer it to generate correct summaries can
+be transformed into the problem of how to com-
+bine texts more effectively with the model prior in
+document summarization. Following this idea, we
+propose a method for incorporating knowledge into
+arXiv:2301.11719v1  [cs.CL]  27 Jan 2023
+
+document summarization based on prefix-tuning.
+Prefix-tuning is a type of prompt-based learn-
+ing. Prompting methods assumes that there is al-
+ready enough knowledge in a pre-trained language
+model, so that we can modify tasks to take advan-
+tage of the model prior and let the model spit out
+the correct answer on its own (Liu et al., 2021a).
+Prefix-tuning is to train a set of continuous task
+specific prefix prompts to steer the model to per-
+form text generation for different generation tasks.
+(Li and Liang, 2021). To incorporate knowledge
+into generation, we simply add a segment of natu-
+ral language prompts extracted by OpenIE (Angeli
+et al., 2015) in front of the original text to empha-
+size the relations we care about. Through experi-
+ments, we found that since the continuous prefix
+is very expressive, after training, the continuous
+prefix can recognize the patterns in the natural lan-
+guage prompts, thus pick out the key information
+from them and then combine the extracted infor-
+mation with the original text to perform summary
+generation, so this is a very direct way to influence
+the summary generation by data.
+To test the effectiveness of this approach, we fi-
+nally use both CoCo (Xie et al., 2021) and ROUGE
+(Lin, 2004) to evaluate the generated summaries.
+CoCo scores the factual consistency of the gener-
+ated summaries by averaging the positional scores
+of important words, while ROUGE is based on the
+overlap of n-grams. Experiments demonstrate that
+since the calculation of the positional score relies
+on the occurrence of important words in the origi-
+nal text, the scoring model will prefer summaries
+generated by other models that contain more words
+from the original text, and instead considers the
+more abstract golden summaries written by humans
+to have lower factual consistency. Nonetheless, the
+ROUGE scores of the summaries generated based
+on this approach are indisputably higher than when
+there is no relation added.
+2
+Literature Rewiew
+Our research mainly related to two directions. One
+is prompt-based learning, the other is controllable
+generation.
+Prompt-based learning: Prompting method is a
+new paradigm for using pre-trained language mod-
+els and was summarized and presented by Liu et al.
+(2021a). The philosophy of Prompt-based learning
+is to modify tasks to fit models instead of modify-
+ing models to fit tasks. According to the survey,
+prompts can be divided into cloze prompts (Petroni
+et al., 2019) and prefix prompts (Li and Liang,
+2021) by location, discrete prompts (Petroni et al.,
+2019), continuous prompts (Li and Liang, 2021)
+and hybrid prompts (Liu et al., 2021b) by shape. To
+find the best prompt for a specific task, we can ei-
+ther manually design different prompts and try out
+the performance of them on the task (Petroni et al.,
+2019), or we can use automated methods such as
+prompt mining (Jiang et al., 2020) or prompt tuning
+(Li and Liang, 2021) to continuously optimize a
+prompt. The studies that can be done using prompt-
+based learning are even more varied. This new
+paradigm has been widely adopted for more than
+20 natural language processing related tasks includ-
+ing text classification, knowledge probing, and so
+forth with good results Liu et al. (2021a). Here
+we mainly refer to prefix-tuning, training a set of
+continuous prefix prompts that can emphasize the
+added information at the beginning of the original
+text, thus achieving controllable generation.
+Controllable generation: Controllable generation
+is a topic that has been studied for a long time,
+and the prefix-tuning proposed in recent years has
+opened up more possibilities in this field. Prefix-
+tuning itself is considered to be a way to steer pre-
+trained language models (Qian et al., 2022). By
+training a set of contrastive prefixes, Qian et al.
+(2022) improved the controllability of the prefixes
+and guided the model for the generation of spe-
+cific sentiments and attributes. Clive et al. (2021)
+then customize prefixes for different documents,
+and achieve controllable generation by combining
+task specific prefix and particular control prefixes.
+A similar study is Tailor (Yang et al., 2022), who
+steers the model to generate sentences with mul-
+tiple attributes at the same time by ‘weaving’ to-
+gether the prefix prompts of different attributes.
+However, most of the above studies have all worked
+on changing prefixes, and few studies have looked
+into the properties of prefix-tuning and what hap-
+pens if we directly change the document, which is
+what we are going to do in this paper.
+3
+Methodology
+In this paper we use OpenIE to extract the relations
+exist in a sentence, and then select the triplet con-
+taining the entities we care about as the relation
+to be added to the original text by part of speech
+tagging (POS) or named entity recognition (NER).
+After adding the triadic relation to the text, we train
+
+Figure 1: An overhead view of the proposed method. The salmon pink block in MLP represents the last layer of
+the MLP. Prefix-tuning reshapes it and passes it to the ‘past_key_values’ parameter of GPT-2, thus achieving the
+purpose of adding continuous prefix prompts to the model. The short text below is an example of Xrel.
+a set of continuous prefix prompts on the modified
+text, hoping that this set of prefixes will recognize
+the information we added and help the model to
+generate factually consistent sentences with the
+added relations in combination with the original
+texts. To evaluate the generation results, we score
+the golden summaries and the generated summaries
+separately using CoCo against the original texts and
+calculate the ROUGE score of the generated sum-
+maries on the golden summaries. The following
+details the techniques we use in this paper.
+3.1
+Preliminary
+Before presenting the solution we propose, we
+would like to first introduce prefix-tuning. Prefix-
+tuning is a kind of prompt-based learning proposed
+by Li and Liang (2021). Compared with tradi-
+tional fine-tuning-based approaches, the most dis-
+tinctive feature of prompt-based learning is that
+it tends to modify tasks to let the model use its
+own knowledge to accomplish the tasks, without
+specifically designing additional structures to adapt
+the model to a task. For example, TL;DR is the
+prompt trained by GPT-2 to prompt the model for
+summarization task. In contrast to TL;DR, prefix-
+tuning does not include natural language prompts
+in the text, but instead uses ‘past_key_values’ to
+concatenate a set of trainable parameters as a pre-
+fix in front of the text X. In training, keeping the
+parameters of the model constant and training only
+the parameters in the prefix, we obtain a set of pre-
+fixes that allow the model to perform a specific
+task. Specifically, since the structure of GPT-2 is
+a multi-layer Transformer decoder (Radford et al.,
+2018), its prediction of the next word is jointly de-
+termined by all the preceding words, so as long as
+we can make targeted changes to the left context,
+then we can control the generation of the model.
+‘past_key_values’ was originally used to store the
+previous computation results of the model to speed
+up the computation, but prefix-tuning cleverly ex-
+ploits this by mapping a set of parameters to the
+size required by ‘past_key_values’ through a fully
+connected neural network (MLP) and then passing
+them to the model’s ‘past_key_values’ parameter
+to achieve the purpose of adding prefix before X.
+Having been passed to the model, this parameter
+will be concatenated with the existing keys and
+values of the model in the dimension of sequence
+length, thus controlling the summary generation.
+3.2
+Knowledge incorporation
+Although language models contain a large amount
+of knowledge, this knowledge is also the reason
+why the summaries generated by the models do not
+always agree with the facts of the original text (Xie
+et al., 2021). Therefore, how to use the knowledge
+we need and eliminate wrong priors becomes a
+problem. To do this, we emphasize the fact we care
+about by adding a set of natural language prompts
+before the text. Intuitively, if we add a structured
+set of relation leading by keywords (such as ‘key
+relation’) in front of the original text, this is equiv-
+alent to giving a model a hint for generation. Then
+first, the model’s prior will assume that some im-
+portant information is contained here. Second, the
+added relation increases the weights of the words
+in the relation and make them more likely to ap-
+
+trainable prefix
+relation + source
+summary
+reshape
+MLP
+Key
+relation
+<end>
+Key relation: I'subject': ' Duchess', 'relation': ' will feature on', 'object': ' cover of British Vogue'? I I Seven photographs taken in :pear in the generated summaries. Third, through
+experiments we found that prefix-tuning is very
+good at extracting structured information, so struc-
+turally adding prefix prompts allows the trainable
+continuous prefix to help the model focus on this
+structured information.
+In our experiments, we first apply this approach
+to the preliminary experiment of sentence extrac-
+tion, giving the model some information about
+a sentence to see if the model could extract that
+sentence accurately, and then apply this idea to
+the knowledge-enhanced document summarization.
+For sentence extraction, we first test whether the
+model can extract the first sentence accurately by
+training the model with the first sentence of the
+source text without any guidance. Then we add
+the first three tokens of any sentence in front of a
+source text and train the model with those sentences
+to see whether the model could extract those sen-
+tences accurately. Finally, we extract any 3 tokens
+from a sentence and train the model, to see whether
+the model can extract that sentence accurately. For
+knowledge-enhance document summarization, we
+use OpenIE to first extract all the triadic relations
+contained in the golden summary, then perform
+NER on the golden summary to pick out all en-
+tities. Adding the triplet containing the entities
+we care about in front of the source text, we have
+a training data Xrel. The overhead view of our
+proposed method is shown as Figure 1
+3.3
+Metrics
+A common metric used in natural language process-
+ing is ROUGE, which scores the generated sum-
+maries based on the overlap of n-grams (Lin, 2004).
+This is effective when evaluating the similarity be-
+tween the generated summary and the golden sum-
+mary, but it is hard to measure how well a sentence
+agrees with the facts of the original. As shown in
+Table 1, perhaps the generated summary and the
+golden summary have a high degree of overlap in
+words, this generated summary is not necessarily
+consistent with the original facts. To overcome this
+issue, in addition to ROUGE, we also use CoCo
+(Xie et al., 2021). CoCo is a new metric based on
+positional scores to evaluate the factual consistency
+between the summary and the original text. The
+basic idea is to first detect the keywords in the sum-
+mary, mask them in the original text to generate
+masked document X
+′, and then use X
+′ and the
+original X to calculate the positional score of each
+ref:
+Hong Kong-based airline Cathay
+Pacific has announced a ban on ship-
+ments of shark fin in a move that has
+been welcomed by conservationists.
+gen: Cathay Pacific is to ban shark fin
+shipments to Hong Kong.
+Table 1: A comparison between a golden summary
+and a generated summary (without relation added). Al-
+though the ROUGE score for the generated summary
+may be high, these two sentences are actually telling
+different stories.
+word in the summary, i.e., the score of each word
+appearing in a specific position in the summary. If
+the positional score of a word appearing at a certain
+position in the summary under X is significantly
+higher than the positional score of the word appear-
+ing at that position under X
+′, then we assume that
+the word at that position in the summary is more in-
+fluenced by the original text, the fact related to this
+word is more likely consistent with the fact of the
+original text. Conversely, if the difference between
+the positional score of a word in the summary un-
+der X and X
+′ is not significant, i.e., whether the
+original text is X or X
+′ has little effect on the posi-
+tional score of the word, then we consider that the
+word is not strongly dependent on the original text,
+the presence of the word in this position is likely
+to cause the summary to be factually inconsistent
+(Xie et al., 2021). When using CoCo we use POS
+or NER to pick the important words to be masked.
+For scoring model, we use BART. We will further
+discuss CoCo in experiments.
+4
+Experiment
+In experiments, we first examine the properties of
+prefix-tuning through a couple of preliminary ex-
+periments, then conduct knowledge-enhance docu-
+ment summarization on this basis. Below we first
+introduce the datasets and baselines we use in our
+experiments, and then present all the experiments
+in detail.
+4.1
+Dataset
+CNN/Daily Mail: CNN/Daily Mail is a dataset
+proposed in 2016 for abstractive summarization
+tasks (Nallapati et al., 2016). The source of each
+data is a long article and the target is the cor-
+responding summary crawled by Nallapati et al.
+(2016).
+The CNN/Daily Mail dataset contains
+
+286,817 training data, 13,368 validation data and
+11,487 test data. The average length of source
+is 766 words, and target is usually three to four
+sentences, with an average length of 53 words (Nal-
+lapati et al., 2016). Since the GPT-2 model has
+a limit on the input length and the parameters re-
+quired to train for prefix-tuning are not many, here
+we only use the first 50,000 data where the sum
+of the length of source and target does not exceed
+800. If the added relation is very long, then we
+will further reduce the data usage according to the
+length of the new text.
+XSum: XSum (Narayan et al., 2018) is another
+commonly used dataset in the field of document
+summarization. The full name of it is extreme sum-
+marization, and its source is BBC’s online articles.
+It contains 204,045 training data, 11,332 validation
+data and 11,334 test data. As its name indicates,
+XSum seeks to summarize an article with a very
+short sentence, so its target is shorter than that of
+CNN/Daily Mail, and also more abstract. The av-
+erage length of XSum’s sources is 431 words, but
+the average length of its targets is only about 23
+words (Narayan et al., 2018). The abstract nature
+of XSum makes the abstractive summarization task
+more challenging, so getting good results on XSum
+can also make a model more convincing. As same
+as CNN/Daily Mail, we only use the first 50,000
+data with the sum of the length of source and target
+less than 800 for training.
+4.2
+Baseline
+To evaluate the performance of the model, we used
+two metrics in our experiments. ROUGE to eval-
+uate the degree of overlap between the generated
+summary and the golden summary, and CoCo to
+evaluate the factual consistency of the generated
+summary with the original text. Since there are
+now many models trained specifically for document
+summarization, it is difficult for GPT-2 to surpass
+the performance of them. Therefore, when com-
+paring ROUGE scores we only compare with the
+ROUGE scores reported by GPT-2 obtained when
+using TL;DR. Given that prefix-tuning is far more
+expressive than this one hard prompt, the scores
+for prefix-tuning should at least exceed those of
+GPT-2, and if relations are added to the text, then
+theoretically the summaries generated by modified
+text should have higher ROUGE scores than if no
+relations are added.
+CoCo, on the other hand, is not a commonly used
+metric like Rouge. It is rare to see articles that use
+CoCo to evaluate the factual consistency of a model
+on summarization tasks, so for the CoCo score, we
+use the summaries generated by the model with-
+out the addition of relations as the baseline. The
+CoCo score with the addition of relations should
+be higher than the CoCo score when there is no
+relation added.
+4.3
+Preliminary Experiment
+Preliminary experiment is of two parts, the first is
+abstractive summarization, and the second part
+is sentence extraction. The experimental results
+show that prefix-tuning is effective for document
+summarization task on GPT-2, and it is good at
+extracting information from structured content in
+texts. This properties of prefix-tuning fit well with
+our purpose.
+4.3.1
+Abstractive summarization
+Abstractive summarization is a preliminary experi-
+ment to examine the performance of prefix tuning
+for document summarization on GPT-2 without the
+addition of relations. The results can be used as a
+baseline for knowledge-enhanced document sum-
+marization.
+CNN/Daily Mail: When using prefix-tuning for
+document summarization task on CNN/Daily Mail,
+the ROUGE-1 is only about 20 if we only use a
+prefix of length 5, but if a longer prefix is used,
+then the performance of the model immediately
+improves significantly. If we use a prefix of length
+100 and allow the model to generate summaries
+with a maximum length of 100, then the ROUGE-1
+for summaries generated on CNN/Daily Mail using
+GPT-2 can reach 30, which is consistent with the
+conclusion reported in Li and Liang (2021) that the
+longer the prefix the more expressive it is. For the
+part of the data we use, the ROUGE scores exceed
+those reported by GPT-2 for document summariza-
+tion on CNN/Daily Mail using TL;DR, see Table
+2.
+ROUGE-1
+ROUGE-2
+ROUGE-L
+TL;DR 29.34
+8.27
+26.58
+Prefix
+32.89
+14.58
+30.78
+Table 2: Results of TL:DR and Prefix-tuning
+However, we found a problem in the generated
+summaries of CNN/Daily Mail using prefix-tuning
+that the original sentences from the source texts
+
+often appear in the summaries. Since there are
+many studies reporting a significant difference in
+the level of abstraction between the summaries of
+CNN/Daily Mail generated by a model and those of
+XSum (Lewis et al., 2020; Xu et al., 2020), we con-
+ducted another abstractive summarization experi-
+ment using XSum in order to determine whether
+prefix-tuning causes sentences and words from the
+original text to appear in the summaries, or whether
+the characteristics of the training data make the
+model behave this way.
+XSum: As described above, the golden summaries
+of the XSum dataset are very concise and refined,
+which poses a higher challenge to a pre-trained lan-
+guage model. If a 100-length prefix is used and
+the model is allowed to generate summaries with
+a maximum length of 100, the ROUGE-1 of the
+summaries generated by GPT-2 is very low, only
+about 20%, but the good news is that the gener-
+ated summaries are indeed more abstract, and al-
+most no original sentences from the original text
+appear. This experiment shows that the appear-
+ance of the original sentence of the article in the
+summaries is a result of the dataset. This does not
+happen if a very abstract dataset is replaced. In
+addition, since prefix-tuning can perfectly pick out
+the key information in data2text task as shown in
+(Li and Liang, 2021) and identifies the range of sen-
+tences in CNN/Daily Mail without any processing
+such as mask on the training data, we found that
+prefix-tuning may be very sensitive to the situation
+that target overlaps with source and can accurately
+recognize the structured information with certain
+pattern from source by only a small amount of
+training. We carried out the preliminary experi-
+ment of sentence extraction with this hypothesis,
+explored whether prefix-tuning can actually iden-
+tify sentence ranges and extract sentences from the
+sources as we expected.
+Another
+interesting
+finding
+in
+abstractive
+summarization is that by taking the prefixes trained
+on CNN/Daily Mail for abstractive summarization
+and having the model generate summaries of
+XSum, the model also exhibits similar behavior
+to generating summaries of CNN/Daily Mail,
+i.e., many times it generates sentences from the
+original text rather than more abstract sentences.
+This shows that trained prefix prompts with certain
+properties can be transferred to other datasets for
+use, and allows texts generated using other datasets
+to exhibit similar properties.
+4.3.2
+Sentence extraction
+Sentence extraction is to further explore the proper-
+ties of prefix tuning. Specifically, this preliminary
+experiment uses the method proposed in methodol-
+ogy to test whether prefix-tuning can truly identify
+the information we want to extract accurately, espe-
+cially their boundaries. The experiments demon-
+strate that, When experimenting with CNN/Daily
+Mail for sentence extraction, if the model is trained
+directly with the first sentence of source without
+any guidance (SenEx1), the model can extract that
+sentence almost perfectly. If the first three tokens
+of a sentence are added to the front of the source
+and then the model is trained (SenEx2), then the
+model can extract that sentence most of the time,
+but with reduced accuracy. If any three tokens from
+a sentence are added to the source and the model is
+trained with that sentence (SenEx3), the model can
+still extract that sentence, but the ROUGE-1 score
+drops to about 65%, as Table 3 shows.
+ROUGE-1
+ROUGE-2
+ROUGE-L
+SenEx1 99.98
+99.79
+99.98
+SenEx2 96.56
+95.64
+96.46
+SenEx3 65.50
+59.38
+64.25
+Table 3: Results of Extracting Sentences. The tokens
+we use here are tokens after byte pair encoding of the
+original text, so it is not guaranteed to be three com-
+plete words.
+It is worth noting that, first of all, the extraction
+is exact to the token. For example, the periods ‘.’
+and ‘ .’ (with a space) are two tokens in GPT-2
+embedding, then if we take the first period ’.’ as the
+criterion for extracting a sentence, the generated
+summary will definitely be bounded by the first
+period, and will never stop at the second kind of
+period ‘ .’. Second, if the model does not correctly
+extract the sentence we want to extract, the result
+is still necessarily a sentence from the original text
+and will not be generated arbitrarily, so we believe
+that when using any 3 tokens of a sentence for
+extraction, the model is still able to recognize the
+boundaries, even though in many cases the model
+does not correctly identify the sentence we are
+extracting. As long as we provide more special
+tokens (e.g., tokens of low-frequency words instead
+of tokens of conjunctions or punctuations), the
+model will be able to identify which sentence we
+are extracting and extract that sentence precisely.
+
+The above preliminary experiments show that
+prefix-tuning can indeed extract key information
+directly from the original text, and its extraction is
+so precise that it can identify the patterns of struc-
+tured information in the text, extract the important
+information with token precision, and filter out the
+frames that contain this information in the origi-
+nal text. Therefore, theoretically we could add the
+knowledge we care about to the text in this way,
+and then let the text combine the added knowledge
+to generate a factually consistent summary. This is
+what knowledge-enhanced document summariza-
+tion does below.
+4.4
+Knowledge-enhanced document
+summarization
+We extracted the relations in the targets of
+CNN/Daily Mail and XSum respectively as de-
+scribed in the methodology section, and then added
+the selected relations to the original text and trained
+prefixes for summary generation. The results show
+that the dataset with the added relations do generate
+summaries that are more relevant to the topic of
+the relations we added, and most of the generated
+summaries maintain factual consistency with the
+added relations. However, there are still some cases
+where this solution can go wrong. We show some
+examples of correct and incorrect generation in the
+next page, and each of these examples is analyzed
+below.
+4.4.1
+CNN/Daily Mail
+We first conducted experiments using CNN/Daily
+Mail. Here we only select the sentence that the
+relation best meets our requirements from the three
+to four sentences of a target, instead of using the
+whole target for training. By analyzing the first ten
+cases of CNN/Daily Mail and comparing the sum-
+maries generated with and without the addition of
+relation, we found that when a) the added relation
+covers thoroughly; b) there is a reference sentence
+in the original text, then the generated summary
+will be correct, as shown in Tables 4(a) and 4(b).
+However, the added relations only ensure that
+the generated summary is consistent with the facts
+described by the relation, and if the summary is not
+generated according to the added relations or the
+generated summary is beyond the coverage of the
+added relation, inconsistency may occur. This is
+manifested in a) the generated summary does not
+refer to the added relation at all, as shown in Ta-
+ble 4(c); b) the generated summary extracts part of
+the content of the added relation, but this changes
+the correct summary, resulting in factual inconsis-
+tency, as shown in Table 4(d). c) the summary
+completely contains the keywords of the added re-
+lation, but factual inconsistency occurs beyond the
+added relation, as Table 4(e) shows.
+Since the model tends to refer to the original text
+or extract the exact sentences from the original text
+when generating summaries with CNN/Daily Mail,
+the effect of this approach is likely to be amplified
+or obscured. To examine the performance of this
+approach on a more abstract dataset, we conducted
+another experiment on XSum.
+4.4.2
+XSum
+Overall, the results of generation using prefix-
+tuning on XSum with added relations are similar
+to CNN/Daily Mail. However, since the results
+of GPT-2 for summarization on XSum are orig-
+inally not good, XSum with added relations im-
+proves the results to a larger extent compared to the
+dataset without added relations. It is experimen-
+tally demonstrated that the generated summaries
+can be correct without referring to the original
+text, as shown in Table 4(f). However, similar
+to CNN/Daily Mail, there is a possibility of error in
+the parts that the added relations are not included
+to, such as time, place, and so forth. But overall,
+the addition of relations has helped the model to
+perform better.
+4.5
+Evaluation
+In this subsection we score the above generated
+results and analyze the scoring results in detail.
+As mentioned above, the CoCo score for the fac-
+tual consistency of the summary with the original
+text is based on the positional scores of important
+words in the target. Specifically, if the original text
+is ‘Biden is the president of the United States.’,
+and the golden summary is the same as the origi-
+nal text, also ‘Biden is the president of the United
+States.’, but the generated text is ‘Obama is the pres-
+ident of the United States.’, then CoCo will first
+use POS or NER to pick out the important words.
+Suppose we use NER and keep all the ‘PERSON’,
+‘TITLE’ and ‘COUNTRY’ in the results, then the
+keywords picked from reference will be ‘Biden’,
+‘president’, ‘United’, ‘States’, and the keywords
+picked from generation will be ‘Obama’, ‘presi-
+dent’, ‘United’, ‘States’. Masking off the corre-
+sponding word in the original text X, we get X′
+ref:
+
+relation:
+Key relation: ‘subject’: ‘ Sally Forrest ’, ‘relation’: ‘ died on ’, ‘object’: ‘ March 15 ’
+reference:
+Sally Forrest , an actress-dancer who graced the silver screen throughout the ’40s and
+’50s in MGM musicals and films died on March 15 .
+X gen:
+Actress: Sally Forrest was in the 1951 Ida Lupino-directed film ‘Hard, Fast and Beauti-
+ful’
+Xrel gen:
+Sally Forrest died on March 15 at her home in Beverly Hills, California.
+(a) A correct case from CNN/Daily Mail. The second half of the sentence is the original article.
+relation:
+Key relation: ‘subject’: ‘ Prince Harry ’, ‘relation’: ‘ is in ’, ‘object’: "
+attendance for England ’s crunch match against France "
+reference:
+Prince Harry in attendance for England’s crunch match against France .
+X gen:
+England beat France 55-35 in ‘Le Crunch’.
+Xrel gen:
+Prince Harry in attendance for England’s crunch match against France .
+(b) A correct case from CNN/Daily Mail. Added relation covers through the sentence.
+relation:
+Key relation: ‘subject’: ‘ valuable stock ’, ‘relation’: ‘ taken from ’, ‘object’: ‘
+his antiques shop in Basingstoke ’
+reference:
+Discovered valuable stock taken from his antiques shop in Basingstoke .
+X gen:
+Alan Stone, 51, arrested on suspicion of theft.
+Xrel gen:
+The father-of-four admitted he had a ‘lump in his throat’
+(c) An incorrect case from CNN/Daily Mail. The generated summary does not follow the added relation at all.
+relation:
+Key relation: ‘subject’: ‘ 1,000 pieces ’, ‘relation’: ‘ is in ’, ‘object’: ‘ last two years ’
+reference:
+Has inked 1,000 pieces of art on leaves in last two years .
+X gen:
+Teacher Wang Lian has drawn hundreds of doodles on leaves for the last 10 years.
+Xrel gen:
+Teacher Wang Lian has drawn hundreds of doodles on leaves for the last two years .
+(d) An incorrect case from CNN/Daily Mail. Added relation wrongly affected the model and causes errors
+relation:
+Key relation: ‘subject’: ‘ could first preventive tool ’, ‘relation’: ‘ is in ’, ‘object’: ‘
+history ’
+reference:
+WHO leader: This vaccine could be "the first preventive tool against Ebola in
+history "
+X gen:
+A vaccine will be tested in a subsequent study.
+Xrel gen:
+The trial could be the first preventive tool in history .
+(e) An incorrect case from CNN/Daily Mail. Error occurs beyond the added relation.
+relation:
+Key relation: ‘subject’: ‘ Former Premier League footballer Sam Sodje ’, ‘relation’: ‘
+has appeared alongside ’, ‘object’: ‘ three brothers accused ’
+reference:
+Former Premier League footballer Sam Sodje has appeared in court alongside
+three brothers accused of charity fraud.
+X gen:
+A former Leeds United defender has been charged with conspiracy to commit fraud.
+Xrel gen:
+Former Premier League footballer Sam Sodje has appeared alongside
+three brothers accused of fraud.
+(f) A correct case from XSum. The word ‘fraudulent’ appears in the original, but ‘fraud’ does not
+Table 4: Analysis
+
+(a) CoCo score for the reference. Score_X and Score_X’ are the positional scores of ‘B’, ‘president’, ‘United’,
+‘States’ on X and X’
+(b) CoCo score for the generated summary. Score_X and Score_X’ are the positional scores of ‘Obama’, ‘presi-
+dent’, ‘United’, ‘States’ on X and X’
+Figure 2: Calculation of CoCo score
+’<mask> is the <mask> of the <mask> <mask>.’
+and X′
+gen ‘Biden is the <mask> of the <mask>
+<mask>.’. This allows us to score reference and
+generated summary using X, X′
+ref and X′
+gen re-
+spectively. When they are put into the scoring
+model BART for scoring, BART will perform byte
+pair encoding (BPE) to these sentences. Regard-
+less of how BPE splits a word, if we only take the
+first token for each important word, then assuming
+that there are N key words picked out, the CoCo
+score is CoCo = (Score_X − Score_X′)/N.
+For example, if we want to calculate the CoCo
+score for the reference, the process can be illus-
+trated as Figure 2(a). The same is true for calcu-
+lating the score of generated summary. Here we
+can see that since the word ‘Biden’ appears in X,
+the Score_X for the first token of that word is
+very high, while in X′, i.e., where the first word is
+‘<mask>’, the score for ‘B’ is considerably lower.
+Thus, the larger the difference between Score_X
+and Score_X′, the more likely the sentence is fac-
+tually consistent. On the contrary, for the score of
+generated summary, since the two positional scores
+for ’Obama’ against ’Biden’ are equivalently small,
+the Score_X − Score_X′ will be small and the
+CoCo score will be low, which means the sentence
+is likely to be factually inconsistent with the origi-
+nal.
+The Pearson correlation and Spearman corre-
+lation between CoCo and human judgments of
+factual consistency on document summarization
+task is higher than that of other matrics (Xie et al.,
+2021), indicating that it does provide a better mea-
+sure than other matrics of how well a sentence is
+factually consistent with the original text. How-
+ever, we found that it is not suitable for compar-
+ing factual consistency of golden summaries and
+generated summaries, because CoCo relies on the
+positional scores of the words in a sentence, but
+the summaries generated from the original texts
+X are often more likely to have words from the
+original text than the golden summaries, which re-
+sults in higher CoCo scores for the generated sum-
+maries and lower scores for the golden summaries.
+And since our relations are drawn from the golden
+summaries, this results in the summaries generated
+from texts with added relations Xrel being closer
+to the golden summaries, thereby making the CoCo
+scores lower than those summaries generated from
+the original texts X, despite the summaries gener-
+ated from Xrel are often more consistent with the
+facts of the original text. This problem is more ob-
+vious on CNN/Daily Mail, because the summaries
+generated on X are mostly the original text, while
+the summaries generated on Xrel are closer to the
+references. Although the summaries generated on
+XSum with and without added relations are sim-
+ilarly abstract, the CoCo score without added re-
+lations is still slightly higher than the CoCo score
+with added relations. Nonetheless, for ROUGE,
+the ROUGE score of the summaries generated with
+the added relations is indisputably higher than the
+ROUGE score of the summaries generated without
+the added relations.CoCo and ROUGE scores are
+shown in Table 5 and Table 6.
+
+X: ['B',"iden',"is','the','president'," of'," the',"United','" States',"']
+Score_X: [0.5603, 0.9279,0.8908,0.9569]
+X': [<mask>',"is',"the','<mask>",of","the','<mask>",'<mask>','.']
+Score_X': [0.0023, 0.0346, 0.0291,0.8516]
+['B',"iden',' is','the',' president','of','the',' United',' States',""']
+CoCo Score=(Score_X - Score_X')/4=0.6046X: ['B',"iden',"is','the','president'," of'," the',"United','" States',"']
+Score_X: [0.0017, 0.3789,0.8725,0.9508]
+X':['B',"iden',"is',"the',"<mask>",'of",'the','<mask>",'<mask>','']
+Score_X': [0.0007,0.0282,0.6997,0.9534]
+['Obama','is',' the',' president','of'," the','United'," States',"']
+CoCo Score=(Score_X - Score_X')/4=0.1305Only Noun
+Noun+Verb
+X
+0.0264
+0.0253
+Xrel
+0.0179
+0.0163
+(a) CNN/Daily Mail
+Only Noun
+Noun+Verb
+X
+0.0086
+0.0074
+Xrel
+0.0074
+0.0063
+(b) XSum
+Table 5: CoCo scores
+ROUGE-1
+ROUGE-2
+ROUGE-L
+X
+25.07
+12.13
+23.46
+Xrel
+62.38
+49.46
+61.44
+(a) CNN/Daily Mail
+ROUGE-1
+ROUGE-2
+ROUGE-L
+X
+23.77
+7.06
+19.73
+Xrel
+45.69
+31.06
+43.01
+(b) XSum
+Table 6: ROUGE scores
+5
+Conclusion
+This paper explores the properties of prefix-tuning,
+proposes a knowledge-enhanced document summa-
+rization method that combines prefix-tuning and
+natural language prompts, and makes a significant
+impact on the summaries generated using GPT-2
+for both CNN/Daily Mail and XSum. However,
+as discussed in the paper, this approach still has
+many shortcomings and can be improved. For ex-
+ample, the relations we add to the source are only
+the relations extracted by OpenIE. The extracted
+content may not be the relations we care about,
+and the relations are often not a good representa-
+tion of the relationship of two entities. In addition,
+we consider that OpenIE many times extracts too
+long content, so that the original texts expose too
+much information to the model, which may have
+an excessive impact on the summary generation.
+To deal with these two issues, in the future, instead
+of adding relations extracted by OpenIE, we can
+add existing knowledge in knowledge bases to the
+model. This ensures the cleanliness and purity of
+the information and avoids the influence of low-
+quality data on the model. Furthermore, since the
+maximum text length that GPT-2 can handle is only
+1024, this not only limits the amount of training
+data that can be used, but also indirectly limits the
+length of prefixes that can be attempted and the
+length of summaries that are allowed to be gen-
+erated. Therefore, if GPT-2 can be replaced with
+a model that can handle infinite length preferably
+trained specifically for summarization task, then
+perhaps we can generate longer summaries using
+more training data with longer prefixes, thus fur-
+ther improving the performance of this approach.
+However, too long prefixes may lead to a decrease
+but not an increase in the performance of the model
+due to inadequate training, or too long prefixes may
+dominate model generation instead of text, so how
+long prefixes should be used in what models and on
+what datasets is also a valuable direction for future
+research.
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf,len=443
+page_content='Incorporating Knowledge into Document Summarization  — an Application of Prefix-Tuning on GPT-2  Chen Chen and Wei Emma Zhang  The University of Adelaide  South Australia 5005, Australia  Abstract  Despite the great development of document  summarization techniques nowadays, factual  inconsistencies between the generated sum-  maries and the original text still occur from  time to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  This paper proposes a prefix-  tuning-based approach that uses a set of train-  able continuous prefix prompt together with  discrete prompts to aid model generation,  which makes a significant impact on both  CNN/Daily Mail and XSum summaries gener-  ated using GPT-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The improvements on fact  preservation in the generated summaries indi-  cates the effectiveness of adopting this prefix-  tuning-based method in knowledge-enhanced  document summarization, and also shows a  great potential on other natural language pro-  cessing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  1  Introduction  With the explosive growth of information on the  Internet in recent years, document summarization  has become an area of increasing interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' There are  two main approaches for document summarization,  one is extractive and the other is abstractive (Qu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Decades ago, there were more studies  on extractive abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' In recent years, benefits  from the proposal of Transformer (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=',  2017) and the emergence of various large-scale  pre-trained language models such as GPT (Radford  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2018), BART (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2020), more and  more researches now focus on using pre-trained  language models to do abstractive summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Abstractive summarization allows us to gener-  ate natural summaries that are more like human-  written ones, rather than simply extracting sen-  tences from the original text, but one problem with  it is that the summaries a model generates are not  always factually consistent (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The  presence of the model prior makes it likely that the  model will generate sentences that do not match  the facts of the original text when we use the pre-  trained model for document summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Al-  though we can fine tune the model using our own  data, fine-tuning a model usually still requires a  large dataset containing tens of thousands of data  (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2020), which often poses a dilemma  for the users of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  To solve this problem, one approach is to incor-  porate knowledge into the model to aid summary  generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' This reduces the need for training data  and ensures that the facts we add will always have  an impact on the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' For example, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  (2021c) proposed KG-BART (knowledge-graph  BART), which can help BART to generate sum-  maries by adding KG-encoder and KG-decoder to  the model, while Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' (2021) designed a graph  attention network (GAT) to take in external knowl-  edge and enhance the factual consistency of the gen-  erated summaries by passing the output of the GAT  to the cross attention layer of decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' However, as  we can see in the above studies, adding knowledge  to a model usually requires not only embedding  knowledge, but also designing new structures to  receive it, so it is not very practical in many cases,  still leaving the problem unresolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Stopping to rethink this sequence-to-sequence  problem of document summarization, we realized  that, in fact, when generating summaries using pre-  trained models, we had actually been trying to fig-  ure out how to generate correct summaries using  the documents together with the model’s prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' If  the model’s prior was correct or the input docu-  ments could effectively influence the summary gen-  eration, then the generated summaries would be  correct, and conversely, the summaries would have  inconsistencies with the original text (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=',  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Thus, the problem of adding knowledge to a  model to steer it to generate correct summaries can  be transformed into the problem of how to com-  bine texts more effectively with the model prior in  document summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Following this idea, we  propose a method for incorporating knowledge into  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='11719v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='CL]  27 Jan 2023  document summarization based on prefix-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Prefix-tuning is a type of prompt-based learn-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Prompting methods assumes that there is al-  ready enough knowledge in a pre-trained language  model, so that we can modify tasks to take advan-  tage of the model prior and let the model spit out  the correct answer on its own (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Prefix-tuning is to train a set of continuous task  specific prefix prompts to steer the model to per-  form text generation for different generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  (Li and Liang, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' To incorporate knowledge  into generation, we simply add a segment of natu-  ral language prompts extracted by OpenIE (Angeli  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2015) in front of the original text to empha-  size the relations we care about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Through experi-  ments, we found that since the continuous prefix  is very expressive, after training, the continuous  prefix can recognize the patterns in the natural lan-  guage prompts, thus pick out the key information  from them and then combine the extracted infor-  mation with the original text to perform summary  generation, so this is a very direct way to influence  the summary generation by data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  To test the effectiveness of this approach, we fi-  nally use both CoCo (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2021) and ROUGE  (Lin, 2004) to evaluate the generated summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  CoCo scores the factual consistency of the gener-  ated summaries by averaging the positional scores  of important words, while ROUGE is based on the  overlap of n-grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Experiments demonstrate that  since the calculation of the positional score relies  on the occurrence of important words in the origi-  nal text, the scoring model will prefer summaries  generated by other models that contain more words  from the original text, and instead considers the  more abstract golden summaries written by humans  to have lower factual consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Nonetheless, the  ROUGE scores of the summaries generated based  on this approach are indisputably higher than when  there is no relation added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  2  Literature Rewiew  Our research mainly related to two directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' One  is prompt-based learning, the other is controllable  generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Prompt-based learning: Prompting method is a  new paradigm for using pre-trained language mod-  els and was summarized and presented by Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The philosophy of Prompt-based learning  is to modify tasks to fit models instead of modify-  ing models to fit tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' According to the survey,  prompts can be divided into cloze prompts (Petroni  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2019) and prefix prompts (Li and Liang,  2021) by location, discrete prompts (Petroni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=',  2019), continuous prompts (Li and Liang, 2021)  and hybrid prompts (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2021b) by shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' To  find the best prompt for a specific task, we can ei-  ther manually design different prompts and try out  the performance of them on the task (Petroni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=',  2019), or we can use automated methods such as  prompt mining (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2020) or prompt tuning  (Li and Liang, 2021) to continuously optimize a  prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The studies that can be done using prompt-  based learning are even more varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' This new  paradigm has been widely adopted for more than  20 natural language processing related tasks includ-  ing text classification, knowledge probing, and so  forth with good results Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Here  we mainly refer to prefix-tuning, training a set of  continuous prefix prompts that can emphasize the  added information at the beginning of the original  text, thus achieving controllable generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Controllable generation: Controllable generation  is a topic that has been studied for a long time,  and the prefix-tuning proposed in recent years has  opened up more possibilities in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Prefix-  tuning itself is considered to be a way to steer pre-  trained language models (Qian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' By  training a set of contrastive prefixes, Qian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  (2022) improved the controllability of the prefixes  and guided the model for the generation of spe-  cific sentiments and attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Clive et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' (2021)  then customize prefixes for different documents,  and achieve controllable generation by combining  task specific prefix and particular control prefixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  A similar study is Tailor (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2022), who  steers the model to generate sentences with mul-  tiple attributes at the same time by ‘weaving’ to-  gether the prefix prompts of different attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  However, most of the above studies have all worked  on changing prefixes, and few studies have looked  into the properties of prefix-tuning and what hap-  pens if we directly change the document, which is  what we are going to do in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  3  Methodology  In this paper we use OpenIE to extract the relations  exist in a sentence, and then select the triplet con-  taining the entities we care about as the relation  to be added to the original text by part of speech  tagging (POS) or named entity recognition (NER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  After adding the triadic relation to the text, we train  Figure 1: An overhead view of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The salmon pink block in MLP represents the last layer of  the MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Prefix-tuning reshapes it and passes it to the ‘past_key_values’ parameter of GPT-2, thus achieving the  purpose of adding continuous prefix prompts to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The short text below is an example of Xrel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  a set of continuous prefix prompts on the modified  text, hoping that this set of prefixes will recognize  the information we added and help the model to  generate factually consistent sentences with the  added relations in combination with the original  texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' To evaluate the generation results, we score  the golden summaries and the generated summaries  separately using CoCo against the original texts and  calculate the ROUGE score of the generated sum-  maries on the golden summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The following  details the techniques we use in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='1  Preliminary  Before presenting the solution we propose, we  would like to first introduce prefix-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Prefix-  tuning is a kind of prompt-based learning proposed  by Li and Liang (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Compared with tradi-  tional fine-tuning-based approaches, the most dis-  tinctive feature of prompt-based learning is that  it tends to modify tasks to let the model use its  own knowledge to accomplish the tasks, without  specifically designing additional structures to adapt  the model to a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' For example, TL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='DR is the  prompt trained by GPT-2 to prompt the model for  summarization task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' In contrast to TL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='DR, prefix-  tuning does not include natural language prompts  in the text, but instead uses ‘past_key_values’ to  concatenate a set of trainable parameters as a pre-  fix in front of the text X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' In training, keeping the  parameters of the model constant and training only  the parameters in the prefix, we obtain a set of pre-  fixes that allow the model to perform a specific  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Specifically, since the structure of GPT-2 is  a multi-layer Transformer decoder (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=',  2018), its prediction of the next word is jointly de-  termined by all the preceding words, so as long as  we can make targeted changes to the left context,  then we can control the generation of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  ‘past_key_values’ was originally used to store the  previous computation results of the model to speed  up the computation, but prefix-tuning cleverly ex-  ploits this by mapping a set of parameters to the  size required by ‘past_key_values’ through a fully  connected neural network (MLP) and then passing  them to the model’s ‘past_key_values’ parameter  to achieve the purpose of adding prefix before X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Having been passed to the model, this parameter  will be concatenated with the existing keys and  values of the model in the dimension of sequence  length, thus controlling the summary generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='2  Knowledge incorporation  Although language models contain a large amount  of knowledge, this knowledge is also the reason  why the summaries generated by the models do not  always agree with the facts of the original text (Xie  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Therefore, how to use the knowledge  we need and eliminate wrong priors becomes a  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' To do this, we emphasize the fact we care  about by adding a set of natural language prompts  before the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Intuitively, if we add a structured  set of relation leading by keywords (such as ‘key  relation’) in front of the original text, this is equiv-  alent to giving a model a hint for generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Then  first, the model’s prior will assume that some im-  portant information is contained here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=" Second, the  added relation increases the weights of the words  in the relation and make them more likely to ap-  trainable prefix  relation + source  summary  reshape  MLP  Key  relation  <end>  Key relation: I'subject': ' Duchess', 'relation': ' will feature on', 'object': ' cover of British Vogue'?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' I I Seven photographs taken in :pear in the generated summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Third, through  experiments we found that prefix-tuning is very  good at extracting structured information, so struc-  turally adding prefix prompts allows the trainable  continuous prefix to help the model focus on this  structured information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  In our experiments, we first apply this approach  to the preliminary experiment of sentence extrac-  tion, giving the model some information about  a sentence to see if the model could extract that  sentence accurately, and then apply this idea to  the knowledge-enhanced document summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  For sentence extraction, we first test whether the  model can extract the first sentence accurately by  training the model with the first sentence of the  source text without any guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Then we add  the first three tokens of any sentence in front of a  source text and train the model with those sentences  to see whether the model could extract those sen-  tences accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Finally, we extract any 3 tokens  from a sentence and train the model, to see whether  the model can extract that sentence accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' For  knowledge-enhance document summarization, we  use OpenIE to first extract all the triadic relations  contained in the golden summary, then perform  NER on the golden summary to pick out all en-  tities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Adding the triplet containing the entities  we care about in front of the source text, we have  a training data Xrel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The overhead view of our  proposed method is shown as Figure 1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='3  Metrics  A common metric used in natural language process-  ing is ROUGE, which scores the generated sum-  maries based on the overlap of n-grams (Lin, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  This is effective when evaluating the similarity be-  tween the generated summary and the golden sum-  mary, but it is hard to measure how well a sentence  agrees with the facts of the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' As shown in  Table 1, perhaps the generated summary and the  golden summary have a high degree of overlap in  words, this generated summary is not necessarily  consistent with the original facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' To overcome this  issue, in addition to ROUGE, we also use CoCo  (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' CoCo is a new metric based on  positional scores to evaluate the factual consistency  between the summary and the original text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The  basic idea is to first detect the keywords in the sum-  mary, mask them in the original text to generate  masked document X  ′, and then use X  ′ and the  original X to calculate the positional score of each  ref:  Hong Kong-based airline Cathay  Pacific has announced a ban on ship-  ments of shark fin in a move that has  been welcomed by conservationists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  gen: Cathay Pacific is to ban shark fin  shipments to Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Table 1: A comparison between a golden summary  and a generated summary (without relation added).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Al-  though the ROUGE score for the generated summary  may be high, these two sentences are actually telling  different stories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  word in the summary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', the score of each word  appearing in a specific position in the summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' If  the positional score of a word appearing at a certain  position in the summary under X is significantly  higher than the positional score of the word appear-  ing at that position under X  ′, then we assume that  the word at that position in the summary is more in-  fluenced by the original text, the fact related to this  word is more likely consistent with the fact of the  original text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Conversely, if the difference between  the positional score of a word in the summary un-  der X and X  ′ is not significant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', whether the  original text is X or X  ′ has little effect on the posi-  tional score of the word, then we consider that the  word is not strongly dependent on the original text,  the presence of the word in this position is likely  to cause the summary to be factually inconsistent  (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' When using CoCo we use POS  or NER to pick the important words to be masked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  For scoring model, we use BART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' We will further  discuss CoCo in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  4  Experiment  In experiments, we first examine the properties of  prefix-tuning through a couple of preliminary ex-  periments, then conduct knowledge-enhance docu-  ment summarization on this basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Below we first  introduce the datasets and baselines we use in our  experiments, and then present all the experiments  in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='1  Dataset  CNN/Daily Mail: CNN/Daily Mail is a dataset  proposed in 2016 for abstractive summarization  tasks (Nallapati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The source of each  data is a long article and the target is the cor-  responding summary crawled by Nallapati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  The CNN/Daily Mail dataset contains  286,817 training data, 13,368 validation data and  11,487 test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The average length of source  is 766 words, and target is usually three to four  sentences, with an average length of 53 words (Nal-  lapati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Since the GPT-2 model has  a limit on the input length and the parameters re-  quired to train for prefix-tuning are not many, here  we only use the first 50,000 data where the sum  of the length of source and target does not exceed  800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' If the added relation is very long, then we  will further reduce the data usage according to the  length of the new text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  XSum: XSum (Narayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2018) is another  commonly used dataset in the field of document  summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The full name of it is extreme sum-  marization, and its source is BBC’s online articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  It contains 204,045 training data, 11,332 validation  data and 11,334 test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' As its name indicates,  XSum seeks to summarize an article with a very  short sentence, so its target is shorter than that of  CNN/Daily Mail, and also more abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The av-  erage length of XSum’s sources is 431 words, but  the average length of its targets is only about 23  words (Narayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The abstract nature  of XSum makes the abstractive summarization task  more challenging, so getting good results on XSum  can also make a model more convincing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' As same  as CNN/Daily Mail, we only use the first 50,000  data with the sum of the length of source and target  less than 800 for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='2  Baseline  To evaluate the performance of the model, we used  two metrics in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' ROUGE to eval-  uate the degree of overlap between the generated  summary and the golden summary, and CoCo to  evaluate the factual consistency of the generated  summary with the original text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Since there are  now many models trained specifically for document  summarization, it is difficult for GPT-2 to surpass  the performance of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Therefore, when com-  paring ROUGE scores we only compare with the  ROUGE scores reported by GPT-2 obtained when  using TL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Given that prefix-tuning is far more  expressive than this one hard prompt, the scores  for prefix-tuning should at least exceed those of  GPT-2, and if relations are added to the text, then  theoretically the summaries generated by modified  text should have higher ROUGE scores than if no  relations are added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  CoCo, on the other hand, is not a commonly used  metric like Rouge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' It is rare to see articles that use  CoCo to evaluate the factual consistency of a model  on summarization tasks, so for the CoCo score, we  use the summaries generated by the model with-  out the addition of relations as the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The  CoCo score with the addition of relations should  be higher than the CoCo score when there is no  relation added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='3  Preliminary Experiment  Preliminary experiment is of two parts, the first is  abstractive summarization, and the second part  is sentence extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The experimental results  show that prefix-tuning is effective for document  summarization task on GPT-2, and it is good at  extracting information from structured content in  texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' This properties of prefix-tuning fit well with  our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='1  Abstractive summarization  Abstractive summarization is a preliminary experi-  ment to examine the performance of prefix tuning  for document summarization on GPT-2 without the  addition of relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The results can be used as a  baseline for knowledge-enhanced document sum-  marization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  CNN/Daily Mail: When using prefix-tuning for  document summarization task on CNN/Daily Mail,  the ROUGE-1 is only about 20 if we only use a  prefix of length 5, but if a longer prefix is used,  then the performance of the model immediately  improves significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' If we use a prefix of length  100 and allow the model to generate summaries  with a maximum length of 100, then the ROUGE-1  for summaries generated on CNN/Daily Mail using  GPT-2 can reach 30, which is consistent with the  conclusion reported in Li and Liang (2021) that the  longer the prefix the more expressive it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' For the  part of the data we use, the ROUGE scores exceed  those reported by GPT-2 for document summariza-  tion on CNN/Daily Mail using TL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='DR, see Table  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  ROUGE-1  ROUGE-2  ROUGE-L  TL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='DR 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='34  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='27  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='58  Prefix  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='89  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='58  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='78  Table 2: Results of TL:DR and Prefix-tuning  However, we found a problem in the generated  summaries of CNN/Daily Mail using prefix-tuning  that the original sentences from the source texts  often appear in the summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Since there are  many studies reporting a significant difference in  the level of abstraction between the summaries of  CNN/Daily Mail generated by a model and those of  XSum (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', 2020), we con-  ducted another abstractive summarization experi-  ment using XSum in order to determine whether  prefix-tuning causes sentences and words from the  original text to appear in the summaries, or whether  the characteristics of the training data make the  model behave this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  XSum: As described above, the golden summaries  of the XSum dataset are very concise and refined,  which poses a higher challenge to a pre-trained lan-  guage model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' If a 100-length prefix is used and  the model is allowed to generate summaries with  a maximum length of 100, the ROUGE-1 of the  summaries generated by GPT-2 is very low, only  about 20%, but the good news is that the gener-  ated summaries are indeed more abstract, and al-  most no original sentences from the original text  appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' This experiment shows that the appear-  ance of the original sentence of the article in the  summaries is a result of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' This does not  happen if a very abstract dataset is replaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' In  addition, since prefix-tuning can perfectly pick out  the key information in data2text task as shown in  (Li and Liang, 2021) and identifies the range of sen-  tences in CNN/Daily Mail without any processing  such as mask on the training data, we found that  prefix-tuning may be very sensitive to the situation  that target overlaps with source and can accurately  recognize the structured information with certain  pattern from source by only a small amount of  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' We carried out the preliminary experi-  ment of sentence extraction with this hypothesis,  explored whether prefix-tuning can actually iden-  tify sentence ranges and extract sentences from the  sources as we expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Another  interesting  finding  in  abstractive  summarization is that by taking the prefixes trained  on CNN/Daily Mail for abstractive summarization  and having the model generate summaries of  XSum, the model also exhibits similar behavior  to generating summaries of CNN/Daily Mail,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', many times it generates sentences from the  original text rather than more abstract sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  This shows that trained prefix prompts with certain  properties can be transferred to other datasets for  use, and allows texts generated using other datasets  to exhibit similar properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='2  Sentence extraction  Sentence extraction is to further explore the proper-  ties of prefix tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Specifically, this preliminary  experiment uses the method proposed in methodol-  ogy to test whether prefix-tuning can truly identify  the information we want to extract accurately, espe-  cially their boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The experiments demon-  strate that, When experimenting with CNN/Daily  Mail for sentence extraction, if the model is trained  directly with the first sentence of source without  any guidance (SenEx1), the model can extract that  sentence almost perfectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' If the first three tokens  of a sentence are added to the front of the source  and then the model is trained (SenEx2), then the  model can extract that sentence most of the time,  but with reduced accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' If any three tokens from  a sentence are added to the source and the model is  trained with that sentence (SenEx3), the model can  still extract that sentence, but the ROUGE-1 score  drops to about 65%, as Table 3 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  ROUGE-1  ROUGE-2  ROUGE-L  SenEx1 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='98  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='79  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='98  SenEx2 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='56  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='64  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='46  SenEx3 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='50  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='38  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='25  Table 3: Results of Extracting Sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The tokens  we use here are tokens after byte pair encoding of the  original text, so it is not guaranteed to be three com-  plete words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  It is worth noting that, first of all, the extraction  is exact to the token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' For example, the periods ‘.’  and ‘ .’ (with a space) are two tokens in GPT-2  embedding, then if we take the first period ’.’ as the  criterion for extracting a sentence, the generated  summary will definitely be bounded by the first  period, and will never stop at the second kind of  period ‘ .’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Second, if the model does not correctly  extract the sentence we want to extract, the result  is still necessarily a sentence from the original text  and will not be generated arbitrarily, so we believe  that when using any 3 tokens of a sentence for  extraction, the model is still able to recognize the  boundaries, even though in many cases the model  does not correctly identify the sentence we are  extracting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' As long as we provide more special  tokens (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', tokens of low-frequency words instead  of tokens of conjunctions or punctuations), the  model will be able to identify which sentence we  are extracting and extract that sentence precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  The above preliminary experiments show that  prefix-tuning can indeed extract key information  directly from the original text, and its extraction is  so precise that it can identify the patterns of struc-  tured information in the text, extract the important  information with token precision, and filter out the  frames that contain this information in the origi-  nal text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Therefore, theoretically we could add the  knowledge we care about to the text in this way,  and then let the text combine the added knowledge  to generate a factually consistent summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' This is  what knowledge-enhanced document summariza-  tion does below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='4  Knowledge-enhanced document  summarization  We extracted the relations in the targets of  CNN/Daily Mail and XSum respectively as de-  scribed in the methodology section, and then added  the selected relations to the original text and trained  prefixes for summary generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The results show  that the dataset with the added relations do generate  summaries that are more relevant to the topic of  the relations we added, and most of the generated  summaries maintain factual consistency with the  added relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' However, there are still some cases  where this solution can go wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' We show some  examples of correct and incorrect generation in the  next page, and each of these examples is analyzed  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='1  CNN/Daily Mail  We first conducted experiments using CNN/Daily  Mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Here we only select the sentence that the  relation best meets our requirements from the three  to four sentences of a target, instead of using the  whole target for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' By analyzing the first ten  cases of CNN/Daily Mail and comparing the sum-  maries generated with and without the addition of  relation, we found that when a) the added relation  covers thoroughly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' b) there is a reference sentence  in the original text, then the generated summary  will be correct, as shown in Tables 4(a) and 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  However, the added relations only ensure that  the generated summary is consistent with the facts  described by the relation, and if the summary is not  generated according to the added relations or the  generated summary is beyond the coverage of the  added relation, inconsistency may occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' This is  manifested in a) the generated summary does not  refer to the added relation at all, as shown in Ta-  ble 4(c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' b) the generated summary extracts part of  the content of the added relation, but this changes  the correct summary, resulting in factual inconsis-  tency, as shown in Table 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' c) the summary  completely contains the keywords of the added re-  lation, but factual inconsistency occurs beyond the  added relation, as Table 4(e) shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Since the model tends to refer to the original text  or extract the exact sentences from the original text  when generating summaries with CNN/Daily Mail,  the effect of this approach is likely to be amplified  or obscured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' To examine the performance of this  approach on a more abstract dataset, we conducted  another experiment on XSum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='2  XSum  Overall, the results of generation using prefix-  tuning on XSum with added relations are similar  to CNN/Daily Mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' However, since the results  of GPT-2 for summarization on XSum are orig-  inally not good, XSum with added relations im-  proves the results to a larger extent compared to the  dataset without added relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' It is experimen-  tally demonstrated that the generated summaries  can be correct without referring to the original  text, as shown in Table 4(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' However, similar  to CNN/Daily Mail, there is a possibility of error in  the parts that the added relations are not included  to, such as time, place, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' But overall,  the addition of relations has helped the model to  perform better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='5  Evaluation  In this subsection we score the above generated  results and analyze the scoring results in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  As mentioned above, the CoCo score for the fac-  tual consistency of the summary with the original  text is based on the positional scores of important  words in the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Specifically, if the original text  is ‘Biden is the president of the United States.’,  and the golden summary is the same as the origi-  nal text, also ‘Biden is the president of the United  States.’, but the generated text is ‘Obama is the pres-  ident of the United States.’, then CoCo will first  use POS or NER to pick out the important words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Suppose we use NER and keep all the ‘PERSON’,  ‘TITLE’ and ‘COUNTRY’ in the results, then the  keywords picked from reference will be ‘Biden’,  ‘president’, ‘United’, ‘States’, and the keywords  picked from generation will be ‘Obama’, ‘presi-  dent’, ‘United’, ‘States’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Masking off the corre-  sponding word in the original text X, we get X′  ref:  relation:  Key relation: ‘subject’: ‘ Sally Forrest ’, ‘relation’: ‘ died on ’, ‘object’: ‘ March 15 ’  reference:  Sally Forrest , an actress-dancer who graced the silver screen throughout the ’40s and  ’50s in MGM musicals and films died on March 15 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  X gen:  Actress: Sally Forrest was in the 1951 Ida Lupino-directed film ‘Hard, Fast and Beauti-  ful’  Xrel gen:  Sally Forrest died on March 15 at her home in Beverly Hills, California.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  (a) A correct case from CNN/Daily Mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The second half of the sentence is the original article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  relation:  Key relation: ‘subject’: ‘ Prince Harry ’, ‘relation’: ‘ is in ’, ‘object’: "  attendance for England ’s crunch match against France "  reference:  Prince Harry in attendance for England’s crunch match against France .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  X gen:  England beat France 55-35 in ‘Le Crunch’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Xrel gen:  Prince Harry in attendance for England’s crunch match against France .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  (b) A correct case from CNN/Daily Mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Added relation covers through the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  relation:  Key relation: ‘subject’: ‘ valuable stock ’, ‘relation’: ‘ taken from ’, ‘object’: ‘  his antiques shop in Basingstoke ’  reference:  Discovered valuable stock taken from his antiques shop in Basingstoke .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  X gen:  Alan Stone, 51, arrested on suspicion of theft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Xrel gen:  The father-of-four admitted he had a ‘lump in his throat’  (c) An incorrect case from CNN/Daily Mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The generated summary does not follow the added relation at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  relation:  Key relation: ‘subject’: ‘ 1,000 pieces ’, ‘relation’: ‘ is in ’, ‘object’: ‘ last two years ’  reference:  Has inked 1,000 pieces of art on leaves in last two years .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  X gen:  Teacher Wang Lian has drawn hundreds of doodles on leaves for the last 10 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Xrel gen:  Teacher Wang Lian has drawn hundreds of doodles on leaves for the last two years .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  (d) An incorrect case from CNN/Daily Mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Added relation wrongly affected the model and causes errors  relation:  Key relation: ‘subject’: ‘ could first preventive tool ’, ‘relation’: ‘ is in ’, ‘object’: ‘  history ’  reference:  WHO leader: This vaccine could be "the first preventive tool against Ebola in  history "  X gen:  A vaccine will be tested in a subsequent study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Xrel gen:  The trial could be the first preventive tool in history .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  (e) An incorrect case from CNN/Daily Mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Error occurs beyond the added relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  relation:  Key relation: ‘subject’: ‘ Former Premier League footballer Sam Sodje ’, ‘relation’: ‘  has appeared alongside ’, ‘object’: ‘ three brothers accused ’  reference:  Former Premier League footballer Sam Sodje has appeared in court alongside  three brothers accused of charity fraud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  X gen:  A former Leeds United defender has been charged with conspiracy to commit fraud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Xrel gen:  Former Premier League footballer Sam Sodje has appeared alongside  three brothers accused of fraud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  (f) A correct case from XSum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The word ‘fraudulent’ appears in the original, but ‘fraud’ does not  Table 4: Analysis  (a) CoCo score for the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Score_X and Score_X’ are the positional scores of ‘B’, ‘president’, ‘United’,  ‘States’ on X and X’  (b) CoCo score for the generated summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Score_X and Score_X’ are the positional scores of ‘Obama’, ‘presi-  dent’, ‘United’, ‘States’ on X and X’  Figure 2: Calculation of CoCo score  ’<mask> is the <mask> of the <mask> <mask>.’  and X′  gen ‘Biden is the <mask> of the <mask>  <mask>.’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' This allows us to score reference and  generated summary using X, X′  ref and X′  gen re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' When they are put into the scoring  model BART for scoring, BART will perform byte  pair encoding (BPE) to these sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Regard-  less of how BPE splits a word, if we only take the  first token for each important word, then assuming  that there are N key words picked out, the CoCo  score is CoCo = (Score_X − Score_X′)/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  For example, if we want to calculate the CoCo  score for the reference, the process can be illus-  trated as Figure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The same is true for calcu-  lating the score of generated summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Here we  can see that since the word ‘Biden’ appears in X,  the Score_X for the first token of that word is  very high, while in X′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=', where the first word is  ‘<mask>’, the score for ‘B’ is considerably lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Thus, the larger the difference between Score_X  and Score_X′, the more likely the sentence is fac-  tually consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' On the contrary, for the score of  generated summary, since the two positional scores  for ’Obama’ against ’Biden’ are equivalently small,  the Score_X − Score_X′ will be small and the  CoCo score will be low, which means the sentence  is likely to be factually inconsistent with the origi-  nal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  The Pearson correlation and Spearman corre-  lation between CoCo and human judgments of  factual consistency on document summarization  task is higher than that of other matrics (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=',  2021), indicating that it does provide a better mea-  sure than other matrics of how well a sentence is  factually consistent with the original text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' How-  ever, we found that it is not suitable for compar-  ing factual consistency of golden summaries and  generated summaries, because CoCo relies on the  positional scores of the words in a sentence, but  the summaries generated from the original texts  X are often more likely to have words from the  original text than the golden summaries, which re-  sults in higher CoCo scores for the generated sum-  maries and lower scores for the golden summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  And since our relations are drawn from the golden  summaries, this results in the summaries generated  from texts with added relations Xrel being closer  to the golden summaries, thereby making the CoCo  scores lower than those summaries generated from  the original texts X, despite the summaries gener-  ated from Xrel are often more consistent with the  facts of the original text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' This problem is more ob-  vious on CNN/Daily Mail, because the summaries  generated on X are mostly the original text, while  the summaries generated on Xrel are closer to the  references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Although the summaries generated on  XSum with and without added relations are sim-  ilarly abstract, the CoCo score without added re-  lations is still slightly higher than the CoCo score  with added relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Nonetheless, for ROUGE,  the ROUGE score of the summaries generated with  the added relations is indisputably higher than the  ROUGE score of the summaries generated without  the added relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='CoCo and ROUGE scores are  shown in Table 5 and Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  X: [\'B\',"iden\',"is\',\'the\',\'president\'," of\'," the\',"United\',\'" States\',"\']  Score_X: [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='5603, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='9279,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='8908,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='9569]  X\': [<mask>\',"is\',"the\',\'<mask>",of","the\',\'<mask>",\'<mask>\',\'.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content="']  Score_X': [0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='0023, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='0346, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='0291,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='8516]  [\'B\',"iden\',\' is\',\'the\',\' president\',\'of\',\'the\',\' United\',\' States\',""\']  CoCo Score=(Score_X - Score_X\')/4=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='6046X: [\'B\',"iden\',"is\',\'the\',\'president\'," of\'," the\',"United\',\'" States\',"\']  Score_X: [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='0017, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='3789,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='8725,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='9508]  X\':[\'B\',"iden\',"is\',"the\',"<mask>",\'of",\'the\',\'<mask>",\'<mask>\',\'\']  Score_X\': [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='0007,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='0282,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='6997,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='9534]  [\'Obama\',\'is\',\' the\',\' president\',\'of\'," the\',\'United\'," States\',"\']  CoCo Score=(Score_X - Score_X\')/4=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='1305Only Noun  Noun+Verb  X  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='0264  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
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+page_content='0063  (b) XSum  Table 5: CoCo scores  ROUGE-1  ROUGE-2  ROUGE-L  X  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='07  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='13  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='46  Xrel  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
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+page_content='44  (a) CNN/Daily Mail  ROUGE-1  ROUGE-2  ROUGE-L  X  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='77  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='06  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='73  Xrel  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='69  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='06  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='01  (b) XSum  Table 6: ROUGE scores  5  Conclusion  This paper explores the properties of prefix-tuning,  proposes a knowledge-enhanced document summa-  rization method that combines prefix-tuning and  natural language prompts, and makes a significant  impact on the summaries generated using GPT-2  for both CNN/Daily Mail and XSum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' However,  as discussed in the paper, this approach still has  many shortcomings and can be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' For ex-  ample, the relations we add to the source are only  the relations extracted by OpenIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' The extracted  content may not be the relations we care about,  and the relations are often not a good representa-  tion of the relationship of two entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' In addition,  we consider that OpenIE many times extracts too  long content, so that the original texts expose too  much information to the model, which may have  an excessive impact on the summary generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  To deal with these two issues, in the future, instead  of adding relations extracted by OpenIE, we can  add existing knowledge in knowledge bases to the  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' This ensures the cleanliness and purity of  the information and avoids the influence of low-  quality data on the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Furthermore, since the  maximum text length that GPT-2 can handle is only  1024, this not only limits the amount of training  data that can be used, but also indirectly limits the  length of prefixes that can be attempted and the  length of summaries that are allowed to be gen-  erated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Therefore, if GPT-2 can be replaced with  a model that can handle infinite length preferably  trained specifically for summarization task, then  perhaps we can generate longer summaries using  more training data with longer prefixes, thus fur-  ther improving the performance of this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  However, too long prefixes may lead to a decrease  but not an increase in the performance of the model  due to inadequate training, or too long prefixes may  dominate model generation instead of text, so how  long prefixes should be used in what models and on  what datasets is also a valuable direction for future  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  References  Gabor Angeli, Melvin Jose Johnson Premkumar, and  Christopher D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Manning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
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+page_content=' The Association for Computer Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Tom B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Brown, Benjamin Mann, Nick Ryder, Melanie  Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind  Neelakantan, Pranav Shyam, Girish Sastry, Amanda  Askell,  Sandhini Agarwal,  Ariel Herbert-Voss,  Gretchen Krueger, Tom Henighan, Rewon Child,  Aditya Ramesh, Daniel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Ziegler, Jeffrey Wu,  Clemens Winter, Christopher Hesse, Mark Chen,  Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin  Chess, Jack Clark, Christopher Berner, Sam Mc-  Candlish, Alec Radford, Ilya Sutskever, and Dario  Amodei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Language models are few-shot learn-  ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' In Advances in Neural Information Processing  Systems 33: Annual Conference on Neural Informa-  tion Processing Systems 2020, NeurIPS 2020, De-  cember 6-12, 2020, virtual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Jordan Clive, Kris Cao, and Marek Rei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Control  prefixes for parameter-efficient text generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Zhengbao Jiang, Frank F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Xu, Jun Araki, and Graham  Neubig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  How can we know what language  models know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Transactions of the Association for  Computational Linguistics, 8:423–438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Mike  Lewis,  Yinhan  Liu,  Naman  Goyal,  Mar-  jan Ghazvininejad, Abdelrahman Mohamed, Omer  Levy, Veselin Stoyanov, and Luke Zettlemoyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' BART: denoising sequence-to-sequence pre-  training for natural language generation, translation,  and comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' In Proceedings of the 58th An-  nual Meeting of the Association for Computational  Linguistics, ACL 2020, Online, July 5-10, 2020,  pages 7871–7880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Association for Computational  Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Xiang Lisa Li and Percy Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Prefix-tuning:  Optimizing continuous prompts for generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' In  Proceedings of the 59th Annual Meeting of the  Association for Computational Linguistics and the  11th International Joint Conference on Natural Lan-  guage Processing, ACL/IJCNLP 2021, (Volume 1:  Long Papers), Virtual Event, August 1-6, 2021,  pages 4582–4597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Association for Computational  Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Chin-Yew Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  ROUGE: A package for auto-  matic evaluation of summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' In Text Summariza-  tion Branches Out, pages 74–81, Barcelona, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Pengfei Liu, Weizhe Yuan, Jinlan Fu, Zhengbao Jiang,  Hiroaki Hayashi, and Graham Neubig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Pre-  train, prompt, and predict: A systematic survey of  prompting methods in natural language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Xiao Liu, Yanan Zheng, Zhengxiao Du, Ming Ding,  Yujie Qian, Zhilin Yang, and Jie Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Gpt  understands, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Ye Liu,  Yao Wan,  Lifang He,  Hao Peng,  and  Philip S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' 2021c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' KG-BART: knowledge graph-  augmented BART for generative commonsense rea-  soning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' In Thirty-Fifth AAAI Conference on Artifi-  cial Intelligence, AAAI 2021, Thirty-Third Confer-  ence on Innovative Applications of Artificial Intelli-  gence, IAAI 2021, The Eleventh Symposium on Ed-  ucational Advances in Artificial Intelligence, EAAI  2021, Virtual Event, February 2-9, 2021, pages  6418–6425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' AAAI Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Ramesh Nallapati, Bowen Zhou, Cícero Nogueira dos  Santos, Çaglar Gülçehre, and Bing Xiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Abstractive text summarization using sequence-to-  sequence rnns and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' In Proceedings of the  20th SIGNLL Conference on Computational Natural  Language Learning, CoNLL 2016, Berlin, Germany,  August 11-12, 2016, pages 280–290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' ACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Shashi Narayan, Shay B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Cohen, and Mirella Lapata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Don’t give me the details, just the summary!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  topic-aware convolutional neural networks for ex-  treme summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' In Proceedings of the 2018  Conference on Empirical Methods in Natural Lan-  guage Processing, Brussels, Belgium, October 31 -  November 4, 2018, pages 1797–1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Association  for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Fabio Petroni, Tim Rocktäschel, Sebastian Riedel,  Patrick S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Lewis, Anton Bakhtin, Yuxiang Wu,  and Alexander H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Miller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Language mod-  els as knowledge bases?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  In Proceedings of the  2019 Conference on Empirical Methods in Natu-  ral Language Processing and the 9th International  Joint Conference on Natural Language Processing,  EMNLP-IJCNLP 2019, Hong Kong, China, Novem-  ber 3-7, 2019, pages 2463–2473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Association for  Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Jing Qian, Li Dong, Yelong Shen, Furu Wei, and  Weizhu Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Controllable natural language  generation with contrastive prefixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  In Findings  of the Association for Computational Linguistics:  ACL 2022, Dublin, Ireland, May 22-27, 2022, pages  2912–2924.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Association for Computational Linguis-  tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Yutong Qu, Wei Emma Zhang, Jian Yang, Lingfei Wu,  and Jia Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Knowledge-aware document  summarization: A survey of knowledge, embedding  methods and architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Alec Radford, Karthik Narasimhan, Tim Salimans, and  Ilya Sutskever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Improving language under-  standing by generative pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob  Uszkoreit, Llion Jones, Aidan N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Gomez, Lukasz  Kaiser, and Illia Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Attention is all  you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Yuexiang Xie, Fei Sun, Yang Deng, Yaliang Li, and  Bolin Ding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Factual consistency evaluation  for text summarization via counterfactual estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  In Findings of the Association for Computational  Linguistics: EMNLP 2021, Virtual Event / Punta  Cana, Dominican Republic, 16-20 November, 2021,  pages 100–110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Association for Computational Lin-  guistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Jiacheng Xu, Shrey Desai, and Greg Durrett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Un-  derstanding neural abstractive summarization mod-  els via uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' In Proceedings of the 2020 Con-  ference on Empirical Methods in Natural Language  Processing, EMNLP 2020, Online, November 16-20,  2020, pages 6275–6281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Association for Computa-  tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Kexin Yang, Dayiheng Liu, Wenqiang Lei, Baosong  Yang, Mingfeng Xue, Boxing Chen, and Jun Xie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Tailor: A prompt-based approach to attribute-  based controlled text generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  Chenguang Zhu, William Hinthorn, Ruochen Xu,  Qingkai Zeng, Michael Zeng, Xuedong Huang, and  Meng Jiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Enhancing factual consistency  of abstractive summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content='  In Proceedings of  the 2021 Conference of the North American Chap-  ter of the Association for Computational Linguistics:  Human Language Technologies, NAACL-HLT 2021,  Online, June 6-11, 2021, pages 718–733.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
+page_content=' Associa-  tion for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf'}
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+Infrared behavior in tame two-field cosmological models
+Elena Mirela Babalic∗ and Calin Iuliu Lazaroiu
+Horia Hulubei National Institute of Physics and Nuclear Engineering, Department of Physics,
+Reactorului 30, Bucharest-Magurele, 077125, Romania
+E-mail: mbabalic@theory.nipne.ro, lcalin@theory.nipne.ro
+We discuss the behavior of cosmological curves and their first order infrared approximants near
+critical ends of the scalar manifold Σ and near interior critical points of the scalar potential
+for tame hyperbolizable two-field cosmological models by determining the universal forms of the
+asymptotic gradient flow of the classical effective potential with respect to the uniformizing metric
+near all these points and ends. We compare the asymptotic behavior of gradient flow curves with
+numerical results for cosmological curves.
+11th International Conference of the Balkan Physical Union (BPU11),
+28 August - 1 September 2022
+Belgrade, Serbia
+∗Speaker
+© Copyright owned by the author(s) under the terms of the Creative Commons
+Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
+https://pos.sissa.it/
+arXiv:2301.00217v1  [hep-th]  31 Dec 2022
+
+IR behavior in tame two-field cosmological models
+Elena Mirela Babalic
+1.
+Introduction
+Two-field cosmological models provide the simplest testing ground for multifield cosmological
+dynamics. The latter are of great importance for connecting cosmology with fundamental theories
+of gravity and matter, since the effective description of generic string and M-theory compactifi-
+cations contains many moduli fields. In particular, multifield models are crucial in cosmological
+applications of the swampland program [1–4], as pointed out for example in [5–7], and may also
+allow for a unified description of inflation, dark matter and dark energy [8].
+A two-field cosmological model is parameterized by a connected borderless smooth surface Σ
+(the target manifold of the scalar fields) endowed with a Riemannian metric G and a scalar potential
+Φ which is a smooth function defined on Σ. We assume that Φ is positive everywhere. In [9],
+we used a dynamical RG (renormaization group) flow analysis and the uniformization theorem of
+Poincaré to show that two-field models whose scalar field metric G has constant Gaussian curvature
+𝐾 equal to −1, 0 or +1 give distinguished representatives for the IR (infrared) universality classes
+of all two-field cosmological models. Hyperbolizable two-field models (which are defined as those
+models for which 𝐾 = −1) comprise all two-field models whose target is of general type as well as
+those models whose target is exceptional (i.e diffeomorphic with R2, the annulus A2 or the Möbius
+strip M2) and for which the metric belongs to a hyperbolizable conformal class. The uniformized
+form of a hyperbolizable model is a two-field generalized 𝛼-attractor model in the sense of [10].
+Some aspects of such models were investigated previously in [11–17] (see [18–21] for brief reviews).
+The infrared behavior of a tractable class of hyperbolizable two field models was studied in
+[22], work which we summarize here together with a brief announcement of further results. We
+will assume that the target manifold Σ is oriented and topologically finite in the sense that it has
+finitely-generated fundamental group. When Σ is non-compact, this condition insures that it has a
+finite number of Freudenthal ends [23] and that its end compactification �Σ is a smooth and oriented
+compact surface. Thus Σ is recovered from �Σ by removing a finite number of points. We also
+assume that the scalar potential Φ admits a smooth extension ˆΦ to �Σ which is a strictly-positive
+Morse function defined on �Σ. A two-field cosmological model is called tame when these conditions
+are satisfied. Thus tame hyperbolizable two-field cosmological models are those classical two-
+field models whose scalar manifold is a connected, oriented and topologically finite hyperbolizable
+Riemann surface (Σ, G) and whose scalar potential Φ admits a positive and Morse extension to the
+end compactification of Σ.
+Notations and conventions.
+All surfaces Σ considered here are connected, smooth, Hausdorff
+and paracompact. If 𝑉 is a smooth real-valued function defined on Σ, we denote by:
+Crit𝑉
+def.
+= {𝑐 ∈ Σ|(d𝑉)(𝑐) = 0}
+the set of its critical points. For any 𝑐 ∈ Crit𝑉, we denote by Hess(𝑉)(𝑐) the Hessian of 𝑉 at 𝑐,
+which is a well-defined and coordinate independent symmetric bilinear form on the tangent space
+𝑇𝑐Σ. Given a metric G on Σ, we define the covariant Hessian tensor of 𝑉 relative to G by:
+HessG(𝑉)
+def.
+= ∇d𝑉 ,
+2
+
+IR behavior in tame two-field cosmological models
+Elena Mirela Babalic
+where ∇ is the Levi-Civita connection of G.
+This symmetric tensor has the following local
+expression in coordinates (𝑥1, 𝑥2) on Σ:
+HessG(𝑉) = (𝜕𝑖𝜕𝑗 − Γ𝑘
+𝑖 𝑗(𝑥)𝜕𝑘)𝑉d𝑥𝑖 ⊗ d𝑥 𝑗 ,
+where Γ𝑘
+𝑖 𝑗(𝑥) are the Christoffel symbols of G.
+Recall that a critical point 𝑐 of 𝑉 is called
+nondegenerate if HessG(𝑉)(𝑐) is a non-degenerate bilinear form. When 𝑉 is a Morse function
+(i.e.
+has only non-degenerate critical points), the set Crit𝑉 is discrete.
+We denote by �Σ the
+end compactification of Σ, which is a compact Hausdorff topological space containing Σ. In the
+topologically finite case, the surface Σ has a finite number of Freudenthal ends and �Σ is a smooth
+compact surface. In this situation, we say that 𝑉 is globally well-behaved on Σ if it admits a smooth
+extension �𝑉 to �Σ. A metric G on Σ is called hyperbolic if it is complete and of constant Gaussian
+curvature 𝐾 = −1.
+2.
+Hyperbolizable two-field models
+Let us recall the global description of two-field cosmological models through a second order
+geometric ODE and the first order infrared approximation introduced in [9]. Such a model is
+parameterized by the rescaled Planck mass 𝑀0
+def.
+= 𝑀
+√︃
+2
+3 (where 𝑀 is the reduced Planck mass)
+and by its scalar triple (Σ, G, Φ), where Σ is the target manifold for the scalar fields (a generally
+non-compact borderless connected surface), G is the scalar field metric and Φ is the scalar potential.
+To ensure conservation of energy, one requires that G is complete. For simplicity, we also assume
+that Φ is strictly positive.
+When neglecting fluctuations, the scalar field 𝜑 : R → Σ coupled to the metric of a Friedmann-
+Lemaître-Robertson-Walker (FLRW) space satisfies the cosmological equation (see (1.4) in [22]):
+∇𝑡 �𝜑(𝑡) + 1
+𝑀0
+�
+|| �𝜑(𝑡)||2
+G + 2Φ(𝜑(𝑡))
+�1/2
+�𝜑(𝑡) + (gradGΦ)(𝜑(𝑡)) = 0 .
+(1)
+Proposition 2.1. The IR behavior (in the sense of [9]) of the cosmological flow of a two-field model
+with scalar triple (Σ, G, Φ) and rescaled Planck mass 𝑀0 is described by the gradient flow of the
+scalar triple (Σ, 𝐺,𝑉), where 𝐺 is the uniformizing metric of G and 𝑉
+def.
+= 𝑀0
+√
+2Φ is the classical
+effective scalar potential of the model.
+Hence the IR behavior is described by the gradient flow equation:
+�𝜑IR(𝑡) + (grad𝐺𝑉)(𝜑IR(𝑡)) = 0 .
+(2)
+The first order IR approximants of cosmological orbits for the model (𝑀0, Σ, G, Φ) coincide with
+those of the model (𝑀0, Σ, 𝐺, Φ). Moreover, these approximants coincide with the gradient flow
+orbits of (Σ, 𝐺,𝑉). In particular, the IR universality classes defined in [9] depend only on the scalar
+triple (Σ, 𝐺,𝑉). This allows for systematic studies of two-field cosmological models belonging to a
+fixed IR universality class by using the infrared expansion of cosmological curves, the first order of
+which is given by the gradient flow of the classical effective potential 𝑉 on the geometrically finite
+hyperbolic surface (Σ, 𝐺). Since the future limit points of cosmological curves and of the gradient
+flow curves of (Σ, 𝐺,𝑉) are critical points of Φ or Freudenthal ends of Σ, the asymptotic behavior
+of such curves for late cosmological times is determined by the form of 𝐺 and 𝑉 near such points.
+3
+
+IR behavior in tame two-field cosmological models
+Elena Mirela Babalic
+2.1 The hyperbolic metric 𝐺 in the vicinity of an end
+In this subsection, we recall the form of the hyperbolic metric 𝐺 in a canonical vicinity of an end
+and extract its asymptotic behavior near each type of end.
+Any end e of Σ admits an open neighborhood 𝑈e ⊂ �Σ diffeomorphic with a disk such that there
+exist semigeodesic polar coordinates (𝑟, 𝜃) ∈ R>0 × S1 defined on �𝑈e
+def.
+= 𝑈e \ {e} ⊂ Σ in which
+the metric 𝐺 has the canonical form:
+d𝑠2
+𝐺| �𝑈e = d𝑟2 + 𝑓e(𝑟)d𝜃2 ,
+𝑓e(𝑟) =
+�������
+�������
+sinh2(𝑟)
+if e = plane end
+1
+(2𝜋)2 𝑒2𝑟
+if e = horn end
+ℓ2
+(2𝜋)2 cosh2(𝑟)
+if e = funnel end of circumference ℓ > 0
+1
+(2𝜋)2 𝑒−2𝑟
+if e = cusp end
+.
+The end corresponds to 𝑟 →∞. Setting 𝜔
+def.
+= 1
+𝑟 , the metric in canonical polar coordinates (𝜔, 𝜃) is:
+d𝑠2
+𝐺| �𝑈e = d𝜔2
+𝜔4 + 𝑓e(1/𝜔)d𝜃2 ,
+where:
+𝑓e(1/𝜔) = ˜𝑐e𝑒
+2𝜀e
+𝜔
+�
+1 + O
+�
+𝑒− 2
+𝜔
+��
+for 𝜔 → 0 ,
+(3)
+with:
+˜𝑐e =
+�������
+�������
+1
+4
+if e = plane end
+1
+(2𝜋)2
+if e = horn end
+ℓ2
+(4𝜋)2
+if e = funnel end of circumference ℓ > 0
+1
+(2𝜋)2
+if e = cusp end
+𝜀e =
+�
++1
+if e = flaring (i.e. plane, horn or funnel) end
+−1
+if e = cusp end
+The term O
+�
+𝑒− 2
+𝜔
+�
+in (3) vanishes identically when e is a cusp or horn end. In particular, the
+constants ˜𝑐e and 𝜖e determine the leading asymptotic behavior of the hyperbolic metric 𝐺 near e.
+The gradient flow equations of ( �𝑈e, 𝐺| �𝑈e,𝑉| �𝑈e) read:
+d𝜔
+d𝑞 = −(grad𝑉)𝜔 ≃ −𝜔4𝜕𝜔𝑉
+d𝜃
+d𝑞 = −(grad𝑉) 𝜃 ≃ − 1
+˜𝑐e
+𝑒− 2𝜖e
+𝜔 𝜕𝜃𝑉 .
+(4)
+We studied these equations in [22] for all ends of Σ. Below, we summarize the results for critical
+ends (see op. cit. for the noncritical ends).
+Recall that 𝑉 is globally well-behaved and �𝑉 is Morse on �Σ. Together with the formulas above,
+this implies that (grad𝐺𝑉)𝜔 tends to zero at all ends while (grad𝐺𝑉) 𝜃 tends to zero exponentially
+at flaring (i.e. non-cusp) ends and to infinity at cusp ends. On the other hand, we have:
+||grad𝐺𝑉||2 = ||d𝑉||2 = 1
+𝜔4 (𝜕𝜔𝑉)2 + 𝑓𝑒(1/𝜔)(𝜕𝜃𝑉)2 ≈ 1
+𝜔4 (𝜕𝜔𝑉)2 + ˜𝑐e𝑒
+2𝜖e
+𝜔 (𝜕𝜃𝑉)2 .
+Thus ||grad𝐺𝑉|| tends to infinity at all ends.
+4
+
+IR behavior in tame two-field cosmological models
+Elena Mirela Babalic
+2.2 Principal canonical coordinates centered at an end e
+Definition 2.2. A canonical Cartesian coordinate system (𝑥, 𝑦) for (Σ, 𝐺) centered at the critical
+end e is called principal for 𝑉 if the tangent vectors 𝜖𝑥 =
+𝜕
+𝜕𝑥
+��
+e and 𝜖𝑦 =
+𝜕
+𝜕𝑦
+��
+e form a principal basis
+for 𝑉 at e.
+Canonical Cartesian coordinates (𝑥, 𝑦) centered at e are given by:
+𝑥 = 𝜔 cos 𝜃 = 1
+𝑟 cos 𝜃
+,
+𝑦 = 𝜔 sin 𝜃 = 1
+𝑟 sin 𝜃 .
+In such coordinates, the end e corresponds to 𝜔 = 0, i.e. (𝑥, 𝑦) = (0, 0). The Taylor expansion
+of �𝑉 at e in principal Cartesian coordinates (𝑥, 𝑦) centered at e and in associated polar coordinates
+(𝜔, 𝜃) reads:
+�𝑉e(𝑥, 𝑦) =
+�𝑉(𝑒) + 1
+2
+�
+𝜆1(e)𝑥2+𝜆2(e)𝑦2�
++ O((𝑥2 + 𝑦2)
+3
+2 )
+�𝑉e(𝜔, 𝜃) = �𝑉(e) + 1
+2𝜔2�
+𝜆1(e) cos2 𝜃+𝜆2(e) sin2 𝜃
+�
++ O(𝜔3) ,
+(5)
+where 𝜔 =
+√︁
+𝑥2 + 𝑦2, 𝜃 = arg(𝑥 + i𝑦) and the real numbers 𝜆1(e) and 𝜆2(e) are the principal values
+of the Hessian of �𝑉(e). When 𝜆1 and 𝜆2 do not both vanish, it is convenient to define:
+Definition 2.3. The critical modulus of (Σ,𝐺,𝑉) at the critical end e is the ratio:
+𝛽e
+def.
+= 𝜆1(e)
+𝜆2(e) ∈ [−1, 1] \ {0} ,
+where 𝜆1(e) and 𝜆2(e) are the principal values of (Σ, 𝐺,𝑉) at e.
+Definition 2.4. The characteristic signs of (Σ, 𝐺,𝑉) at e are:
+𝜀𝑖(e)
+def.
+= sign(𝜆𝑖(e)) ∈ {−1, 1} (𝑖 = 1, 2) .
+The extended scalar potential ˆΦ of the canonical model can be recovered from the extended
+classical effective potential as:
+ˆΦ =
+1
+2𝑀2
+0
+�𝑉2 ≈
+¯𝜆2(e)2
+2
+�
+ˆ¯𝑉(e) + 1
+2𝜔2(𝛽e cos2 𝜃 + sin2 𝜃)
+�2
+,
+(6)
+where we defined
+¯𝜆2(e)
+def.
+= 𝜆2(e)
+𝑀0
+,
+ˆ¯𝑉(e)
+def.
+=
+ˆ𝑉(e)
+𝜆2(e) .
+Solving the gradient flow equation (2) of 𝑉 relative to 𝐺 with the approximations (3) and (5) for
+𝜃 ∉ {0, 𝜋
+2 , 𝜋, 3𝜋
+2 } shows that the unoriented gradient flow orbits of 𝑉 around the end e have implicit
+equation:
+1
+4 [𝜆1(e) − 𝜆2(e)] Γ2
+�2𝜀e
+𝜔
+�
+= 𝐴 + ˜𝑐e [ 𝜆1(e) log | sin 𝜃| − 𝜆2(e) log | cos 𝜃| ] ,
+(7)
+where Γ2 is the lower incomplete Gamma function of order 2 and 𝐴 is an integration constant.
+Below, we compare graphically (making certain choices for 𝛽) the unoriented gradient flow
+orbits given implicitely by equation (7) to the orbits of IR optimal cosmological curves, defined as
+those solutions 𝜑 of the cosmological equation (1) which satisfy �𝜑(0) = −(grad𝐺𝑉)(𝜑(0)).
+5
+
+IR behavior in tame two-field cosmological models
+Elena Mirela Babalic
+2.3 The IR behavior near critical plane ends
+Figure 1 bellow gives the unoriented gradient flow orbits for certain choices of 𝛽, while Figure
+2 gives the numerically computed orbits of the IR optimal cosmological curves for the same choices
+of 𝛽 and various other assumptions mentioned in the description.
+-1.0
+-0.5
+0.0
+0.5
+1.0
+-1.0
+-0.5
+0.0
+0.5
+1.0
+x
+y
+(a) For 𝛽e = −1/2
+-1.0
+-0.5
+0.0
+0.5
+1.0
+-1.0
+-0.5
+0.0
+0.5
+1.0
+x
+y
+(b) For 𝛽e = 1/2
+Figure 1: Gradient flow orbits of 𝑉 (shown in brown) and level sets of 𝑉 (shown in green) near a critical
+plane end e, drawn in principal Cartesian canonical coordinates centered at e for two values of 𝛽e.
+-1.0
+-0.5
+0.0
+0.5
+1.0
+-1.0
+-0.5
+0.0
+0.5
+1.0
+x
+y
+(a) For 𝛽e = −0.5
+y
+-1.0
+-0.5
+0.0
+0.5
+1.0
+-1.0
+-0.5
+0.0
+0.5
+1.0
+(b) For 𝛽e = 0.5
+Figure 2: Numerically computed infrared optimal cosmological orbits of the canonical model (shown in
+brown) and level sets of ˆΦ (shown in green) near a critical plane end e, drawn in principal canonical Cartesian
+coordinates centered at e for two values of 𝛽e. We took ¯𝜆2(e) = 1, ¯ˆ𝑉(e) = 1 and 𝑀0 = 1. The initial point of
+each orbit, i.e. 𝜑(0), is shown as a black dot.
+6
+
+IR behavior in tame two-field cosmological models
+Elena Mirela Babalic
+2.4 The IR behavior near critical horn ends
+We graphically compare Figure 3 bellow, which gives the unoriented gradient flow orbits near
+critical horn ends, with Figure 4 which shows some numerically computed orbits of the IR optimal
+cosmological curves near critical horn ends. The comparison is done for certain choices of 𝛽.
+-1.0
+-0.5
+0.0
+0.5
+1.0
+-1.0
+-0.5
+0.0
+0.5
+1.0
+x
+y
+(a) For 𝛽e = −0.5.
+-1.0
+-0.5
+0.0
+0.5
+1.0
+-1.0
+-0.5
+0.0
+0.5
+1.0
+x
+y
+(b) For 𝛽e = 0.5.
+Figure 3: Gradient flow orbits of 𝑉 (shown in brown) and level sets of 𝑉 (shown in green) near a critical
+horn end e, drawn in principal Cartesian canonical coordinates centered at e for two values of 𝛽e.
+-0.4
+-0.2
+0.0
+0.2
+0.4
+-0.4
+-0.2
+0.0
+0.2
+0.4
+x
+y
+(a) For 𝛽e = −0.5.
+-0.4
+-0.2
+0.0
+0.2
+0.4
+-0.4
+-0.2
+0.0
+0.2
+0.4
+x
+y
+(b) For 𝛽e = 0.5.
+Figure 4: Numerically computed infrared optimal cosmological orbits of the canonical model (shown in
+brown) and level sets of ˆΦ (shown in green) near a critical horn end e, drawn in principal canonical Cartesian
+coordinates centered at e for two values of 𝛽e. We took ¯𝜆2(e) = 1, ¯ˆ𝑉(e) = 1 and 𝑀0 = 1. The initial point of
+each orbit is shown as a black dot.
+7
+
+IR behavior in tame two-field cosmological models
+Elena Mirela Babalic
+2.5 The IR behavior near critical funnel ends
+We visually compare below Figure 5, which gives the unoriented gradient flow orbits near
+critical funnel ends, with Figure 6, which shows some numerically computed orbits of the IR
+optimal cosmological curves near critical funnel ends.
+-1.0
+-0.5
+0.0
+0.5
+1.0
+-1.0
+-0.5
+0.0
+0.5
+1.0
+x
+y
+(a) For 𝛽e = −0.5.
+-1.0
+-0.5
+0.0
+0.5
+1.0
+-1.0
+-0.5
+0.0
+0.5
+1.0
+x
+y
+(b) For 𝛽e = 0.5.
+Figure 5: Gradient flow orbits of 𝑉 (shown in brown) and level sets of 𝑉 (shown in green) near a critical
+funnel end e of circumference ℓ = 1, drawn in principal Cartesian canonical coordinates centered at e for two
+values of 𝛽e.
+-0.3 -0.2 -0.1 0.0
+0.1
+0.2
+0.3
+-0.3
+-0.2
+-0.1
+0.0
+0.1
+0.2
+0.3
+x
+y
+(a) For 𝛽e = −0.5.
+-0.4
+-0.2
+0.0
+0.2
+0.4
+-0.4
+-0.2
+0.0
+0.2
+0.4
+x
+y
+(b) For 𝛽e = 0.5.
+Figure 6: Numerically computed infrared optimal cosmological orbits of the canonical model (shown in
+brown) and level sets of ˆΦ (shown in green) near a critical funnel end e of circumference ℓ = 1, drawn in
+principal canonical coordinates centered at e for two values of 𝛽e. We took ¯𝜆2(e) = 1, ¯ˆ𝑉(e) = 1 and 𝑀0 = 1.
+The initial point of each orbit 𝜑(0) is shown as a black dot.
+8
+
+IR behavior in tame two-field cosmological models
+Elena Mirela Babalic
+2.6 The IR behavior near critical cusp ends
+By graphically comparing Figure 7, which gives some unoriented gradient flow orbits near
+critical cusp ends, and Figure 8, which shows some numerically computed orbits of the IR optimal
+cosmological curves near critical cusp ends, one can assume that higher order corrections are needed
+in the IR expansion to get better approximants for the cusp.
+-4
+-2
+0
+2
+4
+-4
+-2
+0
+2
+4
+x
+y
+(a) For 𝛽e = −0.5.
+-4
+-2
+0
+2
+4
+-4
+-2
+0
+2
+4
+x
+y
+(b) For 𝛽e = 0.5.
+Figure 7: Gradient flow orbits of 𝑉 (shown in brown) and level sets of 𝑉 (shown in green) near a critical
+cusp end e, drawn in principal Cartesian canonical coordinates centered at e for two values of 𝛽e.
+-0.4 -0.2
+0.0
+0.2
+0.4
+-0.4
+-0.2
+0.0
+0.2
+0.4
+x
+y
+(a) For 𝛽e = −0.5.
+-0.4 -0.2
+0.0
+0.2
+0.4
+-0.4
+-0.2
+0.0
+0.2
+0.4
+x
+y
+(b) For 𝛽e = 0.5.
+Figure 8: Numerically computed infrared optimal cosmological orbits of the canonical model (shown in
+brown) and level sets of ˆΦ (shown in green) near a critical cusp end e, drawn in principal canonical coordinates
+centered at e for two values of 𝛽e. We took ¯𝜆2(e) = 1, ¯ˆ𝑉(e) = 1 and 𝑀0 = 1. The initial point of each orbit
+is shown as a black dot.
+9
+
+IR behavior in tame two-field cosmological models
+Elena Mirela Babalic
+2.7 The IR behavior near an interior critical point
+Let c be an interior critical point and (𝑥, 𝑦) be principal Cartesian canonical coordinates centered
+at c. We have the metric:
+d𝑠2
+𝐺 =
+4
+(1 − 𝜔2)2 [d𝜔2 + 𝜔2d𝜃2]
+and:
+𝑉(𝜔, 𝜃) = 𝑉(c) + 1
+2𝜔2 �
+𝜆1(c) cos2 𝜃 + 𝜆2(c) sin2 𝜃
+�
++ O(𝜔3) ,
+where 𝜔
+def.
+=
+√︁
+𝑥2 + 𝑦2 and 𝜃
+def.
+= arg(𝑥 + i𝑦). Thus:
+(grad𝑉)𝜔 ≈ (1 − 𝜔2)2
+4
+𝜕𝜔𝑉 = (1 − 𝜔2)2𝜔
+4
+[𝜆1(c) cos2 𝜃 + 𝜆2(c) sin2 𝜃] ,
+(grad𝑉) 𝜃 ≈ (1 − 𝜔2)2
+4𝜔2
+𝜕𝜃𝑉 = (1 − 𝜔2)2
+4
+[𝜆2(c)−𝜆1(c)] sin(𝜃) cos(𝜃) .
+(8)
+The critical modulus 𝛽c and characteristic signs 𝜖1(c) and 𝜖2(c) of (Σ, 𝐺,𝑉) at c are defined
+through:
+𝛽c
+def.
+= 𝜆1(c)
+𝜆2(c) ∈ [−1, 1] \ {0} , 𝜖𝑖(c)
+def.
+= sign(𝜆𝑖(c)) (𝑖 = 1, 2) .
+Distinguish the cases:
+1. 𝜆1(c) = 𝜆2(c) := 𝜆(c), i.e. 𝛽c = 1. Then 𝜖1(c) = 𝜖2(c) := 𝜖(c) and c is a local minimum of 𝑉
+when 𝜆(c) is positive (i.e. when 𝜖(c) = 1) and a local maximum of 𝑉 when 𝜆(c) is negative
+(i.e. when 𝜖(c) = −1). Relations (8) become:
+(grad𝑉)𝜔 ≈ (1 − 𝜔2)2𝜔
+4
+𝜆(c) , (grad𝑉) 𝜃 ≈ 0
+and the gradient flow equation of (Σ, 𝐺,𝑉) takes the following approximate form near c:
+d𝜔
+d𝑞 = − (1 − 𝜔2)2𝜔
+4
+𝜆(c) ,
+d𝜃
+d𝑞 = 0 .
+This gives 𝜃 = const, i.e. the gradient flow curves near c are approximated by straight lines
+through the origin when drawn in principal Cartesian canonical coordinates (𝑥, 𝑦) at c;
+2. 𝜆1(c) ≠ 𝜆2(c), i.e. 𝛽c ≠ 1. When 𝜃 ∈ {0, 𝜋
+2 , 𝜋, 3𝜋
+2 }, the gradient flow equation reduces to:
+d𝜔
+d𝑞 = (1 − 𝜔2)2𝜔
+4
+×
+�
+𝜆1(c)
+if 𝜃 ∈ {0, 𝜋}
+𝜆2(c)
+if 𝜃 ∈ { 𝜋
+2 , 3𝜋
+2 }
+d𝜃
+d𝑞 = 0 .
+This gives four gradient flow orbits which are approximated near c by the principal geodesic
+orbits. When 𝜃 ∉ {0, 𝜋
+2 , 𝜋, 3𝜋
+2 }, the gradient flow equation takes the form:
+(1 − 𝛽c) d𝜔
+d𝜃 = 𝜔(𝛽c cot 𝜃 + tan 𝜃) ,
+(9)
+with general solution:
+𝜔 = 𝐶 | sin(𝜃)|
+𝛽c
+1−𝛽c
+| cos(𝜃)|
+1
+1−𝛽c
+, 𝜃 ∉ {0, 𝜋
+2 , 𝜋, 3𝜋
+2 } , 𝐶 > 0 .
+(10)
+10
+
+IR behavior in tame two-field cosmological models
+Elena Mirela Babalic
+Below we compare graphically the effective gradient flow orbits given by solutions of equation
+(10), depicted in Figure 9, to the numerically computed orbits of IR optimal cosmological curves,
+represented in Figure 10.
+-0.4
+-0.2
+0.2
+0.4
+x
+-0.4
+-0.2
+0.2
+0.4
+y
+(a) 𝛽c =−0.5. Interior saddle point of 𝑉.
+-0.4
+-0.2
+0.2
+0.4
+x
+-0.4
+-0.2
+0.2
+0.4
+y
+(b) 𝛽c=0.5. Local extremum of 𝑉.
+Figure 9: Unoriented gradient flow orbits of 𝑉 (shown in brown) near an interior critical point superposed
+over the level lines of 𝑉 (shown in green) for two values of 𝛽c, plotted in principal Cartesian canonical
+coordinates centered at the critical point. The figure assumes 𝜔max(c) = 1/2. The principal coordinate axes
+correspond to the principal geodesic orbits at c, which coincide with four special gradient flow orbits.
+-0.4
+-0.2
+0.2
+0.4
+x
+-0.4
+-0.2
+0.2
+0.4
+y
+(a) Saddle point of 𝑉 for 𝛽c = −0.5. The dots are
+initial points 𝜑(0).
+-0.4
+-0.2
+0.2
+0.4
+x
+-0.4
+-0.2
+0.2
+0.4
+y
+(b) Local extremum of 𝑉 for 𝛽c= 0.5. The dots are
+initial points 𝜑(0).
+Figure 10: Numerically computed orbits of infrared optimal cosmological curves 𝜑 of the uniformized
+model (shown in brown) near an interior critical point c, superposed over the level lines of Φ (shown in
+green) for two values of 𝛽c. Here 𝑥, 𝑦 are principal Cartesian canonical coordinates centered at the critical
+point. We assume 𝜔max(c) = 1/2. The initial points 𝜑(0) of these curves are shown as black dots.
+11
+
+IR behavior in tame two-field cosmological models
+Elena Mirela Babalic
+3.
+Brief announcement of further results
+Cosmological curves of two field models can be approximated by mean field curves, using
+an approximation technique which is similar to the mean field approximation of condensed matter
+physics. Cosmological mean field approximations admits an elegant formulation using Ehresmann
+connections defined on the total space of the tangent bundle𝑇Σ, i.e. rank two distributions H ⊂ 𝑇𝑇Σ
+which are complementary to the vertical distribution V ⊂ 𝑇𝑇Σ of the fiber bundle 𝑇Σ → Σ. The
+corresponding mean field approximation replaces cosmological flow curves 𝛾 = �𝜑 : 𝐼 → 𝑇Σ of the
+model with the horizontal lift relative to H of curves in Σ which satisfy the so-called mean field
+curve equation defined by H. This amounts to treating as small the components of �𝜑 : 𝐼 → 𝑇𝑇M
+which are “orthogonal” to H – in a sense which can be made precise.
+The simplest approximations of this type are induced by the choice of a special coordinate
+system on an open subset of the tangent bundle 𝑇Σ, i.e.
+a coordinate system which naturally
+combines a system of coordinate on the base Σ with a coordinate system for the fibers of 𝑇Σ. In this
+case, the corresponding Ehresmann connection is flat and the mean field approximation amounts
+to neglecting the first time derivative of the two fiberwise coordinates, which are thereby being
+treated as “slow”. This parallels the logic of Born-Oppenheimer type approximations in quantum
+mechanics and statistical physics, which separate dynamical variables into “slow” and “fast” and
+treat the dynamics of slow variables approximately.
+In our situation, the first order system of four ODEs which describes the cosmological equation
+in a given special coordinate system is replaced by the algebro-differential system in which the
+time derivatives of the fiberwise coordinates are set to zero. This leads to algebraic consistency
+conditions for the special coordinates called mean field equations, which determine a mean field
+surface inside 𝑇Σ; in many cases, the latter is a semialgebraic multisection of 𝑇Σ defined on an
+open subset of the base Σ.
+Mean field approximations of this type provide a very general procedure for extracting approx-
+imants of cosmological curves in various regimes, where the regime of interest is defined by the
+choice of fiberwise coordinates that one wishes to treat as “slow”. Since fiberwise coordinates are
+pairs of basic observables of the cosmological system which are functionally independent on an
+open subset of 𝑇Σ, each such regime is determined by the choice of a pair of locally independent
+on-shell cosmological observables. As we show in forthcoming work, a careful study of natural
+on-shell observables of two-field cosmological models provides interesting candidates for such
+fiberwise coordinates on 𝑇Σ, thus leading to natural mean field approximation schemes which have
+direct physics significance. The latter can be applied to any two-field model and in particular to
+tame hyperbolizable models.
+One such mean field approximation is the so-called adapted approximation. This uses the
+fiberwise coordinates on 𝑇Σ which are given by the projections of �𝜑 on the direction of the vector
+grad𝐺Φ and on its positive normal direction and provides a mathematical refinement of the proposal
+of [24]. One can also consider the roll-turn approximation, which takes as fiberwise coordinates
+the on-shell second slow roll parameter and turning rate. Finally, one can consider the slow roll
+rate approximation, which uses the first and second on shell slow roll parameters as fiberwise
+coordinates.
+Another kind of approximation which can be considered for two-field cosmological models is
+12
+
+IR behavior in tame two-field cosmological models
+Elena Mirela Babalic
+the so-called angular approximation, which arises by neglecting the first two time derivatives of
+the radial variable in a given semigeodesic coordinate system (𝑟, 𝜃). This provides approximants
+on each semigeodesic coordinate patch, whose accuracy can be characterized theoretically and
+computed numerically.
+In forthcoming work, we study the approximations mentioned above for tame two-field cosmo-
+logical models and the corresponding error terms, which play an important role when ascertaining
+their accuracy (see [8]). In particular, this serves as a test of various proposals made previously
+in the literature. Moreover, we compare these approximations with the IR approximation studied
+in [22] (and summarized above) near interior critical points and near ends of Σ, determining the
+regimes within which the various approximations are accurate.
+4.
+Conclusions
+We studied the first order IR behavior of tame hyperbolizable two-field cosmological models
+by analyzing the asymptotic form of the gradient flow orbits of the classical effective scalar potential
+𝑉 with respect to the uniformizing metric 𝐺 near all interior critical points and ends of Σ. We
+showed that the IR behavior of tame hyperbolizable two field cosmological models is characterized
+by a finite set of parameters associated to their ends and interior critical points. Comparing with
+numerical computations, we found that the first order IR approximation is already quite good for
+all interior critical points and all ends except for cusps, for which one must consider higher order
+corrections in the IR expansion in order to obtain a good approximation. Our results characterize
+the IR universality classes of all tame hyperbolizable two-field models in terms of geometric data
+extracted from the asymptotic behavior of the effective scalar potential and uniformizing metric.
+Since the Morse assumption on the extended potential determines its asymptotic form near all
+points of interest on �Σ, we could derive closed form expressions for the asymptotic gradient flow
+which describes the corresponding infrared phases of such models in the sense of [9]. In particular,
+we found that the asymptotic gradient flow of (Σ, 𝐺,𝑉) near each end which is a critical point of the
+extended potential can be expressed using the incomplete gamma function of order two and certain
+constants which depend on the type of end under consideration and on the principal values of the
+extended effective potential �𝑉 at that end. We also found that flaring ends which are not critical
+points of �𝑉 act like fictitious but non-standard stationary points of the effective gradient flow. While
+the local form near the critical points of 𝑉 is standard (since they are hyperbolic stationary points
+[25, 26] of the cosmological and gradient flow), the asymptotic behavior near Freudenthal ends is
+exotic in that some of the ends act like fictitious stationary points with unusual characteristics.
+We compared these results with numerical computations of cosmological curves near the
+points of interest. We found particularly interesting behavior near cusp ends, around which generic
+cosmological trajectories tend to spiral a large number of times before either “falling into the cusp”
+or being “repelled” back toward the compact core of Σ along principal geodesic orbits determined
+by the classical effective potential 𝑉. In particular, cusp ends lead naturally to “fast turn” behavior
+of cosmological curves.
+13
+
+IR behavior in tame two-field cosmological models
+Elena Mirela Babalic
+References
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf,len=587
+page_content='Infrared behavior in tame two-field cosmological models  Elena Mirela Babalic∗ and Calin Iuliu Lazaroiu  Horia Hulubei National Institute of Physics and Nuclear Engineering, Department of Physics,  Reactorului 30, Bucharest-Magurele, 077125, Romania  E-mail: mbabalic@theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='nipne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='ro, lcalin@theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='nipne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='ro  We discuss the behavior of cosmological curves and their first order infrared approximants near  critical ends of the scalar manifold Σ and near interior critical points of the scalar potential  for tame hyperbolizable two-field cosmological models by determining the universal forms of the  asymptotic gradient flow of the classical effective potential with respect to the uniformizing metric  near all these points and ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' We compare the asymptotic behavior of gradient flow curves with  numerical results for cosmological curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  11th International Conference of the Balkan Physical Union (BPU11),  28 August - 1 September 2022  Belgrade, Serbia  ∗Speaker  © Copyright owned by the author(s) under the terms of the Creative Commons  Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='it/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='00217v1  [hep-th]  31 Dec 2022  IR behavior in tame two-field cosmological models  Elena Mirela Babalic  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Introduction  Two-field cosmological models provide the simplest testing ground for multifield cosmological  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' The latter are of great importance for connecting cosmology with fundamental theories  of gravity and matter, since the effective description of generic string and M-theory compactifi-  cations contains many moduli fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' In particular, multifield models are crucial in cosmological  applications of the swampland program [1–4], as pointed out for example in [5–7], and may also  allow for a unified description of inflation, dark matter and dark energy [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  A two-field cosmological model is parameterized by a connected borderless smooth surface Σ  (the target manifold of the scalar fields) endowed with a Riemannian metric G and a scalar potential  Φ which is a smooth function defined on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' We assume that Φ is positive everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' In [9],  we used a dynamical RG (renormaization group) flow analysis and the uniformization theorem of  Poincaré to show that two-field models whose scalar field metric G has constant Gaussian curvature  𝐾 equal to −1, 0 or +1 give distinguished representatives for the IR (infrared) universality classes  of all two-field cosmological models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Hyperbolizable two-field models (which are defined as those  models for which 𝐾 = −1) comprise all two-field models whose target is of general type as well as  those models whose target is exceptional (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='e diffeomorphic with R2, the annulus A2 or the Möbius  strip M2) and for which the metric belongs to a hyperbolizable conformal class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' The uniformized  form of a hyperbolizable model is a two-field generalized 𝛼-attractor model in the sense of [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Some aspects of such models were investigated previously in [11–17] (see [18–21] for brief reviews).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  The infrared behavior of a tractable class of hyperbolizable two field models was studied in  [22], work which we summarize here together with a brief announcement of further results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' We  will assume that the target manifold Σ is oriented and topologically finite in the sense that it has  finitely-generated fundamental group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' When Σ is non-compact, this condition insures that it has a  finite number of Freudenthal ends [23] and that its end compactification �Σ is a smooth and oriented  compact surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Thus Σ is recovered from �Σ by removing a finite number of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' We also  assume that the scalar potential Φ admits a smooth extension ˆΦ to �Σ which is a strictly-positive  Morse function defined on �Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' A two-field cosmological model is called tame when these conditions  are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Thus tame hyperbolizable two-field cosmological models are those classical two-  field models whose scalar manifold is a connected, oriented and topologically finite hyperbolizable  Riemann surface (Σ, G) and whose scalar potential Φ admits a positive and Morse extension to the  end compactification of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Notations and conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  All surfaces Σ considered here are connected, smooth, Hausdorff  and paracompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' If 𝑉 is a smooth real-valued function defined on Σ, we denote by:  Crit𝑉  def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  = {𝑐 ∈ Σ|(d𝑉)(𝑐) = 0}  the set of its critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' For any 𝑐 ∈ Crit𝑉, we denote by Hess(𝑉)(𝑐) the Hessian of 𝑉 at 𝑐,  which is a well-defined and coordinate independent symmetric bilinear form on the tangent space  𝑇𝑐Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Given a metric G on Σ, we define the covariant Hessian tensor of 𝑉 relative to G by:  HessG(𝑉)  def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  = ∇d𝑉 ,  2  IR behavior in tame two-field cosmological models  Elena Mirela Babalic  where ∇ is the Levi-Civita connection of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  This symmetric tensor has the following local  expression in coordinates (𝑥1, 𝑥2) on Σ:  HessG(𝑉) = (𝜕𝑖𝜕𝑗 − Γ𝑘  𝑖 𝑗(𝑥)𝜕𝑘)𝑉d𝑥𝑖 ⊗ d𝑥 𝑗 ,  where Γ𝑘  𝑖 𝑗(𝑥) are the Christoffel symbols of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Recall that a critical point 𝑐 of 𝑉 is called  nondegenerate if HessG(𝑉)(𝑐) is a non-degenerate bilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' When 𝑉 is a Morse function  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  has only non-degenerate critical points), the set Crit𝑉 is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  We denote by �Σ the  end compactification of Σ, which is a compact Hausdorff topological space containing Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' In the  topologically finite case, the surface Σ has a finite number of Freudenthal ends and �Σ is a smooth  compact surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' In this situation, we say that 𝑉 is globally well-behaved on Σ if it admits a smooth  extension �𝑉 to �Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' A metric G on Σ is called hyperbolic if it is complete and of constant Gaussian  curvature 𝐾 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Hyperbolizable two-field models  Let us recall the global description of two-field cosmological models through a second order  geometric ODE and the first order infrared approximation introduced in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Such a model is  parameterized by the rescaled Planck mass 𝑀0  def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  = 𝑀  √︃  2  3 (where 𝑀 is the reduced Planck mass)  and by its scalar triple (Σ, G, Φ), where Σ is the target manifold for the scalar fields (a generally  non-compact borderless connected surface), G is the scalar field metric and Φ is the scalar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  To ensure conservation of energy, one requires that G is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' For simplicity, we also assume  that Φ is strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  When neglecting fluctuations, the scalar field 𝜑 : R → Σ coupled to the metric of a Friedmann-  Lemaître-Robertson-Walker (FLRW) space satisfies the cosmological equation (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4) in [22]):  ∇𝑡 �𝜑(𝑡) + 1  𝑀0  �  || �𝜑(𝑡)||2  G + 2Φ(𝜑(𝑡))  �1/2  �𝜑(𝑡) + (gradGΦ)(𝜑(𝑡)) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  (1)  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' The IR behavior (in the sense of [9]) of the cosmological flow of a two-field model  with scalar triple (Σ, G, Φ) and rescaled Planck mass 𝑀0 is described by the gradient flow of the  scalar triple (Σ, 𝐺,𝑉), where 𝐺 is the uniformizing metric of G and 𝑉  def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  = 𝑀0  √  2Φ is the classical  effective scalar potential of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Hence the IR behavior is described by the gradient flow equation:  �𝜑IR(𝑡) + (grad𝐺𝑉)(𝜑IR(𝑡)) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  (2)  The first order IR approximants of cosmological orbits for the model (𝑀0, Σ, G, Φ) coincide with  those of the model (𝑀0, Σ, 𝐺, Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Moreover, these approximants coincide with the gradient flow  orbits of (Σ, 𝐺,𝑉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' In particular, the IR universality classes defined in [9] depend only on the scalar  triple (Σ, 𝐺,𝑉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' This allows for systematic studies of two-field cosmological models belonging to a  fixed IR universality class by using the infrared expansion of cosmological curves, the first order of  which is given by the gradient flow of the classical effective potential 𝑉 on the geometrically finite  hyperbolic surface (Σ, 𝐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Since the future limit points of cosmological curves and of the gradient  flow curves of (Σ, 𝐺,𝑉) are critical points of Φ or Freudenthal ends of Σ, the asymptotic behavior  of such curves for late cosmological times is determined by the form of 𝐺 and 𝑉 near such points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  3  IR behavior in tame two-field cosmological models  Elena Mirela Babalic  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='1 The hyperbolic metric 𝐺 in the vicinity of an end  In this subsection, we recall the form of the hyperbolic metric 𝐺 in a canonical vicinity of an end  and extract its asymptotic behavior near each type of end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Any end e of Σ admits an open neighborhood 𝑈e ⊂ �Σ diffeomorphic with a disk such that there  exist semigeodesic polar coordinates (𝑟, 𝜃) ∈ R>0 × S1 defined on �𝑈e  def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  = 𝑈e \\ {e} ⊂ Σ in which  the metric 𝐺 has the canonical form:  d𝑠2  𝐺| �𝑈e = d𝑟2 + 𝑓e(𝑟)d𝜃2 ,  𝑓e(𝑟) =  �������  �������  sinh2(𝑟)  if e = plane end  1  (2𝜋)2 𝑒2𝑟  if e = horn end  ℓ2  (2𝜋)2 cosh2(𝑟)  if e = funnel end of circumference ℓ > 0  1  (2𝜋)2 𝑒−2𝑟  if e = cusp end  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  The end corresponds to 𝑟 →∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Setting 𝜔  def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  = 1  𝑟 , the metric in canonical polar coordinates (𝜔, 𝜃) is:  d𝑠2  𝐺| �𝑈e = d𝜔2  𝜔4 + 𝑓e(1/𝜔)d𝜃2 ,  where:  𝑓e(1/𝜔) = ˜𝑐e𝑒  2𝜀e  𝜔  �  1 + O  �  𝑒− 2  𝜔  ��  for 𝜔 → 0 ,  (3)  with:  ˜𝑐e =  �������  �������  1  4  if e = plane end  1  (2𝜋)2  if e = horn end  ℓ2  (4𝜋)2  if e = funnel end of circumference ℓ > 0  1  (2𝜋)2  if e = cusp end  𝜀e =  �  +1  if e = flaring (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' plane, horn or funnel) end  −1  if e = cusp end  The term O  �  𝑒− 2  𝜔  �  in (3) vanishes identically when e is a cusp or horn end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' In particular, the  constants ˜𝑐e and 𝜖e determine the leading asymptotic behavior of the hyperbolic metric 𝐺 near e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  The gradient flow equations of ( �𝑈e, 𝐺| �𝑈e,𝑉| �𝑈e) read:  d𝜔  d𝑞 = −(grad𝑉)𝜔 ≃ −𝜔4𝜕𝜔𝑉  d𝜃  d𝑞 = −(grad𝑉) 𝜃 ≃ − 1  ˜𝑐e  𝑒− 2𝜖e  𝜔 𝜕𝜃𝑉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  (4)  We studied these equations in [22] for all ends of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Below, we summarize the results for critical  ends (see op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' for the noncritical ends).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Recall that 𝑉 is globally well-behaved and �𝑉 is Morse on �Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Together with the formulas above,  this implies that (grad𝐺𝑉)𝜔 tends to zero at all ends while (grad𝐺𝑉) 𝜃 tends to zero exponentially  at flaring (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' non-cusp) ends and to infinity at cusp ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' On the other hand, we have:  ||grad𝐺𝑉||2 = ||d𝑉||2 = 1  𝜔4 (𝜕𝜔𝑉)2 + 𝑓𝑒(1/𝜔)(𝜕𝜃𝑉)2 ≈ 1  𝜔4 (𝜕𝜔𝑉)2 + ˜𝑐e𝑒  2𝜖e  𝜔 (𝜕𝜃𝑉)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Thus ||grad𝐺𝑉|| tends to infinity at all ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  4  IR behavior in tame two-field cosmological models  Elena Mirela Babalic  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2 Principal canonical coordinates centered at an end e  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' A canonical Cartesian coordinate system (𝑥, 𝑦) for (Σ, 𝐺) centered at the critical  end e is called principal for 𝑉 if the tangent vectors 𝜖𝑥 =  𝜕  𝜕𝑥  ��  e and 𝜖𝑦 =  𝜕  𝜕𝑦  ��  e form a principal basis  for 𝑉 at e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Canonical Cartesian coordinates (𝑥, 𝑦) centered at e are given by:  𝑥 = 𝜔 cos 𝜃 = 1  𝑟 cos 𝜃  ,  𝑦 = 𝜔 sin 𝜃 = 1  𝑟 sin 𝜃 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  In such coordinates, the end e corresponds to 𝜔 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' (𝑥, 𝑦) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' The Taylor expansion  of �𝑉 at e in principal Cartesian coordinates (𝑥, 𝑦) centered at e and in associated polar coordinates  (𝜔, 𝜃) reads:  �𝑉e(𝑥, 𝑦) =  �𝑉(𝑒) + 1  2  �  𝜆1(e)𝑥2+𝜆2(e)𝑦2�  + O((𝑥2 + 𝑦2)  3  2 )  �𝑉e(𝜔, 𝜃) = �𝑉(e) + 1  2𝜔2�  𝜆1(e) cos2 𝜃+𝜆2(e) sin2 𝜃  �  + O(𝜔3) ,  (5)  where 𝜔 =  √︁  𝑥2 + 𝑦2, 𝜃 = arg(𝑥 + i𝑦) and the real numbers 𝜆1(e) and 𝜆2(e) are the principal values  of the Hessian of �𝑉(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' When 𝜆1 and 𝜆2 do not both vanish, it is convenient to define:  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' The critical modulus of (Σ,𝐺,𝑉) at the critical end e is the ratio:  𝛽e  def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  = 𝜆1(e)  𝜆2(e) ∈ [−1, 1] \\ {0} ,  where 𝜆1(e) and 𝜆2(e) are the principal values of (Σ, 𝐺,𝑉) at e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' The characteristic signs of (Σ, 𝐺,𝑉) at e are:  𝜀𝑖(e)  def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  = sign(𝜆𝑖(e)) ∈ {−1, 1} (𝑖 = 1, 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  The extended scalar potential ˆΦ of the canonical model can be recovered from the extended  classical effective potential as:  ˆΦ =  1  2𝑀2  0  �𝑉2 ≈  ¯𝜆2(e)2  2  �  ˆ¯𝑉(e) + 1  2𝜔2(𝛽e cos2 𝜃 + sin2 𝜃)  �2  ,  (6)  where we defined  ¯𝜆2(e)  def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  = 𝜆2(e)  𝑀0  ,  ˆ¯𝑉(e)  def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  =  ˆ𝑉(e)  𝜆2(e) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Solving the gradient flow equation (2) of 𝑉 relative to 𝐺 with the approximations (3) and (5) for  𝜃 ∉ {0, 𝜋  2 , 𝜋, 3𝜋  2 } shows that the unoriented gradient flow orbits of 𝑉 around the end e have implicit  equation:  1  4 [𝜆1(e) − 𝜆2(e)] Γ2  �2𝜀e  𝜔  �  = 𝐴 + ˜𝑐e [ 𝜆1(e) log | sin 𝜃| − 𝜆2(e) log | cos 𝜃| ] ,  (7)  where Γ2 is the lower incomplete Gamma function of order 2 and 𝐴 is an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Below, we compare graphically (making certain choices for 𝛽) the unoriented gradient flow  orbits given implicitely by equation (7) to the orbits of IR optimal cosmological curves, defined as  those solutions 𝜑 of the cosmological equation (1) which satisfy �𝜑(0) = −(grad𝐺𝑉)(𝜑(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  5  IR behavior in tame two-field cosmological models  Elena Mirela Babalic  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='3 The IR behavior near critical plane ends  Figure 1 bellow gives the unoriented gradient flow orbits for certain choices of 𝛽, while Figure  2 gives the numerically computed orbits of the IR optimal cosmological curves for the same choices  of 𝛽 and various other assumptions mentioned in the description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  x  y  (a) For 𝛽e = −1/2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  x  y  (b) For 𝛽e = 1/2  Figure 1: Gradient flow orbits of 𝑉 (shown in brown) and level sets of 𝑉 (shown in green) near a critical  plane end e, drawn in principal Cartesian canonical coordinates centered at e for two values of 𝛽e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  x  y  (a) For 𝛽e = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5  y  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  (b) For 𝛽e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5  Figure 2: Numerically computed infrared optimal cosmological orbits of the canonical model (shown in  brown) and level sets of ˆΦ (shown in green) near a critical plane end e, drawn in principal canonical Cartesian  coordinates centered at e for two values of 𝛽e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' We took ¯𝜆2(e) = 1, ¯ˆ𝑉(e) = 1 and 𝑀0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' The initial point of  each orbit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' 𝜑(0), is shown as a black dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  6  IR behavior in tame two-field cosmological models  Elena Mirela Babalic  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4 The IR behavior near critical horn ends  We graphically compare Figure 3 bellow, which gives the unoriented gradient flow orbits near  critical horn ends, with Figure 4 which shows some numerically computed orbits of the IR optimal  cosmological curves near critical horn ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' The comparison is done for certain choices of 𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
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+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  x  y  (a) For 𝛽e = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
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+page_content='0  x  y  (b) For 𝛽e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Figure 3: Gradient flow orbits of 𝑉 (shown in brown) and level sets of 𝑉 (shown in green) near a critical  horn end e, drawn in principal Cartesian canonical coordinates centered at e for two values of 𝛽e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
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+page_content='4  x  y  (b) For 𝛽e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Figure 4: Numerically computed infrared optimal cosmological orbits of the canonical model (shown in  brown) and level sets of ˆΦ (shown in green) near a critical horn end e, drawn in principal canonical Cartesian  coordinates centered at e for two values of 𝛽e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' We took ¯𝜆2(e) = 1, ¯ˆ𝑉(e) = 1 and 𝑀0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' The initial point of  each orbit is shown as a black dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  7  IR behavior in tame two-field cosmological models  Elena Mirela Babalic  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5 The IR behavior near critical funnel ends  We visually compare below Figure 5, which gives the unoriented gradient flow orbits near  critical funnel ends, with Figure 6, which shows some numerically computed orbits of the IR  optimal cosmological curves near critical funnel ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
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+page_content='  Figure 5: Gradient flow orbits of 𝑉 (shown in brown) and level sets of 𝑉 (shown in green) near a critical  funnel end e of circumference ℓ = 1, drawn in principal Cartesian canonical coordinates centered at e for two  values of 𝛽e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
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+page_content='3  x  y  (a) For 𝛽e = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
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+page_content='4  x  y  (b) For 𝛽e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Figure 6: Numerically computed infrared optimal cosmological orbits of the canonical model (shown in  brown) and level sets of ˆΦ (shown in green) near a critical funnel end e of circumference ℓ = 1, drawn in  principal canonical coordinates centered at e for two values of 𝛽e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' We took ¯𝜆2(e) = 1, ¯ˆ𝑉(e) = 1 and 𝑀0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  The initial point of each orbit 𝜑(0) is shown as a black dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  8  IR behavior in tame two-field cosmological models  Elena Mirela Babalic  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='6 The IR behavior near critical cusp ends  By graphically comparing Figure 7, which gives some unoriented gradient flow orbits near  critical cusp ends, and Figure 8, which shows some numerically computed orbits of the IR optimal  cosmological curves near critical cusp ends, one can assume that higher order corrections are needed  in the IR expansion to get better approximants for the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  4  2  0  2  4  4  2  0  2  4  x  y  (a) For 𝛽e = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
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+page_content='  4  2  0  2  4  4  2  0  2  4  x  y  (b) For 𝛽e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
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+page_content='  Figure 7: Gradient flow orbits of 𝑉 (shown in brown) and level sets of 𝑉 (shown in green) near a critical  cusp end e, drawn in principal Cartesian canonical coordinates centered at e for two values of 𝛽e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
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+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4  x  y  (a) For 𝛽e = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4  x  y  (b) For 𝛽e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Figure 8: Numerically computed infrared optimal cosmological orbits of the canonical model (shown in  brown) and level sets of ˆΦ (shown in green) near a critical cusp end e, drawn in principal canonical coordinates  centered at e for two values of 𝛽e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' We took ¯𝜆2(e) = 1, ¯ˆ𝑉(e) = 1 and 𝑀0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' The initial point of each orbit  is shown as a black dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  9  IR behavior in tame two-field cosmological models  Elena Mirela Babalic  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='7 The IR behavior near an interior critical point  Let c be an interior critical point and (𝑥, 𝑦) be principal Cartesian canonical coordinates centered  at c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' We have the metric:  d𝑠2  𝐺 =  4  (1 − 𝜔2)2 [d𝜔2 + 𝜔2d𝜃2]  and:  𝑉(𝜔, 𝜃) = 𝑉(c) + 1  2𝜔2 �  𝜆1(c) cos2 𝜃 + 𝜆2(c) sin2 𝜃  �  + O(𝜔3) ,  where 𝜔  def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  =  √︁  𝑥2 + 𝑦2 and 𝜃  def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  = arg(𝑥 + i𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Thus:  (grad𝑉)𝜔 ≈ (1 − 𝜔2)2  4  𝜕𝜔𝑉 = (1 − 𝜔2)2𝜔  4  [𝜆1(c) cos2 𝜃 + 𝜆2(c) sin2 𝜃] ,  (grad𝑉) 𝜃 ≈ (1 − 𝜔2)2  4𝜔2  𝜕𝜃𝑉 = (1 − 𝜔2)2  4  [𝜆2(c)−𝜆1(c)] sin(𝜃) cos(𝜃) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  (8)  The critical modulus 𝛽c and characteristic signs 𝜖1(c) and 𝜖2(c) of (Σ, 𝐺,𝑉) at c are defined  through:  𝛽c  def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  = 𝜆1(c)  𝜆2(c) ∈ [−1, 1] \\ {0} , 𝜖𝑖(c)  def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  = sign(𝜆𝑖(c)) (𝑖 = 1, 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Distinguish the cases:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' 𝜆1(c) = 𝜆2(c) := 𝜆(c), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' 𝛽c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Then 𝜖1(c) = 𝜖2(c) := 𝜖(c) and c is a local minimum of 𝑉  when 𝜆(c) is positive (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' when 𝜖(c) = 1) and a local maximum of 𝑉 when 𝜆(c) is negative  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' when 𝜖(c) = −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Relations (8) become:  (grad𝑉)𝜔 ≈ (1 − 𝜔2)2𝜔  4  𝜆(c) , (grad𝑉) 𝜃 ≈ 0  and the gradient flow equation of (Σ, 𝐺,𝑉) takes the following approximate form near c:  d𝜔  d𝑞 = − (1 − 𝜔2)2𝜔  4  𝜆(c) ,  d𝜃  d𝑞 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  This gives 𝜃 = const, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' the gradient flow curves near c are approximated by straight lines  through the origin when drawn in principal Cartesian canonical coordinates (𝑥, 𝑦) at c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' 𝜆1(c) ≠ 𝜆2(c), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' 𝛽c ≠ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' When 𝜃 ∈ {0, 𝜋  2 , 𝜋, 3𝜋  2 }, the gradient flow equation reduces to:  d𝜔  d𝑞 = (1 − 𝜔2)2𝜔  4  ×  �  𝜆1(c)  if 𝜃 ∈ {0, 𝜋}  𝜆2(c)  if 𝜃 ∈ { 𝜋  2 , 3𝜋  2 }  d𝜃  d𝑞 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  This gives four gradient flow orbits which are approximated near c by the principal geodesic  orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' When 𝜃 ∉ {0, 𝜋  2 , 𝜋, 3𝜋  2 }, the gradient flow equation takes the form:  (1 − 𝛽c) d𝜔  d𝜃 = 𝜔(𝛽c cot 𝜃 + tan 𝜃) ,  (9)  with general solution:  𝜔 = 𝐶 | sin(𝜃)|  𝛽c  1−𝛽c  | cos(𝜃)|  1  1−𝛽c  , 𝜃 ∉ {0, 𝜋  2 , 𝜋, 3𝜋  2 } , 𝐶 > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  (10)  10  IR behavior in tame two-field cosmological models  Elena Mirela Babalic  Below we compare graphically the effective gradient flow orbits given by solutions of equation  (10), depicted in Figure 9, to the numerically computed orbits of IR optimal cosmological curves,  represented in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4  x  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4  y  (a) 𝛽c =−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Interior saddle point of 𝑉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4  x  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4  y  (b) 𝛽c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Local extremum of 𝑉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Figure 9: Unoriented gradient flow orbits of 𝑉 (shown in brown) near an interior critical point superposed  over the level lines of 𝑉 (shown in green) for two values of 𝛽c, plotted in principal Cartesian canonical  coordinates centered at the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' The figure assumes 𝜔max(c) = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' The principal coordinate axes  correspond to the principal geodesic orbits at c, which coincide with four special gradient flow orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4  x  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4  y  (a) Saddle point of 𝑉 for 𝛽c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' The dots are  initial points 𝜑(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4  x  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='4  y  (b) Local extremum of 𝑉 for 𝛽c= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' The dots are  initial points 𝜑(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Figure 10: Numerically computed orbits of infrared optimal cosmological curves 𝜑 of the uniformized  model (shown in brown) near an interior critical point c, superposed over the level lines of Φ (shown in  green) for two values of 𝛽c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Here 𝑥, 𝑦 are principal Cartesian canonical coordinates centered at the critical  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' We assume 𝜔max(c) = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' The initial points 𝜑(0) of these curves are shown as black dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  11  IR behavior in tame two-field cosmological models  Elena Mirela Babalic  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Brief announcement of further results  Cosmological curves of two field models can be approximated by mean field curves, using  an approximation technique which is similar to the mean field approximation of condensed matter  physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Cosmological mean field approximations admits an elegant formulation using Ehresmann  connections defined on the total space of the tangent bundle𝑇Σ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' rank two distributions H ⊂ 𝑇𝑇Σ  which are complementary to the vertical distribution V ⊂ 𝑇𝑇Σ of the fiber bundle 𝑇Σ → Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' The  corresponding mean field approximation replaces cosmological flow curves 𝛾 = �𝜑 : 𝐼 → 𝑇Σ of the  model with the horizontal lift relative to H of curves in Σ which satisfy the so-called mean field  curve equation defined by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' This amounts to treating as small the components of �𝜑 : 𝐼 → 𝑇𝑇M  which are “orthogonal” to H – in a sense which can be made precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  The simplest approximations of this type are induced by the choice of a special coordinate  system on an open subset of the tangent bundle 𝑇Σ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  a coordinate system which naturally  combines a system of coordinate on the base Σ with a coordinate system for the fibers of 𝑇Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' In this  case, the corresponding Ehresmann connection is flat and the mean field approximation amounts  to neglecting the first time derivative of the two fiberwise coordinates, which are thereby being  treated as “slow”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' This parallels the logic of Born-Oppenheimer type approximations in quantum  mechanics and statistical physics, which separate dynamical variables into “slow” and “fast” and  treat the dynamics of slow variables approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  In our situation, the first order system of four ODEs which describes the cosmological equation  in a given special coordinate system is replaced by the algebro-differential system in which the  time derivatives of the fiberwise coordinates are set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' This leads to algebraic consistency  conditions for the special coordinates called mean field equations, which determine a mean field  surface inside 𝑇Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' in many cases, the latter is a semialgebraic multisection of 𝑇Σ defined on an  open subset of the base Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Mean field approximations of this type provide a very general procedure for extracting approx-  imants of cosmological curves in various regimes, where the regime of interest is defined by the  choice of fiberwise coordinates that one wishes to treat as “slow”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Since fiberwise coordinates are  pairs of basic observables of the cosmological system which are functionally independent on an  open subset of 𝑇Σ, each such regime is determined by the choice of a pair of locally independent  on-shell cosmological observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' As we show in forthcoming work, a careful study of natural  on-shell observables of two-field cosmological models provides interesting candidates for such  fiberwise coordinates on 𝑇Σ, thus leading to natural mean field approximation schemes which have  direct physics significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' The latter can be applied to any two-field model and in particular to  tame hyperbolizable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  One such mean field approximation is the so-called adapted approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' This uses the  fiberwise coordinates on 𝑇Σ which are given by the projections of �𝜑 on the direction of the vector  grad𝐺Φ and on its positive normal direction and provides a mathematical refinement of the proposal  of [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' One can also consider the roll-turn approximation, which takes as fiberwise coordinates  the on-shell second slow roll parameter and turning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Finally, one can consider the slow roll  rate approximation, which uses the first and second on shell slow roll parameters as fiberwise  coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Another kind of approximation which can be considered for two-field cosmological models is  12  IR behavior in tame two-field cosmological models  Elena Mirela Babalic  the so-called angular approximation, which arises by neglecting the first two time derivatives of  the radial variable in a given semigeodesic coordinate system (𝑟, 𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' This provides approximants  on each semigeodesic coordinate patch, whose accuracy can be characterized theoretically and  computed numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  In forthcoming work, we study the approximations mentioned above for tame two-field cosmo-  logical models and the corresponding error terms, which play an important role when ascertaining  their accuracy (see [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' In particular, this serves as a test of various proposals made previously  in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Moreover, we compare these approximations with the IR approximation studied  in [22] (and summarized above) near interior critical points and near ends of Σ, determining the  regimes within which the various approximations are accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Conclusions  We studied the first order IR behavior of tame hyperbolizable two-field cosmological models  by analyzing the asymptotic form of the gradient flow orbits of the classical effective scalar potential  𝑉 with respect to the uniformizing metric 𝐺 near all interior critical points and ends of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' We  showed that the IR behavior of tame hyperbolizable two field cosmological models is characterized  by a finite set of parameters associated to their ends and interior critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Comparing with  numerical computations, we found that the first order IR approximation is already quite good for  all interior critical points and all ends except for cusps, for which one must consider higher order  corrections in the IR expansion in order to obtain a good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' Our results characterize  the IR universality classes of all tame hyperbolizable two-field models in terms of geometric data  extracted from the asymptotic behavior of the effective scalar potential and uniformizing metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  Since the Morse assumption on the extended potential determines its asymptotic form near all  points of interest on �Σ, we could derive closed form expressions for the asymptotic gradient flow  which describes the corresponding infrared phases of such models in the sense of [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' In particular,  we found that the asymptotic gradient flow of (Σ, 𝐺,𝑉) near each end which is a critical point of the  extended potential can be expressed using the incomplete gamma function of order two and certain  constants which depend on the type of end under consideration and on the principal values of the  extended effective potential �𝑉 at that end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' We also found that flaring ends which are not critical  points of �𝑉 act like fictitious but non-standard stationary points of the effective gradient flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' While  the local form near the critical points of 𝑉 is standard (since they are hyperbolic stationary points  [25, 26] of the cosmological and gradient flow), the asymptotic behavior near Freudenthal ends is  exotic in that some of the ends act like fictitious stationary points with unusual characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content='  We compared these results with numerical computations of cosmological curves near the  points of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' We found particularly interesting behavior near cusp ends, around which generic  cosmological trajectories tend to spiral a large number of times before either “falling into the cusp”  or being “repelled” back toward the compact core of Σ along principal geodesic orbits determined  by the classical effective potential 𝑉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
+page_content=' In particular, cusp ends lead naturally to “fast turn” behavior  of cosmological curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf'}
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+IMPROVING PERFORMANCE IN NEURAL NETWORKS BY
+DENDRITES-ACTIVATED CONNECTIONS
+ARXIV PREPRINT
+Carlo Metta∗
+Marco Fantozzi
+Andrea Papini†
+Gianluca Amato‡
+Matteo Bergamaschi§
+Silvia Giulia Galfr`e¶
+Alessandro Marchetti‡∥
+Michelangelo Vegli`o‡
+Maurizio Parton‡∗∗
+Francesco Morandin§
+CuriosAI Lab
+Code at https://github.com/curiosai/dac-dev
+Contact authors at curiosailab@gmail.com
+ABSTRACT
+Computational units in artificial neural networks follow a simplified model of biological neurons. In
+the biological model, the output signal of a neuron runs down the axon, splits following the many
+branches at its end, and passes identically to all the downward neurons of the network. Each of the
+downward neurons will use their copy of this signal as one of many inputs dendrites, integrate them
+all and fire an output, if above some threshold. In the artificial neural network, this translates to
+the fact that the nonlinear filtering of the signal is performed in the upward neuron, meaning that in
+practice the same activation is shared between all the downward neurons that use that signal as their
+input. Dendrites thus play a passive role.
+We propose a slightly more complex model for the biological neuron, where dendrites play an active
+role: the activation in the output of the upward neuron becomes optional, and instead the signals
+going through each dendrite undergo independent nonlinear filterings, before the linear combination.
+We implement this new model into a ReLU computational unit and discuss its biological plausibility.
+We compare this new computational unit with the standard one and describe it from a geometrical
+point of view. We provide a Keras implementation of this unit into fully connected and convolutional
+layers and estimate their FLOPs and weights change. We then use these layers in ResNet architec-
+tures on CIFAR-10, CIFAR-100, Imagenette, and Imagewoof, obtaining performance improvements
+over standard ResNets up to 1.73%. Finally, we prove a universal representation theorem for con-
+tinuous functions on compact sets and show that this new unit has more representational power than
+its standard counterpart.
+Keywords active dendrite, dendritic neural model, preactivation, multi-bias, image classification, ResNet
+1
+Introduction
+Historically the structure of the perceptron, the artificial neural network’s fundamental computational unit, has rarely
+been questioned. The biological inspiration is straightforward: input signals from the dendrites are accumulated at the
+soma (with a linear combination), and if the result is above the activation threshold (that is, the opposite of the bias)
+there is a nonlinear reaction, as the neuron fires along the axon (with the activation function).
+∗ISTI-CNR Pisa, Italy.
+†Scuola Normale Superiore, Pisa, Italy.
+‡University of Chieti-Pescara, Italy.
+§University
+of Parma, Italy.
+¶University of Pisa, Italy.
+∥Alessandro Marchetti is a PhD student enrolled in the National PhD in
+Artificial Intelligence, XXXVII cycle, course on Health and life sciences, organized by Campus Bio Medico University of Rome.
+∗∗Partially funded by INdAM group GNSAGA.
+arXiv:2301.00924v1  [cs.NE]  3 Jan 2023
+
+Improving Performance in Neural Networks by Dendrites-Activated Connections
+Nevertheless, in time the sigmoid activation function was abandoned in favor of ReLU and variants, and the biolog-
+ical analogy became less stringent, shifting focus on the desirable mathematical properties of the class of functions
+computed by the networks, like representation power and non-vanishing gradients.
+This has brought us to the current situation in which most units output their signal through a nonlinear activation
+function which effectively destroys some information. In fact, ReLU is not invertible, as it collapses to zero all
+negative values. Though some of its variants may be formally invertible (leaky ReLU and ELU for example), the fact
+that they overall perform in a way very similar to ReLU, suggests that their way of compressing negative values with
+small derivatives leads to the same general properties of the latter.
+In this paper, we propose and test a radical rethinking of the standard ReLU-like computational unit, where the output
+brings its full, uncorrupted information to the next units, and the activation function is applied there, with a variable
+bias. One can say that the proposed units are preactivated instead of post-activated, and from the biological point
+of view this is like having dendrites with different thresholds passing the signal to the cell soma. In this sense,
+dendrites are ‘active connections’ with the soma, and correspondingly we call them ‘DAC’, for ‘Dendrites-Activated
+Connections’.
+This kind of reversed view has already proved fruitful in the evolution from ResNets v1 [1] to v2 [2] when a com-
+prehensive ablation study showed that the best choice for residual networks is keeping the information backbone free
+of activations for maximum information propagation, and preactivate the convolutional layers in the residual branch.
+Much earlier in the past, the celebrated LSTM [3] architecture used already an activation-free backbone as long mem-
+ory, even tough inputs and gates have a standard post-activation.
+Here this view is taken forward: if the block bias + activation is better placed at the input of a unit, then the bias itself
+should be specific to each unit, as the weights are. (In fact, this choice turns out to be natural from both mathematical
+and biological points of view, as will be shown in the next sections.)
+In Section 2 we first discuss in detail the idea of dendritic preactivation, which is the core of DAC computational units,
+as compared to post-activation, which is used in standard units. We then motivate DAC units from a geometrical and
+biological point of view.
+In Section 3 we describe in detail our Keras implementation of DAC fully connected and convolutional layers, and
+how we use these layers in ResNet architectures. Moreover, we perform an efficiency and memory analysis.
+In Section 4 we describe our experiments on CIFAR-10, CIFAR-100, Imagenette, and Imagewoof comparing DAC
+and standard models. In particular, Tables 1 and 2 report the main empirical results of the paper.
+Section 5 is devoted to the theoretical properties of DAC units. In particular, we show that every function f ∈
+C0([−1, 1]d) can be approximated by a ReLU fully connected DAC neural network with d DAC layers and (d+2−l)k
+units in layer l = 1, 2, . . . , d − 1, where k is a constant depending on f and the desired approximation accuracy. See
+Theorem 1 and the next remarks.
+Finally, in Section 6 we describe future developments of the DAC framework.
+2
+Motivation
+2.1
+From standard to Dendrites-Activated Connection (DAC) unit
+To describe our idea, it is better to look at a neural network as a directed acyclic computational graph. We denote the
+set of its nodes by I. If i ∈ I is a node, we denote its parents (in-neighbors) by Ii ⊂ I. In what we call the standard
+model for computational units in a neural network, a bias bi, a set of weights wi,j for j ∈ Ii and a nonlinearity ϕi are
+associated with every node i.
+The flow of the network in the standard model can then be described in terms of computational units that, starting from
+input values stored in source nodes, update the value yi stored in node i by the values stored in its parents:
+yi = ϕi
+�
+bi +
+�
+j∈Ii
+wi,j yj
+�
+(standard unit)
+(1)
+In this paper, ϕi is always ϕ = ReLU, and in particular, the theoretical results in Section 5 are strictly dependent on
+this choice. However, using ReLU is not strictly necessary for this paper idea, see Section 6.
+2
+
+Improving Performance in Neural Networks by Dendrites-Activated Connections
+y4 = ϕ(b4 +
+3
+�
+i=1
+w4,iyi)
+y3
+y2
+y1
+I4 = {1, 2, 3}
+w4,1
+w4,2
+w4,3
+y4 = ϕ(b4 + z4)
+z4 =
+3
+�
+i=1
+w4,iyi
+y3 =
+ϕ(b3 + z3)
+y2 =
+ϕ(b2 + z2)
+y1 =
+ϕ(b1 + z1)
+I4 = {1, 2, 3}
+w4,1
+w4,2
+w4,3
+ϕ ◦ b
+Figure 1. A unit with 3 input nodes. The set I of nodes here is {1, 2, 3, 4}. The standard unit on the top is ‘(activation ◦ bias ◦
+linearity)’, represented all together as a single output. On the bottom, the same output is decomposed at a finer scale as
+‘(activation ◦ bias) ◦ linearity’.
+In the literature, according to the purpose of the work at hand, computations may be described at different scales. For
+instance, convolutions can be described on a more granular scale, where three stages (affine map, nonlinearity, and
+pooling) are considered separately, or on a coarser scale, where the composition of the three stages is considered as a
+whole [4, Figure 9.7]. For the purpose of this paper it is convenient to look at the standard unit as a nonlinear filter
+‘activation ◦ bias’ applied to some information linearly aggregated from parents:
+�
+zi = �
+j∈Ii wi,j yj
+linear aggregation
+yi = ϕ(bi + zi)
+nonlinear filter
+Figure 1 exhibits this point of view emphasizing that the values in parent nodes are themselves built from grandparents
+with biases b1, b2, b3 and ReLU. For each parent node, this bias is uniquely determined, meaning that from the point
+of view of children nodes it is shared, in the sense that ‘4’ and other children (not represented here) must use the
+same input with the same parent bias. This forced sharing of biases followed by ReLU, which flattens to zero negative
+values, means that the information arriving at children has in some sense deteriorated.
+To mitigate this loss of information, it seems reasonable to apply the nonlinear filter directly to the parents with a
+non-shared bias, before the linear aggregation, see Figure 2.
+In this paper, we investigate this idea, by studying a new computational unit briefly described as ‘linearity ◦ (activation
+◦ non-shared bias)’. Since the biases in the nonlinear filter can depend on both input and output nodes, that is, on the
+edges of the graph, and since these edges correspond to dendrites in biological neurons, we call this new computational
+unit a Dendrites-Activated Connection unit (DAC unit):
+zi =
+�
+j∈Ii
+wi,j ϕ(bi,j + zj)
+(DAC unit, example in Figure 2)
+(2)
+One can view DAC as a preactivated unit with non-shared biases, in the sense that DAC units sharing the same input
+can use different nonlinearity thresholds.
+3
+
+Improving Performance in Neural Networks by Dendrites-Activated Connections
+3
+�
+i=1
+w4,iϕ(b4,i + zi)
+ϕ(b4,1 + z1)
+z3
+ϕ(b4,2 + z2)
+z2
+ϕ(b4,3 + z3)
+z1
+I4 = {1, 2, 3}
+ϕ ◦ b4,1
+ϕ ◦ b4,2
+ϕ ◦ b4,3
+w4,1
+w4,2
+w4,3
+Figure 2. A DAC unit corresponding to the standard unit in Figure 1. Note that input biases contributing to the output value of this
+unit depend also on the output node i = 4. We call this feature non-shared biases because it allows DAC units sharing
+the same input to use different (non-shared) thresholds instead of a single (shared) input bias.
+2.2
+Comparison of DAC versus standard
+DAC biases must depend on input and output.
+In a DAC unit, it is crucial for the filter to depend on both the input
+and the output nodes (non-shared biases), and not only on the input node (shared biases). In fact, if one considers a
+multilayer perceptron, the equation for standard units gives:
+yi = ϕ
+�
+bi +
+�
+j∈Ii
+wi,j yj
+�
+= ϕ
+�
+bi +
+�
+j∈Ii
+wi,j ϕ
+�
+bj +
+�
+k∈Ij
+wj,k ϕ(. . . )
+��
+(standard unit)
+and the equation for preactivated units with shared biases gives:
+zi =
+�
+j∈Ii
+wi,j ϕ(bj + zj) =
+�
+j∈Ii
+wi,j ϕ
+�
+bj +
+�
+k∈Ij
+wj,k ϕ(. . . )
+�
+(preactivated unit)
+However, if this was the only change, the modification would only be formal, and the resulting network would perform
+the same operations, as can be seen by comparing the right-hand sides of the above equations. For this reason, we
+must use non-shared biases as in (2).
+Last and first layers with DAC.
+This change of perspective has other natural consequences. Consider the last layer
+of a layered network: with standard connections, one would expect a ReLU activation in the output of the last layer,
+but this is typically not desired, as the output of the last layer must be as much informative as possible, and of the
+right type to be plugged into the loss function. So one usually has to remove that last activation, which with a DAC
+connection would not have existed in the first place. Here DAC seems more natural.
+Symmetrically, consider the first layer of a layered network: with a standard connection the input nodes are unaltered
+and have full information – it would not make sense to filter them with ReLU with or without an input-depending
+bias. Nevertheless, with a DAC connection, it is instead very reasonable to apply the nonlinearity to the input nodes,
+because the bias used to filter them may depend on the output nodes, and it is not unlikely that different output nodes
+in the first layer might benefit from node-wise filtering of the input. Here DAC could be more capable of exploiting
+feature diversities in the first layer.
+Remark 1. In this paper, we adhere to the terminology from [4]: whenever the network function f is a composition
+f (L)◦· · ·◦f (1), we say that the network is layered, and composed of L layers f (l), for l = 1, . . . , L. If f (l) : Rm → Rn,
+we say that layer l has m input nodes and n output nodes. When we desire to point out the units in their entirety, and
+not only the output nodes, we say that layer l has n units, each of them with m input nodes.
+Input replication.
+Filtering the input as just described might be even more useful if the input is replicated multiple
+times. Consider the toy example of a one-dimensional input x and a shallow network with only one layer of one
+4
+
+Improving Performance in Neural Networks by Dendrites-Activated Connections
+unit aiming at approximating some function f : R → R. A DAC fully connected layer with one unit ‘0’ and input
+replicated n times x = (x, x, . . . , x) gives:
+ˆfDAC(x) =
+n
+�
+j=1
+w0,j ϕ(b0,j + x)
+(3)
+which is a universal approximator of a large class of functions R → R, for n → ∞ (in particular in Section 5 we
+rigorously prove that ˆf is a universal approximator of C0([−1, 1]), see Theorem 1 and Remark 3).
+On the other hand, a standard fully connected layer with replicated inputs and one unit would give:
+ˆfcl(x) = ϕ
+�
+b0 +
+n
+�
+j=1
+w0,j x
+�
+= ϕ(b0 + ˜w0 x)
+regardless of n. To gain expressivity with the standard connection we can add a hidden layer with n units (with or
+without replicated inputs is the same), obtaining:
+ˆf2×cl(x) = ϕ
+�
+b0 +
+n
+�
+j=1
+w0,j ϕ(bj + ˜wj x)
+�
+.
+To show that ˆf2×cl has a representation power similar to ˆfDAC, we put ˜w0,j = w0,j| ˜wj| and ˜bj = bj/| ˜
+wj| in the above
+expression, which gives:
+ˆf2×cl(x) = ϕ
+�
+b0 +
+n
+�
+j=1
+˜w0,j ϕ
+�˜bj + sign( ˜wj) x
+��
+.
+This discussion shows that, in this toy problem of approximating a function f : R → R, one needs a two-layers
+standard network to get a representation power similar to a single-layer DAC network.
+Input replication inside the computational graph.
+From a biological perspective, replicating the input in a DAC
+unit would correspond to different dendrites of the same downward neuron being connected to different branches of
+the axon of the same upward neuron. This is something that has not been scrutinized in particular in the biological
+literature, but definitely cannot be excluded, given the complexity of dendrite activation, see Section 2.3 for more
+details. This suggests that input replication can be useful in principle at any node in the computational graph, not only
+on its inputs. Thus, when there is a layer structure on the network, one can distinguish between ‘replication’ layers
+and DAC layers. With this terminology, the toy example (3) has one replication layer and one DAC layer.
+A thorough empirical analysis of the effects of replication on DAC performance would first require a baseline without
+replication. For this reason, and because this is a first paper on the DAC paradigm, we have decided to not use
+replication in the experiments described in Section 4. We leave this analysis to future work, see Section 6.
+2.3
+Biological Inspiration
+The proposed extension of the artificial neuron also reflects to some extent a recent shift in the understanding of the
+biological neuron. In fact, the early soma-centric representation of the neuron today has been discarded in favor of a
+more realistic and complex model that incorporates active dendrites [5, 6].
+A typical biological neuron consists of many input branches called dendrites, a main body called soma, and the axon,
+which branches at its end in many terminals, where synapses connect to the dendrites of other neurons. The input
+signals originate in the dendrites, flow through the soma, and are integrated into the region of the soma where the axon
+connects, and if a specific threshold is reached, the neuron fires its signal down the axon, to the synapses.
+Until some years ago, the biological neuron model was soma-centric and essentially modeled by a point neuron where
+dendrites simply pass the signals, and all elaboration happens at the soma. This elementary representation was the
+inspiration of the traditional perceptron in artificial neural networks.
+Current biological models are more complicated and the central role of dendrites in signal modulation is better un-
+derstood [7]. Dendrites in fact present voltage-gated ion channels [5] able to produce local electrical events termed
+dendritic spikes. Dendrites actually present at least four groups of ion channels [6]: the synaptic receptors, activated
+by neurotransmitters, the passive leak channels, the active subthreshold ion channels, able to produce transmembrane
+currents also when the threshold for the action potential is not reached, and supra-threshold ion channel active when
+the threshold is reached. In this way, a dendrite or a group of dendrites can perform the first important local, not linear
+signal integration before reaching the cell axon.
+5
+
+Improving Performance in Neural Networks by Dendrites-Activated Connections
+2.4
+Geometric interpretation
+In a fully connected DAC layer, DAC equation (2) can be written in vectorial form. Let m, n be the number of input
+and output nodes, respectively. Denote by z ∈ Rm the input, by ˆz ∈ Rn the output, by b1, . . . , bn ∈ Rm the DAC
+biases, by w1, . . . , wn ∈ Rm the weights, by ⟨·, ·⟩ the scalar product in Rm and by ϕ component-wise ReLU. DAC
+equation (2) then becomes:
+ˆz(z) = (⟨w1, ϕ(b1 + z)⟩, . . . , ⟨wn, ϕ(bn + z)⟩)T.
+(4)
+Equation (4) says that a DAC layer ˆz factorizes as ˆz = pw◦ˆzb, where ˆzb is a nonlinear embedding into a higher (unless
+n = 1) dimensional space, depending only on DAC biases and not on weights:
+ˆzb : Rm −→
+n times
+�
+��
+�
+Rm × · · · × Rm
+z �−→ (ϕ(b1 + z), . . . , ϕ(bn + z))
+and pw is a multilinear map, depending only on weights and not on DAC biases, projecting back each of the n
+components of Rm × · · · × Rm orthogonally onto the axes wi of Span{w1, . . . , wn} ⊂ Rm × · · · × Rm:
+pw :
+n times
+�
+��
+�
+Rm × · · · × Rm −→ Span{w1, . . . , wn} = Rn
+(y1, . . . , yn) �−→ (⟨w1, y1⟩, . . . , ⟨wn, yn⟩)T
+The multilinear map pw can be seen also as the corresponding linear map p⊗
+w on the tensor product Rm ⊗ · · · ⊗ Rm,
+and the whole factorization can be visualized with the commutative diagram (2.4):
+Rm
+Rn = Span{w1, . . . , wn}
+Rm × · · · × Rm
+�
+��
+�
+n times
+Rm ⊗ · · · ⊗ Rm
+�
+��
+�
+n times
+ˆzb
+⊗
+p⊗
+w
+ˆz
+pw
+Thus, DAC preactivation can be seen as an intermediate step: a nonlinear embedding into a space of equal (if n = 1)
+or increased (if n > 1) dimension, followed by a projection. Note that a standard layer is a linear map Rm → Rn
+followed by the nonlinearity, and as such does not have a similar geometric interpretation.
+In general, maintaining or increasing the dimension, before a last classification layer, can make a problem easier.
+In Figure 3 we provide a toy example of a binary classification problem that cannot be solved by a standard layer
+R2 → R, but can be solved by its DAC counterpart, thanks to the intermediate embedding step maintaining the input
+dimension. In Figure 4 another toy example, where a DAC layer R2 → R2 can separate a dataset thanks to the
+intermediate embedding doubling the input dimension.
+3
+Methods
+ResNets [1, 2], SE-Nets [8] and Transformers [9] are some popular architectures that can be implemented in a modular
+way, using standard layers from highly optimized frameworks like PyTorch or Keras. On the other hand, DAC needs
+to be implemented within standard layers, and as such cannot easily be optimized, see implementation issues in
+Section 3.5.
+In this section, we describe equations, complexity, and implementation issues for DAC versions of fully connected
+and convolutional layers, and for ResNet architectures.
+3.1
+DAC fully connected layers
+A fully connected layer with dendrites-activated input connections, m inputs, and n outputs, in its most general form,
+is the following:
+dDAC(y) = dDAC
+i
+(y1, . . . , ym) = ψ
+�
+bi +
+m
+�
+j=1
+wi,j ϕ(bi,j + yj)
+�
+,
+i = 1, . . . , n
+(5)
+6
+
+Improving Performance in Neural Networks by Dendrites-Activated Connections
+(a)
+(b)
+bias
+(c)
+ReLu
+(d)
+DAC
+Figure 3. (a) We want to separate the black point from the white points. For this, we learn a DAC one-layer f : R2 → R. (b) DAC
+can learn a translation (DAC biases) that moves the leftmost point to the second quadrant (red line). (c) ReLU projects
+all points in the second quadrant onto the vertical axis (red line), making the dataset linearly separable. (d) DAC can now
+learn a direction (blue line) onto which to project the dataset. The black point is now separated from the white points.
+(a)
+1
+1
+-1
+-1
+(b)
+1
+1
+-1
+-1
+Figure 4. Given a square like the one on the left, the combination of a DAC layer f : R2 → R2 and a linear layer g : R2 → R can
+have positive values inside and negative outside, being able to separate the blue and orange points on the right. In fact,
+let [f(x)]i = �
+j=1,2 wijϕ(bij + xj) with ϕ denoting ReLU, then it is enough to set wij = 1, b1j = 1, b2j = 0 and
+g(y) = y1 − 2y2 − 1 to get g(f(x)) = ϕ(1 + x1) + ϕ(1 + x2) − 2ϕ(x1) − 2ϕ(x2) − 1, which equals 1 − |x1| − |x2|
+inside the square and is negative outside. On the contrary, it is easy to see that if f was a standard fully connected layer,
+then for all choices of f and g the set where g(f(x)) ≥ 0 would always be unbounded or empty.
+where ψ is an optional output activation, bi is an optional output bias, ϕ is the ReLU input activation and bi,j are the
+DAC biases. When stacking consecutive DAC layers, ψ and bi will typically not be used, but they might in general be
+needed, for example, if the subsequent operation is a global average pooling.
+This layer will add m × n new biases bi,j, for a roughly doubled total number of weights, hence one must use care in
+comparing DAC and regular networks. The increase in the number of FLOPs is about 50% (see Section 3.4).
+3.2
+DAC convolutional layers
+Consider a 2D convolutional layer with dendrites-activated input connections, m input channels, and n output L × L
+kernels. In its most general form, the DAC biases would depend on the m input channels, on the n output kernels,
+and also on the position inside the kernel. However, this seems extreme, and in this paper, we opted for the following
+version of a DAC convolutional layer, a better trade-off between representation power and number of parameters (for
+ease of notation, we consider only odd L = 2l + 1 as kernel size):
+cDAC(y) = cDAC
+h,k,i(y) = ψ
+�
+bi +
+l
+�
+a,b=−l
+m
+�
+j=1
+wa,b,i,j ϕ(bi,j + yh+a,k+b,j)
+�
+,
+i = 1, . . . , n
+(6)
+7
+
+Improving Performance in Neural Networks by Dendrites-Activated Connections
+where (h, k) varies in the set of coordinates of the output and where, as before, ψ is an optional output activation, bi is
+an optional output bias, ϕ is the ReLU input activation and bi,j are the DAC biases. Thus, instead of having the DAC
+biases ba,b,i,j depending on channel, kernel and position in the kernel, we have one single DAC bias bi,j for every
+input channel and output kernel.
+This layer will need more operations due to the m × n new biases bi,j, for an increase in the total number of FLOPs
+and weights of roughly 1/2L2 and 1/L2 respectively (i.e. for a standard 3 × 3 kernels, the FLOPs will increase by about
+5.5% while the weights by about 11%). For an in-depth efficiency and memory analysis, refer to Section 3.4.
+3.3
+DAC ResNet architecture
+For our experiments, we used DAC convolutional layers with the original ResNet architectures [1, 2], because ResNets
+are the building blocks of the most popular convolutional neural network architectures around.
+In particular, we replaced all the 3×3 convolutions with corresponding DAC convolutions (6) without output activation
+ψ and bias bi. Moreover, the convolutions inside residual blocks were preceded by a batch normalization layer and a
+ReLU activation: we removed the activation (because DAC includes it in the input) and removed the trainable shift
+parameter β of the batch normalization (because DAC includes its own biases in the input).
+In the case of ResNet v1, this has the collateral effect of moving the activation from the information backbone to the
+residual branch. This implies that DAC turns ResNet v1 into something similar to the ResNet v2 architecture, but with
+DAC biases instead of regular biases, and with the batch normalization layers moved after the convolutions. Apart
+from the biases, this is the ReLU-only preactivation that was tested in [2, Fig. 4(d)].
+Again in the case of ResNet v1, after the last residual block and before the global average pooling, we added a
+traditional bias layer and a ReLU activation, since they were removed from the residual block output and no DAC
+layer follows.
+Remark 2. In the case of ResNet v2, we explored also the bottleneck architecture, but with inconclusive results that
+would need further research to improve. The difficulty here is that with the bottleneck architecture, about half of
+the weights and operations are devoted to 1 × 1 convolutions, and while the DAC version of the 3 × 3 convolutions
+increases the number of weights and operations by about 1/9, the same quantities approximately double for 1 × 1
+convolutions. So, to compare DAC version with the standard one, there is a need to fine-tune hyperparameters like the
+number of channels, to get fair testing and this is beyond the scope of this work.
+3.4
+Efficiency and memory analysis
+In this section, we estimate how FLOPs and weights change when modifying a standard layer into its DAC counter-
+part, for fully connected and convolutional layers. The total number of floating point operations FLOPs provides an
+estimate of the amount of work performed by a computational process, and weights are correlated with the memory
+footprint. Thus, FLOPs and weights allow us to compare the change in computation and memory burden due to the
+adoption of the DAC paradigm. The use of FLOPs as an effective measure of efficiency has been advocated in [10].
+We will follow the convention used in the analysis of standard layers that the computational cost of the activation
+functions can be safely ignored: despite the large number of activations involved in the DAC paradigm, it is still fine
+to ignore the activation costs since we consider only the ReLU activation, that requires no expensive calculations on
+any architecture. However, this might need to be amended in case someone wants to use other, more costly, activation
+functions in the dendrites.
+3.4.1
+Fully connected layers
+The equation for a standard fully connected layer with m inputs and n outputs is:
+dstd(y) = dstd
+i (y1, . . . , ym) = ψ
+�
+bi +
+m
+�
+j=1
+wi,j yj
+�
+,
+i = 1, . . . , n.
+Inside the parenthesis, we get m multiplications and then m additions plus one for the bias. This must be repeated for
+all the n outputs, giving us a total of:
+FLOP(dstd) = (2m + 1)n ≈ 2mn = 2(#inputs)(#outputs).
+The weights are:
+weights(dstd) = (m + 1)n ≈ mn = (#inputs)(#outputs).
+8
+
+Improving Performance in Neural Networks by Dendrites-Activated Connections
+We consider now a fully connected layer with dendrites-activated input connections, with m inputs and n outputs,
+see (5):
+dDAC(y) = dDAC
+i
+(y1, . . . , ym) = ψ
+�
+bi +
+m
+�
+j=1
+wi,j ϕ(bi,j + yj)
+�
+,
+i = 1, . . . , n.
+Inside the parenthesis we get m bias additions, then m multiplications and m additions plus one for the general bias.
+This must be repeated for all the n outputs, giving us a total of:
+FLOP(dDAC) = (3m + 1)n ≈ 3mn = 3(#inputs)(#outputs).
+The weights are:
+weights(dDAC) = (2m + 1)n ≈ 2mn = 2(#inputs)(#outputs).
+Thus, adopting the DAC paradigm in a fully connected layer roughly increases the number of FLOPs by 50% and
+roughly doubles the number of weights.
+3.4.2
+Convolutional layers
+The equation for a standard 2D convolutional layer, with m input channels and n output L×L kernels, with L = 2l+1,
+is:
+cstd(y) = cstd
+h,k,i(y) = ψ
+�
+bi +
+l
+�
+a,b=−l
+m
+�
+j=1
+wa,b,i,j yh+a,k+b,j
+�
+,
+i = 1, . . . , n
+where (h, k) indicates a point in the 2D, s × t input shape.
+Inside the parenthesis we get m multiplications and then m additions repeated through the square kernel of size L,
+giving a factor of L2, then we add once the bias. This is done for all the n output kernels and for all the s · t points of
+the geometric grid, giving us a total of:
+FLOP(cstd) = (2L2m + 1)nst ≈ 2L2mnst = 2(KernelSize)2(#inputs)(#outputs)(#area).
+The weights are:
+weights(cstd) = (L2m + 1)n ≈ L2mn = (KernelSize)2(#inputs)(#outputs).
+We consider now a 2D convolutional layer with dendrites-activated input connections, m input channels, and n output
+L × L kernels, with L = 2l + 1, see (6):
+cDAC(y) = cDAC
+h,k,i(y) = ψ
+�
+bi +
+l
+�
+a,b=−l
+m
+�
+j=1
+wa,b,i,j ϕ(bi,j + yh+a,k+b,j)
+�
+,
+i = 1, . . . , n
+where (h, k) indicates a point in the 2D, s × t input shape.
+For the FLOPs calculation, we first notice that, since the biases bi,j do not depend on the particular kernel position
+a, b, it is possible to cache the results of the initial activations ϕ(bi,j + y·,·,j) that require m additions for each output
+and each grid point. Thus, starting from what it is left inside the parenthesis, we get m multiplications and then m
+additions repeated through the square kernel of size L, giving a factor of L2, then we add once the out bias. This is
+done for all the n output kernels and all s · t points of the grid, giving us a total of:
+FLOP(cDAC) = mnst + (2L2m + 1)nst ≈ (1 + 2L2)mnst = (2(KernelSize)2 + 1)(#inputs)(#outputs)(#area).
+The weights are:
+weights(cDAC) = (m(1 + L2) + 1)n ≈ (1 + L2)mn = ((KernelSize)2 + 1)(#inputs)(#outputs).
+Thus, adopting the DAC paradigm in a convolutional layer roughly increases the number of FLOPs and weights by a
+fraction 1/2L2 and 1/L2, respectively. This is a much smaller increase, with respect to the fully connected case.
+We notice that, for the calculation of FLOPs in a DAC layer implemented as in GitHub, one can use the keras-flops
+module [11] without any modification. This is in fact what we used for the plots in Section 4.
+9
+
+Improving Performance in Neural Networks by Dendrites-Activated Connections
+3.5
+Implementation
+DAC fully connected and convolutional layers, as detailed at the beginning of this section, can be easily implemented
+on modern deep learning platforms, like PyTorch and TensorFlow that we used for our experiments. Unfortunately,
+such a high-level approach results in a much heavier memory footprint and computation time than its non-DAC anal-
+ogous. In fact, we performed comprehensive experiments to assess the theoretical soundness of dendrites-active
+connections, but to make DAC layers practical to use, one would need to implement them at a low level inside the
+deep learning platform of choice.
+To understand the difference between a high-level and a low-level implementation, recall the form of the internal
+computation of a DAC layer from (5) and (6):
+(FC)
+m
+�
+j=1
+wi,j ϕ(bi,j + yj),
+(Conv)
+l
+�
+a,b=−l
+m
+�
+j=1
+wa,b,i,j ϕ(bi,j + yh+a,k+b,j).
+At a high level, one needs to replicate the output y of the preceding layer for each of the i = 1, . . . , n units/filter of the
+current layer, then apply ReLU using a different bias for each input channel/output unit combination, before applying
+the usual layer calculations separately to each replica block to generate the output units.
+For fully connected layers we used standard matrix multiplications after reshaping the replicated input in the appropri-
+ate manner. For convolutions, using TensorFlow, we exploited the grouping feature of the Conv2D layers, to operate
+the convolution separately on each of the replicated inputs. This allowed us to use a single layer to handle all the
+convolutions at once, saving a lot of computational time as the grouping feature is implemented at a low level.
+The code is available on GitHub, and, after our best efforts, we achieved training the DAC layers using up to 10 times
+more memory and time with respect to the non-DAC analogous. The situation is somewhat better in the case of the
+forward calculation, but still far from the non-DAC case.
+At a low level, consider as an example the forward computation of a fully connected layer. The standard version
+needs one matrix multiplication [w y]i = �m
+j=1 wi,j yj, for i = 1, . . . , n. This is typically implemented with parallel
+programming, with many units that share access to w and y and that compute fragments of the m × n multiplications
+and m × n additions needed (+wi,j · yj). The DAC version would be very similar, with the units also sharing access
+to b and computing fragments of the corresponding DAC operations which require one more addition and one more
+maximum for each of the m × n indices (+wi,j · ϕ(bi,j + yj)), recall that ϕ(x) = max(0, x) is ReLU). Hence, a
+low-level implementation of DAC would have less than twice the complexity of the corresponding standard layer.
+Such a low-level implementation is outside the scope of this paper and is planned as a possible future development,
+see Section 6.
+4
+Experiments and results
+We first evaluate our method on a ResNet architecture for the CIFAR-10 and CIFAR-100 datasets [12]. Both datasets
+consist of 50k training images and 10k test images, divided into 10 and 100 different classes, respectively. We replicate
+each experiment 5 times, using a 5-fold cross-validation scheme for splitting the 50k training dataset into a 10k
+validation set and a 40k training set.
+Since there is no settled ResNet architecture for CIFAR-10 and CIFAR-100, for our experiments we chose the designs
+detailed in the original papers [1, 2] for these two datasets. Our implementation can be found on GitHub and to our
+knowledge, it is the most faithful public Keras implementation of these architectures.
+4.1
+Error rate estimation
+While we always trained for the same number of steps, we evaluated the best error rate by simulating early stopping.
+Let vk,j and tk,j denote the validation and test errors for replicate k and epoch j. We select the epoch m corresponding
+to the minimum validation error (averaging on the 5 replicates and on a moving window of 5 epochs), and then compute
+the average test error of the same 5 epochs and replicates:
+m := arg min
+i
+1
+25
+5
+�
+k=1
+2
+�
+j=−2
+vk,i+j,
+T := 1
+25
+5
+�
+k=1
+2
+�
+j=−2
+tk,m+j.
+This approach allows for evaluation of the statistical error of the estimator and it is more robust and reliable than
+simply taking the minimum of the test error, as in a real application one would be able to choose the early stopping on
+the validation set, but not on the test set.
+10
+
+Improving Performance in Neural Networks by Dendrites-Activated Connections
+Assuming that tk,m+j = µm + σ Zk + τ Zk,j, with µm the true value, Zk and Zk,j independent standard Gaussian
+noises, and σ, τ coefficients measuring randomness in replicates and epochs, the square of the standard error of T is
+Var(T) = 1
+5σ2 + 1
+25τ 2. Here the two terms were conservatively estimated using respectively:
+σ2 ≤ σ2 + τ 2 ≈ 1
+4
+5
+�
+k=1
+(tk,m − T)2
+and
+1
+5τ 2 ≈ 1
+4
+2
+�
+j=−2
+(t∗,m+j − T)2.
+4.2
+Baseline ResNets
+We experiment with 20, 32, 44, and 56 layers (n = 3, 5, 7, 9) both with the v1 post-activated [1] architecture and the
+v2 preactivated [2] architecture. Following the original paper scheme, we do not use bottleneck architectures since
+they give a significant improvement only for deeper networks (see also Remark 2). We mimic faithfully the training
+performed in the seminal paper [1] using the same settings, hyperparameters and data augmentation, except for the
+regularization: the original authors used weight decay of 10−4, corresponding to an L2-regularization of 0.5 · 10−4 on
+all weights, while we applied L2-regularization only on the kernel weights, fine-tuning the best coefficient on baseline
+ResNets, and then using the same coefficients for the DAC versions. We found that optimal values are 2 · 10−4 for
+CIFAR-10 and 3 · 10−4 for CIFAR-100.
+In order to explore the possibility of further performance improvement on lower learning rate levels, we have added
+an extra window to the learning rate schedule compared to the original implementation: the training is thus extended
+from 64k to 80k iterations with a further cut of 0.1 to the previous learning rate.
+4.3
+DAC networks
+Next, we train the corresponding DAC ResNet architectures as introduced in Section 3.3. The hyperparameters and
+network settings are the same as for baseline networks.
+As for the extra weights, namely the biases of each active dendrite connection, we adopt the same choices as the
+baseline biases, i.e. initialization to zero and no regularization.
+4.4
+Results
+Figure 5 compares the performances of the baseline networks and the corresponding ones in the DAC version. In
+order to identify a more robust and less noisy metric than simply choosing the least error on the test set, we report the
+average test error (over 5 replicates and 5 epochs) as described in Section 4.1. The error bars measure one standard
+error.
+The metric used to plot the x-axis are the FLOPs needed to complete a forward pass within the entire network. This
+was a key point in comparing architectures with different structures and computational costs such as baseline networks
+and their versions with the DAC implementation.
+To provide also a comparison with the overall best accuracy as is usually done in the literature, Table 1 shows the
+least error rates obtained on the test dataset (minimum of the average). Both the figure and the table describe the same
+picture. DAC networks show a significant performance improvement in almost all versions and benchmark datasets.
+Figures 6 and 7 show the training and test evolution graphs along all 256 epochs (in the case of Resnet20 v1). Sudden
+improvements in performance can be noted in correspondence with the cuts in the learning rate. It should be noted that
+in the central part of the training there is an accentuated overfitting phenomenon, probably due to the choice of hy-
+perparameters, including the high amount of regularization injected. Despite this phenomenon, the final performances
+correspond to those of the original papers [1, 2].
+To better appreciate the difference between baseline networks and the corresponding DAC versions, Figure 7 shows the
+same image restricted to the last 100 epochs. Deeper DAC networks outperform the corresponding baseline networks
+even with 12 fewer layers.
+4.4.1
+Imagenette and Imagewoof
+Imagenette and Imagewoof [13] are two subsets of ImageNet often used for model benchmarking because they provide
+a simple and faster alternative to ImageNet while preserving many of the inherent challenges. Imagenette is a subset of
+10 well-distinguished classes, while Imagewoof consists of 10 classes very similar to each other because they picture
+10 different dog breeds. For each dataset, we opted for its 160 pixels version (that is, the shortest side is resized to
+11
+
+Improving Performance in Neural Networks by Dendrites-Activated Connections
+Figure 5. Compared performances of ResNet baseline and DAC networks. DAC networks of the same number of layers have about
+1/9 more weights and 1/18 more FLOPs (on the horizontal axis), but in most cases it is apparent that the orange line is
+well below the blue line, meaning that the increase in performance between baseline and DAC is much larger than what
+would be justified by the increase in complexity.
+Best Performance Comparison
+Dataset
+CIFAR-10
+CIFAR-100
+Version
+v1
+v2
+v1
+v2
+Model
+Base
+DAC
+Base
+DAC
+Base
+DAC
+Base
+DAC
+20 layers
+8.64%
+8.27%
+8.73%
+8.32%
+33.50%
+32.53%
+33.49%
+32.62%
+32 layers
+7.90%
+7.56%
+7.84%
+7.55%
+31.65%
+31.09%
+31.79%
+31.04%
+44 layers
+7.68%
+7.26%
+7.14%
+7.09%
+30.39%
+30.42%
+30.50%
+29.92%
+56 layers
+7.33%
+7.12%
+7.01%
+6.88%
+30.53%
+30.32%
+29.89%
+29.31%
+Table 1. Comparison table of models best performances.
+160 pixels, with the aspect ratio maintained), which is further processed to a final size of 80x80 pixels. 5-fold cross-
+validation is performed, similarly to what we did for CIFAR-10 and CIFAR-100, although with different cardinality
+for training, validation, and test sets. After validation of the regularization parameter we chose 0.0002, that is, the
+value that maximized baseline performances. We then trained baseline and DAC ResNets with 20 and 32 layers. All
+other hyperparameters were the same as in previous sections.
+In Table 2 we display a comparison of the best performances. The baselines obtained were similar to those available in
+the literature for networks of similar complexity. Networks with DAC implementation outperformed baseline networks
+even with fewer layers, and the improvement was higher than in the case of CIFAR-10 and CIFAR-100.
+12
+
+Test error Cifarl0
+Resnet vl baseline
+0.090
+20 layers
+Resnet vl DAC
+0.085
+32 layers
+Test error
+44 layers
+0.080
+56 layers
+0.075
+0.070
+0.075
+0.100 0.125 0.150 0.175 0.200 0.225 0.250
+0.275
+(GIGA) FlopTest error Cifarl0
+20 layers
+Resnet v2 baseline
+0.090
+Resnet v2 DAC
+0.085
+32 layers
+Test error
+0.080
+0.075
+44 layers
+56 layers
+0.070
+0.075
+0.100 0.125 0.150 0.175 0.200 0.225
+0.250
+0.275
+(GIGA) FlopTest error Cifarloo
+20 layers
+Resnet v1 baseline
+0.34
+Resnet vl DAC
+0.33
+32 layers
+ error
+0.32
+Test
+56 layers
+44 layers
+0.31
+0.30
+0.075
+0.100 0.125 0.150 0.175 0.200 0.225 0.250
+0.275
+(GIGA) FlopTest error Cifarloo
+20 layers
+Resnet v2 baseline
+0.34
+Resnet v2 DAC
+0.33
+32 layers
+ error
+0.32
+Test
+44 layers
+0.31
+56 layers
+0.30
+0.075
+0.100 0.125 0.150 0.175 0.200 0.225 0.250
+0.275
+(GIGA) FlopImproving Performance in Neural Networks by Dendrites-Activated Connections
+Figure 6. Training evolution for ResNet v1 baseline and DAC networks for CIFAR-10 dataset. The dashed lines represent the
+training error, while the continuous lines represent the corresponding test error.
+Figure 7. The same evolution plots of Figure 6, but restricted to the last 100 epochs. DAC networks outperform the corresponding
+baselines, even with fewer layers. Compare, for example, the DAC network with 32 layers and the baseline with 44
+layers.
+Best Performance Comparison
+Dataset
+Imagenette
+Imagewoof
+Version
+v1
+v2
+v1
+v2
+Model
+Base
+DAC
+Base
+DAC
+Base
+DAC
+Base
+DAC
+20 layers
+13.41%
+11.88%
+11.97%
+11.78%
+23.19%
+22.60%
+22.61%
+21.70%
+32 layers
+13.13%
+11.75%
+11.40%
+11.27%
+23.05%
+21.32%
+21.75%
+20.65%
+Giga Flop
+0.509
+0.542
+0.509
+0.542
+0.865
+0.918
+0.865
+0.918
+Table 2. Comparison table for Imagenette and Imagewoof datasets. Giga Flops do not depend on the version of Resnet or the
+chosen dataset, as long as one chooses image with the same size.
+13
+
+Baseline models - Cifar-10
+0.30
+Resnetv1 20
+Resnetv1 32
+0.25
+Resnetv1 44
+Resnetv1 56
+0.20
+error
+0.15
+0.10
+0.05
+0.00
+0
+50
+100
+150
+200
+250
+epochsDAC models - Cifar-10
+0.30
+Resnetv1 20
+T
+Resnetv1 32
+0.25
+Resnetv1 44
+Resnetv1 56
+0.20
+error
+0.15
+0.10
+0.05
+0.00
+0
+50
+100
+150
+200
+250
+epochsTest error - Cifar-10. ResNet v1
+0.090
+base 20
+0.085
+base 32
+base 44
+error
+0.080
+base 56
+DAC 20
+DAC 32
+0.075
+DAC 44
+DAC 56
+0.070
+160
+180
+200
+220
+240
+epochsImproving Performance in Neural Networks by Dendrites-Activated Connections
+5
+Universal approximation
+In this section, we follow the idea proposed in [14] and the direct construction in [15] to prove a density result for the
+set of functions representable by a DAC fully connected neural network. In particular, we exploit the fact that ReLU
+can be used to build a narrow spike function, to show that its convolutions can approximate the identity operator, and
+hence continuous function defined on a compact set of Rd can be approximated by a DAC fully connected neural
+network having d DAC layers (and one replication layer, see Section 2.2, paragraphs on input replication), with a
+number of units in each layer growing at most linearly in the dimension d. This result is analogous to the one in [16].
+Theorem 1 (Universal approximation for DAC). Let Gd be the set of functions g : [−1, 1]d → R representable
+by a DAC fully connected neural network with ReLU preactivation and d layers, for d ≥ 1. Then Gd is dense in
+C0([−1, 1]d). In particular, for every function f ∈ C0([−1, 1]d) and every ε > 0, there exist a ReLU fully connected
+DAC neural network g ∈ Gd with d DAC layers such that ∥f − g∥∞ := supx |f(x) − g(x)| < ε. Moreover, for d ≥ 2,
+g is sparse, in the sense that there exists k := k(ε, f) ∈ N such that layer l = 1, 2, . . . , d − 1 of g has 2k + d − l units
+with no more than 4 input nodes each.
+Proof. The general idea to prove the theorem is to first build a family of densities ψd,δ that can approximate the
+Dirac delta in dimension d, and show that ψd,δ is actually in Gd, see (15). Then one can define the convolution
+operator Td,δ(f) := f ∗ ψd,δ and show that Td,δ approximates the identity on C0([−1, 1]d) when δ is small. Since the
+convolution is a d-dimensional integral, one can then approximate it with a sum over a finite partition of the domain,
+maintaining the desired accuracy by choosing an appropriate size of the mesh. Finally, one shows that the sum itself
+corresponds to a DAC network that merges one subnet for each of the terms of the summation.
+We start with the proof in the simplest case d = 1, which will be used for the induction in the general case.
+d = 1.
+Consider the set G1 of functions R → R given by a single-layer, one unit DAC neural network, with input
+replication as described in Section 2.2 and ReLU preactivation: g ∈ G1 if:
+g(x) =
+n
+�
+i=1
+wi ϕ(bi + x)
+for some integer n and parameters wi, bi for i = 1, . . . , n. Aim of this paragraph is to show that G1 is dense in
+C0([−1, 1]), giving a bound on the ‘size’ n of the DAC layer needed to approximate to some level a given function.
+Note that in this case there is only one unit in the layer, and n is the replication factor of the single input x ∈ R.
+To prove the claim one needs to find, for every continuous f and every desired accuracy ε, some g ∈ G1 such that
+∥f − g∥∞ < ε. To do this we start by building a spike function ψ1 : R → R:
+ψ1(x) :=
+�0
+|x| ≥ 1
+1 − |x|
+|x| < 1
+(7)
+Note that ψ1 ∈ G1, with the input x replicated 3 times:
+ψ1(x) = ϕ(−1 + x) − 2ϕ(x) + ϕ(1 + x) ∈ G1.
+(8)
+Linear rescalings of ψ1 are also in G1, in particular for δ > 0, we introduce:
+ψ1,δ(x) := δ−1ψ(δ−1x) = δ−2�
+ϕ(−δ + x) − 2ϕ(x) + ϕ(δ + x)
+�
+.
+(9)
+Notice that
+� 1
+−1 ψ1,δ(x)dx = 1 and ψ1,δ(x) = 0 for |x| > δ, so if we consider the convolution operator T1,δ(f) :=
+f ∗ ψ1,δ, defined by:
+f ∗ ψ1,δ(x) :=
+� 1
+−1
+f(t)ψ1,δ(x − t)dt
+(10)
+then by the approximation property of the convolution, T1,δ(f) → f uniformly for δ → 0, meaning that:
+∀f ∈ C0([−1, 1]), ∀ε > 0, ∃δ > 0 s.t. ∥f − f ∗ ψ1,δ∥∞ < ε/2.
+The next step is approximating the integral in (10) with a finite sum that happens to be an element of G1. The integrand
+function t �→ h(t) := f(t)ψ1,δ(x − t) is continuous, so there exists an integer k such that:
+����
+� 1
+−1
+h(t)dt −
+k
+�
+j=1
+h
+�2j − 1
+k
+− 1
+� 2
+k
+���� < ε/2.
+14
+
+Improving Performance in Neural Networks by Dendrites-Activated Connections
+In principle, this k might depend on x, but since we are in a compact set, it is clear that one can choose a k that satisfies
+the inequality for all x. For the sake of notation simplicity, let tj := 2j−1
+k
+− 1 and define g(x) := �k
+j=1
+2
+k h(tj),
+yielding ∥f ∗ ψ1,δ − g∥∞ < ε/2 and so ∥f − g∥∞ < ε as desired.
+We are only left to show that g ∈ G1. In fact, expanding the definitions of h and ψ1,δ, and letting wj := 2
+k f(tj)δ−2,
+we get:
+g(x) =
+k
+�
+j=1
+2
+k f(tj)ψ1,δ(x − tj) =
+k
+�
+j=1
+�
+wjϕ(−tj − δ + x) − 2wjϕ(−tj + x) + wjϕ(−tj + δ + x)
+�
+.
+(11)
+The right side of (11) shows that g ∈ G1 with the input x replicated 3k times, that is, g is a single-layer, one unit DAC
+neural network with 3k (replicated) input nodes approximating f uniformly with accuracy ε. The weights and DAC
+biases of g are a combination of those of ψ1,δ from (9) with wj and tj. This is explicitly written in the right-hand side
+of (11), where we can see that n = 3k, the weights are wj, −2wj, wj and the DAC biases are −tj − δ, −tj, −tj + δ,
+for j = 1, . . . , k. This concludes the proof in the case d = 1.
+d ≥ 2.
+The first step is to build a spike function ψd : Rd → R in dimension d. As in (7), we can define:
+ψd(x) :=
+�0
+∥x∥1 ≥ 1
+1 − ∥x∥1
+∥x∥1 < 1
+(12)
+where ∥x∥1 := �d
+j=1 |xj|. Consider ˜ψ : R × R+ → R defined by:
+˜ψ(x, t) := ϕ(x − t) − 2ϕ(x) + ϕ(x + t) = (t − |x|)+.
+If x = (x1, . . . , xd), let ˜x = (x1, . . . , xd−1). Then:
+ψd(x) =
+�
+1 −
+d
+�
+j=1
+|xj|
+�+
+= ˜ψ
+�
+xd,
+�
+1 −
+d−1
+�
+j=1
+|xj|
+�+�
+= ˜ψ(xd, ψd−1(˜x)),
+so that:
+ψd(x) = ϕ(xd − ψd−1(˜x)) − 2ϕ(xd) + ϕ(xd + ψd−1(˜x)).
+(13)
+The latter is a recursive expression for ψd, starting from ψ0 = 1 that can be used to construct a DAC neural network
+computing this function.
+We have already seen that ψ1 can be obtained as a 1 layer, 1 unit DAC network with 3 (replicated) input nodes. Then
+x2 + 1 = ϕ(x2 + 1) can also be obtained as 1 unit on the first layer (with one input node); by linearity, the same holds
+for the two terms ϕ(x2 + 1) ± ψ1(x1), with 4 input nodes (x1 replicated thrice and x2 taken once). Then:
+ψ2(x1, x2) = ϕ(−1 + x2 + 1 − ψ1(x1)) − 2ϕ(−1 + x2 + 1) + ϕ(−1 + x2 + 1 + ψ1(x1))
+can be obtained as a 2-layer DAC network, with 1 unit (with 3 inputs and all DAC biases equal to −1) in layer 2 and
+3 units (with 4, 1, and 4 input nodes) in layer 1.
+One can generalize this approach to a higher dimension d ≥ 3 as follows. Let τi denote ψi(x1, . . . , xi), for the sake
+of simplicity. In the generic layer i = 2, . . . , d − 1 there will be two DAC units (with 4 input nodes and three biases
+equal to −1) computing:
+�xi+1 + 1 + τi = ϕ(xi+1 + 1) + ϕ(−1 + xi + 1 − τi−1) − 2ϕ(−1 + xi + 1) + ϕ(−1 + xi + 1 + τi−1)
+xi+1 + 1 − τi = ϕ(xi+1 + 1) − ϕ(−1 + xi + 1 − τi−1) + 2ϕ(−1 + xi + 1) − ϕ(−1 + xi + 1 + τi−1)
+(14)
+and d − i additional units to pass on the variables xi+1, . . . , xd, through xj + 1 = ϕ(xj + 1). Layer 1 will be similar
+but with τ0 = 1. Layer d will have just 1 unit, computing:
+ψd(x) = τd = ϕ(−1 + xd + 1 − τd−1) − 2ϕ(−1 + xd + 1) + ϕ(−1 + xd + 1 + τd−1).
+It’s straightforward to see that this DAC network computes ψd(x) following the recursion given by (13), using d layers
+with d + 1, d, . . . , 4, 3, 1 units in this order (see also Figure 8).
+Later we will need to merge shifted copies ψd(x − c) of this function, for several values of c, and this can be done
+efficiently by sharing the d − i additional units of layer i and duplicating just the two units of equation (14). Let
+τi := ψi(x1 − c1, . . . , xi − ci) and τ0 = 1. Then at layer i, we can have a DAC unit with inputs and biases
+input
+bias
+xi+1 + 1
+0
+xi + 1 − τi−1
+−1 − ci
+xi + 1
+−1 − ci
+xi + 1 + τi−1
+−1 − ci
+15
+
+Improving Performance in Neural Networks by Dendrites-Activated Connections
+DAC
+DAC
+DAC
+x1
+d = 1
+x1
+x1
+ψ1(x1)+x2 + 1
+x1
+d = 2
+x2
+x2
+−ψ1(x1) + x2 + 1
+ψ2(x1, x2)+x3 + 1
+x2 + 1
+d = 3
+x3
+x3
+x3 + 1
+−ψ2(x1, x2) + x3 + 1
+ψ3(x1, x2, x3)
+x3 + 1
+Figure 8. This figure illustrates how the d-layers network corresponding to ψd can be inductively built from the d − 1-layers
+network corresponding to ψd−1, for d = 1, 2, 3. When d = 1, only the red elements in the figure exist, corresponding
+to (8): after replication of x1, the three red arrows perform preactivation with biases −1, 1, 0 and weights 1, 1, −2, to
+produce ψ1. When d = 2, the green elements are included on top of the red ones. The new variable x2 is taken forward
+as ϕ(x2 + 1) = x2 + 1 and combined with the appropriate weights to produce 3 ReLU arguments needed in (13) for
+d = 2. These are the new layer 1 outputs, that are combined with weights 1, 1, −2 (and DAC biases −1, −1, −1 to
+recover the original value of x2), to produce the layer 2 output ψ2. When d = 3, the blue elements are included on top
+of the red and green ones. The same operations performed for d = 2 with the red network nodes are now performed with
+the green network nodes, but 1 layer forward. This produces the layer 3 output ψ3.
+computing:
+xi+1 + 1 + τi = ϕ(xi+1 + 1) + ϕ(−1 − ci + xi + 1 − τi−1) − 2ϕ(−1 − ci + xi + 1) + ϕ(−1 − ci + xi + 1 + τi−1)
+and analogously a second DAC unit will compute xi+1 + 1 − τi. In particular, these two units compute functions of x
+that depend also on c and hence must be duplicated for different c, but the units transporting xj +1 for j = i+1, . . . , d
+should be shared.
+The second step is rescaling ψd with a positive constant δ > 0 and the normalization coefficient Cd = 2−d(d + 1)!
+that makes the integral equal to 1, yielding an approximation of the Dirac delta ψd,δ : Rd → R:
+ψd,δ(x) := Cdδ−dψd(δ−1x).
+(15)
+To ensure that also ψd,δ can be realized as a DAC network of the same structure as ψd, we observe that for a DAC unit
+g defined by g(x) = �n
+j=1 wj ϕ(bj + xj) and for λ > 0:
+g(λx) =
+n
+�
+j=1
+wj ϕ(bj + λxj) =
+n
+�
+j=1
+λwj ϕ(bj/λ + xj) =: ˜g(x)
+so that given any multi-layer DAC network h, by simply changing the coefficients of the units in the first layer, it is
+possible to obtain another DAC network ˜h such that ˜h(x) = h(λx). By linearity, the outer coefficient Cdδ−d can be
+absorbed into the weights of the last layer, and we get ψd,δ ∈ Gd with the same structure of ψd.
+The third and last step of the proof is similar to the case d = 1. First, we consider the convolution operator Td,δ(f) :=
+f ∗ ψd,δ, which by the approximation property of the convolution ensures that:
+∀f ∈ C0([−1, 1]d), ∀ε > 0, ∃δ > 0 s.t. ∥f − f ∗ ψd,δ∥∞ < ε/2.
+(16)
+Then, fixed f, we approximate the integral in (16) with a finite sum. To this aim, let h(y, x) := f(y)ψd,δ(x − y),
+so that f ∗ ψd,δ(x) =
+�
+h(y, x)dy. Choose a partition of [−1, 1]d by a large number k of small sets (e.g. cubes) of
+average volume 2d/k. Consider a set C of k points, one for each of the sets (e.g. the center points of the cubes). Since
+h is continuous and the hyper-cube is compact, there exists a partition fine enough that for all x:
+�����
+�
+[−1,1]d h(y, x)dy −
+�
+c∈C
+h(c, x)2d
+k
+����� < ε/2.
+16
+
+Improving Performance in Neural Networks by Dendrites-Activated Connections
+Therefore we define g(x) := �
+c∈C 2dk−1h(c, x), yielding ∥f ∗ ψd,δ − g∥∞ < ε/2 and so ∥f − g∥∞ < ε.
+Finally, g can be realized as a DAC neural network with d layers, since the same holds for ψδ,d, and we need only to
+compute a linear combination of the output of k translated versions of the latter:
+g(x) :=
+�
+c∈C
+2dk−1f(c)ψd,δ(x − c) =:
+�
+c∈C
+wcψd,δ(x − c).
+By linearity, g ∈ Gd, can be realized as a DAC network with d layers and recalling that only two units in each layer
+must be duplicated to account for different values of c, while the others can be shared, we get that the layers 1, 2, . . . , d
+have respectively 2k + d − 1, 2k + d − 2, . . . , 2k + 2, 2k + 1, 1 units.
+Remark 3. Case d = 1 in Theorem 1 shows that a single-layer DAC neural network is a universal approximator in
+C0([−1, 1]). This is not true for a standard single-layer neural network, where input replication has no effect.
+Remark 4. Proof of Theorem 1 can be easily extended to functions with compact support.
+Remark 5. We observe that the d-dimensional spike function ψd appearing in the proof, can also be realized as a
+2-layer DAC network as follows:
+ψd(x) = ϕ
+�
+1 − d +
+d
+�
+j=1
+�
+ϕ(xj + 1) − 2ϕ(xj)
+��
+.
+In fact, ϕ(x + 1) − 2ϕ(x) is positive only for |x| < 1, and equals 1 − |x| on that interval. Therefore, the sum above is
+equal to d − ∥x∥1 inside [−1, 1]d and lesser than d − 1 outside, so thanks to the outer ϕ, we obtain ψd everywhere.
+Using this construction, one could prove a result similar to Theorem 1, but with a DAC network with 2 layers, regard-
+less of d. This network won’t be sparse, as here we are trading a d-layers-deep representation with 4 or fewer input
+nodes per unit, for a shallow one with 2d input nodes per unit.
+6
+Future work and limitations
+In our opinion, the most important limitation and possible future development of DAC, as of now, is its current high-
+level implementation. As already mentioned in Section 3.5, an efficient low-level implementation of DAC could
+potentially yield just a modest computational complexity increase with respect to the corresponding non-DAC coun-
+terpart (of the same order as the increase in the number of weights). Currently, the memory needs and training times
+of DAC architectures are 5-10 times larger than that. Low-level implementation would allow experimenting DAC with
+larger network structures, like the ones that are needed for datasets like ImageNet. Moreover, this would be the first
+step for integrating DAC in frameworks like PyTorch and Keras, making it an option for standard layers, similar to
+what we did in our high-level implementation.
+Experimenting with activation functions other than ReLU is another possible development. One should test whether
+the performance benefits from a different dendrites activation, and quantify the corresponding increase in computa-
+tional complexity. A similar possible improvement is the following: we described in Section 3.2 a DAC implementa-
+tion for convolutional layers with DAC biases that do not depend on the position in the kernel but only on the input
+and output channels. One could test whether the increase in computational complexity given by position-dependent
+biases is overcome by a better performance. Different activations and position-dependent biases could be reasonably
+tested only after a low-level and efficient DAC implementation.
+A natural question is in which way DAC improves expressivity. To answer this question, one could estimate some
+model complexity measure, like the classical VC dimension, the recently introduced geometric complexity [17] or the
+effective model complexity proposed in [18]. In this analysis, one could also measure how much input replication
+within layers is effective in terms of expressivity.
+Visualizing the loss landscape using techniques like in [19] could also be useful for understanding dendrites activation
+effects on a neural network.
+In this paper, we did not use any activation other than DAC preactivation. However, in the biological neuron, after
+the dendritic preactivation, the signals flow through the soma and are jointly activated where the axon connects. This
+suggests as a further development of DAC that using an additional post-activation after DAC preactivation could be
+useful.
+Another future work we are planning is the analysis of the filters learned by a DAC convolutional tower in the first
+layers: are these filters similar to the ones learned in the non-DAC case? If yes, what are the differences?
+17
+
+Improving Performance in Neural Networks by Dendrites-Activated Connections
+Finally, the experimental part of this paper is concerned with the performance of DAC-ResNet neural networks. One
+could analyze how DAC could be used to improve the performance in other convolution-based architectures like
+Inception, Xception, SENet and ResNeXt. The same analysis could be performed on a wider range of architectures,
+like recurrent neural networks, Transformers, GANs and autoencoders, to name a few.
+Acknowledgements.
+Computational resources for this work were provided by CLAI laboratory of Chieti-Pescara.
+The authors wish to thank Rosa Gini for her important intellectual contribution.
+References
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+putational Explorations,” Neuroscience, vol. 489, pp. 111–142, May 2022.
+[7] P. Poirazi and A. Papoutsi, “Illuminating dendritic function with computational models,” Nature Reviews
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+edu/∼kriz/cifar.html.
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+Improving Performance in Neural Networks by Dendrites-Activated Connections
+[19] H. Li, Z. Xu, G. Taylor, C. Studer, and T. Goldstein, “Visualizing the loss landscape of neural nets,”
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+
diff --git a/odAzT4oBgHgl3EQfAPrQ/content/tmp_files/load_file.txt b/odAzT4oBgHgl3EQfAPrQ/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..c16e654d32e4ffa2534e41f48b69850beae24adc
--- /dev/null
+++ b/odAzT4oBgHgl3EQfAPrQ/content/tmp_files/load_file.txt
@@ -0,0 +1,833 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf,len=832
+page_content='IMPROVING PERFORMANCE IN NEURAL NETWORKS BY  DENDRITES-ACTIVATED CONNECTIONS  ARXIV PREPRINT  Carlo Metta∗  Marco Fantozzi  Andrea Papini†  Gianluca Amato‡  Matteo Bergamaschi§  Silvia Giulia Galfr`e¶  Alessandro Marchetti‡∥  Michelangelo Vegli`o‡  Maurizio Parton‡∗∗  Francesco Morandin§  CuriosAI Lab  Code at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='com/curiosai/dac-dev  Contact authors at curiosailab@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='com  ABSTRACT  Computational units in artificial neural networks follow a simplified model of biological neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In  the biological model, the output signal of a neuron runs down the axon, splits following the many  branches at its end, and passes identically to all the downward neurons of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Each of the  downward neurons will use their copy of this signal as one of many inputs dendrites, integrate them  all and fire an output, if above some threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In the artificial neural network, this translates to  the fact that the nonlinear filtering of the signal is performed in the upward neuron, meaning that in  practice the same activation is shared between all the downward neurons that use that signal as their  input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Dendrites thus play a passive role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  We propose a slightly more complex model for the biological neuron, where dendrites play an active  role: the activation in the output of the upward neuron becomes optional, and instead the signals  going through each dendrite undergo independent nonlinear filterings, before the linear combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  We implement this new model into a ReLU computational unit and discuss its biological plausibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  We compare this new computational unit with the standard one and describe it from a geometrical  point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' We provide a Keras implementation of this unit into fully connected and convolutional  layers and estimate their FLOPs and weights change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' We then use these layers in ResNet architec-  tures on CIFAR-10, CIFAR-100, Imagenette, and Imagewoof, obtaining performance improvements  over standard ResNets up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='73%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Finally, we prove a universal representation theorem for con-  tinuous functions on compact sets and show that this new unit has more representational power than  its standard counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Keywords active dendrite, dendritic neural model, preactivation, multi-bias, image classification, ResNet  1  Introduction  Historically the structure of the perceptron, the artificial neural network’s fundamental computational unit, has rarely  been questioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The biological inspiration is straightforward: input signals from the dendrites are accumulated at the  soma (with a linear combination), and if the result is above the activation threshold (that is, the opposite of the bias)  there is a nonlinear reaction, as the neuron fires along the axon (with the activation function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  ∗ISTI-CNR Pisa, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  †Scuola Normale Superiore, Pisa, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  ‡University of Chieti-Pescara, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  §University  of Parma, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  ¶University of Pisa, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  ∥Alessandro Marchetti is a PhD student enrolled in the National PhD in  Artificial Intelligence, XXXVII cycle, course on Health and life sciences, organized by Campus Bio Medico University of Rome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  ∗∗Partially funded by INdAM group GNSAGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='00924v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='NE]  3 Jan 2023  Improving Performance in Neural Networks by Dendrites-Activated Connections  Nevertheless, in time the sigmoid activation function was abandoned in favor of ReLU and variants, and the biolog-  ical analogy became less stringent, shifting focus on the desirable mathematical properties of the class of functions  computed by the networks, like representation power and non-vanishing gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  This has brought us to the current situation in which most units output their signal through a nonlinear activation  function which effectively destroys some information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In fact, ReLU is not invertible, as it collapses to zero all  negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Though some of its variants may be formally invertible (leaky ReLU and ELU for example), the fact  that they overall perform in a way very similar to ReLU, suggests that their way of compressing negative values with  small derivatives leads to the same general properties of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  In this paper, we propose and test a radical rethinking of the standard ReLU-like computational unit, where the output  brings its full, uncorrupted information to the next units, and the activation function is applied there, with a variable  bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' One can say that the proposed units are preactivated instead of post-activated, and from the biological point  of view this is like having dendrites with different thresholds passing the signal to the cell soma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In this sense,  dendrites are ‘active connections’ with the soma, and correspondingly we call them ‘DAC’, for ‘Dendrites-Activated  Connections’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  This kind of reversed view has already proved fruitful in the evolution from ResNets v1 [1] to v2 [2] when a com-  prehensive ablation study showed that the best choice for residual networks is keeping the information backbone free  of activations for maximum information propagation, and preactivate the convolutional layers in the residual branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Much earlier in the past, the celebrated LSTM [3] architecture used already an activation-free backbone as long mem-  ory, even tough inputs and gates have a standard post-activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Here this view is taken forward: if the block bias + activation is better placed at the input of a unit, then the bias itself  should be specific to each unit, as the weights are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' (In fact, this choice turns out to be natural from both mathematical  and biological points of view, as will be shown in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=')  In Section 2 we first discuss in detail the idea of dendritic preactivation, which is the core of DAC computational units,  as compared to post-activation, which is used in standard units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' We then motivate DAC units from a geometrical and  biological point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  In Section 3 we describe in detail our Keras implementation of DAC fully connected and convolutional layers, and  how we use these layers in ResNet architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Moreover, we perform an efficiency and memory analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  In Section 4 we describe our experiments on CIFAR-10, CIFAR-100, Imagenette, and Imagewoof comparing DAC  and standard models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In particular, Tables 1 and 2 report the main empirical results of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Section 5 is devoted to the theoretical properties of DAC units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In particular, we show that every function f ∈  C0([−1, 1]d) can be approximated by a ReLU fully connected DAC neural network with d DAC layers and (d+2−l)k  units in layer l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , d − 1, where k is a constant depending on f and the desired approximation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' See  Theorem 1 and the next remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Finally, in Section 6 we describe future developments of the DAC framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  2  Motivation  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='1  From standard to Dendrites-Activated Connection (DAC) unit  To describe our idea, it is better to look at a neural network as a directed acyclic computational graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' We denote the  set of its nodes by I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' If i ∈ I is a node, we denote its parents (in-neighbors) by Ii ⊂ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In what we call the standard  model for computational units in a neural network, a bias bi, a set of weights wi,j for j ∈ Ii and a nonlinearity ϕi are  associated with every node i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  The flow of the network in the standard model can then be described in terms of computational units that, starting from  input values stored in source nodes, update the value yi stored in node i by the values stored in its parents:  yi = ϕi  �  bi +  �  j∈Ii  wi,j yj  �  (standard unit)  (1)  In this paper, ϕi is always ϕ = ReLU, and in particular, the theoretical results in Section 5 are strictly dependent on  this choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' However, using ReLU is not strictly necessary for this paper idea, see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  2  Improving Performance in Neural Networks by Dendrites-Activated Connections  y4 = ϕ(b4 +  3  �  i=1  w4,iyi)  y3  y2  y1  I4 = {1, 2, 3}  w4,1  w4,2  w4,3  y4 = ϕ(b4 + z4)  z4 =  3  �  i=1  w4,iyi  y3 =  ϕ(b3 + z3)  y2 =  ϕ(b2 + z2)  y1 =  ϕ(b1 + z1)  I4 = {1, 2, 3}  w4,1  w4,2  w4,3  ϕ ◦ b  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' A unit with 3 input nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The set I of nodes here is {1, 2, 3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The standard unit on the top is ‘(activation ◦ bias ◦  linearity)’, represented all together as a single output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' On the bottom, the same output is decomposed at a finer scale as  ‘(activation ◦ bias) ◦ linearity’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  In the literature, according to the purpose of the work at hand, computations may be described at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' For  instance, convolutions can be described on a more granular scale, where three stages (affine map, nonlinearity, and  pooling) are considered separately, or on a coarser scale, where the composition of the three stages is considered as a  whole [4, Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' For the purpose of this paper it is convenient to look at the standard unit as a nonlinear filter  ‘activation ◦ bias’ applied to some information linearly aggregated from parents:  �  zi = �  j∈Ii wi,j yj  linear aggregation  yi = ϕ(bi + zi)  nonlinear filter  Figure 1 exhibits this point of view emphasizing that the values in parent nodes are themselves built from grandparents  with biases b1, b2, b3 and ReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' For each parent node, this bias is uniquely determined, meaning that from the point  of view of children nodes it is shared, in the sense that ‘4’ and other children (not represented here) must use the  same input with the same parent bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' This forced sharing of biases followed by ReLU, which flattens to zero negative  values, means that the information arriving at children has in some sense deteriorated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  To mitigate this loss of information, it seems reasonable to apply the nonlinear filter directly to the parents with a  non-shared bias, before the linear aggregation, see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  In this paper, we investigate this idea, by studying a new computational unit briefly described as ‘linearity ◦ (activation  non-shared bias)’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Since the biases in the nonlinear filter can depend on both input and output nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' on the  edges of the graph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' and since these edges correspond to dendrites in biological neurons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' we call this new computational  unit a Dendrites-Activated Connection unit (DAC unit):  zi =  �  j∈Ii  wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='j ϕ(bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='j + zj)  (DAC unit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' example in Figure 2)  (2)  One can view DAC as a preactivated unit with non-shared biases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' in the sense that DAC units sharing the same input  can use different nonlinearity thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  3  Improving Performance in Neural Networks by Dendrites-Activated Connections  3  �  i=1  w4,iϕ(b4,i + zi)  ϕ(b4,1 + z1)  z3  ϕ(b4,2 + z2)  z2  ϕ(b4,3 + z3)  z1  I4 = {1, 2, 3}  ϕ ◦ b4,1  ϕ ◦ b4,2  ϕ ◦ b4,3  w4,1  w4,2  w4,3  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' A DAC unit corresponding to the standard unit in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Note that input biases contributing to the output value of this  unit depend also on the output node i = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' We call this feature non-shared biases because it allows DAC units sharing  the same input to use different (non-shared) thresholds instead of a single (shared) input bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='2  Comparison of DAC versus standard  DAC biases must depend on input and output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  In a DAC unit, it is crucial for the filter to depend on both the input  and the output nodes (non-shared biases), and not only on the input node (shared biases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In fact, if one considers a  multilayer perceptron, the equation for standard units gives:  yi = ϕ  �  bi +  �  j∈Ii  wi,j yj  �  = ϕ  �  bi +  �  j∈Ii  wi,j ϕ  �  bj +  �  k∈Ij  wj,k ϕ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' )  ��  (standard unit)  and the equation for preactivated units with shared biases gives:  zi =  �  j∈Ii  wi,j ϕ(bj + zj) =  �  j∈Ii  wi,j ϕ  �  bj +  �  k∈Ij  wj,k ϕ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' )  �  (preactivated unit)  However, if this was the only change, the modification would only be formal, and the resulting network would perform  the same operations, as can be seen by comparing the right-hand sides of the above equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' For this reason, we  must use non-shared biases as in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Last and first layers with DAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  This change of perspective has other natural consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Consider the last layer  of a layered network: with standard connections, one would expect a ReLU activation in the output of the last layer,  but this is typically not desired, as the output of the last layer must be as much informative as possible, and of the  right type to be plugged into the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' So one usually has to remove that last activation, which with a DAC  connection would not have existed in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Here DAC seems more natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Symmetrically, consider the first layer of a layered network: with a standard connection the input nodes are unaltered  and have full information – it would not make sense to filter them with ReLU with or without an input-depending  bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Nevertheless, with a DAC connection, it is instead very reasonable to apply the nonlinearity to the input nodes,  because the bias used to filter them may depend on the output nodes, and it is not unlikely that different output nodes  in the first layer might benefit from node-wise filtering of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Here DAC could be more capable of exploiting  feature diversities in the first layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In this paper, we adhere to the terminology from [4]: whenever the network function f is a composition  f (L)◦· · ·◦f (1), we say that the network is layered, and composed of L layers f (l), for l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' If f (l) : Rm → Rn,  we say that layer l has m input nodes and n output nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' When we desire to point out the units in their entirety, and  not only the output nodes, we say that layer l has n units, each of them with m input nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Input replication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Filtering the input as just described might be even more useful if the input is replicated multiple  times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Consider the toy example of a one-dimensional input x and a shallow network with only one layer of one  4  Improving Performance in Neural Networks by Dendrites-Activated Connections  unit aiming at approximating some function f : R → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' A DAC fully connected layer with one unit ‘0’ and input  replicated n times x = (x, x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , x) gives:  ˆfDAC(x) =  n  �  j=1  w0,j ϕ(b0,j + x)  (3)  which is a universal approximator of a large class of functions R → R, for n → ∞ (in particular in Section 5 we  rigorously prove that ˆf is a universal approximator of C0([−1, 1]), see Theorem 1 and Remark 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  On the other hand, a standard fully connected layer with replicated inputs and one unit would give:  ˆfcl(x) = ϕ  �  b0 +  n  �  j=1  w0,j x  �  = ϕ(b0 + ˜w0 x)  regardless of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' To gain expressivity with the standard connection we can add a hidden layer with n units (with or  without replicated inputs is the same), obtaining:  ˆf2×cl(x) = ϕ  �  b0 +  n  �  j=1  w0,j ϕ(bj + ˜wj x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  To show that ˆf2×cl has a representation power similar to ˆfDAC, we put ˜w0,j = w0,j| ˜wj| and ˜bj = bj/| ˜  wj| in the above  expression, which gives:  ˆf2×cl(x) = ϕ  �  b0 +  n  �  j=1  ˜w0,j ϕ  �˜bj + sign( ˜wj) x  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  This discussion shows that, in this toy problem of approximating a function f : R → R, one needs a two-layers  standard network to get a representation power similar to a single-layer DAC network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Input replication inside the computational graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  From a biological perspective, replicating the input in a DAC  unit would correspond to different dendrites of the same downward neuron being connected to different branches of  the axon of the same upward neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' This is something that has not been scrutinized in particular in the biological  literature, but definitely cannot be excluded, given the complexity of dendrite activation, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='3 for more  details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' This suggests that input replication can be useful in principle at any node in the computational graph, not only  on its inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Thus, when there is a layer structure on the network, one can distinguish between ‘replication’ layers  and DAC layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' With this terminology, the toy example (3) has one replication layer and one DAC layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  A thorough empirical analysis of the effects of replication on DAC performance would first require a baseline without  replication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' For this reason, and because this is a first paper on the DAC paradigm, we have decided to not use  replication in the experiments described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' We leave this analysis to future work, see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='3  Biological Inspiration  The proposed extension of the artificial neuron also reflects to some extent a recent shift in the understanding of the  biological neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In fact, the early soma-centric representation of the neuron today has been discarded in favor of a  more realistic and complex model that incorporates active dendrites [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  A typical biological neuron consists of many input branches called dendrites, a main body called soma, and the axon,  which branches at its end in many terminals, where synapses connect to the dendrites of other neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The input  signals originate in the dendrites, flow through the soma, and are integrated into the region of the soma where the axon  connects, and if a specific threshold is reached, the neuron fires its signal down the axon, to the synapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Until some years ago, the biological neuron model was soma-centric and essentially modeled by a point neuron where  dendrites simply pass the signals, and all elaboration happens at the soma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' This elementary representation was the  inspiration of the traditional perceptron in artificial neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Current biological models are more complicated and the central role of dendrites in signal modulation is better un-  derstood [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Dendrites in fact present voltage-gated ion channels [5] able to produce local electrical events termed  dendritic spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Dendrites actually present at least four groups of ion channels [6]: the synaptic receptors, activated  by neurotransmitters, the passive leak channels, the active subthreshold ion channels, able to produce transmembrane  currents also when the threshold for the action potential is not reached, and supra-threshold ion channel active when  the threshold is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In this way, a dendrite or a group of dendrites can perform the first important local, not linear  signal integration before reaching the cell axon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  5  Improving Performance in Neural Networks by Dendrites-Activated Connections  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='4  Geometric interpretation  In a fully connected DAC layer, DAC equation (2) can be written in vectorial form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Let m, n be the number of input  and output nodes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Denote by z ∈ Rm the input, by ˆz ∈ Rn the output, by b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , bn ∈ Rm the DAC  biases, by w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , wn ∈ Rm the weights, by ⟨·, ·⟩ the scalar product in Rm and by ϕ component-wise ReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' DAC  equation (2) then becomes:  ˆz(z) = (⟨w1, ϕ(b1 + z)⟩, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , ⟨wn, ϕ(bn + z)⟩)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  (4)  Equation (4) says that a DAC layer ˆz factorizes as ˆz = pw◦ˆzb, where ˆzb is a nonlinear embedding into a higher (unless  n = 1) dimensional space, depending only on DAC biases and not on weights:  ˆzb : Rm −→  n times  �  ��  �  Rm × · · · × Rm  z �−→ (ϕ(b1 + z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , ϕ(bn + z))  and pw is a multilinear map, depending only on weights and not on DAC biases, projecting back each of the n  components of Rm × · · · × Rm orthogonally onto the axes wi of Span{w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , wn} ⊂ Rm × · · · × Rm:  pw :  n times  �  ��  �  Rm × · · · × Rm −→ Span{w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , wn} = Rn  (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , yn) �−→ (⟨w1, y1⟩, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , ⟨wn, yn⟩)T  The multilinear map pw can be seen also as the corresponding linear map p⊗  w on the tensor product Rm ⊗ · · · ⊗ Rm,  and the whole factorization can be visualized with the commutative diagram (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='4):  Rm  Rn = Span{w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , wn}  Rm × · · · × Rm  �  ��  �  n times  Rm ⊗ · · · ⊗ Rm  �  ��  �  n times  ˆzb  ⊗  p⊗  w  ˆz  pw  Thus, DAC preactivation can be seen as an intermediate step: a nonlinear embedding into a space of equal (if n = 1)  or increased (if n > 1) dimension, followed by a projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Note that a standard layer is a linear map Rm → Rn  followed by the nonlinearity, and as such does not have a similar geometric interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  In general, maintaining or increasing the dimension, before a last classification layer, can make a problem easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  In Figure 3 we provide a toy example of a binary classification problem that cannot be solved by a standard layer  R2 → R, but can be solved by its DAC counterpart, thanks to the intermediate embedding step maintaining the input  dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In Figure 4 another toy example, where a DAC layer R2 → R2 can separate a dataset thanks to the  intermediate embedding doubling the input dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  3  Methods  ResNets [1, 2], SE-Nets [8] and Transformers [9] are some popular architectures that can be implemented in a modular  way, using standard layers from highly optimized frameworks like PyTorch or Keras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' On the other hand, DAC needs  to be implemented within standard layers, and as such cannot easily be optimized, see implementation issues in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  In this section, we describe equations, complexity, and implementation issues for DAC versions of fully connected  and convolutional layers, and for ResNet architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='1  DAC fully connected layers  A fully connected layer with dendrites-activated input connections, m inputs, and n outputs, in its most general form,  is the following:  dDAC(y) = dDAC  i  (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , ym) = ψ  �  bi +  m  �  j=1  wi,j ϕ(bi,j + yj)  �  ,  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , n  (5)  6  Improving Performance in Neural Networks by Dendrites-Activated Connections  (a)  (b)  bias  (c)  ReLu  (d)  DAC  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' (a) We want to separate the black point from the white points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' For this, we learn a DAC one-layer f : R2 → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' (b) DAC  can learn a translation (DAC biases) that moves the leftmost point to the second quadrant (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' (c) ReLU projects  all points in the second quadrant onto the vertical axis (red line), making the dataset linearly separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' (d) DAC can now  learn a direction (blue line) onto which to project the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The black point is now separated from the white points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  (a)  1  1  1  1  (b)  1  1  1  1  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Given a square like the one on the left, the combination of a DAC layer f : R2 → R2 and a linear layer g : R2 → R can  have positive values inside and negative outside, being able to separate the blue and orange points on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In fact,  let [f(x)]i = �  j=1,2 wijϕ(bij + xj) with ϕ denoting ReLU, then it is enough to set wij = 1, b1j = 1, b2j = 0 and  g(y) = y1 − 2y2 − 1 to get g(f(x)) = ϕ(1 + x1) + ϕ(1 + x2) − 2ϕ(x1) − 2ϕ(x2) − 1, which equals 1 − |x1| − |x2|  inside the square and is negative outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' On the contrary, it is easy to see that if f was a standard fully connected layer,  then for all choices of f and g the set where g(f(x)) ≥ 0 would always be unbounded or empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  where ψ is an optional output activation, bi is an optional output bias, ϕ is the ReLU input activation and bi,j are the  DAC biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' When stacking consecutive DAC layers, ψ and bi will typically not be used, but they might in general be  needed, for example, if the subsequent operation is a global average pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  This layer will add m × n new biases bi,j, for a roughly doubled total number of weights, hence one must use care in  comparing DAC and regular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The increase in the number of FLOPs is about 50% (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='2  DAC convolutional layers  Consider a 2D convolutional layer with dendrites-activated input connections, m input channels, and n output L × L  kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In its most general form, the DAC biases would depend on the m input channels, on the n output kernels,  and also on the position inside the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' However, this seems extreme, and in this paper, we opted for the following  version of a DAC convolutional layer, a better trade-off between representation power and number of parameters (for  ease of notation, we consider only odd L = 2l + 1 as kernel size):  cDAC(y) = cDAC  h,k,i(y) = ψ  �  bi +  l  �  a,b=−l  m  �  j=1  wa,b,i,j ϕ(bi,j + yh+a,k+b,j)  �  ,  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , n  (6)  7  Improving Performance in Neural Networks by Dendrites-Activated Connections  where (h, k) varies in the set of coordinates of the output and where, as before, ψ is an optional output activation, bi is  an optional output bias, ϕ is the ReLU input activation and bi,j are the DAC biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Thus, instead of having the DAC  biases ba,b,i,j depending on channel, kernel and position in the kernel, we have one single DAC bias bi,j for every  input channel and output kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  This layer will need more operations due to the m × n new biases bi,j, for an increase in the total number of FLOPs  and weights of roughly 1/2L2 and 1/L2 respectively (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' for a standard 3 × 3 kernels, the FLOPs will increase by about  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='5% while the weights by about 11%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' For an in-depth efficiency and memory analysis, refer to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='3  DAC ResNet architecture  For our experiments, we used DAC convolutional layers with the original ResNet architectures [1, 2], because ResNets  are the building blocks of the most popular convolutional neural network architectures around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  In particular, we replaced all the 3×3 convolutions with corresponding DAC convolutions (6) without output activation  ψ and bias bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Moreover, the convolutions inside residual blocks were preceded by a batch normalization layer and a  ReLU activation: we removed the activation (because DAC includes it in the input) and removed the trainable shift  parameter β of the batch normalization (because DAC includes its own biases in the input).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  In the case of ResNet v1, this has the collateral effect of moving the activation from the information backbone to the  residual branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' This implies that DAC turns ResNet v1 into something similar to the ResNet v2 architecture, but with  DAC biases instead of regular biases, and with the batch normalization layers moved after the convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Apart  from the biases, this is the ReLU-only preactivation that was tested in [2, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' 4(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Again in the case of ResNet v1, after the last residual block and before the global average pooling, we added a  traditional bias layer and a ReLU activation, since they were removed from the residual block output and no DAC  layer follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In the case of ResNet v2, we explored also the bottleneck architecture, but with inconclusive results that  would need further research to improve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The difficulty here is that with the bottleneck architecture, about half of  the weights and operations are devoted to 1 × 1 convolutions, and while the DAC version of the 3 × 3 convolutions  increases the number of weights and operations by about 1/9, the same quantities approximately double for 1 × 1  convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' So, to compare DAC version with the standard one, there is a need to fine-tune hyperparameters like the  number of channels, to get fair testing and this is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='4  Efficiency and memory analysis  In this section, we estimate how FLOPs and weights change when modifying a standard layer into its DAC counter-  part, for fully connected and convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The total number of floating point operations FLOPs provides an  estimate of the amount of work performed by a computational process, and weights are correlated with the memory  footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Thus, FLOPs and weights allow us to compare the change in computation and memory burden due to the  adoption of the DAC paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The use of FLOPs as an effective measure of efficiency has been advocated in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  We will follow the convention used in the analysis of standard layers that the computational cost of the activation  functions can be safely ignored: despite the large number of activations involved in the DAC paradigm, it is still fine  to ignore the activation costs since we consider only the ReLU activation, that requires no expensive calculations on  any architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' However, this might need to be amended in case someone wants to use other, more costly, activation  functions in the dendrites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='1  Fully connected layers  The equation for a standard fully connected layer with m inputs and n outputs is:  dstd(y) = dstd  i (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , ym) = ψ  �  bi +  m  �  j=1  wi,j yj  �  ,  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Inside the parenthesis, we get m multiplications and then m additions plus one for the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' This must be repeated for  all the n outputs, giving us a total of:  FLOP(dstd) = (2m + 1)n ≈ 2mn = 2(#inputs)(#outputs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  The weights are:  weights(dstd) = (m + 1)n ≈ mn = (#inputs)(#outputs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  8  Improving Performance in Neural Networks by Dendrites-Activated Connections  We consider now a fully connected layer with dendrites-activated input connections, with m inputs and n outputs,  see (5):  dDAC(y) = dDAC  i  (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , ym) = ψ  �  bi +  m  �  j=1  wi,j ϕ(bi,j + yj)  �  ,  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Inside the parenthesis we get m bias additions, then m multiplications and m additions plus one for the general bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  This must be repeated for all the n outputs, giving us a total of:  FLOP(dDAC) = (3m + 1)n ≈ 3mn = 3(#inputs)(#outputs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  The weights are:  weights(dDAC) = (2m + 1)n ≈ 2mn = 2(#inputs)(#outputs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Thus, adopting the DAC paradigm in a fully connected layer roughly increases the number of FLOPs by 50% and  roughly doubles the number of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='2  Convolutional layers  The equation for a standard 2D convolutional layer, with m input channels and n output L×L kernels, with L = 2l+1,  is:  cstd(y) = cstd  h,k,i(y) = ψ  �  bi +  l  �  a,b=−l  m  �  j=1  wa,b,i,j yh+a,k+b,j  �  ,  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , n  where (h, k) indicates a point in the 2D, s × t input shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Inside the parenthesis we get m multiplications and then m additions repeated through the square kernel of size L,  giving a factor of L2, then we add once the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' This is done for all the n output kernels and for all the s · t points of  the geometric grid, giving us a total of:  FLOP(cstd) = (2L2m + 1)nst ≈ 2L2mnst = 2(KernelSize)2(#inputs)(#outputs)(#area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  The weights are:  weights(cstd) = (L2m + 1)n ≈ L2mn = (KernelSize)2(#inputs)(#outputs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  We consider now a 2D convolutional layer with dendrites-activated input connections, m input channels, and n output  L × L kernels, with L = 2l + 1, see (6):  cDAC(y) = cDAC  h,k,i(y) = ψ  �  bi +  l  �  a,b=−l  m  �  j=1  wa,b,i,j ϕ(bi,j + yh+a,k+b,j)  �  ,  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , n  where (h, k) indicates a point in the 2D, s × t input shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  For the FLOPs calculation, we first notice that, since the biases bi,j do not depend on the particular kernel position  a, b, it is possible to cache the results of the initial activations ϕ(bi,j + y·,·,j) that require m additions for each output  and each grid point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Thus, starting from what it is left inside the parenthesis, we get m multiplications and then m  additions repeated through the square kernel of size L, giving a factor of L2, then we add once the out bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' This is  done for all the n output kernels and all s · t points of the grid, giving us a total of:  FLOP(cDAC) = mnst + (2L2m + 1)nst ≈ (1 + 2L2)mnst = (2(KernelSize)2 + 1)(#inputs)(#outputs)(#area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  The weights are:  weights(cDAC) = (m(1 + L2) + 1)n ≈ (1 + L2)mn = ((KernelSize)2 + 1)(#inputs)(#outputs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Thus, adopting the DAC paradigm in a convolutional layer roughly increases the number of FLOPs and weights by a  fraction 1/2L2 and 1/L2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' This is a much smaller increase, with respect to the fully connected case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  We notice that, for the calculation of FLOPs in a DAC layer implemented as in GitHub, one can use the keras-flops  module [11] without any modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' This is in fact what we used for the plots in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  9  Improving Performance in Neural Networks by Dendrites-Activated Connections  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='5  Implementation  DAC fully connected and convolutional layers, as detailed at the beginning of this section, can be easily implemented  on modern deep learning platforms, like PyTorch and TensorFlow that we used for our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Unfortunately,  such a high-level approach results in a much heavier memory footprint and computation time than its non-DAC anal-  ogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In fact, we performed comprehensive experiments to assess the theoretical soundness of dendrites-active  connections, but to make DAC layers practical to use, one would need to implement them at a low level inside the  deep learning platform of choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  To understand the difference between a high-level and a low-level implementation, recall the form of the internal  computation of a DAC layer from (5) and (6):  (FC)  m  �  j=1  wi,j ϕ(bi,j + yj),  (Conv)  l  �  a,b=−l  m  �  j=1  wa,b,i,j ϕ(bi,j + yh+a,k+b,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  At a high level, one needs to replicate the output y of the preceding layer for each of the i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , n units/filter of the  current layer, then apply ReLU using a different bias for each input channel/output unit combination, before applying  the usual layer calculations separately to each replica block to generate the output units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  For fully connected layers we used standard matrix multiplications after reshaping the replicated input in the appropri-  ate manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' For convolutions, using TensorFlow, we exploited the grouping feature of the Conv2D layers, to operate  the convolution separately on each of the replicated inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' This allowed us to use a single layer to handle all the  convolutions at once, saving a lot of computational time as the grouping feature is implemented at a low level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  The code is available on GitHub, and, after our best efforts, we achieved training the DAC layers using up to 10 times  more memory and time with respect to the non-DAC analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The situation is somewhat better in the case of the  forward calculation, but still far from the non-DAC case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  At a low level, consider as an example the forward computation of a fully connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The standard version  needs one matrix multiplication [w y]i = �m  j=1 wi,j yj, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' This is typically implemented with parallel  programming, with many units that share access to w and y and that compute fragments of the m × n multiplications  and m × n additions needed (+wi,j · yj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The DAC version would be very similar, with the units also sharing access  to b and computing fragments of the corresponding DAC operations which require one more addition and one more  maximum for each of the m × n indices (+wi,j · ϕ(bi,j + yj)), recall that ϕ(x) = max(0, x) is ReLU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Hence, a  low-level implementation of DAC would have less than twice the complexity of the corresponding standard layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Such a low-level implementation is outside the scope of this paper and is planned as a possible future development,  see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  4  Experiments and results  We first evaluate our method on a ResNet architecture for the CIFAR-10 and CIFAR-100 datasets [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Both datasets  consist of 50k training images and 10k test images, divided into 10 and 100 different classes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' We replicate  each experiment 5 times, using a 5-fold cross-validation scheme for splitting the 50k training dataset into a 10k  validation set and a 40k training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Since there is no settled ResNet architecture for CIFAR-10 and CIFAR-100, for our experiments we chose the designs  detailed in the original papers [1, 2] for these two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Our implementation can be found on GitHub and to our  knowledge, it is the most faithful public Keras implementation of these architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='1  Error rate estimation  While we always trained for the same number of steps, we evaluated the best error rate by simulating early stopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Let vk,j and tk,j denote the validation and test errors for replicate k and epoch j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' We select the epoch m corresponding  to the minimum validation error (averaging on the 5 replicates and on a moving window of 5 epochs), and then compute  the average test error of the same 5 epochs and replicates:  m := arg min  i  1  25  5  �  k=1  2  �  j=−2  vk,i+j,  T := 1  25  5  �  k=1  2  �  j=−2  tk,m+j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  This approach allows for evaluation of the statistical error of the estimator and it is more robust and reliable than  simply taking the minimum of the test error, as in a real application one would be able to choose the early stopping on  the validation set, but not on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  10  Improving Performance in Neural Networks by Dendrites-Activated Connections  Assuming that tk,m+j = µm + σ Zk + τ Zk,j, with µm the true value, Zk and Zk,j independent standard Gaussian  noises, and σ, τ coefficients measuring randomness in replicates and epochs, the square of the standard error of T is  Var(T) = 1  5σ2 + 1  25τ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Here the two terms were conservatively estimated using respectively:  σ2 ≤ σ2 + τ 2 ≈ 1  4  5  �  k=1  (tk,m − T)2  and  1  5τ 2 ≈ 1  4  2  �  j=−2  (t∗,m+j − T)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='2  Baseline ResNets  We experiment with 20, 32, 44, and 56 layers (n = 3, 5, 7, 9) both with the v1 post-activated [1] architecture and the  v2 preactivated [2] architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Following the original paper scheme, we do not use bottleneck architectures since  they give a significant improvement only for deeper networks (see also Remark 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' We mimic faithfully the training  performed in the seminal paper [1] using the same settings, hyperparameters and data augmentation, except for the  regularization: the original authors used weight decay of 10−4, corresponding to an L2-regularization of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='5 · 10−4 on  all weights, while we applied L2-regularization only on the kernel weights, fine-tuning the best coefficient on baseline  ResNets, and then using the same coefficients for the DAC versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' We found that optimal values are 2 · 10−4 for  CIFAR-10 and 3 · 10−4 for CIFAR-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  In order to explore the possibility of further performance improvement on lower learning rate levels, we have added  an extra window to the learning rate schedule compared to the original implementation: the training is thus extended  from 64k to 80k iterations with a further cut of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='1 to the previous learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='3  DAC networks  Next, we train the corresponding DAC ResNet architectures as introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The hyperparameters and  network settings are the same as for baseline networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  As for the extra weights, namely the biases of each active dendrite connection, we adopt the same choices as the  baseline biases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' initialization to zero and no regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='4  Results  Figure 5 compares the performances of the baseline networks and the corresponding ones in the DAC version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In  order to identify a more robust and less noisy metric than simply choosing the least error on the test set, we report the  average test error (over 5 replicates and 5 epochs) as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The error bars measure one standard  error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  The metric used to plot the x-axis are the FLOPs needed to complete a forward pass within the entire network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' This  was a key point in comparing architectures with different structures and computational costs such as baseline networks  and their versions with the DAC implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  To provide also a comparison with the overall best accuracy as is usually done in the literature, Table 1 shows the  least error rates obtained on the test dataset (minimum of the average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Both the figure and the table describe the same  picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' DAC networks show a significant performance improvement in almost all versions and benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Figures 6 and 7 show the training and test evolution graphs along all 256 epochs (in the case of Resnet20 v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Sudden  improvements in performance can be noted in correspondence with the cuts in the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' It should be noted that  in the central part of the training there is an accentuated overfitting phenomenon, probably due to the choice of hy-  perparameters, including the high amount of regularization injected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Despite this phenomenon, the final performances  correspond to those of the original papers [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  To better appreciate the difference between baseline networks and the corresponding DAC versions, Figure 7 shows the  same image restricted to the last 100 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Deeper DAC networks outperform the corresponding baseline networks  even with 12 fewer layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='1  Imagenette and Imagewoof  Imagenette and Imagewoof [13] are two subsets of ImageNet often used for model benchmarking because they provide  a simple and faster alternative to ImageNet while preserving many of the inherent challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Imagenette is a subset of  10 well-distinguished classes, while Imagewoof consists of 10 classes very similar to each other because they picture  10 different dog breeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' For each dataset, we opted for its 160 pixels version (that is, the shortest side is resized to  11  Improving Performance in Neural Networks by Dendrites-Activated Connections  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Compared performances of ResNet baseline and DAC networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' DAC networks of the same number of layers have about  1/9 more weights and 1/18 more FLOPs (on the horizontal axis), but in most cases it is apparent that the orange line is  well below the blue line, meaning that the increase in performance between baseline and DAC is much larger than what  would be justified by the increase in complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Best Performance Comparison  Dataset  CIFAR-10  CIFAR-100  Version  v1  v2  v1  v2  Model  Base  DAC  Base  DAC  Base  DAC  Base  DAC  20 layers  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='64%  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='27%  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='73%  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='32%  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='50%  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='53%  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='49%  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='62%  32 layers  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='90%  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='56%  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='84%  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='55%  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='65%  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='09%  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='79%  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='04%  44 layers  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='68%  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='26%  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='14%  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='09%  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='39%  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='42%  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='50%  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='92%  56 layers  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='33%  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='12%  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='01%  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='88%  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='53%  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='32%  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='89%  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='31%  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Comparison table of models best performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  160 pixels, with the aspect ratio maintained), which is further processed to a final size of 80x80 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' 5-fold cross-  validation is performed, similarly to what we did for CIFAR-10 and CIFAR-100, although with different cardinality  for training, validation, and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' After validation of the regularization parameter we chose 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='0002, that is, the  value that maximized baseline performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' We then trained baseline and DAC ResNets with 20 and 32 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' All  other hyperparameters were the same as in previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  In Table 2 we display a comparison of the best performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The baselines obtained were similar to those available in  the literature for networks of similar complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Networks with DAC implementation outperformed baseline networks  even with fewer layers, and the improvement was higher than in the case of CIFAR-10 and CIFAR-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  12  Test error Cifarl0  Resnet vl baseline  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='090  20 layers  Resnet vl DAC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='085  32 layers  Test error  44 layers  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='080  56 layers  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='075  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='070  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='075  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='250  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='275  (GIGA) FlopTest error Cifarl0  20 layers  Resnet v2 baseline  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='090  Resnet v2 DAC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='085  32 layers  Test error  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='080  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='075  44 layers  56 layers  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='070  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='075  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='225  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='250  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='275  (GIGA) FlopTest error Cifarloo  20 layers  Resnet v1 baseline  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='34  Resnet vl DAC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='33  32 layers  error  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='32  Test  56 layers  44 layers  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='075  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='250  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='275  (GIGA) FlopTest error Cifarloo  20 layers  Resnet v2 baseline  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='34  Resnet v2 DAC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='33  32 layers  error  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='32  Test  44 layers  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='31  56 layers  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='075  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='250  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='275  (GIGA) FlopImproving Performance in Neural Networks by Dendrites-Activated Connections  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Training evolution for ResNet v1 baseline and DAC networks for CIFAR-10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The dashed lines represent the  training error, while the continuous lines represent the corresponding test error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The same evolution plots of Figure 6, but restricted to the last 100 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' DAC networks outperform the corresponding  baselines, even with fewer layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Compare, for example, the DAC network with 32 layers and the baseline with 44  layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Best Performance Comparison  Dataset  Imagenette  Imagewoof  Version  v1  v2  v1  v2  Model  Base  DAC  Base  DAC  Base  DAC  Base  DAC  20 layers  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='41%  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='88%  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='97%  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='78%  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='19%  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='60%  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='61%  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='70%  32 layers  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='13%  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='75%  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='40%  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='27%  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='05%  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='32%  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='75%  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='65%  Giga Flop  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='509  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='542  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='509  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='542  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='865  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='918  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='865  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='918  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Comparison table for Imagenette and Imagewoof datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Giga Flops do not depend on the version of Resnet or the  chosen dataset, as long as one chooses image with the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  13  Baseline models - Cifar-10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='30  Resnetv1 20  Resnetv1 32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='25  Resnetv1 44  Resnetv1 56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='20  error  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='00  0  50  100  150  200  250  epochsDAC models - Cifar-10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='30  Resnetv1 20  T  Resnetv1 32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='25  Resnetv1 44  Resnetv1 56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='20  error  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='00  0  50  100  150  200  250  epochsTest error - Cifar-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' ResNet v1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='090  base 20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='085  base 32  base 44  error  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='080  base 56  DAC 20  DAC 32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='075  DAC 44  DAC 56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='070  160  180  200  220  240  epochsImproving Performance in Neural Networks by Dendrites-Activated Connections  5  Universal approximation  In this section, we follow the idea proposed in [14] and the direct construction in [15] to prove a density result for the  set of functions representable by a DAC fully connected neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In particular, we exploit the fact that ReLU  can be used to build a narrow spike function, to show that its convolutions can approximate the identity operator, and  hence continuous function defined on a compact set of Rd can be approximated by a DAC fully connected neural  network having d DAC layers (and one replication layer, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='2, paragraphs on input replication), with a  number of units in each layer growing at most linearly in the dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' This result is analogous to the one in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Theorem 1 (Universal approximation for DAC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Let Gd be the set of functions g : [−1, 1]d → R representable  by a DAC fully connected neural network with ReLU preactivation and d layers, for d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Then Gd is dense in  C0([−1, 1]d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In particular, for every function f ∈ C0([−1, 1]d) and every ε > 0, there exist a ReLU fully connected  DAC neural network g ∈ Gd with d DAC layers such that ∥f − g∥∞ := supx |f(x) − g(x)| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Moreover, for d ≥ 2,  g is sparse, in the sense that there exists k := k(ε, f) ∈ N such that layer l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , d − 1 of g has 2k + d − l units  with no more than 4 input nodes each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The general idea to prove the theorem is to first build a family of densities ψd,δ that can approximate the  Dirac delta in dimension d, and show that ψd,δ is actually in Gd, see (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Then one can define the convolution  operator Td,δ(f) := f ∗ ψd,δ and show that Td,δ approximates the identity on C0([−1, 1]d) when δ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Since the  convolution is a d-dimensional integral, one can then approximate it with a sum over a finite partition of the domain,  maintaining the desired accuracy by choosing an appropriate size of the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Finally, one shows that the sum itself  corresponds to a DAC network that merges one subnet for each of the terms of the summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  We start with the proof in the simplest case d = 1, which will be used for the induction in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Consider the set G1 of functions R → R given by a single-layer, one unit DAC neural network, with input  replication as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='2 and ReLU preactivation: g ∈ G1 if:  g(x) =  n  �  i=1  wi ϕ(bi + x)  for some integer n and parameters wi, bi for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Aim of this paragraph is to show that G1 is dense in  C0([−1, 1]), giving a bound on the ‘size’ n of the DAC layer needed to approximate to some level a given function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Note that in this case there is only one unit in the layer, and n is the replication factor of the single input x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  To prove the claim one needs to find, for every continuous f and every desired accuracy ε, some g ∈ G1 such that  ∥f − g∥∞ < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' To do this we start by building a spike function ψ1 : R → R:  ψ1(x) :=  �0  |x| ≥ 1  1 − |x|  |x| < 1  (7)  Note that ψ1 ∈ G1, with the input x replicated 3 times:  ψ1(x) = ϕ(−1 + x) − 2ϕ(x) + ϕ(1 + x) ∈ G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  (8)  Linear rescalings of ψ1 are also in G1, in particular for δ > 0, we introduce:  ψ1,δ(x) := δ−1ψ(δ−1x) = δ−2�  ϕ(−δ + x) − 2ϕ(x) + ϕ(δ + x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  (9)  Notice that  � 1  −1 ψ1,δ(x)dx = 1 and ψ1,δ(x) = 0 for |x| > δ, so if we consider the convolution operator T1,δ(f) :=  f ∗ ψ1,δ, defined by:  f ∗ ψ1,δ(x) :=  � 1  −1  f(t)ψ1,δ(x − t)dt  (10)  then by the approximation property of the convolution, T1,δ(f) → f uniformly for δ → 0, meaning that:  ∀f ∈ C0([−1, 1]), ∀ε > 0, ∃δ > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' ∥f − f ∗ ψ1,δ∥∞ < ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  The next step is approximating the integral in (10) with a finite sum that happens to be an element of G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The integrand  function t �→ h(t) := f(t)ψ1,δ(x − t) is continuous, so there exists an integer k such that:  ����  � 1  −1  h(t)dt −  k  �  j=1  h  �2j − 1  k  − 1  � 2  k  ���� < ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  14  Improving Performance in Neural Networks by Dendrites-Activated Connections  In principle, this k might depend on x, but since we are in a compact set, it is clear that one can choose a k that satisfies  the inequality for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' For the sake of notation simplicity, let tj := 2j−1  k  − 1 and define g(x) := �k  j=1  2  k h(tj),  yielding ∥f ∗ ψ1,δ − g∥∞ < ε/2 and so ∥f − g∥∞ < ε as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  We are only left to show that g ∈ G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In fact, expanding the definitions of h and ψ1,δ, and letting wj := 2  k f(tj)δ−2,  we get:  g(x) =  k  �  j=1  2  k f(tj)ψ1,δ(x − tj) =  k  �  j=1  �  wjϕ(−tj − δ + x) − 2wjϕ(−tj + x) + wjϕ(−tj + δ + x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  (11)  The right side of (11) shows that g ∈ G1 with the input x replicated 3k times, that is, g is a single-layer, one unit DAC  neural network with 3k (replicated) input nodes approximating f uniformly with accuracy ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The weights and DAC  biases of g are a combination of those of ψ1,δ from (9) with wj and tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' This is explicitly written in the right-hand side  of (11), where we can see that n = 3k, the weights are wj, −2wj, wj and the DAC biases are −tj − δ, −tj, −tj + δ,  for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' This concludes the proof in the case d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  The first step is to build a spike function ψd : Rd → R in dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' As in (7), we can define:  ψd(x) :=  �0  ∥x∥1 ≥ 1  1 − ∥x∥1  ∥x∥1 < 1  (12)  where ∥x∥1 := �d  j=1 |xj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Consider ˜ψ : R × R+ → R defined by:  ˜ψ(x, t) := ϕ(x − t) − 2ϕ(x) + ϕ(x + t) = (t − |x|)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  If x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , xd), let ˜x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , xd−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Then:  ψd(x) =  �  1 −  d  �  j=1  |xj|  �+  = ˜ψ  �  xd,  �  1 −  d−1  �  j=1  |xj|  �+�  = ˜ψ(xd, ψd−1(˜x)),  so that:  ψd(x) = ϕ(xd − ψd−1(˜x)) − 2ϕ(xd) + ϕ(xd + ψd−1(˜x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  (13)  The latter is a recursive expression for ψd, starting from ψ0 = 1 that can be used to construct a DAC neural network  computing this function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  We have already seen that ψ1 can be obtained as a 1 layer, 1 unit DAC network with 3 (replicated) input nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Then  x2 + 1 = ϕ(x2 + 1) can also be obtained as 1 unit on the first layer (with one input node);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' by linearity, the same holds  for the two terms ϕ(x2 + 1) ± ψ1(x1), with 4 input nodes (x1 replicated thrice and x2 taken once).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Then:  ψ2(x1, x2) = ϕ(−1 + x2 + 1 − ψ1(x1)) − 2ϕ(−1 + x2 + 1) + ϕ(−1 + x2 + 1 + ψ1(x1))  can be obtained as a 2-layer DAC network, with 1 unit (with 3 inputs and all DAC biases equal to −1) in layer 2 and  3 units (with 4, 1, and 4 input nodes) in layer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  One can generalize this approach to a higher dimension d ≥ 3 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Let τi denote ψi(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , xi), for the sake  of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In the generic layer i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , d − 1 there will be two DAC units (with 4 input nodes and three biases  equal to −1) computing:  �xi+1 + 1 + τi = ϕ(xi+1 + 1) + ϕ(−1 + xi + 1 − τi−1) − 2ϕ(−1 + xi + 1) + ϕ(−1 + xi + 1 + τi−1)  xi+1 + 1 − τi = ϕ(xi+1 + 1) − ϕ(−1 + xi + 1 − τi−1) + 2ϕ(−1 + xi + 1) − ϕ(−1 + xi + 1 + τi−1)  (14)  and d − i additional units to pass on the variables xi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , xd, through xj + 1 = ϕ(xj + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Layer 1 will be similar  but with τ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Layer d will have just 1 unit, computing:  ψd(x) = τd = ϕ(−1 + xd + 1 − τd−1) − 2ϕ(−1 + xd + 1) + ϕ(−1 + xd + 1 + τd−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  It’s straightforward to see that this DAC network computes ψd(x) following the recursion given by (13), using d layers  with d + 1, d, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , 4, 3, 1 units in this order (see also Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Later we will need to merge shifted copies ψd(x − c) of this function, for several values of c, and this can be done  efficiently by sharing the d − i additional units of layer i and duplicating just the two units of equation (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Let  τi := ψi(x1 − c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , xi − ci) and τ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Then at layer i, we can have a DAC unit with inputs and biases  input  bias  xi+1 + 1  0  xi + 1 − τi−1  −1 − ci  xi + 1  −1 − ci  xi + 1 + τi−1  −1 − ci  15  Improving Performance in Neural Networks by Dendrites-Activated Connections  DAC  DAC  DAC  x1  d = 1  x1  x1  ψ1(x1)+x2 + 1  x1  d = 2  x2  x2  −ψ1(x1) + x2 + 1  ψ2(x1, x2)+x3 + 1  x2 + 1  d = 3  x3  x3  x3 + 1  −ψ2(x1, x2) + x3 + 1  ψ3(x1, x2, x3)  x3 + 1  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' This figure illustrates how the d-layers network corresponding to ψd can be inductively built from the d − 1-layers  network corresponding to ψd−1, for d = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' When d = 1, only the red elements in the figure exist, corresponding  to (8): after replication of x1, the three red arrows perform preactivation with biases −1, 1, 0 and weights 1, 1, −2, to  produce ψ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' When d = 2, the green elements are included on top of the red ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The new variable x2 is taken forward  as ϕ(x2 + 1) = x2 + 1 and combined with the appropriate weights to produce 3 ReLU arguments needed in (13) for  d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' These are the new layer 1 outputs, that are combined with weights 1, 1, −2 (and DAC biases −1, −1, −1 to  recover the original value of x2), to produce the layer 2 output ψ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' When d = 3, the blue elements are included on top  of the red and green ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The same operations performed for d = 2 with the red network nodes are now performed with  the green network nodes, but 1 layer forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' This produces the layer 3 output ψ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  computing:  xi+1 + 1 + τi = ϕ(xi+1 + 1) + ϕ(−1 − ci + xi + 1 − τi−1) − 2ϕ(−1 − ci + xi + 1) + ϕ(−1 − ci + xi + 1 + τi−1)  and analogously a second DAC unit will compute xi+1 + 1 − τi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In particular, these two units compute functions of x  that depend also on c and hence must be duplicated for different c, but the units transporting xj +1 for j = i+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , d  should be shared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  The second step is rescaling ψd with a positive constant δ > 0 and the normalization coefficient Cd = 2−d(d + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  that makes the integral equal to 1, yielding an approximation of the Dirac delta ψd,δ : Rd → R:  ψd,δ(x) := Cdδ−dψd(δ−1x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  (15)  To ensure that also ψd,δ can be realized as a DAC network of the same structure as ψd, we observe that for a DAC unit  g defined by g(x) = �n  j=1 wj ϕ(bj + xj) and for λ > 0:  g(λx) =  n  �  j=1  wj ϕ(bj + λxj) =  n  �  j=1  λwj ϕ(bj/λ + xj) =: ˜g(x)  so that given any multi-layer DAC network h, by simply changing the coefficients of the units in the first layer, it is  possible to obtain another DAC network ˜h such that ˜h(x) = h(λx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' By linearity, the outer coefficient Cdδ−d can be  absorbed into the weights of the last layer, and we get ψd,δ ∈ Gd with the same structure of ψd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  The third and last step of the proof is similar to the case d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' First, we consider the convolution operator Td,δ(f) :=  f ∗ ψd,δ, which by the approximation property of the convolution ensures that:  ∀f ∈ C0([−1, 1]d), ∀ε > 0, ∃δ > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' ∥f − f ∗ ψd,δ∥∞ < ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  (16)  Then, fixed f, we approximate the integral in (16) with a finite sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' To this aim, let h(y, x) := f(y)ψd,δ(x − y),  so that f ∗ ψd,δ(x) =  �  h(y, x)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Choose a partition of [−1, 1]d by a large number k of small sets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' cubes) of  average volume 2d/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Consider a set C of k points, one for each of the sets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' the center points of the cubes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Since  h is continuous and the hyper-cube is compact, there exists a partition fine enough that for all x:  �����  �  [−1,1]d h(y, x)dy −  �  c∈C  h(c, x)2d  k  ����� < ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  16  Improving Performance in Neural Networks by Dendrites-Activated Connections  Therefore we define g(x) := �  c∈C 2dk−1h(c, x), yielding ∥f ∗ ψd,δ − g∥∞ < ε/2 and so ∥f − g∥∞ < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Finally, g can be realized as a DAC neural network with d layers, since the same holds for ψδ,d, and we need only to  compute a linear combination of the output of k translated versions of the latter:  g(x) :=  �  c∈C  2dk−1f(c)ψd,δ(x − c) =:  �  c∈C  wcψd,δ(x − c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  By linearity, g ∈ Gd, can be realized as a DAC network with d layers and recalling that only two units in each layer  must be duplicated to account for different values of c, while the others can be shared, we get that the layers 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , d  have respectively 2k + d − 1, 2k + d − 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' , 2k + 2, 2k + 1, 1 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Case d = 1 in Theorem 1 shows that a single-layer DAC neural network is a universal approximator in  C0([−1, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' This is not true for a standard single-layer neural network, where input replication has no effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Proof of Theorem 1 can be easily extended to functions with compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' We observe that the d-dimensional spike function ψd appearing in the proof, can also be realized as a  2-layer DAC network as follows:  ψd(x) = ϕ  �  1 − d +  d  �  j=1  �  ϕ(xj + 1) − 2ϕ(xj)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  In fact, ϕ(x + 1) − 2ϕ(x) is positive only for |x| < 1, and equals 1 − |x| on that interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Therefore, the sum above is  equal to d − ∥x∥1 inside [−1, 1]d and lesser than d − 1 outside, so thanks to the outer ϕ, we obtain ψd everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Using this construction, one could prove a result similar to Theorem 1, but with a DAC network with 2 layers, regard-  less of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' This network won’t be sparse, as here we are trading a d-layers-deep representation with 4 or fewer input  nodes per unit, for a shallow one with 2d input nodes per unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  6  Future work and limitations  In our opinion, the most important limitation and possible future development of DAC, as of now, is its current high-  level implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' As already mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='5, an efficient low-level implementation of DAC could  potentially yield just a modest computational complexity increase with respect to the corresponding non-DAC coun-  terpart (of the same order as the increase in the number of weights).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Currently, the memory needs and training times  of DAC architectures are 5-10 times larger than that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Low-level implementation would allow experimenting DAC with  larger network structures, like the ones that are needed for datasets like ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Moreover, this would be the first  step for integrating DAC in frameworks like PyTorch and Keras, making it an option for standard layers, similar to  what we did in our high-level implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Experimenting with activation functions other than ReLU is another possible development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' One should test whether  the performance benefits from a different dendrites activation, and quantify the corresponding increase in computa-  tional complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' A similar possible improvement is the following: we described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='2 a DAC implementa-  tion for convolutional layers with DAC biases that do not depend on the position in the kernel but only on the input  and output channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' One could test whether the increase in computational complexity given by position-dependent  biases is overcome by a better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' Different activations and position-dependent biases could be reasonably  tested only after a low-level and efficient DAC implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  A natural question is in which way DAC improves expressivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' To answer this question, one could estimate some  model complexity measure, like the classical VC dimension, the recently introduced geometric complexity [17] or the  effective model complexity proposed in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' In this analysis, one could also measure how much input replication  within layers is effective in terms of expressivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Visualizing the loss landscape using techniques like in [19] could also be useful for understanding dendrites activation  effects on a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  In this paper, we did not use any activation other than DAC preactivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' However, in the biological neuron, after  the dendritic preactivation, the signals flow through the soma and are jointly activated where the axon connects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' This  suggests as a further development of DAC that using an additional post-activation after DAC preactivation could be  useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Another future work we are planning is the analysis of the filters learned by a DAC convolutional tower in the first  layers: are these filters similar to the ones learned in the non-DAC case?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' If yes, what are the differences?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  17  Improving Performance in Neural Networks by Dendrites-Activated Connections  Finally, the experimental part of this paper is concerned with the performance of DAC-ResNet neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' One  could analyze how DAC could be used to improve the performance in other convolution-based architectures like  Inception, Xception, SENet and ResNeXt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content=' The same analysis could be performed on a wider range of architectures,  like recurrent neural networks, Transformers, GANs and autoencoders, to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  Computational resources for this work were provided by CLAI laboratory of Chieti-Pescara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
+page_content='  The authors wish to thank Rosa Gini for her important intellectual contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
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+page_content=' Sebe, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
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+page_content='html  19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf'}
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+HOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER
+1
+Grid-Based Decimation for Wavelet Transforms
+with Stably Invertible Implementation
+Nicki Holighaus, Günther Koliander, Clara Hollomey, and Friedrich Pillichshammer
+Abstract—The constant center frequency to bandwidth ratio
+(Q-factor) of wavelet transforms provides a very natural rep-
+resentation for audio data. However, invertible wavelet trans-
+forms have either required non-uniform decimation—leading to
+irregular data structures that are cumbersome to work with—or
+require excessively high oversampling with unacceptable com-
+putational overhead. Here, we present a novel decimation strat-
+egy for wavelet transforms that leads to stable representations
+with oversampling rates close to one and uniform decimation.
+Specifically, we show that finite implementations of the resulting
+representation are energy-preserving in the sense of frame theory.
+The obtained wavelet coefficients can be stored in a time-
+frequency matrix with a natural interpretation of columns as time
+frames and rows as frequency channels. This matrix structure
+immediately grants access to a large number of algorithms that
+are successfully used in time-frequency audio processing, but
+could not previously be used jointly with wavelet transforms. We
+demonstrate the application of our method in processing based
+on nonnegative matrix factorization, in onset detection, and in
+phaseless reconstruction.
+Index Terms—wavelet transforms, low-discrepancy sequences,
+sampling methods, signal reconstruction, audio applications,
+shift-invariant systems, uniform decimation
+I. INTRODUCTION
+The wavelet transform is one of the most important and
+well-studied time-frequency filter banks, only rivaled by the
+short-time Fourier transform. Due to its constant center fre-
+quency to bandwidth ratio, or Q-factor [1], wavelets provide
+a natural and physically meaningful representation of audio:
+The impulse response of every filter captures an equal number
+of wavelengths of its center frequency. The constant Q-factor
+ensures that equal intervals on (Western) musical scales,
+which consist of geometrically spaced tones, are resolved
+equally well, independent of their absolute position. Finally,
+at frequencies above 500 Hz, the peripheral auditory system in
+Manuscript received XXX; revised August XXX.
+N. Holighaus (corresponding author) and C. Hollomey are with the
+Acoustics
+Research
+Institute
+(ARI),
+Austrian
+Academy
+of
+Sciences,
+Wohllebengasse
+12–14,
+1040
+Vienna,
+Austria.
+G.
+Koliander
+is
+with
+ARI and the Faculty of Mathematics, University of Vienna, Austria.
+F. Pillichshammer is with the Institute of Financial Mathematics and
+Applied Number Theory, Johannes Kepler University Linz, Austria. e-mail:
+{nicki.holighaus,guenther.koliander,clara.hollomey}
+@oeaw.ac.at, friedrich.pillichshammer@jku.ac.at
+Extended results, audio files and code for reproducing the presented
+experiments is available at: ltfat.org/notes/057
+This work is supported by the Austrian Science Fund (FWF): I 3067–N30
+(N.H.), Y 1199 “Time-Frequency Analysis, Randomness and Sampling.”
+(G.K.), and F5509-N26, which is a part of the Special Research Program
+“Quasi-Monte Carlo Methods: Theory and Applications.” (F.P.).
+Copyright (c) 2023 IEEE. Personal use of this material is permitted.
+However, permission to use this material for any other purposes must be
+obtained from the IEEE by sending a request to pubs-permissions@ieee.org.
+humans is well-approximated by a constant-Q filter bank [2].
+For a short discussion, see, e.g., [3]. Consequently, constant-
+Q filter banks in general [4]–[6] and wavelet transforms in
+particular [7]–[10] have been used to great success in the
+analysis of speech and audio signals. For the longest time,
+however, constant-Q filter banks have been inaccessible, or at
+least inconvenient, for processing audio at sufficient fidelity:
+Wavelet bases [11] and undecimated, discrete wavelet systems
+[12], [13] possess a very low Q-factor that is not suitable
+for processing audio. Modern implementations of constant-
+Q filter banks allow for the tuning of the Q-factor and the
+oversampling rate, but computationally efficient, numerically
+stable, and invertible constant-Q filter banks have only recently
+been constructed: In [14] this is achieved by purposeful
+modification of Mallat’s algorithm for the fast wavelet trans-
+form, and in [15], [16] by means of mathematical frame the-
+ory [17]. To achieve perfect reconstruction, these constructions
+rely on non-uniform decimation, choosing sufficiently small
+decimation factors inversely proportional to the bandwidth
+or, equivalently, center frequency of the constant-Q filters.1
+Consequently, every channel produces coefficients at different,
+usually unmatched, rates. Alternatively, it is possible to employ
+no in-channel decimation whatsoever, resulting, however, in
+prohibitively high oversampling at large Q-factors.
+Although the availability of invertible representations with
+appropriate frequency resolution, i.e., a sufficiently large Q-
+factor, presents an important step towards making constant-
+Q filter banks more attractive for audio processing, another
+fundamental issue remains, previously discussed in [3], along-
+side a partial solution: Employing non-uniform decimation, the
+filter bank coefficients form an irregular data structure that
+is fundamentally more difficult to work with than matrices,
+both computationally and conceptually. Such coefficients are
+incompatible with any method that relies on matrix manip-
+ulation, and substantial work is required to integrate them
+into readily available audio processing toolboxes such as,
+e.g., [18] and [19], usually implemented under the assumption
+of matrix-structured coefficients. This effect is most readily
+apparent in algorithms for the ubiquitous short-time Fourier
+transform (STFT), which often rely on the segmentation of the
+STFT coefficients into time frames, see [20] and [21]. These
+time frames are processed individually (or in small groups)
+and often in real-time, but the segmentation relies on the
+assumption that the coefficients are time-aligned. More pre-
+cisely, the between channel coefficient relations are assumed
+1This approach for choosing decimation factors is not exclusive to invert-
+ible constant-Q transforms, but shared across most constant-Q and wavelet
+transform implementations.
+arXiv:2301.01640v1  [eess.AS]  4 Jan 2023
+
+HOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER
+2
+to be position-independent, which is clearly not the case for
+non-uniform decimation. Hence, the adaptation of successful
+processing schemes for STFT coefficients to the constant-Q
+setting remains challenging. Nonuniform decimation also has
+implications concerning the reconstruction procedure itself,
+see Section II-A. Most importantly, even efficient reconstruc-
+tion algorithms [22], [23] involve costly iteration, unless all
+filters are strictly band-limited [24], precluding the use of finite
+impulse response filters.
+In this work, we present a family of wavelet filter banks
+based on quasi-random sampling of the continuous wavelet
+transform that are computationally efficient, with tunable Q-
+factor through unrestricted choice of the mother wavelet, and
+flexible oversampling rate. With oversampling rates close to
+one, the proposed filter banks achieve perfect reconstruction of
+discrete signals. Uniquely, they do so while employing uniform
+decimation and thereby overcoming the difficulties induced by
+the irregularity of previous invertible constant-Q transforms.
+Previous work on random and quasi-random sampling of time-
+frequency representations: This work is not the first to consider
+(quasi-)randomized sampling of time-frequency representa-
+tions or more general integral transforms. In a series of recent
+works, Levie et al. consider Monte Carlo [25], [26] and Quasi-
+Monte Carlo [27] style random sampling of time-frequency
+integral operators. Their work is concerned with the approx-
+imation of continuous domain time-frequency processing by
+means of Monte Carlo integration with (quasi-)random time-
+frequency samples. By invoking prior results on Monte Carlo
+integration, the authors demonstrate that the approximation
+error can be controlled when a technical structure condition
+is satisfied. They further show that this condition is satisfied
+by the STFT, the wavelet transform, and a custom blending
+of the two, referred to as localizing time-frequency transform.
+The invertibility or stability of the sampled representation is
+not investigated, however. In the context of random sampling,
+these properties are considered in the literature on relevant
+sampling, introduced by Bass and Gröchenig for bandlimited
+functions [28] and later generalized to various settings [29]–
+[31], including time-frequency representations [32]. Relevant
+sampling provides a probabilistic framework for stable sam-
+pling of functions that are localized in a domain of finite
+volume, e.g., bandlimited signals that have only negligible
+energy outside a finite interval. Our work differs from these
+prior works in multiple ways: Our proposed sampling sets
+are not fully (quasi-)random, but correspond to a uniform
+time-frequency grid, up to the introduction of a small, quasi-
+random delay in every wavelet channel. Further, we consider
+perfect reconstruction of arbitrary signals, without localization
+assumptions.
+Contribution:
+Previously, grid-based decimation strategies
+were considered ill-suited for the continuous wavelet transform
+and not expected to provide efficient and stably invertible
+representations even for discrete signals. Indeed, known con-
+structions did not admit perfect reconstruction at moderate, or
+even low, oversampling rates. In this paper, we propose the first
+grid-based decimation strategy for wavelet transforms with a
+tunable Q-factor that allows perfect reconstruction at oversam-
+pling rates close to 1 and that provides excellent numerical
+stability, in the sense of energy preservation,2 at moderate
+oversampling rates, ranging from 2 to 8, as commonly used
+for audio processing with the STFT. To achieve this, we
+combine shift-invariant systems [33], [34] with ideas from
+quasi-random sampling using low-discrepancy sequences [35],
+[36]. Similar to wavelet bases and prior invertible constant-Q
+implementations, we use a set of compensation filters to cover
+an arbitrarily small low-frequency region. Our construction
+is validated in the finite domain by computing exact frame
+bound ratios and accumulated spectrograms of the decimated
+wavelet systems across a range of system parameters covering
+variations of the mother wavelet, the number of frequency
+channels, the decimation factor, and the oversampling rate. As
+a proof of concept, we apply the proposed wavelet decimation
+to several audio applications. We replicate an experiment on
+signal enhancement based on Itakura-Saito nonnegative matrix
+factorization (NMF) as proposed by Févotte et al. [37], which
+relies heavily on the natural interpretation of the representa-
+tion coefficients as a time-frequency matrix. Furthermore, we
+illustrate the use of our proposed scheme in onset detection
+based on a straightforward adaptation of the classic spectral
+flux method [38, Sec. 3-A]. As an indication that the pro-
+posed scheme performs on par with established, non-uniform
+constant-Q transforms in tasks that do not require matrix
+structure, we further evaluate the suitability of the proposed
+method for phaseless reconstruction with the fast Griffin-Lim
+algorithm [39]. In addition to the standard, irregular wavelet
+decimation, we also compare to the STFT.
+Paper structure:
+A short introduction to wavelet systems
+and quasi-random sequences is given in Section II, before
+presenting the proposed decimation scheme in Section III.
+In Section IV, we discuss details of practical implementation
+and complexity, and evaluate our construction numerically.
+We further apply the proposed scheme in three illustrating
+experiments in audio processing (Section V). Specifically,
+we consider the decomposition of a signal with nonnegative
+matrix factorization, onset detection based on an adaptation
+of spectral flux, and phaseless reconstruction from time-
+frequency coefficients based on the fast Griffin-Lim algorithm.
+The paper concludes with a summary of the results and an
+outlook towards related future work (Section VI).
+II. TECHNICAL BACKGROUND
+Before introducing our novel decimation strategy, we review
+some basics of wavelet systems and quasi-random sequences.
+A. The Wavelet Transform
+A wavelet system is a collection of functions (or vectors)
+generated from a single prototype, the mother wavelet, by
+translation and dilation. Since we are interested in real-valued
+signals, audio signals in particular, we consider a mother
+wavelet ψ such that its Fourier transform vanishes for negative
+frequencies, i.e., ˆψ(ξ) = 0 for ξ ∈ (−∞, 0]. Such mother
+wavelets are often called analytic, although the terminology
+2For more details on stability and energy preservation, see Section II-A.
+
+HOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER
+3
+analytic wavelet transform has been used in at least two dif-
+ferent manners in the past [40], [41]. The continuous wavelet
+system is generated via dilation by s > 0 and translation by
+x ∈ R of the mother wavelet:
+ψ(x,s) := s−1/2ψ
+�• − x
+s
+�
+.
+A signal f can now be filtered using this system resulting in
+the continuous wavelet transform
+Wψf(x, s) = 1
+√s
+�
+f(t)ψ
+�t − x
+s
+�
+dt
+= ⟨f, ψ(x,s)⟩L2
+=
+�
+f ∗ ψ (− • /s)
+�
+(x) .
+(1)
+The mathematical study of wavelet transforms usually con-
+siders Wψf in terms of the inner product representation
+of the transform coefficients, but the final equality justifies
+the interpretation as a filter bank. Importantly, the wavelet
+transform is shift-invariant, i.e., the wavelet transform of a
+delayed signal f(•−y) equals the delayed (in the first variable)
+wavelet transform of f. Also note that the scale s is inversely
+proportional to the filter center frequency: If ψ has its passband
+around frequency ξ1, then ψ(•/s) has its passband around
+ξs := ξ1/s.
+In applications, only a discrete subset of the continuous
+wavelet system can be considered; the system, or equiva-
+lently the transform, is decimated. Commonly, the discrete
+subset
+�
+(xl,j, sj)
+�
+l∈Z,j∈I ⊂ R × [0, ∞) of translation-dilation
+pairs is generated by certain decimation rules. We denote
+the corresponding decimated wavelet system by {ψl,j}l∈Z,j∈I
+with ψl,j := ψ(xl,j,sj). Regarding the choice of decimation
+�
+(xl,j, sj)
+�
+l∈Z,j∈I, it is desirable that any function f can
+be stably recovered from the decimated transform and the
+energy of f is accurately represented by its coefficients.
+Mathematically, these properties are equivalent to the discrete
+subset (ψl,j)l,j of the continuous wavelet system constituting
+a frame [17], i.e., it satisfies the energy equivalence relation
+A∥f∥2
+2 ≤
+�
+l,j
+|⟨f, ψ(l,j)⟩|2 ≤ B∥f∥2
+2,
+(2)
+for all f and some constants 0 < A ≤ B < ∞. Implicitly,
+we always assume that A, B are the optimal constants such
+that (2) holds. The ratio RFB := B/A of the so-called upper
+(B) and lower (A) frame bounds quantifies how well the
+decimated transform preserves signal energy, i.e., how stable
+it is in numerical computation. If RFB ̸= 1, then error-free
+reconstruction of f from the decimated transform requires
+a dual frame, which can be efficiently precomputed under
+certain conditions on mother wavelet and decimation [15],
+[24], [42], and realized by iterative schemes otherwise [22],
+[23], [43]. If the transform coefficients are modified, the
+energy of the synthesized function ˜f can nevertheless be
+bounded by A−1 times the energy of the modified coefficients.
+Note that the dual frame of a wavelet frame is not necessarily
+a wavelet frame, or even a filter bank. In fact, that is the
+exception rather than the norm.
+Regarding the choice of the scales sj, recall that the
+bandwidth of wavelet filters increases linearly with their center
+Figure 1. Time-frequency geometry of decimated wavelet systems: (a) The
+classical nonuniform wavelet decimation provides a mostly even covering
+of time-frequency space, at the cost of introducing nonuniform decimation.
+(b) Wavelet decimation on a uniform grid results in an increasingly uneven
+covering away from a small frequency strip. The depicted hyperbolic circles
+have identical radius across both panels.
+frequency ξs, which is inversely proportional to their scale s.
+Thus, it is natural to decimate the frequency channels by taking
+integer powers of a fixed base scale a > 1, i.e., sj = a−j.
+Conversely, the width of the wavelet impulse responses is
+proportional to the scale, suggesting in-channel decimation
+according to xl,j = a−j ·lb for some fixed b > 0, see also [44,
+Chapter 3]. Intuitively, the time-frequency region covered by
+the individual ψl,j corresponds roughly to ellipses of constant
+area that grow narrower as j increases, or, more accurately, to
+hyperbolic circles of constant radius,3 centered at (xl,j, ξ1/sj),
+see Fig. 1(a).
+This decimation strategy choosing (xl,j, sj)
+=
+(a−j ·
+lb, a−j), with geometric frequency spacing and channel-
+dependent decimation factor (hop size) dj = a−j · b, is
+customary and well-studied for wavelet bases, overcomplete
+wavelet systems, and constant-Q transforms, e.g., [15], [44].
+We will refer to this convention as classical wavelet deci-
+mation. If ψ is not strictly bandlimited or the decimation
+factors are too large, then the dual frame required for error-
+free reconstruction will only be a filter bank in exceptional
+cases. Furthermore, the dependence of the decimation factor
+dj on the scale leads to filter bank coefficients that are not
+time-aligned, and, for a lack of position-independent relations
+3A hyperbolic circle contains all time-scale pairs that have a hy-
+perbolic
+distance less
+than
+a
+given
+radius to
+the
+center
+(xl,j, sj).
+Here, hyperbolic distance between (x, s) and (x′, s′) is measured as
+2 arsinh
+√
+(x′−x)2+(s′−s)2
+2
+√
+ss′
+. Note that we depict these circles in the time-
+frequency rather than the time-scale plane.
+
+(a)
+Frequency
+Time(b)
+Frequency
+TimeHOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER
+4
+between coefficients in different bands, cannot be represented
+as a time-frequency matrix or segmented into time frames of
+identical structure. In many applications and implementations,
+this irregularity encumbers the workflow.
+In contrast to the
+STFT, the time-frequency geometry of the wavelet transform
+described above is ill-suited for decimation on a uniform,
+rectangular grid, independent of whether the center frequencies
+are spaced linearly or logarithmically, see Fig. 1(b). It is
+easy to see and, in fact, straightforward to prove, that such
+a decimation strategy cannot lead to a numerically stable
+system with perfect reconstruction. Considering that many
+successful processing schemes for the STFT rely on both
+perfect reconstruction and the representation of the coefficients
+as a time-frequency matrix, this presents a notable obstruction
+to the adoption of wavelets in (audio) signal processing.
+B. Quasi-Random Sequences
+Quasi-random sequences, also known as low-discrepancy
+sequences [35], are deterministic sequences of numbers or k-
+dimensional coordinates that share some properties of uni-
+formly distributed random numbers and can be used as
+a replacement for the latter in various applications, most
+prominently the quasi-Monte Carlo method for numerical
+integration [35]. The term discrepancy refers to one of several
+related measures that quantify how uniformly distributed a
+set of points is. This property is important, e.g., to ob-
+tain error bounds in quasi-Monte Carlo integration. While
+most modern theoretical studies are concerned with quasi-
+random sequences of high dimensionality, we only require
+one-dimensional sequences. Specifically, any N consecutive
+elements of a low-discrepancy sequence in dimension D = 1
+provides N numbers that are particularly well distributed in
+the unit interval.
+Decimation strategies for wavelets can be derived from any
+low-discrepancy sequence. Here, we focus on two carefully
+chosen examples based on two prominent classes of quasi-
+random sequences.
+Kronecker sequences: A deceptively simple construction
+are so-called Kronecker sequences which are of the form
+({αl})l≥0
+= (0, {α}, {2α}, . . .) for some real α, where
+{x} := x − ⌊x⌋ denotes the fractional part of a real x. It
+is well-known that Kronecker sequences are particularly well
+distributed if α is a badly approximable number. These are
+irrational numbers that are particularly poorly approximated
+by rationals. Formally, a number α ∈ R is badly approximable
+if there is a constant c > 0, such that
+|α − p/q| > c/q2,
+for all nonzero integers p, q ∈ Z \ {0}. It is known that an
+irrational number is badly approximable if and only if the
+coefficients of its continued fraction expansion are bounded.
+Among all the badly approximable numbers, the golden ratio
+φ = 1+
+√
+5
+2
+maximizes the optimal constant c. Since this prop-
+erty is shared exactly with all so-called equivalent numbers
+of the form aφ+b
+cφ+d, for integers a, b, c, d with ad − bc = ±1,
+we may likewise consider the Kronecker sequence with α =
+1 − 1/φ, i.e., a, c = 1, b = −1, and d = 0. For more
+information on Kronecker sequences we refer to the books
+[45], [46].
+Digital (0, 1)-sequences over Z2: A (0, 1)-sequence in base
+2 is an infinite sequence (xl)l≥0 = (x0, x1, . . .) in the unit-
+interval [0, 1) with the following property: for every m ∈ N0
+and every k ∈ {0, 1, . . . , 2m − 1}, the elementary interval
+[ k
+2m , k+1
+2m ) contains exactly one element of the point set
+{xl : p2m ≤ l ≤ (p + 1)2m − 1}
+for every p ∈ N0.
+One example for such a sequence is the well-known van der
+Corput sequence [47] (see also [48]), which is related to the
+bit-reversal permutation. Typically, (0, 1)-sequences in base 2
+are constructed by the so-called digital method over the finite
+field Z2 of order two, i.e., the integers modulo 2. To this end,
+let C = (cr,k)r,k≥1 be an ∞ × ∞-matrix over Z2, i.e., with
+entries cr,k from {0, 1}, such that for every m ∈ N the left-
+upper m × m sub-matrix is non-singular. Then the sequence
+(xl)l≥0 defined by
+xl =
+∞
+�
+r=0
+ηl,r
+2r+1
+where ηl,r =
+∞
+�
+k=0
+cr+1,k+1lk
+(mod 2),
+where lk ∈ {0, 1} are the binary digits of the index l, i.e.,
+l = l0 +l12+l222 +· · · (which are obviously 0 from a certain
+index on), is a (0, 1)-sequence in base 2. In this context, the
+van der Corput sequence is covered by choosing the identity
+matrix for C. We will use the specific digital (0, 1)-sequence
+which is obtained from the infinite matrix
+C =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+0
+0
+0
+0
+. . .
+1
+1
+0
+0
+0
+. . .
+0
+1
+1
+0
+0
+. . .
+0
+0
+1
+1
+0
+. . .
+0
+0
+0
+1
+1
+. . .
+...
+...
+...
+...
+...
+...
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+For more information on (digital) (0, 1)-sequences we refer to
+the books [35], [36] or the survey article [48, Section 3.2].
+III. A NEW CONVENTION FOR WAVELET SAMPLING
+Due to the geometric intuition outlined in Section II-A,
+linear spacing of wavelet center frequencies ξj ∼ s−1
+j
+has
+hardly been considered in the literature. For the same reason,
+uniform decimation, i.e., dj = d for some fixed d > 0,
+is usually disregarded. We will now describe a decimation
+scheme that follows both of these conventions and which can
+be used to construct numerically stable, perfect reconstruction
+wavelet transforms with moderate, or even marginal, oversam-
+pling. By virtue of being a uniform filter bank frame, standard
+results [33], [34] ensure that the dual frame is a uniform filter
+bank as well, with the same decimation factor. The dual filter
+bank is, however, not necessarily a wavelet system. In particu-
+lar, this implies that said dual frame can be precomputed using
+optimized factorization algorithms and, thus, synthesis from
+transform coefficients is highly efficient. The proposed scheme
+results in an almost time-aligned decimated representation.
+More precisely, the coefficients across bands are in a position-
+independent, fixed relation, thus providing a meaningful notion
+
+HOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER
+5
+Figure 2. Time-frequency geometry of decimated wavelet systems with quasi-
+random delay: (a) Decimation on a uniform grid with delays chosen according
+to the golden Kronecker sequence provides an even covering comparable to
+the classic wavelet decimation (Fig. 1(a)). (b) Deriving the delays instead
+from the digital sequence described in Section II-B, we obtain a covering that
+is slightly less even. In both cases, the resulting covering will become uneven
+below a certain frequency, similar to the bottom region in Fig. 1(b). Note that
+the radius of the depicted hyperbolic circles is chosen as in Fig. 1 to allow
+comparison. A mild increase in radius is sufficient to remove the blind spots
+in panel (b), whereas even a minor decrease would introduce similar blind
+spots in panel (a) and Fig. 1(a).
+of time frames and suggesting a natural arrangement in a time-
+frequency matrix.
+Given a largest scale of interest b > 0 (or equivalently a
+minimal frequency of interest ξmin > 0), a decimation factor
+d > 0, a third parameter q > 0 that determines the step
+size in the frequency direction, and finally a low-discrepancy
+sequence (δ0, δ1, δ2, . . .) determining channel specific delays,
+we select the translation-dilation pairs
+(xl,j, sj) =
+�
+d(l + δj),
+1
+b−1 + q−1j
+�
+,
+(3)
+for all integers l ∈ Z and nonnegative integers j ∈ N0. In other
+words, we consider the discrete wavelet system {ψl,j}l∈Z,j∈N0,
+with
+ψl,j(t) =
+�
+1
+b + j
+q · ψ
+�
+( 1
+b + j
+q) · (t − d(l + δj)
+�
+.
+(4)
+In contrast to the rectangular grid shown in Fig. 1(b), which
+can be generated by the above construction with delays δj = 0
+for all j, choosing the δj as the elements of a quasi-random
+sequence provides a wavelet system that covers the time-
+frequency plane surprisingly evenly, see Fig. 2. Intriguingly,
+if we choose the δj as the elements of a Kronecker sequence,
+the points {(xl,j, ξj)}l∈Z,j∈N0 form a uniform (skewed) time-
+frequency grid.
+Figure 3. Proposed covering of the low-frequency region by modulated and
+delayed versions of the largest scale wavelet filters for the decimated wavelet
+system with quasi-random delays chosen according to the golden Kronecker
+sequence. The frequency axis is rescaled compared to Fig. 2(a) for better
+visibility.
+A. Numerically Stable Wavelet Transforms with Perfect Re-
+construction and Uniform Decimation
+As indicated by Fig. 2, the proposed wavelet decimation
+yields a surprisingly uniform covering of the time-frequency
+plane. However, due to the uniform spacing in frequency, this
+is only true as long as the scale of the wavelets is small enough
+(relative to the frequency step parameter q). Since the wavelet
+bandwidth is proportional to its center frequency, we find that
+the lower frequency region is insufficiently covered, similar
+to Fig. 1(b), for arbitrarily large base scale b. The size of
+this region depends on the chosen wavelet and the frequency
+step parameter q. To compensate for this lack of coverage,
+we introduce additional compensation filters covering the low-
+frequency region
+ψl,j(t) = 1
+√
+b
+ψ
+�t − d(l + δj)
+b
+�
+e
+2πiξ1· j(t−d(l+δj))
+q
+,
+(5)
+for negative integers j ≥ −q/b. In words, we demodulate
+the wavelet at the base scale b in uniform steps for as long
+as the center frequency remains nonnegative. Note that the
+construction above implies that the index of the quasi-random
+sequence of delays is shifted accordingly, i.e., we denote the
+j-th element by δj−⌊q/b⌋ instead of δj. This construction is
+illustrated in Fig. 3, where we rescaled the frequency axis
+compared to Fig. 2(a) (but not the time axis) for better
+visibility. Clearly, this is only one of many possibilities for
+the construction of suitable filters covering the low-frequency
+region, chosen here because it preserves the filter prototype,
+or mother wavelet, and the uniform decimation of the system
+in time and frequency.
+The full system {ψl,j}l∈Z,j∈Z∩[−q/b,∞) remains uniformly
+decimated and its properties, in particular the perfect recon-
+struction property and numerical stability can be studied using
+frame theory for uniform filter banks [34] or, equivalently,
+shift-invariant systems [33], which also provides highly effi-
+cient algorithms for reconstruction from the filter bank coeffi-
+cients. We observe that, in practice, choosing the decimation
+parameters d and 1/q small enough is sufficient to obtain an
+invertible, numerically stable filter bank. In Section IV-A, we
+validate this assessment for finite wavelet systems in CL.
+
+(a)
+Frequency
+Time(b)
+Frequency
+Time Wavelets
+Frequency
+Compensation 1
+TimeHOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER
+6
+IV. NUMERICAL EVALUATION, COMPLEXITY, AND
+IMPLEMENTATION
+Although the parametrization of {ψl,j}l,j introduced in
+the previous section in terms of b > 0 and q > 0 arises
+naturally, it is inconvenient for practical construction. Instead,
+we subsequently adopt the following parametrization, where
+we restrict to the case q/b ∈ N. We select a mother wavelet ψ,
+the desired number M +1 of frequency channels equidistantly
+spaced at center frequencies in [0, ξsamp/2], where ξsamp is
+the sampling rate, and the number MC of compensation
+filters. The relation to the parametrization introduced above
+is given by MC = q/b and ξ1/q = ξsamp/(2M). In the
+provided implementation, the wavelet is automatically scaled
+such that the center frequency of ψl,0 equals ξsamp·MC/(2M).
+The redundancy of the resulting system is then controlled
+by selecting the decimation factor d, similar to a common
+approach for parametrizing the STFT.
+A. Numerical Evaluation
+To validate that the proposed decimation strategy indeed
+leads to invertible, numerically stable wavelet systems, we
+compute the frame bound ratio RFB for various wavelet filter
+banks so decimated. Here, we consider filter banks acting on
+finite sequences, in which case, this ratio equals the condition
+number of the matrix that realizes the composition of filter
+bank analysis and synthesis and can be computed directly. To
+do so, we rely on the Large Time-Frequency Analysis Toolbox
+(LTFAT, ltfat.org), which achieves this computation by means
+of an efficient factorization of said matrix. Recall that the filter
+bank is invertible if and only if this RFB is finite and stable if
+it is small, with perfect stability if RFB = 1. The evaluation
+presented here can be reproduced using the code available at
+ltfat.org/notes/057.
+In a pre-test for the proposed evaluation, we noticed that for
+fixed M, the frame bound ratio RFB decreases monotonically
+with increasing MC, up to a certain value of MC after which
+a further increase yields no benefit. Furthermore, we found
+that this value of MC does not depend on the choice of
+M. Increasing MC stabilizes the frequency response of the
+filter bank, which equals Ψ = �
+j |�
+ψ0,j|2 up to a positive
+multiplicative constant, especially in the low-frequency region.
+Strong fluctuations of Ψ have detrimental effect on RFB, and
+increasing MC reduces these fluctuation, explaining the first
+observed effect. The lack of a dependence on M can be
+explained by the fact that the frequency response of two such
+filter banks that only differ in the choice of M are equal up
+to a dilation and a positive multiplicative constant. Altogether,
+these observations allow us to greatly reduce the number of
+tested configurations.
+With the above considerations in mind, we perform an opti-
+mization of the frame bounds by first determining, for a fixed
+mother wavelet and oversampling rate, the smallest value of
+MC that yields optimal RFB. We then proceed to determine the
+value M
+∈ {128, 256, 384, 512, 640, 768, 1024, 1280, 1536,
+2048} that optimizes RFB, with the additional restriction that
+M ≥ 50MC, by an exhaustive search. Finally, we further refine
+the choice of M by a divide and conquer approach starting
+from the determined optimizer and its two neighbors in the
+sequence of tested values for M. The restriction MC ≤ M/50
+ensures that the center frequency of ψl,0 is no larger than
+ξsamp/100, e.g., 441 Hz for a sampling rate of 44.1 kHz.
+Results:
+In Table I, we show the obtained optimal frame
+bounds for oversampling rates 1.2, 2, 4, and 8. As mother
+wavelet, we consider Cauchy wavelets [49], defined (up to a
+normalization constant) by �ψ(ξ) = ξ
+α−1
+2 e−2πξ, for α equal to
+100, 300, 900, and 2700, where higher α implies a higher Q-
+factor.4 As an example of a compactly supported wavelet, we
+consider the modulated fourth order B-spline defined (up to a
+normalization constant) by �ψ(ξ) = sin(π(ξ − ξfm))4/(π(ξ −
+ξfm))4, with ξfm equal to 3, 6, and 10, implying Q-factors that
+are roughly equivalent to Cauchy wavelets with α equal to
+257, 1024, and 2842, respectively5, see [50].
+As is to be expected, the optimal frame bound ratio de-
+creases with higher oversampling rate. For fixed oversampling,
+however, there is only limited dependence on the mother
+wavelet and Q-factor. Specifically, we obtain RFB close to
+14, 3, 1.65, and 1.3 for oversampling rates 1.2, 2, 4,
+and 8 across almost all conditions with Kronecker-sequence
+based decimation. For redundancy 4 and below, the results
+for the decimation based on a digital-(0,1) sequence yield
+slightly worse stability, i.e., larger RFB. Interestingly, for factor
+8 oversampling, RFB is improved by using this decimation
+scheme, for all but one condition, indicating that a decimation
+scheme based on digital-(0,1) sequences is beneficial when
+large oversampling rates are used.
+Note that our divide and conquer refinement procedure is
+only guaranteed to find the global minimum of RFB as a
+function in M if this function is convex. This is usually not
+the case as illustrated in Fig. 4, where we plot the values of
+RFB at oversampling rate 2, for all choices of M considered in
+the exhaustive search described above and all seven considered
+mother wavelets. Thus, it may still be possible to improve the
+reported bounds. Furthermore, this is an explanation why the
+reported optimal parameter M does not depend monotonically
+on the Q-factor of the mother wavelet which is larger for larger
+choices of α or ξfm.
+To validate that the wavelet coverings illustrated in Figs. 1
+and 2 above do indeed conform to the time-frequency domain
+being well covered by the proposed wavelet systems, we
+calculated accumulated spectrograms [51] of these systems
+at an oversampling rate of approximately 2, see Fig. 5.
+More specifically, we calculated the spectrogram for each
+wavelet in the system using a short-time Fourier transform
+with a Gaussian window and summed all of them. Since
+the spectrogram of a signal is a representation of its time-
+frequency energy localization, this sum illustrates the time-
+frequency area that is well represented by the different wavelet
+systems. Our simulations essentially confirm the findings of
+4The Q-factors for these hyperparameters have been estimated as the ratio
+of the mother wavelets’ center frequency and their bandwidth at −3dB height
+relative to their maximum. They correspond to 2.9999, 5.2053, 9.0212, and
+15.6282, respectively.
+5The estimated Q-factors for the B-spline wavelets correspond to 4.1600,
+8.3200, and 13.8666, while those for the Cauchy wavelets correspond to
+4.8171, 9.6229, and 16.0340, respectively.
+
+HOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER
+7
+Table I
+OPTIMIZED FRAME BOUND RATIOS FOR THE PROPOSED WAVELET SYSTEMS. TABLE VALUES ARE RFB (MC, M), I.E., THE OPTIMAL FRAME BOUND
+RATIO RFB IS ACHIEVED WITH MC COMPENSATION CHANNELS AND M TOTAL CHANNELS.
+Kronecker-sequence delays
+Cauchy
+B-Spline
+Oversampling
+α = 100
+α = 300
+α = 900
+α = 2700
+ξfm = 3
+ξfm = 6
+ξfm = 10
+1.2
+15.06 (2, 102)
+14.17 (4, 202)
+13.61 (6, 307)
+13.74 (11, 550)
+15.04 (4, 202)
+14.08 (7, 449)
+14.08 (12, 620)
+2
+3.22 (3, 448)
+2.98 (5, 253)
+2.92 (9, 501)
+2.94 (15, 764)
+3.01 (5, 253)
+2.93 (9, 511)
+2.97 (15, 764)
+4
+1.72 (4, 768)
+1.62 (7, 350)
+1.60 (12, 768)
+1.59 (20, 1012)
+1.60 (6, 363)
+1.59 (12, 768)
+1.59 (20, 1024)
+8
+1.31 (4, 214)
+1.25 (8, 404)
+1.23 (14, 702)
+1.24 (26, 1306)
+1.25 (7, 473)
+1.24 (15, 757)
+1.25 (25, 1250)
+Digital (0,1)-sequence delays
+Cauchy
+B-Spline
+Oversampling
+α = 100
+α = 300
+α = 900
+α = 2700
+ξfm = 3
+ξfm = 6
+ξfm = 10
+1.2
+20.38 (2, 127)
+19.27 (4, 255)
+17.48 (6, 384)
+16.02 (11, 1023)
+20.67 (4, 256)
+17.26 (7, 384)
+16.98 (12, 640)
+2
+3.80 (3, 260)
+3.73 (5, 384)
+3.66 (8, 656)
+3.57 (14, 771)
+3.72 (5, 388)
+3.62 (9, 771)
+3.62 (14, 771)
+4
+1.74 (4, 255)
+1.67 (7, 383)
+1.63 (12, 639)
+1.62 (21, 1279)
+1.78 (6, 640)
+1.62 (12, 639)
+1.60 (20, 1023)
+8
+1.27 (5, 256)
+1.20 (9, 511)
+1.20 (16, 1535)
+1.21 (27, 1791)
+1.21 (8, 511)
+1.27 (16, 1408)
+1.22 (27, 1791)
+128 256 384 512 640 768
+1024
+1280
+1536
+2048
+Number of channels (M)
+2.5
+3
+3.5
+4
+Frame bound ratio (RFB)
+Dependence of frame bound ratio on M
+Cauchy 
+=100
+Cauchy 
+=300
+Cauchy 
+=900
+Cauchy
+=2700
+B-Spline
+fm =3
+B-Spline 
+fm =6
+B-Spline 
+fm =10
+Figure 4. Examples for the dependence of RFB on M for Kronecker-sequence
+delays and oversampling rate 2.
+Fig. 1 and 2. The classical wavelet system in Fig. 5(a) and
+the delay shifted system using the golden Kronecker sequence
+in Fig. 5(c) show the best uniformity. Clearly, the uniform
+system without delays in Fig. 5(b) overemphasizes certain
+time-frequency regions while completely missing the area
+in between. The uniform system with delays based on a
+digital (0, 1)-sequence in Fig. 5(d) does not cover the area
+quite as uniformly. However, as the accumulated spectrograms
+presented here were obtained with two-fold oversampling, this
+finding matches the computed frame bounds: At low oversam-
+pling rates, the Kronecker-sequence delays are superior to the
+digital (0, 1)-sequence delays.
+B. Block-Processing and Complexity
+The most straightforward, efficient implementation of the
+proposed wavelet decimation implements the forward and the
+backward transform as FFT filter banks [52], similar to [16],
+[53]. Compared to these references, this type of implementa-
+tion of the proposed decimation implies a moderately higher
+computational load, due to its large number of overlapping
+frequency channels. By construction, FFT filter banks process
+the entire input signal at once. Nonetheless, it is possible to
+achieve block-wise processing by adopting the slicing scheme
+proposed in [15]. Sliced processing requires the segmentation
+of an incoming data stream into large blocks and the incurred
+delay may not be acceptable for certain applications. The im-
+plementation of wavelet filter banks with short block-length is
+often quite intricate, not least due to their usually non-uniform
+decimation, see, e.g., [54]. In our setting, however, shorter
+block-length, and thus reduced delay, can be achieved with a
+straightforward time-domain implementation, using a mother
+wavelet with finite support. Although real-time implementation
+is not our main objective, we outline the computational cost
+of such an implementation.
+The proposed uniform decimation admits an implementation
+with fixed block length, where each block corresponds to
+a time frame that contains one filter bank coefficient per
+channel. The relative temporal positions of all coefficients
+are identical in each time frame. Such an implementation is
+computationally straightforward and achieved by computing
+the inner products between the input signal and the wavelet
+atoms directly, see (1). Due to the constant Q-factor of the
+wavelet transform, the computational cost of doing so depends
+only logarithmically on the number of channels. Specifically,
+we compute the wavelet coefficients with a decimation factor
+d for M + 1 equidistant frequency channels, the first MC of
+which are compensation filters. We assume that, at the largest
+considered scale, the wavelet impulse response has a length
+of LW samples. Since the delays are in the interval [0, d),
+we can segment the input into time frames, or blocks, of
+length LB = LW + d, with an overlap of LW samples. In
+each time frame, we compute one coefficient per channel.
+The total cost per time frame for computing the compensation
+filters is MC · LW multiplications and additions each. For the
+
+HOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER
+8
+Figure 5.
+Accumulated spectrograms of decimated wavelet systems: (a) classical nonuniform decimation, (b) uniform grid, (c) uniform grid with delays
+chosen according to the golden Kronecker sequence, (d) delays derived from the digital sequence described in Section II-B.
+remaining filters, the impulse response length decreases as
+scales decrease and one can easily see that
+LW ·
+M
+�
+j=MC
+MC
+j
+≤ MCLW ·
+� M
+MC−1
+s−1 ds
+= MCLW · ln(M/(MC − 1))
+(6)
+multiplications and additions are required per time frame.
+Here, we used the standard integral estimate for partial sums
+of harmonic series. Note that the above computation ignores
+the rounding of impulse response lengths to the next integer
+and assumes that MC ≥ 2. Overall, a direct implementation
+of the forward transform amounts to approximately
+MCLW · (1 + ln(M/(MC − 1))
+(7)
+multiplications and additions per time frame and introduces
+a delay of LB samples. A more sophisticated implementation
+employing low-pass filtering and subsequent decimation in the
+spirit of [3] can introduce significant optimization at the cost
+of a small error in the coefficient computation.
+Computation of the backwards transform with perfect re-
+construction relies on the dual filters (see Section II-A). For
+the wavelet configurations in the present work, the essential
+impulse response length of the dual filters is of the same order
+as that of the corresponding original filter. This implies that the
+backward transform can be achieved at similar computational
+cost and delay as the forward transform. However, since the
+dual impulse responses are not expected to have finite support,
+they must be approximated, introducing a trade-off between
+accuracy and delay.
+C. Wavelets in the Large Time-Frequency Analysis Toolbox
+All computations in Sections IV-A and V rely on the imple-
+mentation of wavelet filter banks in the Large Time-Frequency
+Analysis Toolbox (LTFAT), updated to support the proposed
+wavelet decimation with the release of LTFAT 2.5.0. In partic-
+ular, the function waveletfilters supplies the frequency
+responses of the wavelet filters, as specified by the given
+input parameters, alongside a set of decimation factors and
+the filters’ center frequencies. By default, waveletfilters
+accepts the filter bank length L and a vector of wavelet
+scales, where the unit scale s = 1 corresponds to a center
+frequency of ξsamp/20, as input arguments. Additionally, a
+number of increasingly specific, optional input parameters can
+be used to customize the wavelet filters and cover a large
+number of use cases, including those in [40], [50] and in
+the present work. As an alternative to providing the wavelet
+scales directly, waveletfilters supports the automatic
+allocation of filters in a specified frequency range, spaced
+either geometrically with a set number of bins per octave,
+or linearly with a set number of channels.
+Most important for the present work, waveletfilters
+provides options for controlling the mother wavelet and
+decimation settings, as well as the oversampling rate. By
+default, waveletfilters selects a Cauchy-type mother
+wavelet with α = 300 and non-uniform, integer decima-
+tion factors adapted to the wavelet bandwidth. Currently, the
+Cauchy [55], Morse [56], Morlet, frequency B-spline [57],
+analytic spline [58], and complex spline wavelets are im-
+plemented for positive and negative scales. The individual
+filters are generated by the separate function freqwavelet,
+enabling the future addition of further wavelet prototypes.
+waveletfilters supports uniform decimation and non-
+uniform decimation, with integer or fractional decimation
+factors. To construct perfect reconstruction filter banks,
+waveletfilters supplies two options for covering the
+frequency range from 0 Hz to the center frequency of the
+largest wavelet scale: By a single low-pass filter, or by several
+frequency-shifted copies of the filter corresponding to the
+largest wavelet scale, as described in Section III-A.
+For example, the call [g,a,fc]=waveletfilters(
+Ls,’linear’,2,MC/M,1,M−MC+1,{’cauchy’,900},
+’uniform’,’redtar’,4,’repeat’,’delay’,@dly)
+supplies a wavelet filter bank for signals of length Ls with
+linear frequency spacing. Considering a sampling rate of
+2, M−MC+1 wavelet scales are spaced between frequencies
+MC/M and 1 (Nyquist). The filter bank uses a Cauchy wavelet
+with α = 900, uniform decimation and a target oversampling
+rate of 4. Finally, the flag repeat adds MC compensation
+filters between frequency 0 and MC/M, resulting in M+1 total
+channels. The function @dly is used to generate the desired
+sequence of delays, and will be called internally, with the
+number of required sequence elements M+1 and the vector
+of decimation factors a as input arguments. The output
+
+(a)
+1.6
+1.4
+Frequency
+1.2
+1
+0.8
+0.6
+Time(b)
+1.6
+1.4
+Frequency
+1.2
+1
+0.8
+0.6
+Time(c)
+1.6
+1.4
+Frequency
+1.2
+1
+0.8
+0.6
+Time(d)
+1.6
+1.4
+Frequency
+1.2
+1
+0.8
+0.6
+TimeHOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER
+9
+[g,a,fc] comprises a cell array g of filter frequency
+responses, a vector a of decimation factors and a vector fc
+of center frequencies.
+The filterbank module of LTFAT provides a host of func-
+tions for working with and analyzing a filter bank so created.
+The function filterbank calculates the filter bank coeffi-
+cients via fast FFT-based convolution. It accepts as input a
+target signal, as well as the wavelet filters and their associated
+decimation factors, as provided by waveletfilters. Syn-
+thesis from filter bank coefficients is realized by ifilter-
+bank and their visualization can be achieved via plotfil-
+terbank. If analysis-synthesis filter bank pairs with perfect
+reconstruction are desired, a dual filter bank can be obtained
+by applying filterbankdual to the output of wavelet-
+filters, respectively filterbankrealdual for filter
+banks that cover only positive frequencies. Note that this is
+only possible for uniform filter banks or under strict conditions
+on non-uniform filter banks. Generally, perfect reconstruction
+can be achieved with an iterative method, implemented in
+ifilterbankiter, provided that the analysis filter bank is
+invertible. The frame bounds of a filter bank can be obtained
+by filterbankbounds and filterbankrealbounds
+respectively, e.g., to verify invertibility. The LTFAT filterbank
+module provides selected methods for advanced functionality,
+such as phaseless reconstruction [39], [59] and time-frequency
+reassignment [60].
+Finally, the output of waveletfilters is compatible
+with the block processing framework in LTFAT, which enables
+experimental real-time application by implementing a variant
+of the sliced processing proposed in [15].
+The code used in this work, found at ltfat.org/notes/057,
+illustrates the construction of several wavelet filter banks, as
+well as the use of some of the functions outlined above. A
+more detailed demo will be integrated into the next LTFAT
+release.
+V. EXPERIMENTS IN AUDIO PROCESSING
+We present three applications of the proposed wavelet
+decimation scheme. In the first application, we replicate ex-
+periments from prior work on NMF-based signal decompo-
+sition [37], which originally relied on the STFT. We simply
+substitute the wavelet transform for the STFT in a plug-and-
+play manner, leaving all other parameters unchanged. Only the
+parameters of the wavelet transform, i.e., the mother wavelet
+ψ, the number of channels M and compensation channels MC,
+were adjusted. Given that the considered processing scheme
+was originally conceived for the STFT, and not adapted in
+any way, it is not expected that using the wavelet transform
+will outperform the original methods using the STFT. Instead,
+the goal is to demonstrate that even a naive plug-and-play
+approach can achieve comparable results.
+In the second application, we consider onset detection in
+musical signals. Using the idea of spectral flux [38, Sec. 3-
+A] for the wavelet coefficients in the proposed decimation
+scheme instead of the commonly used STFT coefficients, we
+calculate a detection function for onsets. The results for a small
+annotated test set [61] are compared to onsets detected by the
+same method using STFT coefficients as well as the output of
+a readily available onset detector [19].
+In a third application, we compare the performance of
+phaseless reconstruction with the fast Griffin-Lim algorithm
+(FGLA) [39], [62] between the proposed decimation scheme
+and classical decimation. Since FGLA does not rely on the
+segmentation of the transform coefficients into time frames,
+this allows us to compare the processing performance of
+our method to classical wavelet decimation. Additionally,
+results on phaseless reconstruction with FGLA from STFT
+coefficients are provided as a reference. Note that real-time
+variants of the Griffin-Lim algorithm do rely on frame-wise
+processing, e.g., [63], and could be adapted to the proposed
+decimation scheme.
+Accompanying audio files and code for reproducing the
+presented experiments are available at: ltfat.org/notes/057.
+A. Signal Decomposition with Nonnegative Matrix Factoriza-
+tion
+In [37], Févotte et al. showed that nonnegative matrix
+factorization with the Itakura-Saito cost function achieves a
+meaningful decomposition of STFT spectrograms of audio
+data as S = W · H, with W and H being nonnegative
+component and activation matrices, respectively. The authors
+presented the decomposition, denoising, and upmixing of a
+1920s recording of “My Heart (Will Always Lead Me Back
+to You)” by Louis Armstrong and His Hot Five. According to
+the subjective analysis in [37], the recording contains trumpet,
+clarinet, trombone, double bass, and piano tracks, as well as
+significant hiss noise and crackling. The signal used was an
+excerpt of length 108 s, sampled at ξsamp = 11 025 Hz, with
+a total length of L = 1 191 735 samples. In the original
+contribution, an Itakura-Saito NMF decomposition with 10
+components was computed from an STFT spectrogram at
+oversampling rate 2, with M = 129 channels from the zero
+to Nyquist frequencies. An inverse Gamma prior was used to
+regularize the component activation matrix H.
+In our experiment, we adapt the code provided with the
+follow-up contribution by Févotte [64], which uses a different
+smoothing prior, but is otherwise identical. As in that con-
+tribution, the regularization parameter is set to λ = 25. We
+use a wavelet transform with a Cauchy wavelet [40], [49],
+with hyperparameter α = 450. Decimation is based on the
+Kronecker sequence and the oversampling rate is set to 2.
+The parameters are optimized6 as in Section IV-A, leading to
+a total of M + 1 = 449 channels and MC = 6 compensation
+channels, i.e., the center frequency of ψl,0 is at 74 Hz. To
+achieve the desired oversampling rate, we set the decimation
+factor to d = 448. The resulting NMF components and signal
+decomposition are shown in Fig. 6.
+Results:
+Comparing to the results shown in [37], we see
+that the harmonic structure of the components in W is not
+as pronounced when the wavelet transform is used, but still
+present. This is not entirely surprising, considering the large
+6Due to memory constraints, we restrict the optimization to choose no more
+than 769 channels.
+
+HOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER
+10
+Figure 6. NMF decomposition of music excerpt. (Left) dB-scaled columns of W, i.e., NMF components. (Right) Reconstructed signal components. Components
+4, and 6–8 capture most of the piano and double bass tracks. Components 9, 10 characterize large parts of the hiss and crackling noise. The remaining
+components contain most of the trumpet and clarinet lead track, as well as the trombone.
+bandwidth of high frequency wavelets and the non-aligned
+phase space covering of the Kronecker sequence based deci-
+mation. Nonetheless, the decomposition achieves a separation
+into lead, accompaniment, and noise comparable to the results
+presented for the STFT in [37], [64]. The main difference
+is that the trombone track is not as clearly separated, but
+mostly mixed with the lead track. Audio examples, including
+individual components, as well as denoised and upmixed
+versions of the original signal, are provided on the website.
+B. Onset Detection in Music
+In our second experiment, we use the proposed wavelet
+decimation for the detection of onsets in audio signals. More
+specifically, we use a spectral flux method as, e.g., described in
+[38, Sec. 3-A] but replace the STFT with the proposed wavelet
+system. Spectral flux measures the increase in magnitude or
+energy in different frequency bands (a decrease is set to 0)
+and takes the sum of these increments at each time frame. The
+resulting time dependent spectral flux function is then used as
+a basis for a peak-picking procedure based on the assumption
+that local maxima of spectral flux are onsets of new musical
+events.
+In our experiment, we use in place of the frequency channels
+of an STFT the channels of our proposed wavelet system. The
+resulting spectral flux for a signal f is defined as
+S(l) =
+�
+j
+H
+�
+|Wψf(xl,j, sj)| − |Wψf(xl−1,j, sj)|
+�
+(8)
+where H(x) = (x + |x|)/2 is a rectifier and the sum is
+only over the channels corresponding to our novel decimation
+scheme ignoring the MC compensation channels. We choose
+a system with oversampling factor 4, a Cauchy wavelet with
+α = 2700, and the number of channels resulting in the best
+frame bound, i.e., M = 1012 and MC = 20 (see Table I).
+To avoid spurious local maxima, an additional time-dependent
+thresholding step is implemented. We use here a multiple λ of
+the local median which has been suggested as a robust choice
+in [38, Sec. 4-B].
+
+w
+Temporalcomponents
+-8:3
+8:2
+-8:2
+m
+0.4
+0.2
+K
+-0.4
+0.4
+0.2
+-
+K
+0.4
+0.4
+-8:3
+6
+8:2
+=
+:8:2
+0.4
+0.2
+=
+K
+0.4
+8:2
+8
+K
+-0.4
+9
+8:2
+-8:2
+K
+4
+10
+8:3
+4
+:8:2
+K
+.6
+100
+200
+300
+400
+2
+4
+6
+8
+10
+×105HOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER
+11
+0
+0.2
+0.4
+0.6
+0.8
+1
+P
+R
+F
+STFT
+Proposed WL
+MIR Onsets
+Figure 7.
+Whisker plots showing the minimal, median, and maximal F-
+measure F, recall R, and precision P, respectively, for 19 signals and
+three onset detectors. The first two onset detectors are based on spectral
+flux calculated for the proposed wavelet transform or an STFT, respectively.
+The third method is the standard method from the MIRtoolbox based on an
+amplitude envelope.
+For our experiment, we use the annotated onset detection
+database provided by the Pattern Recognition and Artificial
+Intelligence Group - University of Alicante (PRAIg-UA) [61].
+In the performance evaluation, we consider an onset to be
+correctly detected if the estimated onset is within 50ms of the
+annotated onset (this is a common measure in the literature
+[38, Sec. 5-A]). We then calculate the precision P as the
+quotient of correctly detected onsets and total estimated onsets,
+and the recall R as the quotient of correctly detected onsets
+and total annotated onsets. As a single performance measure,
+we further calculate the F-measure as F = 2PR/(P + R).
+We compare our method with spectral flux based on the STFT
+with a Hann window, i.e., replacing the wavelet coefficients in
+(8) with STFT coefficients. Here, we use the same decimation
+factor as in the wavelet case and adapt the number of channels
+and window length to obtain a tight frame with oversampling
+factor 4. Based on an optimization of the F-measure for the
+first audio sample in [61], we choose the threshold factor
+λ to be 1.24 for the STFT and 1.34 for the wavelet case.
+This sample is excluded in the results below. We compare our
+results to the basic onset detection algorithm implemented in
+the MIRtoolbox [19] which is based on an amplitude envelope.
+Results:
+Our results are illustrated in Fig. 7. The methods
+based on spectral flux are on average slightly better than the
+reference method based on an amplitude envelope. However,
+no method turned out to be universally best or worst over
+all signals. The difference between the STFT-based and the
+wavelet-based spectral flux is on average very small although
+for specific signals quite substantial differences are observed.
+This hints at the possibility that based on the signal class either
+method might be superior and a detailed analysis of their
+respective benefits and drawbacks is an interesting direction
+for further research. Finally, in both spectral flux cases the
+comparatively large recall and small precision suggests that
+the threshold factor λ was actually chosen too small for a
+good balance between wrong and missed detections.
+C. Phaseless Reconstruction with the Fast Griffin-Lim Algo-
+rithm
+In this experiment, we evaluate phaseless reconstruction,
+i.e., the reconstruction of an audio signal from magnitude-only
+time-frequency coefficients in our proposed wavelet decima-
+tion scheme. The Griffin-Lim algorithm [62] (GLA) remains
+the most popular iterative method for phaseless reconstruction
+from STFT or general time-frequency spectrograms. Here,
+we consider the fast Griffin-Lim (FGLA) variant proposed
+by Perraudin et al. [39], which introduces a Nesterov-like
+acceleration term. The reconstruction error is measured as
+relative spectral error, often referred to as spectral convergence
+and given by
+errMS(f, fr) = 10 log10
+∥|Wψfr| − |Wψf|∥2
+∥Wψf∥2
+,
+(9)
+with the target signal f and the reconstructed signal fr. In the
+literature, it is not always clear with respect to which time-
+frequency representation the quantity errMS is computed.
+Since we compare results across different representations and
+parameter choices, we fix the representation for computing
+errMS, as in [65]. Here, we choose a highly oversampled
+wavelet transform using a Cauchy-type mother wavelet with
+α = 1000, with M +1 = 181 geometrically spaced frequency
+channels and d = 7, i.e., approximately 50-fold oversampling,
+as a reference representation.
+The experimental setup is similar to [50, Section 4.2]: We
+consider the same 15 signals from the EBU SQAM dataset7
+and test Cauchy-type mother wavelets with α = 1000, at
+oversampling rates8 of 3, 5, and 10. For the proposed method,
+we choose M = 750, 968, and 1369 for low, medium, and
+high oversampling, respectively. We further use 11, 13, and 17
+compensation channels. The number of compensation channels
+is determined as in Section IV-A. After promising results
+were obtained for M = 750 at oversampling rate 3, the
+parameters for higher rates are chosen by isotropic scaling
+of the sampling grid, i.e., the product dM remains constant.9
+As references, we first consider wavelet transforms at the same
+oversampling rates, but with geometric frequency spacing.
+Since the uniform decimation scheme used in [50, Section 4.2]
+leads to unstable systems at low and medium oversampling,
+we further use channel-dependent decimation as discussed in
+Section II-A. The number of channels is set to M = 90, 125,
+and 180 for low, medium, and high oversampling, respectively.
+Furthermore, we consider STFTs, with a 1536 sample Hann
+window, at the same redundancies with M = 1536, 1920,
+and 2880 channels and uniform decimation. To prevent issues
+with poor initialization, we first compute 20 FGLA iterations
+for initial phase 0 and for five random uniformly distributed
+phase initializations. The best of these six candidates is used
+7The first 5 seconds of signals 01, 02, 04, 14, 15, 16, 27, 39, 49, 50, 51,
+52, 53, 54, and 70.
+8Note that the value M/a considered in [50, Section 4.2] corresponds to
+roughly half the oversampling rate, considering that coefficients are complex-
+valued.
+9Note that the requirement that d, M ∈ N leads to round-off errors.
+Moreover, the implementation we use chooses the largest value d such that
+M/(2d) is at least as large as the desired oversampling rate. Hence, the actual
+oversampling rate can be slightly larger.
+
+HOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER
+12
+−45
+−40
+−35
+−30
+−25
+−20
+−15
+−10
+−5
+10
+5
+3
+Spectral Error (dB)
+Oversampling Rate
+Classical WL
+Proposed WL
+STFT
+Figure 8. Whisker plots showing the minimal, median, and maximal spectral
+error after phaseless reconstruction for 15 signals and three transforms,
+namely, the classical wavelet transform, the proposed wavelet transform
+with Kronecker sequence based decimation, and the STFT. The different
+oversampling rates are arranged vertically.
+to compute the final solution by applying another 130 FGLA
+iterations.
+Results:
+In Fig. 8, our results are summarized. Overall, we
+see a clear trend that an increase in oversampling improves
+performance, matching the results obtained in [65]. On aver-
+age, the classical wavelet decimation is superior to the STFT
+at medium and high oversampling and to the proposed method
+at high oversampling rates. At medium oversampling both
+wavelet methods are roughly on par. At low oversampling,
+the performance of the proposed method is superior to both
+reference methods. On the associated website, we provide au-
+dio examples and additional results obtained when computing
+errMS with respect to a reference STFT. As observed in [65],
+errMS is biased towards representations that are similar to the
+reference, and we found that errMS with STFT reference,
+when compared to Fig. 8, favors the STFT over the wavelet
+transform in general, and Kronecker decimation over classical
+wavelet decimation in particular. However, the overall findings
+are similar, such that we do not include these results here.
+VI. CONCLUSION AND OUTLOOK
+We proposed a novel, uniform wavelet decimation scheme
+with quasi-random delays. Conceptually, and through numer-
+ical evaluation, we demonstrated that the proposed scheme
+is suitable for constructing wavelet systems with the perfect
+reconstruction property, even at oversampling rates close to
+1. In an audio decomposition application using Itakura-Saito
+NMF, we have shown that processing schemes previously
+proposed for the STFT are easily adapted to our setting by
+interpreting the filter bank coefficients as a time-frequency
+matrix, in contrast to classical, non-uniform wavelet decima-
+tion. Furthermore, we illustrated on a small test set of audio
+samples that our method can be used for onset detection by
+a straightforward adaptation of the spectral flux method origi-
+nally defined for the STFT. Finally, we observed promising
+performance of our method in phaseless reconstruction for
+diverse audio signals.
+We expect that the proposed decimation strategy will be-
+come a valuable asset for future work in different applications
+in audio and beyond. Furthermore, whereas the analysis of the
+proposed method in this work is restricted to finite systems,
+we expect the proposed construction to yield invertible wavelet
+transforms in the continuous domain as well. A rigorous study
+of its formal, mathematical properties in that setting is in
+preparation, considering frame and function space theory.
+ACKNOWLEDGMENTS
+We would like to thank Cédric Févotte for kindly providing
+their code for Itakura-Saito NMF and some guidance on its
+use. We further thank Georg Tauböck for fruitful discussion
+on potential applications of the proposed method.
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+
diff --git a/qdAzT4oBgHgl3EQfrP0p/content/tmp_files/load_file.txt b/qdAzT4oBgHgl3EQfrP0p/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf,len=1140
+page_content='HOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER  1  Grid-Based Decimation for Wavelet Transforms  with Stably Invertible Implementation  Nicki Holighaus, Günther Koliander, Clara Hollomey, and Friedrich Pillichshammer  Abstract—The constant center frequency to bandwidth ratio  (Q-factor) of wavelet transforms provides a very natural rep-  resentation for audio data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' However, invertible wavelet trans-  forms have either required non-uniform decimation—leading to  irregular data structures that are cumbersome to work with—or  require excessively high oversampling with unacceptable com-  putational overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Here, we present a novel decimation strat-  egy for wavelet transforms that leads to stable representations  with oversampling rates close to one and uniform decimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Specifically, we show that finite implementations of the resulting  representation are energy-preserving in the sense of frame theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  The obtained wavelet coefficients can be stored in a time-  frequency matrix with a natural interpretation of columns as time  frames and rows as frequency channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' This matrix structure  immediately grants access to a large number of algorithms that  are successfully used in time-frequency audio processing, but  could not previously be used jointly with wavelet transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' We  demonstrate the application of our method in processing based  on nonnegative matrix factorization, in onset detection, and in  phaseless reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Index Terms—wavelet transforms, low-discrepancy sequences,  sampling methods, signal reconstruction, audio applications,  shift-invariant systems, uniform decimation  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' INTRODUCTION  The wavelet transform is one of the most important and  well-studied time-frequency filter banks, only rivaled by the  short-time Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Due to its constant center fre-  quency to bandwidth ratio, or Q-factor [1], wavelets provide  a natural and physically meaningful representation of audio:  The impulse response of every filter captures an equal number  of wavelengths of its center frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The constant Q-factor  ensures that equal intervals on (Western) musical scales,  which consist of geometrically spaced tones, are resolved  equally well, independent of their absolute position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Finally,  at frequencies above 500 Hz, the peripheral auditory system in  Manuscript received XXX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' revised August XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Holighaus (corresponding author) and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Hollomey are with the  Acoustics  Research  Institute  (ARI),  Austrian  Academy  of  Sciences,  Wohllebengasse  12–14,  1040  Vienna,  Austria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Koliander  is  with  ARI and the Faculty of Mathematics, University of Vienna, Austria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Pillichshammer is with the Institute of Financial Mathematics and  Applied Number Theory, Johannes Kepler University Linz, Austria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' e-mail:  {nicki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='holighaus,guenther.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='koliander,clara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='hollomey}  @oeaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='at, friedrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='pillichshammer@jku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='at  Extended results, audio files and code for reproducing the presented  experiments is available at: ltfat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='org/notes/057  This work is supported by the Austrian Science Fund (FWF): I 3067–N30  (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='), Y 1199 “Time-Frequency Analysis, Randomness and Sampling.”  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='), and F5509-N26, which is a part of the Special Research Program  “Quasi-Monte Carlo Methods: Theory and Applications.” (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Copyright (c) 2023 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Personal use of this material is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  However, permission to use this material for any other purposes must be  obtained from the IEEE by sending a request to pubs-permissions@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  humans is well-approximated by a constant-Q filter bank [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  For a short discussion, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Consequently, constant-  Q filter banks in general [4]–[6] and wavelet transforms in  particular [7]–[10] have been used to great success in the  analysis of speech and audio signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' For the longest time,  however, constant-Q filter banks have been inaccessible, or at  least inconvenient, for processing audio at sufficient fidelity:  Wavelet bases [11] and undecimated, discrete wavelet systems  [12], [13] possess a very low Q-factor that is not suitable  for processing audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Modern implementations of constant-  Q filter banks allow for the tuning of the Q-factor and the  oversampling rate, but computationally efficient, numerically  stable, and invertible constant-Q filter banks have only recently  been constructed: In [14] this is achieved by purposeful  modification of Mallat’s algorithm for the fast wavelet trans-  form, and in [15], [16] by means of mathematical frame the-  ory [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' To achieve perfect reconstruction, these constructions  rely on non-uniform decimation, choosing sufficiently small  decimation factors inversely proportional to the bandwidth  or, equivalently, center frequency of the constant-Q filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='1  Consequently, every channel produces coefficients at different,  usually unmatched, rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Alternatively, it is possible to employ  no in-channel decimation whatsoever, resulting, however, in  prohibitively high oversampling at large Q-factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Although the availability of invertible representations with  appropriate frequency resolution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', a sufficiently large Q-  factor, presents an important step towards making constant-  Q filter banks more attractive for audio processing, another  fundamental issue remains, previously discussed in [3], along-  side a partial solution: Employing non-uniform decimation, the  filter bank coefficients form an irregular data structure that  is fundamentally more difficult to work with than matrices,  both computationally and conceptually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Such coefficients are  incompatible with any method that relies on matrix manip-  ulation, and substantial work is required to integrate them  into readily available audio processing toolboxes such as,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', [18] and [19], usually implemented under the assumption  of matrix-structured coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' This effect is most readily  apparent in algorithms for the ubiquitous short-time Fourier  transform (STFT), which often rely on the segmentation of the  STFT coefficients into time frames, see [20] and [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' These  time frames are processed individually (or in small groups)  and often in real-time, but the segmentation relies on the  assumption that the coefficients are time-aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' More pre-  cisely, the between channel coefficient relations are assumed  1This approach for choosing decimation factors is not exclusive to invert-  ible constant-Q transforms, but shared across most constant-Q and wavelet  transform implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='01640v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='AS]  4 Jan 2023  HOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER  2  to be position-independent, which is clearly not the case for  non-uniform decimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Hence, the adaptation of successful  processing schemes for STFT coefficients to the constant-Q  setting remains challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Nonuniform decimation also has  implications concerning the reconstruction procedure itself,  see Section II-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Most importantly, even efficient reconstruc-  tion algorithms [22], [23] involve costly iteration, unless all  filters are strictly band-limited [24], precluding the use of finite  impulse response filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  In this work, we present a family of wavelet filter banks  based on quasi-random sampling of the continuous wavelet  transform that are computationally efficient, with tunable Q-  factor through unrestricted choice of the mother wavelet, and  flexible oversampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' With oversampling rates close to  one, the proposed filter banks achieve perfect reconstruction of  discrete signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Uniquely, they do so while employing uniform  decimation and thereby overcoming the difficulties induced by  the irregularity of previous invertible constant-Q transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Previous work on random and quasi-random sampling of time-  frequency representations: This work is not the first to consider  (quasi-)randomized sampling of time-frequency representa-  tions or more general integral transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' In a series of recent  works, Levie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' consider Monte Carlo [25], [26] and Quasi-  Monte Carlo [27] style random sampling of time-frequency  integral operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Their work is concerned with the approx-  imation of continuous domain time-frequency processing by  means of Monte Carlo integration with (quasi-)random time-  frequency samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' By invoking prior results on Monte Carlo  integration, the authors demonstrate that the approximation  error can be controlled when a technical structure condition  is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' They further show that this condition is satisfied  by the STFT, the wavelet transform, and a custom blending  of the two, referred to as localizing time-frequency transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  The invertibility or stability of the sampled representation is  not investigated, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' In the context of random sampling,  these properties are considered in the literature on relevant  sampling, introduced by Bass and Gröchenig for bandlimited  functions [28] and later generalized to various settings [29]–  [31], including time-frequency representations [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Relevant  sampling provides a probabilistic framework for stable sam-  pling of functions that are localized in a domain of finite  volume, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', bandlimited signals that have only negligible  energy outside a finite interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Our work differs from these  prior works in multiple ways: Our proposed sampling sets  are not fully (quasi-)random, but correspond to a uniform  time-frequency grid, up to the introduction of a small, quasi-  random delay in every wavelet channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Further, we consider  perfect reconstruction of arbitrary signals, without localization  assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Contribution:  Previously, grid-based decimation strategies  were considered ill-suited for the continuous wavelet transform  and not expected to provide efficient and stably invertible  representations even for discrete signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Indeed, known con-  structions did not admit perfect reconstruction at moderate, or  even low, oversampling rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' In this paper, we propose the first  grid-based decimation strategy for wavelet transforms with a  tunable Q-factor that allows perfect reconstruction at oversam-  pling rates close to 1 and that provides excellent numerical  stability, in the sense of energy preservation,2 at moderate  oversampling rates, ranging from 2 to 8, as commonly used  for audio processing with the STFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' To achieve this, we  combine shift-invariant systems [33], [34] with ideas from  quasi-random sampling using low-discrepancy sequences [35],  [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Similar to wavelet bases and prior invertible constant-Q  implementations, we use a set of compensation filters to cover  an arbitrarily small low-frequency region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Our construction  is validated in the finite domain by computing exact frame  bound ratios and accumulated spectrograms of the decimated  wavelet systems across a range of system parameters covering  variations of the mother wavelet, the number of frequency  channels, the decimation factor, and the oversampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' As  a proof of concept, we apply the proposed wavelet decimation  to several audio applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' We replicate an experiment on  signal enhancement based on Itakura-Saito nonnegative matrix  factorization (NMF) as proposed by Févotte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' [37], which  relies heavily on the natural interpretation of the representa-  tion coefficients as a time-frequency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Furthermore, we  illustrate the use of our proposed scheme in onset detection  based on a straightforward adaptation of the classic spectral  flux method [38, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 3-A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' As an indication that the pro-  posed scheme performs on par with established, non-uniform  constant-Q transforms in tasks that do not require matrix  structure, we further evaluate the suitability of the proposed  method for phaseless reconstruction with the fast Griffin-Lim  algorithm [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' In addition to the standard, irregular wavelet  decimation, we also compare to the STFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Paper structure:  A short introduction to wavelet systems  and quasi-random sequences is given in Section II, before  presenting the proposed decimation scheme in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  In Section IV, we discuss details of practical implementation  and complexity, and evaluate our construction numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  We further apply the proposed scheme in three illustrating  experiments in audio processing (Section V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Specifically,  we consider the decomposition of a signal with nonnegative  matrix factorization, onset detection based on an adaptation  of spectral flux, and phaseless reconstruction from time-  frequency coefficients based on the fast Griffin-Lim algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  The paper concludes with a summary of the results and an  outlook towards related future work (Section VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' TECHNICAL BACKGROUND  Before introducing our novel decimation strategy, we review  some basics of wavelet systems and quasi-random sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The Wavelet Transform  A wavelet system is a collection of functions (or vectors)  generated from a single prototype, the mother wavelet, by  translation and dilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Since we are interested in real-valued  signals, audio signals in particular, we consider a mother  wavelet ψ such that its Fourier transform vanishes for negative  frequencies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', ˆψ(ξ) = 0 for ξ ∈ (−∞, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Such mother  wavelets are often called analytic, although the terminology  2For more details on stability and energy preservation, see Section II-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  HOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER  3  analytic wavelet transform has been used in at least two dif-  ferent manners in the past [40], [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The continuous wavelet  system is generated via dilation by s > 0 and translation by  x ∈ R of the mother wavelet:  ψ(x,s) := s−1/2ψ  �• − x  s  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  A signal f can now be filtered using this system resulting in  the continuous wavelet transform  Wψf(x, s) = 1  √s  �  f(t)ψ  �t − x  s  �  dt  = ⟨f, ψ(x,s)⟩L2  =  �  f ∗ ψ (− • /s)  �  (x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  (1)  The mathematical study of wavelet transforms usually con-  siders Wψf in terms of the inner product representation  of the transform coefficients, but the final equality justifies  the interpretation as a filter bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Importantly, the wavelet  transform is shift-invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', the wavelet transform of a  delayed signal f(•−y) equals the delayed (in the first variable)  wavelet transform of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Also note that the scale s is inversely  proportional to the filter center frequency: If ψ has its passband  around frequency ξ1, then ψ(•/s) has its passband around  ξs := ξ1/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  In applications, only a discrete subset of the continuous  wavelet system can be considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' the system, or equiva-  lently the transform, is decimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Commonly, the discrete  subset  �  (xl,j, sj)  �  l∈Z,j∈I ⊂ R × [0, ∞) of translation-dilation  pairs is generated by certain decimation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' We denote  the corresponding decimated wavelet system by {ψl,j}l∈Z,j∈I  with ψl,j := ψ(xl,j,sj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Regarding the choice of decimation  �  (xl,j, sj)  �  l∈Z,j∈I, it is desirable that any function f can  be stably recovered from the decimated transform and the  energy of f is accurately represented by its coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Mathematically, these properties are equivalent to the discrete  subset (ψl,j)l,j of the continuous wavelet system constituting  a frame [17], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', it satisfies the energy equivalence relation  A∥f∥2  2 ≤  �  l,j  |⟨f, ψ(l,j)⟩|2 ≤ B∥f∥2  2,  (2)  for all f and some constants 0 < A ≤ B < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Implicitly,  we always assume that A, B are the optimal constants such  that (2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The ratio RFB := B/A of the so-called upper  (B) and lower (A) frame bounds quantifies how well the  decimated transform preserves signal energy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', how stable  it is in numerical computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' If RFB ̸= 1, then error-free  reconstruction of f from the decimated transform requires  a dual frame, which can be efficiently precomputed under  certain conditions on mother wavelet and decimation [15],  [24], [42], and realized by iterative schemes otherwise [22],  [23], [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' If the transform coefficients are modified, the  energy of the synthesized function ˜f can nevertheless be  bounded by A−1 times the energy of the modified coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Note that the dual frame of a wavelet frame is not necessarily  a wavelet frame, or even a filter bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' In fact, that is the  exception rather than the norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Regarding the choice of the scales sj, recall that the  bandwidth of wavelet filters increases linearly with their center  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Time-frequency geometry of decimated wavelet systems: (a) The  classical nonuniform wavelet decimation provides a mostly even covering  of time-frequency space, at the cost of introducing nonuniform decimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  (b) Wavelet decimation on a uniform grid results in an increasingly uneven  covering away from a small frequency strip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The depicted hyperbolic circles  have identical radius across both panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  frequency ξs, which is inversely proportional to their scale s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Thus, it is natural to decimate the frequency channels by taking  integer powers of a fixed base scale a > 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', sj = a−j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Conversely, the width of the wavelet impulse responses is  proportional to the scale, suggesting in-channel decimation  according to xl,j = a−j ·lb for some fixed b > 0, see also [44,  Chapter 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Intuitively, the time-frequency region covered by  the individual ψl,j corresponds roughly to ellipses of constant  area that grow narrower as j increases, or, more accurately, to  hyperbolic circles of constant radius,3 centered at (xl,j, ξ1/sj),  see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  This decimation strategy choosing (xl,j, sj)  =  (a−j ·  lb, a−j), with geometric frequency spacing and channel-  dependent decimation factor (hop size) dj = a−j · b, is  customary and well-studied for wavelet bases, overcomplete  wavelet systems, and constant-Q transforms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', [15], [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  We will refer to this convention as classical wavelet deci-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' If ψ is not strictly bandlimited or the decimation  factors are too large, then the dual frame required for error-  free reconstruction will only be a filter bank in exceptional  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Furthermore, the dependence of the decimation factor  dj on the scale leads to filter bank coefficients that are not  time-aligned, and, for a lack of position-independent relations  3A hyperbolic circle contains all time-scale pairs that have a hy-  perbolic  distance less  than  a  given  radius to  the  center  (xl,j, sj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Here, hyperbolic distance between (x, s) and (x′, s′) is measured as  2 arsinh  √  (x′−x)2+(s′−s)2  2  √  ss′  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Note that we depict these circles in the time-  frequency rather than the time-scale plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  (a)  Frequency  Time(b)  Frequency  TimeHOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER  4  between coefficients in different bands, cannot be represented  as a time-frequency matrix or segmented into time frames of  identical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' In many applications and implementations,  this irregularity encumbers the workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  In contrast to the  STFT, the time-frequency geometry of the wavelet transform  described above is ill-suited for decimation on a uniform,  rectangular grid, independent of whether the center frequencies  are spaced linearly or logarithmically, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' It is  easy to see and, in fact, straightforward to prove, that such  a decimation strategy cannot lead to a numerically stable  system with perfect reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Considering that many  successful processing schemes for the STFT rely on both  perfect reconstruction and the representation of the coefficients  as a time-frequency matrix, this presents a notable obstruction  to the adoption of wavelets in (audio) signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Quasi-Random Sequences  Quasi-random sequences, also known as low-discrepancy  sequences [35], are deterministic sequences of numbers or k-  dimensional coordinates that share some properties of uni-  formly distributed random numbers and can be used as  a replacement for the latter in various applications, most  prominently the quasi-Monte Carlo method for numerical  integration [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The term discrepancy refers to one of several  related measures that quantify how uniformly distributed a  set of points is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' This property is important, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', to ob-  tain error bounds in quasi-Monte Carlo integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' While  most modern theoretical studies are concerned with quasi-  random sequences of high dimensionality, we only require  one-dimensional sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Specifically, any N consecutive  elements of a low-discrepancy sequence in dimension D = 1  provides N numbers that are particularly well distributed in  the unit interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Decimation strategies for wavelets can be derived from any  low-discrepancy sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Here, we focus on two carefully  chosen examples based on two prominent classes of quasi-  random sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Kronecker sequences: A deceptively simple construction  are so-called Kronecker sequences which are of the form  ({αl})l≥0  = (0, {α}, {2α}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=') for some real α, where  {x} := x − ⌊x⌋ denotes the fractional part of a real x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' It  is well-known that Kronecker sequences are particularly well  distributed if α is a badly approximable number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' These are  irrational numbers that are particularly poorly approximated  by rationals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Formally, a number α ∈ R is badly approximable  if there is a constant c > 0, such that  |α − p/q| > c/q2,  for all nonzero integers p, q ∈ Z \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' It is known that an  irrational number is badly approximable if and only if the  coefficients of its continued fraction expansion are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Among all the badly approximable numbers, the golden ratio  φ = 1+  √  5  2  maximizes the optimal constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Since this prop-  erty is shared exactly with all so-called equivalent numbers  of the form aφ+b  cφ+d, for integers a, b, c, d with ad − bc = ±1,  we may likewise consider the Kronecker sequence with α =  1 − 1/φ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', a, c = 1, b = −1, and d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' For more  information on Kronecker sequences we refer to the books  [45], [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Digital (0, 1)-sequences over Z2: A (0, 1)-sequence in base  2 is an infinite sequence (xl)l≥0 = (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=') in the unit-  interval [0, 1) with the following property: for every m ∈ N0  and every k ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' , 2m − 1}, the elementary interval  [ k  2m , k+1  2m ) contains exactly one element of the point set  {xl : p2m ≤ l ≤ (p + 1)2m − 1}  for every p ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  One example for such a sequence is the well-known van der  Corput sequence [47] (see also [48]), which is related to the  bit-reversal permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Typically, (0, 1)-sequences in base 2  are constructed by the so-called digital method over the finite  field Z2 of order two, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', the integers modulo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' To this end,  let C = (cr,k)r,k≥1 be an ∞ × ∞-matrix over Z2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', with  entries cr,k from {0, 1}, such that for every m ∈ N the left-  upper m × m sub-matrix is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Then the sequence  (xl)l≥0 defined by  xl =  ∞  �  r=0  ηl,r  2r+1  where ηl,r =  ∞  �  k=0  cr+1,k+1lk  (mod 2),  where lk ∈ {0, 1} are the binary digits of the index l, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=',  l = l0 +l12+l222 +· · · (which are obviously 0 from a certain  index on), is a (0, 1)-sequence in base 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' In this context, the  van der Corput sequence is covered by choosing the identity  matrix for C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' We will use the specific digital (0, 1)-sequence  which is obtained from the infinite matrix  C =  �  �  �  �  �  �  �  �  �  1  0  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  1  1  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  0  1  1  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  0  0  1  1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  0  0  0  1  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  �  �  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  For more information on (digital) (0, 1)-sequences we refer to  the books [35], [36] or the survey article [48, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' A NEW CONVENTION FOR WAVELET SAMPLING  Due to the geometric intuition outlined in Section II-A,  linear spacing of wavelet center frequencies ξj ∼ s−1  j  has  hardly been considered in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' For the same reason,  uniform decimation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', dj = d for some fixed d > 0,  is usually disregarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' We will now describe a decimation  scheme that follows both of these conventions and which can  be used to construct numerically stable, perfect reconstruction  wavelet transforms with moderate, or even marginal, oversam-  pling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' By virtue of being a uniform filter bank frame, standard  results [33], [34] ensure that the dual frame is a uniform filter  bank as well, with the same decimation factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The dual filter  bank is, however, not necessarily a wavelet system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' In particu-  lar, this implies that said dual frame can be precomputed using  optimized factorization algorithms and, thus, synthesis from  transform coefficients is highly efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The proposed scheme  results in an almost time-aligned decimated representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  More precisely, the coefficients across bands are in a position-  independent, fixed relation, thus providing a meaningful notion  HOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER  5  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Time-frequency geometry of decimated wavelet systems with quasi-  random delay: (a) Decimation on a uniform grid with delays chosen according  to the golden Kronecker sequence provides an even covering comparable to  the classic wavelet decimation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' (b) Deriving the delays instead  from the digital sequence described in Section II-B, we obtain a covering that  is slightly less even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' In both cases, the resulting covering will become uneven  below a certain frequency, similar to the bottom region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Note that  the radius of the depicted hyperbolic circles is chosen as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 1 to allow  comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' A mild increase in radius is sufficient to remove the blind spots  in panel (b), whereas even a minor decrease would introduce similar blind  spots in panel (a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  of time frames and suggesting a natural arrangement in a time-  frequency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Given a largest scale of interest b > 0 (or equivalently a  minimal frequency of interest ξmin > 0), a decimation factor  d > 0, a third parameter q > 0 that determines the step  size in the frequency direction, and finally a low-discrepancy  sequence (δ0, δ1, δ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=') determining channel specific delays,  we select the translation-dilation pairs  (xl,j, sj) =  �  d(l + δj),  1  b−1 + q−1j  �  ,  (3)  for all integers l ∈ Z and nonnegative integers j ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' In other  words, we consider the discrete wavelet system {ψl,j}l∈Z,j∈N0,  with  ψl,j(t) =  �  1  b + j  q · ψ  �  ( 1  b + j  q) · (t − d(l + δj)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  (4)  In contrast to the rectangular grid shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 1(b), which  can be generated by the above construction with delays δj = 0  for all j, choosing the δj as the elements of a quasi-random  sequence provides a wavelet system that covers the time-  frequency plane surprisingly evenly, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Intriguingly,  if we choose the δj as the elements of a Kronecker sequence,  the points {(xl,j, ξj)}l∈Z,j∈N0 form a uniform (skewed) time-  frequency grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Proposed covering of the low-frequency region by modulated and  delayed versions of the largest scale wavelet filters for the decimated wavelet  system with quasi-random delays chosen according to the golden Kronecker  sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The frequency axis is rescaled compared to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 2(a) for better  visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Numerically Stable Wavelet Transforms with Perfect Re-  construction and Uniform Decimation  As indicated by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 2, the proposed wavelet decimation  yields a surprisingly uniform covering of the time-frequency  plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' However, due to the uniform spacing in frequency, this  is only true as long as the scale of the wavelets is small enough  (relative to the frequency step parameter q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Since the wavelet  bandwidth is proportional to its center frequency, we find that  the lower frequency region is insufficiently covered, similar  to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 1(b), for arbitrarily large base scale b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The size of  this region depends on the chosen wavelet and the frequency  step parameter q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' To compensate for this lack of coverage,  we introduce additional compensation filters covering the low-  frequency region  ψl,j(t) = 1  √  b  ψ  �t − d(l + δj)  b  �  e  2πiξ1· j(t−d(l+δj))  q  ,  (5)  for negative integers j ≥ −q/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' In words, we demodulate  the wavelet at the base scale b in uniform steps for as long  as the center frequency remains nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Note that the  construction above implies that the index of the quasi-random  sequence of delays is shifted accordingly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', we denote the  j-th element by δj−⌊q/b⌋ instead of δj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' This construction is  illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 3, where we rescaled the frequency axis  compared to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 2(a) (but not the time axis) for better  visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Clearly, this is only one of many possibilities for  the construction of suitable filters covering the low-frequency  region, chosen here because it preserves the filter prototype,  or mother wavelet, and the uniform decimation of the system  in time and frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  The full system {ψl,j}l∈Z,j∈Z∩[−q/b,∞) remains uniformly  decimated and its properties, in particular the perfect recon-  struction property and numerical stability can be studied using  frame theory for uniform filter banks [34] or, equivalently,  shift-invariant systems [33], which also provides highly effi-  cient algorithms for reconstruction from the filter bank coeffi-  cients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' We observe that, in practice, choosing the decimation  parameters d and 1/q small enough is sufficient to obtain an  invertible, numerically stable filter bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' In Section IV-A, we  validate this assessment for finite wavelet systems in CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  (a)  Frequency  Time(b)  Frequency  Time Wavelets  Frequency  Compensation 1  TimeHOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER  6  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' NUMERICAL EVALUATION, COMPLEXITY, AND  IMPLEMENTATION  Although the parametrization of {ψl,j}l,j introduced in  the previous section in terms of b > 0 and q > 0 arises  naturally, it is inconvenient for practical construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Instead,  we subsequently adopt the following parametrization, where  we restrict to the case q/b ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' We select a mother wavelet ψ,  the desired number M +1 of frequency channels equidistantly  spaced at center frequencies in [0, ξsamp/2], where ξsamp is  the sampling rate, and the number MC of compensation  filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The relation to the parametrization introduced above  is given by MC = q/b and ξ1/q = ξsamp/(2M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' In the  provided implementation, the wavelet is automatically scaled  such that the center frequency of ψl,0 equals ξsamp·MC/(2M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  The redundancy of the resulting system is then controlled  by selecting the decimation factor d, similar to a common  approach for parametrizing the STFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Numerical Evaluation  To validate that the proposed decimation strategy indeed  leads to invertible, numerically stable wavelet systems, we  compute the frame bound ratio RFB for various wavelet filter  banks so decimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Here, we consider filter banks acting on  finite sequences, in which case, this ratio equals the condition  number of the matrix that realizes the composition of filter  bank analysis and synthesis and can be computed directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' To  do so, we rely on the Large Time-Frequency Analysis Toolbox  (LTFAT, ltfat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='org), which achieves this computation by means  of an efficient factorization of said matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Recall that the filter  bank is invertible if and only if this RFB is finite and stable if  it is small, with perfect stability if RFB = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The evaluation  presented here can be reproduced using the code available at  ltfat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='org/notes/057.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  In a pre-test for the proposed evaluation, we noticed that for  fixed M, the frame bound ratio RFB decreases monotonically  with increasing MC, up to a certain value of MC after which  a further increase yields no benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Furthermore, we found  that this value of MC does not depend on the choice of  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Increasing MC stabilizes the frequency response of the  filter bank, which equals Ψ = �  j |�  ψ0,j|2 up to a positive  multiplicative constant, especially in the low-frequency region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Strong fluctuations of Ψ have detrimental effect on RFB, and  increasing MC reduces these fluctuation, explaining the first  observed effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The lack of a dependence on M can be  explained by the fact that the frequency response of two such  filter banks that only differ in the choice of M are equal up  to a dilation and a positive multiplicative constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Altogether,  these observations allow us to greatly reduce the number of  tested configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  With the above considerations in mind, we perform an opti-  mization of the frame bounds by first determining, for a fixed  mother wavelet and oversampling rate, the smallest value of  MC that yields optimal RFB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' We then proceed to determine the  value M  ∈ {128, 256, 384, 512, 640, 768, 1024, 1280, 1536,  2048} that optimizes RFB, with the additional restriction that  M ≥ 50MC, by an exhaustive search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Finally, we further refine  the choice of M by a divide and conquer approach starting  from the determined optimizer and its two neighbors in the  sequence of tested values for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The restriction MC ≤ M/50  ensures that the center frequency of ψl,0 is no larger than  ξsamp/100, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', 441 Hz for a sampling rate of 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='1 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Results:  In Table I, we show the obtained optimal frame  bounds for oversampling rates 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='2, 2, 4, and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' As mother  wavelet, we consider Cauchy wavelets [49], defined (up to a  normalization constant) by �ψ(ξ) = ξ  α−1  2 e−2πξ, for α equal to  100, 300, 900, and 2700, where higher α implies a higher Q-  factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='4 As an example of a compactly supported wavelet, we  consider the modulated fourth order B-spline defined (up to a  normalization constant) by �ψ(ξ) = sin(π(ξ − ξfm))4/(π(ξ −  ξfm))4, with ξfm equal to 3, 6, and 10, implying Q-factors that  are roughly equivalent to Cauchy wavelets with α equal to  257, 1024, and 2842, respectively5, see [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  As is to be expected, the optimal frame bound ratio de-  creases with higher oversampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' For fixed oversampling,  however, there is only limited dependence on the mother  wavelet and Q-factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Specifically, we obtain RFB close to  14, 3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='65, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='3 for oversampling rates 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='2, 2, 4,  and 8 across almost all conditions with Kronecker-sequence  based decimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' For redundancy 4 and below, the results  for the decimation based on a digital-(0,1) sequence yield  slightly worse stability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', larger RFB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Interestingly, for factor  8 oversampling, RFB is improved by using this decimation  scheme, for all but one condition, indicating that a decimation  scheme based on digital-(0,1) sequences is beneficial when  large oversampling rates are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Note that our divide and conquer refinement procedure is  only guaranteed to find the global minimum of RFB as a  function in M if this function is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' This is usually not  the case as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 4, where we plot the values of  RFB at oversampling rate 2, for all choices of M considered in  the exhaustive search described above and all seven considered  mother wavelets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Thus, it may still be possible to improve the  reported bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Furthermore, this is an explanation why the  reported optimal parameter M does not depend monotonically  on the Q-factor of the mother wavelet which is larger for larger  choices of α or ξfm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  To validate that the wavelet coverings illustrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 1  and 2 above do indeed conform to the time-frequency domain  being well covered by the proposed wavelet systems, we  calculated accumulated spectrograms [51] of these systems  at an oversampling rate of approximately 2, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  More specifically, we calculated the spectrogram for each  wavelet in the system using a short-time Fourier transform  with a Gaussian window and summed all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Since  the spectrogram of a signal is a representation of its time-  frequency energy localization, this sum illustrates the time-  frequency area that is well represented by the different wavelet  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Our simulations essentially confirm the findings of  4The Q-factors for these hyperparameters have been estimated as the ratio  of the mother wavelets’ center frequency and their bandwidth at −3dB height  relative to their maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' They correspond to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='9999, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='2053, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='0212, and  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='6282, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  5The estimated Q-factors for the B-spline wavelets correspond to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='1600,  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='3200, and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='8666, while those for the Cauchy wavelets correspond to  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='8171, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='6229, and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='0340, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  HOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER  7  Table I  OPTIMIZED FRAME BOUND RATIOS FOR THE PROPOSED WAVELET SYSTEMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' TABLE VALUES ARE RFB (MC, M), I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', THE OPTIMAL FRAME BOUND  RATIO RFB IS ACHIEVED WITH MC COMPENSATION CHANNELS AND M TOTAL CHANNELS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Kronecker-sequence delays  Cauchy  B-Spline  Oversampling  α = 100  α = 300  α = 900  α = 2700  ξfm = 3  ξfm = 6  ξfm = 10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
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+page_content='20 (16, 1535)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='21 (27, 1791)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='21 (8, 511)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='27 (16, 1408)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='22 (27, 1791)  128 256 384 512 640 768  1024  1280  1536  2048  Number of channels (M)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='5  4  Frame bound ratio (RFB)  Dependence of frame bound ratio on M  Cauchy  =100  Cauchy  =300  Cauchy  =900  Cauchy  =2700  B-Spline  fm =3  B-Spline  fm =6  B-Spline  fm =10  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Examples for the dependence of RFB on M for Kronecker-sequence  delays and oversampling rate 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The classical wavelet system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 5(a) and  the delay shifted system using the golden Kronecker sequence  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 5(c) show the best uniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Clearly, the uniform  system without delays in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 5(b) overemphasizes certain  time-frequency regions while completely missing the area  in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The uniform system with delays based on a  digital (0, 1)-sequence in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 5(d) does not cover the area  quite as uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' However, as the accumulated spectrograms  presented here were obtained with two-fold oversampling, this  finding matches the computed frame bounds: At low oversam-  pling rates, the Kronecker-sequence delays are superior to the  digital (0, 1)-sequence delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Block-Processing and Complexity  The most straightforward, efficient implementation of the  proposed wavelet decimation implements the forward and the  backward transform as FFT filter banks [52], similar to [16],  [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Compared to these references, this type of implementa-  tion of the proposed decimation implies a moderately higher  computational load, due to its large number of overlapping  frequency channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' By construction, FFT filter banks process  the entire input signal at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Nonetheless, it is possible to  achieve block-wise processing by adopting the slicing scheme  proposed in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Sliced processing requires the segmentation  of an incoming data stream into large blocks and the incurred  delay may not be acceptable for certain applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The im-  plementation of wavelet filter banks with short block-length is  often quite intricate, not least due to their usually non-uniform  decimation, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' In our setting, however, shorter  block-length, and thus reduced delay, can be achieved with a  straightforward time-domain implementation, using a mother  wavelet with finite support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Although real-time implementation  is not our main objective, we outline the computational cost  of such an implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  The proposed uniform decimation admits an implementation  with fixed block length, where each block corresponds to  a time frame that contains one filter bank coefficient per  channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The relative temporal positions of all coefficients  are identical in each time frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Such an implementation is  computationally straightforward and achieved by computing  the inner products between the input signal and the wavelet  atoms directly, see (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Due to the constant Q-factor of the  wavelet transform, the computational cost of doing so depends  only logarithmically on the number of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Specifically,  we compute the wavelet coefficients with a decimation factor  d for M + 1 equidistant frequency channels, the first MC of  which are compensation filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' We assume that, at the largest  considered scale, the wavelet impulse response has a length  of LW samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Since the delays are in the interval [0, d),  we can segment the input into time frames, or blocks, of  length LB = LW + d, with an overlap of LW samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' In  each time frame, we compute one coefficient per channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  The total cost per time frame for computing the compensation  filters is MC · LW multiplications and additions each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' For the  HOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER  8  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Accumulated spectrograms of decimated wavelet systems: (a) classical nonuniform decimation, (b) uniform grid, (c) uniform grid with delays  chosen according to the golden Kronecker sequence, (d) delays derived from the digital sequence described in Section II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  remaining filters, the impulse response length decreases as  scales decrease and one can easily see that  LW ·  M  �  j=MC  MC  j  ≤ MCLW ·  � M  MC−1  s−1 ds  = MCLW · ln(M/(MC − 1))  (6)  multiplications and additions are required per time frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Here, we used the standard integral estimate for partial sums  of harmonic series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Note that the above computation ignores  the rounding of impulse response lengths to the next integer  and assumes that MC ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Overall, a direct implementation  of the forward transform amounts to approximately  MCLW · (1 + ln(M/(MC − 1))  (7)  multiplications and additions per time frame and introduces  a delay of LB samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' A more sophisticated implementation  employing low-pass filtering and subsequent decimation in the  spirit of [3] can introduce significant optimization at the cost  of a small error in the coefficient computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Computation of the backwards transform with perfect re-  construction relies on the dual filters (see Section II-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' For  the wavelet configurations in the present work, the essential  impulse response length of the dual filters is of the same order  as that of the corresponding original filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' This implies that the  backward transform can be achieved at similar computational  cost and delay as the forward transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' However, since the  dual impulse responses are not expected to have finite support,  they must be approximated, introducing a trade-off between  accuracy and delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Wavelets in the Large Time-Frequency Analysis Toolbox  All computations in Sections IV-A and V rely on the imple-  mentation of wavelet filter banks in the Large Time-Frequency  Analysis Toolbox (LTFAT), updated to support the proposed  wavelet decimation with the release of LTFAT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' In partic-  ular, the function waveletfilters supplies the frequency  responses of the wavelet filters, as specified by the given  input parameters, alongside a set of decimation factors and  the filters’ center frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' By default, waveletfilters  accepts the filter bank length L and a vector of wavelet  scales, where the unit scale s = 1 corresponds to a center  frequency of ξsamp/20, as input arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Additionally, a  number of increasingly specific, optional input parameters can  be used to customize the wavelet filters and cover a large  number of use cases, including those in [40], [50] and in  the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' As an alternative to providing the wavelet  scales directly, waveletfilters supports the automatic  allocation of filters in a specified frequency range, spaced  either geometrically with a set number of bins per octave,  or linearly with a set number of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Most important for the present work, waveletfilters  provides options for controlling the mother wavelet and  decimation settings, as well as the oversampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' By  default, waveletfilters selects a Cauchy-type mother  wavelet with α = 300 and non-uniform, integer decima-  tion factors adapted to the wavelet bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Currently, the  Cauchy [55], Morse [56], Morlet, frequency B-spline [57],  analytic spline [58], and complex spline wavelets are im-  plemented for positive and negative scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The individual  filters are generated by the separate function freqwavelet,  enabling the future addition of further wavelet prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  waveletfilters supports uniform decimation and non-  uniform decimation, with integer or fractional decimation  factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' To construct perfect reconstruction filter banks,  waveletfilters supplies two options for covering the  frequency range from 0 Hz to the center frequency of the  largest wavelet scale: By a single low-pass filter, or by several  frequency-shifted copies of the filter corresponding to the  largest wavelet scale, as described in Section III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  For example, the call [g,a,fc]=waveletfilters(  Ls,’linear’,2,MC/M,1,M−MC+1,{’cauchy’,900},  ’uniform’,’redtar’,4,’repeat’,’delay’,@dly)  supplies a wavelet filter bank for signals of length Ls with  linear frequency spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Considering a sampling rate of  2, M−MC+1 wavelet scales are spaced between frequencies  MC/M and 1 (Nyquist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The filter bank uses a Cauchy wavelet  with α = 900, uniform decimation and a target oversampling  rate of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Finally, the flag repeat adds MC compensation  filters between frequency 0 and MC/M, resulting in M+1 total  channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The function @dly is used to generate the desired  sequence of delays, and will be called internally, with the  number of required sequence elements M+1 and the vector  of decimation factors a as input arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The output  (a)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='4  Frequency  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='2  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='6  Time(b)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='4  Frequency  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='2  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='6  Time(c)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='4  Frequency  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='2  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='6  Time(d)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='4  Frequency  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='2  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='6  TimeHOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER  9  [g,a,fc] comprises a cell array g of filter frequency  responses, a vector a of decimation factors and a vector fc  of center frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  The filterbank module of LTFAT provides a host of func-  tions for working with and analyzing a filter bank so created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  The function filterbank calculates the filter bank coeffi-  cients via fast FFT-based convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' It accepts as input a  target signal, as well as the wavelet filters and their associated  decimation factors, as provided by waveletfilters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Syn-  thesis from filter bank coefficients is realized by ifilter-  bank and their visualization can be achieved via plotfil-  terbank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' If analysis-synthesis filter bank pairs with perfect  reconstruction are desired, a dual filter bank can be obtained  by applying filterbankdual to the output of wavelet-  filters, respectively filterbankrealdual for filter  banks that cover only positive frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Note that this is  only possible for uniform filter banks or under strict conditions  on non-uniform filter banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Generally, perfect reconstruction  can be achieved with an iterative method, implemented in  ifilterbankiter, provided that the analysis filter bank is  invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The frame bounds of a filter bank can be obtained  by filterbankbounds and filterbankrealbounds  respectively, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', to verify invertibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The LTFAT filterbank  module provides selected methods for advanced functionality,  such as phaseless reconstruction [39], [59] and time-frequency  reassignment [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Finally, the output of waveletfilters is compatible  with the block processing framework in LTFAT, which enables  experimental real-time application by implementing a variant  of the sliced processing proposed in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  The code used in this work, found at ltfat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='org/notes/057,  illustrates the construction of several wavelet filter banks, as  well as the use of some of the functions outlined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' A  more detailed demo will be integrated into the next LTFAT  release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' EXPERIMENTS IN AUDIO PROCESSING  We present three applications of the proposed wavelet  decimation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' In the first application, we replicate ex-  periments from prior work on NMF-based signal decompo-  sition [37], which originally relied on the STFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' We simply  substitute the wavelet transform for the STFT in a plug-and-  play manner, leaving all other parameters unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Only the  parameters of the wavelet transform, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', the mother wavelet  ψ, the number of channels M and compensation channels MC,  were adjusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Given that the considered processing scheme  was originally conceived for the STFT, and not adapted in  any way, it is not expected that using the wavelet transform  will outperform the original methods using the STFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Instead,  the goal is to demonstrate that even a naive plug-and-play  approach can achieve comparable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  In the second application, we consider onset detection in  musical signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Using the idea of spectral flux [38, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 3-  A] for the wavelet coefficients in the proposed decimation  scheme instead of the commonly used STFT coefficients, we  calculate a detection function for onsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The results for a small  annotated test set [61] are compared to onsets detected by the  same method using STFT coefficients as well as the output of  a readily available onset detector [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  In a third application, we compare the performance of  phaseless reconstruction with the fast Griffin-Lim algorithm  (FGLA) [39], [62] between the proposed decimation scheme  and classical decimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Since FGLA does not rely on the  segmentation of the transform coefficients into time frames,  this allows us to compare the processing performance of  our method to classical wavelet decimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Additionally,  results on phaseless reconstruction with FGLA from STFT  coefficients are provided as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Note that real-time  variants of the Griffin-Lim algorithm do rely on frame-wise  processing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', [63], and could be adapted to the proposed  decimation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Accompanying audio files and code for reproducing the  presented experiments are available at: ltfat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='org/notes/057.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Signal Decomposition with Nonnegative Matrix Factoriza-  tion  In [37], Févotte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' showed that nonnegative matrix  factorization with the Itakura-Saito cost function achieves a  meaningful decomposition of STFT spectrograms of audio  data as S = W · H, with W and H being nonnegative  component and activation matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The authors  presented the decomposition, denoising, and upmixing of a  1920s recording of “My Heart (Will Always Lead Me Back  to You)” by Louis Armstrong and His Hot Five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' According to  the subjective analysis in [37], the recording contains trumpet,  clarinet, trombone, double bass, and piano tracks, as well as  significant hiss noise and crackling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The signal used was an  excerpt of length 108 s, sampled at ξsamp = 11 025 Hz, with  a total length of L = 1 191 735 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' In the original  contribution, an Itakura-Saito NMF decomposition with 10  components was computed from an STFT spectrogram at  oversampling rate 2, with M = 129 channels from the zero  to Nyquist frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' An inverse Gamma prior was used to  regularize the component activation matrix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  In our experiment, we adapt the code provided with the  follow-up contribution by Févotte [64], which uses a different  smoothing prior, but is otherwise identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' As in that con-  tribution, the regularization parameter is set to λ = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' We  use a wavelet transform with a Cauchy wavelet [40], [49],  with hyperparameter α = 450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Decimation is based on the  Kronecker sequence and the oversampling rate is set to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  The parameters are optimized6 as in Section IV-A, leading to  a total of M + 1 = 449 channels and MC = 6 compensation  channels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', the center frequency of ψl,0 is at 74 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' To  achieve the desired oversampling rate, we set the decimation  factor to d = 448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The resulting NMF components and signal  decomposition are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Results:  Comparing to the results shown in [37], we see  that the harmonic structure of the components in W is not  as pronounced when the wavelet transform is used, but still  present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' This is not entirely surprising, considering the large  6Due to memory constraints, we restrict the optimization to choose no more  than 769 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  HOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER  10  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' NMF decomposition of music excerpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' (Left) dB-scaled columns of W, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', NMF components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' (Right) Reconstructed signal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Components  4, and 6–8 capture most of the piano and double bass tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Components 9, 10 characterize large parts of the hiss and crackling noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The remaining  components contain most of the trumpet and clarinet lead track, as well as the trombone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  bandwidth of high frequency wavelets and the non-aligned  phase space covering of the Kronecker sequence based deci-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Nonetheless, the decomposition achieves a separation  into lead, accompaniment, and noise comparable to the results  presented for the STFT in [37], [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The main difference  is that the trombone track is not as clearly separated, but  mostly mixed with the lead track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Audio examples, including  individual components, as well as denoised and upmixed  versions of the original signal, are provided on the website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Onset Detection in Music  In our second experiment, we use the proposed wavelet  decimation for the detection of onsets in audio signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' More  specifically, we use a spectral flux method as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', described in  [38, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 3-A] but replace the STFT with the proposed wavelet  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Spectral flux measures the increase in magnitude or  energy in different frequency bands (a decrease is set to 0)  and takes the sum of these increments at each time frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The  resulting time dependent spectral flux function is then used as  a basis for a peak-picking procedure based on the assumption  that local maxima of spectral flux are onsets of new musical  events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  In our experiment, we use in place of the frequency channels  of an STFT the channels of our proposed wavelet system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The  resulting spectral flux for a signal f is defined as  S(l) =  �  j  H  �  |Wψf(xl,j, sj)| − |Wψf(xl−1,j, sj)|  �  (8)  where H(x) = (x + |x|)/2 is a rectifier and the sum is  only over the channels corresponding to our novel decimation  scheme ignoring the MC compensation channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' We choose  a system with oversampling factor 4, a Cauchy wavelet with  α = 2700, and the number of channels resulting in the best  frame bound, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', M = 1012 and MC = 20 (see Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  To avoid spurious local maxima, an additional time-dependent  thresholding step is implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' We use here a multiple λ of  the local median which has been suggested as a robust choice  in [38, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 4-B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  w  Temporalcomponents  8:3  8:2  8:2  m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='2  K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='2 K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='4  8:3  6  8:2  =  :8:2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='2  =  K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='4  8:2  8  K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='4  9  8:2  8:2  K  4  10  8:3  4  :8:2  K  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='6  100  200  300  400  2  4  6  8  10  ×105HOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER  11  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='8  1  P  R  F  STFT  Proposed WL  MIR Onsets  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Whisker plots showing the minimal, median, and maximal F-  measure F, recall R, and precision P, respectively, for 19 signals and  three onset detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The first two onset detectors are based on spectral  flux calculated for the proposed wavelet transform or an STFT, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  The third method is the standard method from the MIRtoolbox based on an  amplitude envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  For our experiment, we use the annotated onset detection  database provided by the Pattern Recognition and Artificial  Intelligence Group - University of Alicante (PRAIg-UA) [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  In the performance evaluation, we consider an onset to be  correctly detected if the estimated onset is within 50ms of the  annotated onset (this is a common measure in the literature  [38, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 5-A]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' We then calculate the precision P as the  quotient of correctly detected onsets and total estimated onsets,  and the recall R as the quotient of correctly detected onsets  and total annotated onsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' As a single performance measure,  we further calculate the F-measure as F = 2PR/(P + R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  We compare our method with spectral flux based on the STFT  with a Hann window, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', replacing the wavelet coefficients in  (8) with STFT coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Here, we use the same decimation  factor as in the wavelet case and adapt the number of channels  and window length to obtain a tight frame with oversampling  factor 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Based on an optimization of the F-measure for the  first audio sample in [61], we choose the threshold factor  λ to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='24 for the STFT and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='34 for the wavelet case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  This sample is excluded in the results below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' We compare our  results to the basic onset detection algorithm implemented in  the MIRtoolbox [19] which is based on an amplitude envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Results:  Our results are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The methods  based on spectral flux are on average slightly better than the  reference method based on an amplitude envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' However,  no method turned out to be universally best or worst over  all signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The difference between the STFT-based and the  wavelet-based spectral flux is on average very small although  for specific signals quite substantial differences are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  This hints at the possibility that based on the signal class either  method might be superior and a detailed analysis of their  respective benefits and drawbacks is an interesting direction  for further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Finally, in both spectral flux cases the  comparatively large recall and small precision suggests that  the threshold factor λ was actually chosen too small for a  good balance between wrong and missed detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Phaseless Reconstruction with the Fast Griffin-Lim Algo-  rithm  In this experiment, we evaluate phaseless reconstruction,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', the reconstruction of an audio signal from magnitude-only  time-frequency coefficients in our proposed wavelet decima-  tion scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The Griffin-Lim algorithm [62] (GLA) remains  the most popular iterative method for phaseless reconstruction  from STFT or general time-frequency spectrograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Here,  we consider the fast Griffin-Lim (FGLA) variant proposed  by Perraudin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' [39], which introduces a Nesterov-like  acceleration term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The reconstruction error is measured as  relative spectral error, often referred to as spectral convergence  and given by  errMS(f, fr) = 10 log10  ∥|Wψfr| − |Wψf|∥2  ∥Wψf∥2  ,  (9)  with the target signal f and the reconstructed signal fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' In the  literature, it is not always clear with respect to which time-  frequency representation the quantity errMS is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Since we compare results across different representations and  parameter choices, we fix the representation for computing  errMS, as in [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Here, we choose a highly oversampled  wavelet transform using a Cauchy-type mother wavelet with  α = 1000, with M +1 = 181 geometrically spaced frequency  channels and d = 7, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', approximately 50-fold oversampling,  as a reference representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  The experimental setup is similar to [50, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='2]: We  consider the same 15 signals from the EBU SQAM dataset7  and test Cauchy-type mother wavelets with α = 1000, at  oversampling rates8 of 3, 5, and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' For the proposed method,  we choose M = 750, 968, and 1369 for low, medium, and  high oversampling, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' We further use 11, 13, and 17  compensation channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The number of compensation channels  is determined as in Section IV-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' After promising results  were obtained for M = 750 at oversampling rate 3, the  parameters for higher rates are chosen by isotropic scaling  of the sampling grid, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=', the product dM remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='9  As references, we first consider wavelet transforms at the same  oversampling rates, but with geometric frequency spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Since the uniform decimation scheme used in [50, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='2]  leads to unstable systems at low and medium oversampling,  we further use channel-dependent decimation as discussed in  Section II-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The number of channels is set to M = 90, 125,  and 180 for low, medium, and high oversampling, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Furthermore, we consider STFTs, with a 1536 sample Hann  window, at the same redundancies with M = 1536, 1920,  and 2880 channels and uniform decimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' To prevent issues  with poor initialization, we first compute 20 FGLA iterations  for initial phase 0 and for five random uniformly distributed  phase initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The best of these six candidates is used  7The first 5 seconds of signals 01, 02, 04, 14, 15, 16, 27, 39, 49, 50, 51,  52, 53, 54, and 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  8Note that the value M/a considered in [50, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='2] corresponds to  roughly half the oversampling rate, considering that coefficients are complex-  valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  9Note that the requirement that d, M ∈ N leads to round-off errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Moreover, the implementation we use chooses the largest value d such that  M/(2d) is at least as large as the desired oversampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Hence, the actual  oversampling rate can be slightly larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  HOLIGHAUS, KOLIANDER, HOLLOMEY, AND PILLICHSHAMMER  12  −45  −40  −35  −30  −25  −20  −15  −10  −5  10  5  3  Spectral Error (dB)  Oversampling Rate  Classical WL  Proposed WL  STFT  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Whisker plots showing the minimal, median, and maximal spectral  error after phaseless reconstruction for 15 signals and three transforms,  namely, the classical wavelet transform, the proposed wavelet transform  with Kronecker sequence based decimation, and the STFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' The different  oversampling rates are arranged vertically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  to compute the final solution by applying another 130 FGLA  iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  Results:  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 8, our results are summarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Overall, we  see a clear trend that an increase in oversampling improves  performance, matching the results obtained in [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' On aver-  age, the classical wavelet decimation is superior to the STFT  at medium and high oversampling and to the proposed method  at high oversampling rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' At medium oversampling both  wavelet methods are roughly on par.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' At low oversampling,  the performance of the proposed method is superior to both  reference methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' On the associated website, we provide au-  dio examples and additional results obtained when computing  errMS with respect to a reference STFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' As observed in [65],  errMS is biased towards representations that are similar to the  reference, and we found that errMS with STFT reference,  when compared to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' 8, favors the STFT over the wavelet  transform in general, and Kronecker decimation over classical  wavelet decimation in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' However, the overall findings  are similar, such that we do not include these results here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' CONCLUSION AND OUTLOOK  We proposed a novel, uniform wavelet decimation scheme  with quasi-random delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Conceptually, and through numer-  ical evaluation, we demonstrated that the proposed scheme  is suitable for constructing wavelet systems with the perfect  reconstruction property, even at oversampling rates close to  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' In an audio decomposition application using Itakura-Saito  NMF, we have shown that processing schemes previously  proposed for the STFT are easily adapted to our setting by  interpreting the filter bank coefficients as a time-frequency  matrix, in contrast to classical, non-uniform wavelet decima-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Furthermore, we illustrated on a small test set of audio  samples that our method can be used for onset detection by  a straightforward adaptation of the spectral flux method origi-  nally defined for the STFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Finally, we observed promising  performance of our method in phaseless reconstruction for  diverse audio signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  We expect that the proposed decimation strategy will be-  come a valuable asset for future work in different applications  in audio and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' Furthermore, whereas the analysis of the  proposed method in this work is restricted to finite systems,  we expect the proposed construction to yield invertible wavelet  transforms in the continuous domain as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' A rigorous study  of its formal, mathematical properties in that setting is in  preparation, considering frame and function space theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  We would like to thank Cédric Févotte for kindly providing  their code for Itakura-Saito NMF and some guidance on its  use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
+page_content=' We further thank Georg Tauböck for fruitful discussion  on potential applications of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAzT4oBgHgl3EQfrP0p/content/2301.01640v1.pdf'}
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diff --git a/rdAzT4oBgHgl3EQf5_7j/content/tmp_files/2301.01870v1.pdf.txt b/rdAzT4oBgHgl3EQf5_7j/content/tmp_files/2301.01870v1.pdf.txt
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+arXiv:2301.01870v1  [math.AP]  5 Jan 2023
+A nonlinear elasticity problem with no local but many
+global minimizers unrelated by symmetry
+Yury Grabovsky
+Lev Truskinovsky
+January 6, 2023
+Abstract
+In this paper, dedicated to 85th birthday of R. Fosdick, we touch upon two issues
+which attracted considerable attention in his own research: nonuniqueness in geomet-
+rically linear elasticity and the Clapeyron theorem. To this end we consider a simple
+model of a solid-solid phase transition with incompatible energy wells which exhibits
+multiplicity of global minimizers in a hard device, unrelated to either objectivity or
+material symmetry. At the same time, the same example shows the absence of strong
+local Lipschitz minimizers which are not global minimizers.
+Our proof that every
+strong local minimizer in the corresponding variational problem is also a global one
+hinges on the new general sufficiency theorem, proved by means of the novel nonlinear
+generalization of the Clapeyron theorem.
+1
+Introduction
+The phenomenon of metastability in elastostatics, manifesting itself through the existence of
+strong local minimizers which are not global, is usually associated with a Neumann problem
+(soft device) and is linked to the incompatibility of the energy wells [7]. In this paper we
+present an analytically transparent example of an energy with incompatible wells for which
+one can prove the absence of strong local minimizers which are not global on any domain
+and for any Dirichlet boundary conditions (hard device).
+Moreover, for the same energy one can prove dramatic nonuniqueness of global mini-
+mizers. Previously, it has been understood that to ensure uniqueness, the use of Dirichlet
+boundary conditions and topological, or even geometric simplicity of the domain are essential
+[41, 48, 46]. Uniqueness has been established for star-shaped domains, affine displacement
+boundary conditions, and strictly quasiconvex stored energy functions [29, 47]. Whether
+uniqueness holds may also depend on the regularity class in which one looks for a minimizer
+[30], or even on its integrability class [4, 44, 23]. For mixed boundary-value problems of
+nonlinear elasticity nonuniqueness is common with the most familiar examples being those
+associated with buckling, due to the emergence of multiple symmetry-related energy minima
+[14, 17]. The possibility of nonuniqueness with Dirichlet boundary conditions was shown for
+non-homogeneous problems [12, 34, 50]. Our simple example shows that the multiplicity of
+1
+
+f( )
+Figure 1: Double-well nonlinearity in a geometrically linear bi-quadratic Hadamard material.
+global minimizers in a hard device problem can be obtained even in the absence of geomet-
+rical complexity of the domain, non-homogeneity, and can be unrelated to either objectivity
+or material symmetry. It is very similar to examples related to the nonuniqueness of optimal
+microstructures in composites [3, 38, 16, 35].
+More specifically, we consider a “geometrically linearized” Hadamard material which is
+a simplification of the fully nonlinear Hadamard material [21, 25] whose energy density
+function is of the form
+W(F ) = µ|F |2 + h(det F ),
+(1.1)
+where µ is the measure of rigidity and the non-negative function h(d) is defined on (0, +∞)
+and has the property that h(d) → ∞, as d → 0+. If we use the “geometric” approximation
+det F ≈ 1 + Tr (F − I), which is valid in the limit F → I, such formal asymptotic expan-
+sion with respect to a small parameter would also induce physical linearization and would
+trivialize the problem. To retain physical non-linearity we consider the energy
+W(H) = g(Tr H) + µ|H|2,
+H = F − I.
+(1.2)
+We may now perform geometric linearization in the second term as well [27, 9, 26, 1, 2],
+replacing H with its symmetric part ε = (H +Ht)/2 and, to emphasise its isotropic nature,
+writing the energy density as
+W(ε) = f(Tr ε) + µ|dev(ε)|2,
+dev(ε) = ε − 1
+3Tr (ε)I.
+(1.3)
+Mathematically, the analyses of (1.2) and (1.3) are very similar. We will therefore, focus on
+(1.3), because it preserves at least the linearized version of the frame indifference property.
+We assume that µ > 0 and the function f has the “double-well” shape. For analytical
+transparency we use in our explicit constructions the bi-quadratic potential
+f(θ) = min{κ0θ2, κ0(θ − θp)2 + f0},
+(1.4)
+illustrated in Fig. 1. The ensuing model describes a material capable of undergoing a purely
+dilatational phase transformation between two phases which are both linearly elastic. They
+have the same moduli and differ only by the transformation strain and the reference (chemi-
+cal) energy. The relaxation of such energies has been always considered as a benchmark case
+for all theories of elastic phase transitions [27, 31, 40].
+2
+
+We recall that if the wells are rank-one connected, a quasiconvex energy would have to
+be convex, due to the rank-one conexity of quasiconvex functions [39]. On the other hand,
+the incompatibility of the wells can be responsible for quasiconvexity of a non-convex energy
+density, as we will see in our example, where the energy wells are not rank one connected.
+In our example, despite the fact that the non-convexity enters (1.3) only through a scalar
+potential f(θ), the energy density (1.3) is non-quasiconvex for all µ > 0, if f(θ) is given
+by (1.4), [31]. Therefore, even if subjected to homogeneous loading in a hard device, the
+materials described by (1.4) must undergo non-homogeneous deformations by developing mi-
+crostructures. In the presence of local minima, such systems can be expected to experience
+hysteresis [5, 24, 42, 43, 52]. However, as we have already mentioned, the local minima in
+this model are absent. Then, the macroscopic quasistatic mechanical response in the hard
+device corresponding to global minimization of the total energy at each value of the loading
+parameter is defined uniquely, and as a result, the deformation path is reversible. The asso-
+ciated stress-strain relation can be obtained from the knowledge of the quasiconvex envelope
+of the energy (1.3) that can be characterized explicitly in the whole range of parameters and
+for all double-well shaped potentials f(θ). In this paper we perform the implied relaxation
+of the energy (1.3) using simple laminates as the optimal microstructure; a different proof
+based on matching of upper and lower bounds can be found in [20].
+One of the general results of this paper is the formulation of the necessary and sufficient
+conditions for global minimizers in a star-shaped domain with affine boundary conditions.
+This creates a tool to characterize energy minimizing configurations in examples where the
+energy relaxation, or at least the elastic binodal, is explicitly known. An important technical
+tool in this analysis is a fundamental nonlinear generalization of the Clapeyron theorem
+which is presented here for the first time. These results can be viewed as a generalization of
+optimality conditions for extremal microstructures in composites (e.g. [16]).
+We show that using our necessary and sufficient conditions one can reduce the problem of
+finding a global minimizer to the solution of a nonlinear free boundary problem involving a
+system of linear PDEs in a finite domain. It is remarkable that for the energy density (1.3) the
+solution of the ensuing problem can be found explicitly, for any double-well scalar potential
+f(θ), providing us with an explicit example of multiplicity of global minimizers. In particular,
+we were able to compute explicitly a one parameter family of low surface energy non-affine
+configurations in the square domain, whose boundary is piecewise smooth. In this way we
+showed the possibility to always generate in this class of problems an optimal microstructure
+with low surface area, which is physically advantageous comparing, for instance, to multiscale
+lamination with infinitely large surface area [32].
+The analysis of optimal microstructures suggests that, despite the absence of the total
+energy hysteresis in this system, the direct and the reverse transformation may follow dif-
+ferent transformation paths in the configuration space. Therefore in this problem, in the
+absence of metastability and “constitutive hysteresis”, we encounter a phenomenon of “mor-
+phological hysteresis”, whereby the forward and reverse transformations occur along different
+morphological paths, while traversing energetically equivalent configurations.
+The paper is organized as follows. In Section 2 we discuss the restrictions on the structure
+of strong local minimizers. The necessary and sufficient conditions for global minimizers are
+formulated in Section 3. The issue of attainment and the multiplicity of global minimizers
+3
+
+in our model are discussed in Section 4.
+An explicit construction of a nontrivial global
+minimizer in a square domain which has expressly low surface area is presented in Section 5.
+The last Section 6 contains our conclusions. An Appendix A contains a technical discussion
+of the generic loss of ellipticity for the energy represented by a rank one convex envelope.
+2
+Local minimizers
+Given that the case of interest involves hard device loading, our goal is to study strong local
+and global minima of the functional
+E0(u) =
+�
+Ω
+�
+f(∇ · u) + µ
+����e(u) − 1
+3(∇ · u)I
+����
+2�
+dx
+(2.1)
+among all Lipschitz displacement vector fields u ∈ W 1,∞(Ω; R3) subject to the constraint
+u(x) = u0(x),
+x ∈ ∂Ω,
+(2.2)
+where u0(x) is a given Lipschitz function on ∂Ω. Restricting attention to Lipschitz minimiz-
+ers allows us to focus on instabilities caused by failure of quasiconvexity [39, 10] and exclude
+some other instabilities related, for instance, to the mismatch between the integrability of
+the minimizing sequences and the growth of the energy density at infinity [6].
+First we recall that one of the necessary conditions of metastability, understood here as
+the lack of strong local Lipschitz minimizers which are not also global minimizers, is stability
+with respect to nucleation of a coherent precipitate of the new phase in the interior of the
+old phase. That means that we must have ∇u(x) ∈ A, for a.e. x ∈ Ω, [48, Proposition 4.1],
+where
+A = {H ∈ M : W(H) = QW(H)},
+(2.3)
+where M denotes the set of all 3 × 3 matrices and QW(H) is the quasiconvexification of W
+[10].
+In this paper we show that if u(x) is a weak solution of the Euler-Lagrange equation
+for the energy (2.1) with the boundary conditions (2.2) and satisfies ∇u(x) ∈ A for a.e.
+x ∈ Ω then, it is necessarily a global minimizer for E0(u). However, before we move to the
+actual analysis, it is appropriate to mention that we are not aware of any examples of proper
+metastable states in a hard device in domains with trivial topology. Moreover it has been
+proved for general energies that every strong local minimizer is global in the hard device with
+affine Dirichlet boundary conditions in star-shaped domains [47]. In our special example,
+the same result turns out to be true for all Dirichlet boundary conditions in domains with
+piecewise smooth boundaries and arbitrary topology.
+The first step of the analysis is to characterize the set A of admissible displacement
+gradients for the chosen material model. It can result from the computation of QW(H)
+which in our case can be done explicitly.
+Theorem 2.1. Suppose that W(H) is given by (1.3). Then,
+QW(H) = Φ∗∗(Tr H) + µ
+4|H − Ht|2 − 2µJ2(H),
+(2.4)
+4
+
+where
+Φ(θ) = f(θ) + 2
+3µθ2,
+(2.5)
+2J2(H) = (Tr H)2 − Tr (H2) = 2
+�
+i<j
+����
+hii
+hij
+hji
+hjj
+����
+(2.6)
+is a quadratic null-Lagrangian, and Φ∗∗ denotes the convexification of Φ(θ).
+Proof. It will be convenient to use the definition of quasiconvexification based on periodic
+functions [10]:
+QW(H) =
+inf
+φ−Q-periodic −
+�
+Q
+W(H + ∇φ(x))dx,
+(2.7)
+where Q = [0, 1]3. The computation of QW is based on the following formula
+����e(u) − 1
+3(∇ · u)I
+����
+2
+= 2
+3(∇ · u)2 + 1
+2|∇ × u|2 − 2J2(∇u).
+(2.8)
+Let u(x) = Hx + φ(x). Then
+−
+�
+Q
+W(∇u)dx = −2µJ2(H) + −
+�
+Q
+�
+Φ(∇ · u) + µ
+2|∇ × u|2�
+dx.
+(2.9)
+Since Φ∗∗(θ) ≤ Φ(θ) for all θ ∈ R, we obtain the inequality
+−
+�
+Q
+W(∇u)dx ≥ −2µJ2(H) + −
+�
+Q
+�
+Φ∗∗(∇ · u) + µ
+2|∇ × u|2�
+dx.
+Now applying the Jensen’s inequality we get:
+−
+�
+Q
+W(∇u)dx ≥ −2µJ2(H) + Φ∗∗(Tr H) + µ
+4 |H − Ht|2.
+(2.10)
+Therefore,
+QW(H) ≥ −2µJ2(H) + Φ∗∗(Tr H) + µ
+4 |H − Ht|2.
+(2.11)
+In order to prove equality, we need to exhibit a periodic function φ such that ∇ × φ = 0
+and
+−
+�
+Q
+Φ(∇ · φ(z) + Tr H)dz = Φ∗∗(Tr H).
+(2.12)
+If Φ(θ), given by (2.5) is convex1, then the original energy W(H) is quasiconvex (polyconvex,
+actually), and formula (2.4) holds. The case of interest is therefore when Φ(θ) retains the
+double-well shape, in which case Φ∗∗(θ) = Φ(θ) if and only if θ ̸∈ (θ1, θ2), as in Fig. 2. For
+example, for the potential (1.4) we find
+θ1 =
+f0
+2κ0θp
++
+µθp
+3κ0 + 2µ,
+θ2 = θp +
+f0
+2κ0θp
+−
+µθp
+3κ0 + 2µ.
+(2.13)
+1This is never the case for the potential (1.4).
+5
+
+1
+2
+( )
+Figure 2: Definition of θ1 < θ2 via the common tangent construction.
+Thus, if Tr H ̸∈ (θ1, θ2), then φ = 0 attains equality in (2.10). If Tr H ∈ (θ1, θ2), then there
+exists ω ∈ (0, 1) such that
+Tr H = ωθ1 + (1 − ω)θ2.
+(2.14)
+In order to attain equality in (2.10) we need to split the period cell Q = [0, 1]3 into two
+subsets A1, of volume ω and A2 of volume 1 − ω such that
+∇ · u(x) = θ1χA1(x) + θ2χA2(x)
+(2.15)
+for all x ∈ Q. This is easily achieved with a laminate construction with A1 = [0, ω] × [0, 1]2
+and A2 = [1 − ω, 1] × [0, 1]2 and function ∇φ being symmetric and piecewise constant:
+∇φ(x) = M1χA1(x) + M2χA2(x).
+The symmetric matrices M1 and M2 are easily found from the periodicity of φ(x), that is
+equivalent to −�
+Q ∇φ(x)dx = 0, and (2.15):
+M1 = (1 − ω)[[θ]]e1 ⊗ e1,
+M2 = −ω[[θ]]e1 ⊗ e1,
+where [[θ]] = θ1 − θ2. These choices guarantee equality in (2.12) and the symmetry of ∇φ
+guarantees ∇ × φ = 0.
+Corollary 2.2.
+A = {H ∈ M : Tr H ̸∈ (θ1, θ2)}.
+(2.16)
+The constructed energy density QW(H) is obviously polyconvex, and coincides with
+RW(H), the rank-1 convexification of W(H). Indeed, as we see from the proof of The-
+orem 2.1, the infimum in (2.7) can be achieved by a simple laminate.
+In view of the
+simplicity of the optimal microstructures, one can post factum conclude that computing
+QW(H) explicitly was not really necessary for the characterization of the set A. We could
+also delineate it by using inequality (2.10) and relying on the knowledge and properties of
+6
+
+the jump set introduced in [18, 19]. Indeed, inequality (2.10) implies that all H satisfying
+Φ∗∗(Tr H) = Φ(Tr H) are admissible, i.e. {H ∈ M : Tr H ̸∈ (θ1, θ2)} ⊂ A. We can then
+show that all H with θ1 < Tr H < θ2 fail rank-one convexity by looking for pairs H± of the
+displacement gradient values that can occur on the two sides of a smooth jump discontinuity
+representing a phase boundary. We recall, [18], that these matrices satisfy the equations
+[[H]] = a ⊗ n,
+[[WH]]n = 0,
+[[WH]]ta = 0,
+[[W]] = ⟨{{WH }}, [[H]]⟩,
+(2.17)
+where
+[[H]] = H+ − H−,
+{{WH }} = 1
+2(WH(H+) + WH(H−)).
+We have shown in [20] that the jump set for the energy (1.3) consists of the union of two
+hyperplanes
+J− = {H ∈ M : Tr H = θ1},
+J+ = {H ∈ M : Tr H = θ2},
+so that if Tr H− = θ1, then the set of corresponding H+ is a projective plane worth of points
+H+ = H− + (θ2 − θ1)n ⊗ n,
+|n| = 1.
+As we have already remarked, inequality (2.10) implies that J± ⊂ A.
+In this case the
+following lemma shows that our results from [19] imply that all matrices Ht = tH+ + (1 −
+t)H− would also fail to be rank-one convex for all t ∈ (0, 1), proving equality (2.16), as the
+line segments Ht cover all points H, satisfying θ1 < Tr H < θ2.
+Lemma 2.3. Suppose H± is a corresponding pair of points on the jump set J of the energy
+density function W(H). Suppose that
+(i) W(H) ≥ W(H−) + ⟨P−, H − H−⟩ for any matrix H, such that rank(H − H−) = 1,
+where P− = WH(H−);
+(ii) None of the matrices Ht = tH+ +(1−t)H−, t ∈ (0, 1) satisfies the jump set equations
+(2.17) with H− as the corresponding pair.
+Then all of the matrices Ht, t ∈ (0, 1) must fail rank-one convexity in the sense that they
+must fail to satisfy the inequality
+W(H) ≥ W(Ht) + ⟨Pt, H − Ht⟩
+∀H : rank(H − Ht) = 1,
+(2.18)
+where Pt = WH(Ht).
+Proof. Suppose, with the goal of getting a contradiction, that (2.18) holds. Then, since
+matrices H−, Ht and H+ all lie on the same rank-one line, we obtain
+W(H−) + t⟨P−, [[H]]⟩ ≤ W(Ht) ≤ W(H+) − (1 − t)⟨Pt, [[H]]⟩.
+Now, using the jump set equations (2.17), we obtain
+(1 − t)(⟨P−, [[H]]⟩ − ⟨Pt, [[H]]⟩) ≥ 0.
+7
+
+Using the fact that Ht − H− = t[[H]] we obtain
+⟨Pt − P−, Ht − H−⟩ ≤ 0.
+(2.19)
+The variational significance of inequality (2.19), called the phase interchange stability in-
+equality, was understood in [19], where, according to [19, Lemma 4.1], the pair of rank-one
+related matrices H− and Ht would have to satisfy the jump set equations (2.17), in con-
+tradiction with the assumption (ii) of the lemma. Thus, the assumption that (2.18) holds
+cannot be true, and all matrices Ht, t ∈ (0, 1), must fail to be rank-one convex.
+Remark 2.4. The energy density QW(H) = RW(H) is non-convex as it has a double-well
+shape along the multiples of I, specifically, QW(εI) = Φ∗∗(3ε)−6µε2. This is clear from the
+fact that its bulk modulus is negative in the interval θ ∈ (θ1, θ2), where θ1 and θ2 are given
+by (2.13) for the bi-quadratic material model (1.4). Moreover, the value of this modulus is
+constant and equal to −(4/3)µ, cf. [9], which corresponds exactly to the threshold for the
+loss of strong ellipticity of the equilibrium equations (saturation of the Legendre-Hadamard
+conditions). This is associated with the generic degeneration of the acoustic tensor along
+rank one envelopes as we explain in our Appendix A. The vanishing of the velocity of the
+longitudinal waves in our example turns the phase-transforming material into an elastic
+aether [13].
+We are now ready to show that for a class of energies to which our example belongs all
+strong local minimizers in a hard device (Dirichlet boundary conditions) must be global.
+The first observation, leading to this result is that any two energies W and W ′ that have one
+and the same quasiconvex envelope are equivalent as far as existence of metastable states
+are concerned. This is because for any metastable state u
+�
+Ω
+W(∇u)dx =
+�
+Ω
+QW(∇u)dx.
+Thus, any metastable configuration for W is also metastable for QW. Theorem 2.1 shows
+that our energy has the property
+QW(H) = N(H) + C(H),
+(2.20)
+where N(H) is a null-Lagrangian and C(H) is convex and C1 smooth.
+Theorem 2.5. Let W(H) be the energy density satisfying (2.20).
+Then any Lipschitz
+strong local minimizer of
+�
+Ω W(∇u)dx with prescribed Dirichlet boundary conditions is a
+global minimizer.
+Proof. Let v be a Lipschitz competitor that agrees with u on ∂Ω. Then
+�
+Ω
+N(∇v)dx =
+�
+Ω
+N(∇u)dx,
+C(∇v) ≥ C(∇u) + ⟨CF (∇u), ∇(v − u)⟩.
+(2.21)
+Since u is an equilibrium we have
+0 = ∇ · WF (∇u) = ∇ · QWF (∇u) = ∇ · NF (∇u) + ∇ · CF (∇u) = ∇ · CF (∇u).
+8
+
+The equality of WF (H) = QWF (H) for any H ∈ A follows from the fact that the function
+W(H) − QW(H) is nonnegative, C1 smooth and attains its minimum value of 0 at all
+F ∈ A. Now, the integration by parts in the second term on the right-hand side in the
+inequality in (2.21) implies
+�
+Ω C(∇v)dx ≥
+�
+Ω C(∇u)dx. It follows that
+�
+Ω
+QW(∇v)dx ≥
+�
+Ω
+QW(∇u)dx.
+Remark 2.6. An important observation was made by Sivaloganathan and Spector in [45]
+that C(H) does not need to be convex for inequality in (2.21) to hold. In fact, as the authors
+show, it is only required that C∗∗(∇u(x)) = C(∇u(x)) for all x ∈ Ω, i.e. C(F ) agrees with
+its convex hull at all values of ∇u.
+3
+Global minimizers
+Consider next the problem of attainment in the definition of the quasiconvex envelope
+QW(F ) =
+inf
+y∈F x+W 1,∞
+0
+(Ω;Rm)
+−
+�
+Ω
+W(∇y)dx,
+(3.1)
+where Ω ⊂ Rd is a star-shaped Lipschitz domain. Below we show that Lipschitz equilib-
+rium configurations for (3.1) must be global minimizers, provided they satisfy the necessary
+conditions for metastability and are sufficiently regular near ∂Ω.
+Recall first the well-known necessary conditions for Lipschitz strong local minima of
+variational functionals.
+The first classical necessary condition is the Euler-Lagrange equation
+∇ · P (∇y) = 0,
+P (F ) = WF (F ),
+(3.2)
+which even weak local minimizers have to satisfy. The second, is the Noether equation
+∇ · P ∗(∇y) = 0,
+P ∗(F ) = W(F )Id − F tP (F ).
+(3.3)
+We recall that if y(x) is of class C2 then (3.3) is a consequence of (3.2), while there are
+Lipschitz configurations satisfying (3.2), but not (3.3). For instance, the Maxwell relation,
+which is the last equation in (2.17), is a consequence of (3.3), but not of (3.2).
+Definition 3.1. A configuration y ∈ W 1,∞(Ω; Rm) is called a stationary (or an equilibrium)
+state if it satisfies both (3.2) and (3.3).
+The second classical necessary condition is quasiconvexity ∇y ∈ A for a.e. x ∈ Ω, where
+A is defined in (2.3). It was shown in [19] that for configurations, whose only singularities are
+smooth phase boundaries the combination of (3.2) and quasiconvexity implies stationarity.
+In general, however, an example in [33] shows that there could be weak local minimizers
+9
+
+for strictly quasiconvex energies that are not strong local minimizers; most probably, the
+configuration in [33] fails stationarity.
+It turns out that in the case affine boundary conditions in a star-shaped domain, the
+two necessary conditions formulated above are sufficient for y(x) to be a global minimizer,
+provided we assume its regularity near the boundary of the domain.
+Theorem 3.2. Let Ω ⊂ Rd be a star-shaped domain with a Lipschitz boundary. Assume
+that
+(i) y : Ω → Rm is Lipschitz continuous,
+(ii) y solves (3.2) and (3.3) in the sense of distributions,
+(iii) ∇y(x) is locally stable for a.e. x ∈ Ω,
+(iv) y(x) = F x for all x ∈ ∂Ω,
+(v) There exists ǫ > 0, such that y ∈ C1(Ωǫ; Rm), where Ωǫ = {x ∈ Ω : dist(x, ∂Ω) < ǫ.
+Then y(x) is the global minimizer in (3.1), i.e.
+�
+Ω
+W(∇y)dx = |Ω|QW(F ).
+(3.4)
+Proof. While each ingredient of the proof presented below was in one way or another already
+present in the uniqueness proof of Knops and Stuart [29], to express the energy of an equilib-
+rium configuration as a boundary integral, we use a far reaching generalization of a related
+relation in [29] which, by itself, can be viewed as a nontrivial nonlinear generalization of the
+Clapeyron theorem [15].
+Lemma 3.3. Suppose y(x) is a Lipschitz stationary equilibrium in a Lipschitz domain Ω.
+Then
+E[y] =
+�
+Ω
+W(∇y) = 1
+d
+�
+∂Ω
+{P n · y + P ∗n · x}dx,
+(3.5)
+where P n and P ∗n can be regarded as trace functionals, since they act on Lipschitz functions
+y(x) and x, respectively.
+Proof.
+�
+∂Ω
+P ∗n · xdS =
+�
+Ω
+⟨P ∗, ∇x⟩dx =
+�
+Ω
+Tr (P ∗)dx = d
+�
+Ω
+W(∇y)dx −
+�
+Ω
+⟨P , ∇y⟩dx.
+Equation (3.5) follows from
+�
+Ω
+⟨P , ∇y⟩dx =
+�
+∂Ω
+P n · ydS.
+10
+
+We now apply formula (3.5) for the energy of y(x). Using the boundary condition (iv)
+and the assumption (v) we have at all x ∈ ∂Ω
+∇y = F + a ⊗ n,
+a = ∂y
+∂n − F n.
+We therefore compute
+dE[y] =
+�
+∂Ω
+(W(F + a ⊗ n) − P (F + a ⊗ n)n · a)(n · x)dS
+next, we use the local stability condition (iii) and appeal to [19, Lemma 4.2]. We quote the
+part of the lemma we need for the sake of completeness.
+Lemma 3.4. Let V (F ) be a rank-one convex function such that V (F ) ≤ W(F ). Let
+AV = {F ∈ O : W(F ) = V (F )},
+where O is an open subset of M on which W(F ) is of class C1. Then for every F ∈ AV ,
+u ∈ Rm, and v ∈ Rd
+V (F + u ⊗ v) ≥ W(F ) + P (F )v · u.
+(3.6)
+We now apply Lemma 3.4 by choosing V (F ) to be QW(F ). By assumption (iii) for each
+x ∈ ∂Ω the field ∇y(x) = F + a ⊗ n is in AV . Choosing u ⊗ v = −a ⊗ n, inequality (3.6)
+becomes
+QW(F ) ≥ W(F + a ⊗ n) − P (F + a ⊗ n)n · a.
+(3.7)
+Finally, we use the assumption of Ω being star-shaped. If we choose the origin at the star
+point, then the function n(x) · x is always non-negative at all points on ∂Ω. Therefore,
+inequality (3.7) implies
+dE[y] ≤ QW(F )
+�
+∂Ω
+(n · x)dS = d|Ω|QW(F ).
+Since |Ω|QW(F ) is the minimal value of E[y] we conclude that E[y] = QW(F ), and y(x)
+is the global minimizer.
+4
+Multiplicity of global minimizers
+We remark that the construction of the periodic laminate in Section 2 provides us with a
+minimizing sequence in (3.1) in arbitrary domains. In this section we raise the question of
+attainability of the minimum in (3.1). In this regard, laminates can no longer be used as
+the boundary conditions (4.2) are inconsistent with ∇u taking exactly two specific values
+everywhere in Ω. We will therefore attempt to find energy minimizers using the sufficiency
+conditions from Theorem 3.2. Accordingly, we only need to find solutions u(x) of (3.2) that
+satisfies (4.2) while respecting the stability condition
+∇ · u(x) ̸∈ (θ1, θ2) for a.e. x ∈ Ω.
+(4.1)
+11
+
+In what follows, in the interest of analytical transparency, we restrict our analysis to
+specific categories of bounded domains Ω and assume affine boundary conditions
+u(x) = H0x,
+x ∈ ∂Ω.
+(4.2)
+If Tr H0 ̸∈ [θ1, θ2] then, at least for star-shaped domain the theorem of Knops and Stuart
+[29] (see also [47]) ensures that the solution u(x) = H0x is the unique minimizer. Therefore
+we only need to consider the case Tr H0 ∈ (θ1, θ2).
+Let φ(x) = u(x) − H0x. Then (3.2) can be written in terms of φ as follows
+µ∆φ + ∇Φ′(∇ · u) = µ∇(∇ · φ),
+x ∈ Ω.
+(4.3)
+Integrating the dot product of φ(x) and (4.3) by parts we obtain
+�
+Ω
+�
+µ(|∇ · φ|2 − |∇φ|2) − Φ′(∇ · u)∇ · φ
+�
+dx = 0.
+Now we use the algebraic identity
+|∇ · φ|2 − |∇φ|2 = 2J2(∇φ) − |∇ × φ|2
+and the fact that J2(∇φ) given by (2.6) is a null-Lagrangian to obtain
+�
+Ω
+�
+µ|∇ × φ|2 + Φ′(∇ · u)∇ · φ
+�
+dx = 0.
+(4.4)
+Now let us show that (4.1) implies that
+�
+Ω
+Φ′(∇ · u)∇ · φ(x)dx ≥ 0.
+(4.5)
+Indeed, from the geometric interpretation of θ1 and θ2 (as the points of common tangency )
+we conclude that in view of (4.1) we can write
+Φ(∇ · u(x)) = Φ∗∗(∇ · u(x))
+(4.6)
+for a.e. x ∈ Ω. For a convex function Φ∗∗(θ) the tangent line to the graph of that function
+lies always below the graph which means that
+Φ∗∗(η) ≥ Φ′(θ)(η − θ) + Φ(θ)
+for any η ∈ R and any θ ̸∈ (θ1, θ2). Substituting η = Tr H0 and θ = ∇ · u(x) we obtain
+Φ′(∇ · u(x))∇ · φ(x) ≥ Φ(∇ · u(x)) − Φ∗∗(Tr H0) = Φ∗∗(∇ · u(x)) − Φ∗∗(Tr H0)
+for a.e. x ∈ Ω. Now (4.5) follows from Jensen’s inequality for a convex function Φ∗∗(θ) and
+the fact that
+1
+|Ω|
+�
+Ω
+∇ · u(x)dx = Tr H0.
+(4.7)
+12
+
+Now, inequality (4.5) together with (4.4) imply that ∇ × φ = 0, and in simply connected
+regions we may use the representation φ(x) = ∇h(x), where h ∈ W 2,2
+0 (Ω) is some scalar
+potential. Substituting u = H0x + ∇h into (4.3), we obtain ∇Φ′(∇ · u) = 0. Thus, there
+exists a constant P0, such that
+Φ′(∇ · u) = P0.
+(4.8)
+The constraint (4.1) then implies that either ∇ · u = Tr H0 + ∆h is constant in Ω, which
+means that u(x) = H0x + a0, or that P0 = Φ′(θ1) = Φ′(θ2), in which case there exists a
+subset A of Ω such that
+∇ · u(x) = θ1χA(x) + θ2χΩ\A(x),
+(4.9)
+where θ1 < θ2 are the endpoints of the interval {θ ∈ R : Φ∗∗(θ) < Φ(θ)} (see Fig. 2).
+In terms of the potential h we obtain a free boundary problem
+�
+∆h = (θ1 − θ2)(χA(x) − ω),
+x ∈ Ω
+h ∈ W 2,2
+0 (Ω),
+(4.10)
+where
+ω = |A|
+|Ω| = Tr H0 − θ2
+θ1 − θ2
+(4.11)
+is the volume fraction of the phase in which ∇ · u(x) = θ1. As formula (4.11) indicates, the
+volume fraction ω of the precipitate is uniquely determined by the hard device loading H0,
+whose shear component has no effect on the precipitate morphology.
+We remark that only the divergence of the displacement gradient is constrained by the
+requirement of optimality. This explains a virtual cornucopia of optimal microstructures, the
+simplest of which we are going to exhibit. A solution of (4.10) can be easily found in explicit
+form for some particularly simple domain geometries. For instance, it is easy to construct
+a radially symmetric solution when Ω is the unit ball, taking as the set A to be either the
+concentric ball of radius r0 = ω1/3, or its complement, in which case r0 = (1 − ω)1/3. Then
+(4.10) is solved by h(x) = hω(|x|), given by
+hω(r) =
+
+
+
+
+
+
+
+
+
+1
+6(θ1 − θ2)(1 − ω)
+�
+r2 − 3ω(1 − ω1/3)
+ω1/3(1 − ω)
+�
+,
+0 ≤ r ≤ ω1/3
+−1
+6(θ1 − θ2)ω
+�
+r2 + 2
+r − 3
+�
+,
+ω1/3 ≤ r ≤ 1,
+(4.12)
+in the former case. In the latter case, h(x) is given by the same formula (4.12) where θ1 and
+θ2 are interchanged and ω is replaced by 1 − ω, i.e. h(x) = −h1−ω(|x|).
+It is not clear from the free boundary problem (4.10) if solutions exist in less symmetric
+domains.
+In fact, Hashin’s “concentric sphere” construction, [22], shows that (4.10) has
+solutions in any Lipschitz domain Ω. In this construction the set A is a countable union
+of variously scaled copies of the radially symmetric solution (4.12) filling Ω up to a set of
+Lebesgue measure zero. Let B(xi, ai) ⊂ B(xi, Ri), i = 1, 2, . . . are the concentric balls used
+in Hashin’s construction, where
+a3
+i
+R3
+i
+= ω.
+13
+
+The inner balls B(xi, ai) in the construction belong to the set A, while the spherical shells
+B(xi, Ri) \ B(xi, ai) belong to Ω \ A. The function h(x) restricted to the ball B(xi, Ri) is
+given by
+h(x) = R2
+i hω
+�|x − xi|
+Ri
+�
+,
+x ∈ B(xi, Ri),
+where hω is given by (4.12). The function h(x) defined like this on each of the concentric
+balls does indeed solve (4.10), since on the boundary of each ball B(xi, Ri) both h(x) and
+∇h(x) are zero.
+The problem with the above “concentric sphere” construction is the same one as with
+the laminate-based optimal microstructures: they all have an infinite surface area (for the
+“concentric spheres” see the formal proof in [36, 51]), which is unacceptable in physical
+problems of interest. However, we also expect (4.10) to have solutions with finite surface
+area in all piecewise smooth domains2. For example, if Ω is an ellipsoid, then A would be a
+confocal ellipsoid (or its complement) [8, 37, 49, 53, 16]. In the next section we present a more
+complex solution with finite surface area in a two dimensional domain with piecewise-smooth
+boundary.
+5
+A finite perimeter global minimizer
+For a square, the 2D version of (4.10) can be solved using complex variables. The actual
+geometry of the phase boundary can be then found numerically.
+To obtain the shape of phase boundary we used the conventional complex potentials.
+Indeed, in view of (4.10) the functions
+�h+(x, y) = h(x, y) + 1
+4(θ1 − θ2)ω(x2 + y2),
+�h−(x, y) = h(x, y) − 1
+4(θ1 − θ2)(1 − ω)(x2 + y2)
+are harmonic in Ω\A and A respectively. We can then conclude that the functions ∂�h±/∂x−
+i∂�h±/∂y are analytic in the complex variable z = x + iy on their respective domains. Thus,
+the functions
+H+(z) =
+2
+θ1 − θ2
+�∂h
+∂x − i∂h
+∂y
+�
++ ωz,
+z ∈ Ω \ A,
+H−(z) =
+2
+θ1 − θ2
+�∂h
+∂x − i∂h
+∂y
+�
+− (1 − ω)z,
+z ∈ A
+are also analytic. The boundary conditions in (4.10) and the continuity of ∇h across ∂A,
+representing the kinematic compatibility of the displacement, imply that
+�
+H+(z) = ωz,
+z ∈ ∂Ω,
+H+(z) − H−(z) = z,
+z ∈ ∂A.
+(5.1)
+The knowledge of Ω allows one to determine H+. If Ω is a square centered at the origin
+with diagonal of length 2, then along the bottom side of the square we have H+(x−i/
+√
+2) =
+2We can say that this is a conjecture, as we have no formal proof of this statement.
+14
+
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+Figure 3: A coherent precipitate in the square.
+ω(x + i/
+√
+2) = ω(z + i
+√
+2). Therefore H+(z) = ω(z + i
+√
+2), while along the right side of the
+square we have H+(1/
+√
+2+iy) = ω(1/
+√
+2−iy) = ω(
+√
+2−z). Thus, H+(z) = ω(
+√
+2−z). These
+contradictory expressions for H+(z) can only be reconciled by a structure with topology
+indicated in Figure 3, where the set A, in which ∇ · u = θ1, is shaded.
+The square symmetry of a problem also suggests that we should look for a structure with
+square symmetry. Thus we just need to find the curve Γ that joins the top and bottom ends
+of the right side of the square, for example. Knowing H+(z) in all four regions adjacent to
+the sides of the square gives the boundary values for H−(z) on ∂A. The square symmetry
+of A together with (5.1) implies that
+H−(iz) = −iH−(z),
+z ∈ ∂A.
+(5.2)
+There is a holomorphic function H− in A with given boundary values if and only if for every
+n ≥ 0
+�
+∂A
+H−(z)zndz = 0.
+(5.3)
+The integral in (5.3) can easily be written as an integral along Γ because of (5.2). If n is
+not a multiple of 4 then the integral in (5.3) will evaluate to zero by virtue of (5.2) alone. If
+n = 4k, then we require that
+�
+Γ
+H−(z)z4kdz = 0,
+k ≥ 0.
+(5.4)
+On Γ we have H−(z) = ω(
+√
+2 − z) − z. Therefore, (5.4) is equivalent to
+�
+Γ
+zz4kdz =
+(−1)kiω
+(4k + 1)(2k + 1),
+k ≥ 0.
+(5.5)
+If we integrate by parts in (5.5):
+�
+Γ
+zz4kdz =
+�
+Γ
+z d
+� z4k+1
+4k + 1
+�
+= −
+�
+Γ
+z4k+1
+4k + 1dz.
+and parametrize the curve Γ by z(t) = a(t) + it, t ∈ [−1/
+√
+2, 1/
+√
+2]. Then z′(t) = z′(t) − 2i,
+we obtain
+�
+Γ
+zz4kdz =
+(−1)k+1i
+(4k + 1)(2k + 1) +
+2i
+4k + 1
+� 1/
+√
+2
+−1/
+√
+2
+z(t)4k+1dt.
+15
+
+Thus, the equation (5.5) becomes
+� 1/
+√
+2
+−1/
+√
+2
+z(t)4k+1dt = (−1)k(ω + 1)
+2(2k + 1)
+.
+Due to the symmetry we have z(−t) = z(t). Therefore, we finally obtain
+ℜe
+�� 1/
+√
+2
+0
+(a(t) + it)4k+1dt
+�
+= (−1)k(ω + 1)
+4(2k + 1)
+,
+k ≥ 0.
+(5.6)
+Observe that |a(t) + it| < 1 for all t ∈ [0, 1/
+√
+2) and |a(t) + it| = 1 for t = 1/
+√
+2. Thus, for
+large k the principal contribution to the integral (5.6) comes only from the neighborhood of
+t = 1/
+√
+2. Therefore, we approximate
+a(t) = 1
+√
+2
++ m(t − 1
+√
+2
+) + O((t − 1
+√
+2
+)2), as t → 1
+√
+2
+.
+(5.7)
+Substituting the leading term in (5.6) and computing the integral explicitly we obtain
+(−1)k
+(4k + 2)(m2 + 1) −
+(1 − m)4k+2m
+22k+1(4k + 2)(m2 + 1) ≈ (−1)k(ω + 1)
+2(4k + 2)
+.
+(5.8)
+If we choose
+m =
+�
+1 − ω
+1 + ω
+(5.9)
+then for large k we have equality in (5.6) up to an exponentially small error. Thus, we have
+determined the slope m with which the curve Γ enters the corner of the square.
+In order to find the function a(t) numerically, we approximate it by even polynomials
+at t = 0 and enforce (5.7) at every approximation.
+Notice that the error term in (5.8)
+decays very fast with k. Hence, satisfying (5.6) only for small values of k should give a very
+good approximate solution. With the knowledge of this, we approximate a(t) by an even
+polynomial
+an(t) =
+n
+�
+j=0
+ajt2j.
+In order to satisfy (5.8) we require that
+n
+�
+j=0
+aj
+2j = 1
+√
+2,
+n
+�
+j=1
+jaj
+2j−1 = m
+√
+2.
+(5.10)
+Observe that equation (5.6) for k = 0 is linear. Hence,
+n
+�
+j=0
+aj
+2j(2j + 1) = ω + 1
+2
+√
+2 .
+(5.11)
+16
+
+To solve the system (5.6) for k = 0, 1, . . . , K we set n = K +2 which leaves us with an easily
+solvable system of K polynomial equations in K unknowns.
+The configurations of the sets A and Ω \ A for ω = 0.3 (left panel) and ω = 0.8 (right
+panel) are shown in Fig. 3. When ω is almost zero, we effectively deal with phase nucleation
+associated with the “direct” (1 → 2) transformation. Phase “1” appears first in the form of
+two infinitely thin intersecting segments; the subsequent thickening of these segments starts
+first around their intersection point at the center of the square domain. As ω increases, the
+segments turn into a 4-cornered star, which eventually takes over the whole body. Instead,
+at ω almost one, we deal with the nucleation of phase “2” which appears first in the form of
+a thin “film” along the boundary of the body. In other words, during the “reverse” (1 → 2)
+transformation a new phase grows away from the boundary and eventually takes over the
+whole body leaving phase “1” only as a star shaped subdomain centered at the origin.
+However, it is important to keep in mind that phases “1” and “2” in these scenaria can be
+always interchanged. The complete constitutive symmetry between the two phases (the fact
+that θ1 < θ2 is irrelevant) implies that the new phase may first appear either as an infinitely
+thin coating of the sides of the square (unshaded regions in the right panel in Fig. 3) or as
+a cross located at the center of the domain and extending its arms to its corners (shaded
+regions in the left panel in Fig. 3).
+6
+Conclusions
+In this paper we considered a classical problem of nonlinear elastostatics for a material
+undergoing phase transition. Mathematically it reduces to a non-convex vectorial problem
+of the calculus of variations. In this context we presented a simple, yet non-trivial example
+of an energy density (of a material) for which a large class of local minimizers could be made
+explicit. In particular, we exhibited a material model, where the absence of metastability
+coexists with a wild nonuniqueness of global minimizers.
+Metastability in elastostatics, understood as the existence of strong local but not global
+minimizers, is an important problem in nonlinear elasticity because of the ubiquity of hys-
+teresis in martensitic phase transitions. While in Neumann problem the existence of such
+local minimizers has been linked to the incompatibility of the energy wells, here we showed
+that in Dirichlet problem the incompatibility of the wells is not sufficient for metastability.
+More specifically, we presented an example of the energy density of a hyperelastic material
+with non-rank-one convex, double well energy, for which we could prove the lack of strong
+local minimizers which are not global on any domain and for any Dirichlet boundary con-
+ditions. The analytical transparency of our arguments was due to the choice of the energy
+density which is geometrically linear and isotropic.
+For the same material model we could fully characterize the necessary and sufficient
+conditions for global energy minima in hard device loading, which allowed us to discover
+the multiplicity of global minimizers with nonuniqueness unrelated to either objectivity or
+material symmetry. As an important element of this analysis we used a novel way of ex-
+pressing the energy of equilibrium configurations as boundary integrals which can be viewed
+as a nontrivial nonlinear generalization of the classical Clapeyron theorem. First of all, we
+17
+
+showed that the quasiconvex envelope in our model can be achieved by simple lamination
+which can be characterized explicitly. The problem with this construction is that in real
+physical situations surface energy plays the role of the selection mechanism of the otherwise
+energetically equivalent configurations [28, 32, 11], ruling out such well-known optimal mi-
+crogeometrs as laminates and coated sphere constructions. Therefore, of particular interest
+in physical applications are global minimizers with finite surface area, and we exhibited their
+existence for our material model.
+An interesting aspect of our highly degenerate problem with elastically indistinguish-
+able phases is a sensitivity of the morphology of the interface between the phases to the
+global shape of the transforming body. To illustrate this effect, we explicitly computed a
+one-parameter family of non-affine energy minimizing configurations for the case of a finite
+domain with piecewise-smooth boundary. Our analysis of this example suggests that de-
+spite the absence of metastability and the associated constitutive hysteresis in this model,
+the direct and the reverse transformation may follow different morphological paths while
+traversing energetically equivalent configurations.
+Acknowledgments.
+YG was supported by the National Science Foundation under
+Grant No. DMS-2005538. The work of LT was supported by the French grant ANR-10-
+IDEX-0001-02 PSL. YG is grateful to Pavel Etingof who pointed out the references for the
+proof of the infinity of the surface area of Hashin’s concentric sphere construction in 1992.
+A
+The degeneracy of acoustic tensors of rank-1 en-
+velopes
+Here we prove that the acoustic tensor of a rank-one convex envelope RW of the non rank-one
+convex energy must have a degenerate direction at all points F0 where RW(F0) < W(F0).
+We also show that in particularly simple situations, such as the one discussed in this paper,
+this property may be even sufficient to compute the whole rank-one convex envelope.
+We recall that the acoustic tensor of the energy W(F ) at F = F0 in the direction n is a
+quadratic form A(n) defined by
+A(n)a · a = ⟨WF F (F0)(a ⊗ n), a ⊗ n⟩.
+Theorem A.1. Let F0 be fixed and suppose RW(F0) < W(F0), where RW denotes the
+rank-1 convexification of W. Assume further that W and RW are C2 near F0. Let A0(n)
+be the acoustic tensor of RW at F0. Then there is a direction n such that det A0(n) = 0.
+Proof. Assume that there is no such direction n. Since RW is necessarily rank-1 convex we
+can conclude that for any direction n the matrix A0(n) is positive semidefinite. Since the
+function n �→ A0(n) is continuous there exists a positive number α such that for every unit
+vector n
+A0(n) ≥ αI.
+Since RW ∈ C2 near F0 there exists a number δ > 0 such that for every F satisfying
+|F − F0| < δ the following inequalities hold:
+18
+
+1. A(n) ≥ 1
+2αI, where A(n) is the acoustic tensor of RW at F .
+2. W(F ) − RW(F ) ≥ 1
+2(W(F0) − RW(F0)) = β > 0.
+Now consider a function
+Tǫ(F ) = RW(F ) + ǫφ(F − F0
+4√ǫ
+),
+where φ is a smooth nonnegative function supported on the unit ball in M and such that
+φ(0) = 1. We can choose ǫ so small that the following inequalities hold:
+1. ∥ǫφ∥L∞ < 1
+2β,
+2. ∥√ǫ∇F ∇F φ∥L∞ ≤ 1
+4α,
+3. supp φ( F −F0
+4√ǫ ) ⊂ B(F0, δ), where B(F0, δ) is the ball of radius δ around F0 in M.
+Then Tǫ(F ) ≤ W(F ) and the acoustic tensor Aǫ(n) of Tǫ(F ) satisfies
+Aǫ(n) ≥ 1
+4αI
+in the sense of quadratic forms.
+Thus Tǫ(F ) is rank-1 convex and Tǫ(F ) ≤ W(F ) but
+Tǫ(F0) > RW(F0). Contradiction.
+Thus our assumption is false and at every point F0
+where RW(F0) < W(F0) there is a direction n such that the acoustic tensor A0(n) is
+degenerate.
+Remark A.2. It follows that materials that transform by forming microstructures with sharp
+phase boundaries to accommodate deformations produced by the propagation of sound waves
+will have a direction with a zero sound speed.
+We can apply Theorem A.1 to the energy (1.3).
+Since our material is isotropic, the
+degeneration of the acoustic tensor may be either through µ = 0 or through λ + 2µ = 0 with
+the latter also meaning that the bulk modulus κ = −(4/3)µ. The possibility that µ = 0 is
+excluded because the tangential shear modulus is the same at every deformation.
+Suppose that we have somehow guessed that if
+W(H) = f(Tr ε) + µ|dev(ε)|2,
+then
+RW(H) = F(Tr ε) + µ|dev(ε)|2
+for some function F, yet to be determined. In that case Theorem A.1 will let us determine
+the function F(θ). It is easy to compute that for any unit vector n and any vector a ∈ R3
+⟨RWHH(H)(n ⊗ a), n ⊗ a⟩ = F ′′(Tr ε)(a, n)2 + 2µ
+����
+1
+2(n ⊗ a + a ⊗ n) − 1
+3(a, n)I
+����
+2
+.
+19
+
+Therefore, we get a formula for the acoustic tensor A(n).
+A(n) = (F ′′(Tr H) + 1
+3µ)n ⊗ n + µI.
+(A.1)
+We see that det A(n) = µ2(F ′′(Tr ε) + 4µ/3) for all directions n. Thus, for all H for which
+RW(H) < W(H) we get F ′′(Tr H) = −4µ/3. The continuity of RWH implies that at
+the boundary points θ1 and θ2 of the binodal region we have Φ′(θ1) = Φ′(θ2). Further, the
+continuity of W implies that the affine function3 y(θ) = F(θ)+2µθ2/3 would be the equation
+of the common tangent to the graph of Φ(θ). Thus, we obtain that F(θ) + 2µθ2/3 = Φ∗∗(θ),
+and we recover the rank-one convex envelope of W(H), which in this case is seen to coincide
+with its quasiconvexification (2.4).
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+23
+
diff --git a/rdAzT4oBgHgl3EQf5_7j/content/tmp_files/load_file.txt b/rdAzT4oBgHgl3EQf5_7j/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf,len=933
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='01870v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='AP]  5 Jan 2023  A nonlinear elasticity problem with no local but many  global minimizers unrelated by symmetry  Yury Grabovsky  Lev Truskinovsky  January 6, 2023  Abstract  In this paper, dedicated to 85th birthday of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Fosdick, we touch upon two issues  which attracted considerable attention in his own research: nonuniqueness in geomet-  rically linear elasticity and the Clapeyron theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' To this end we consider a simple  model of a solid-solid phase transition with incompatible energy wells which exhibits  multiplicity of global minimizers in a hard device, unrelated to either objectivity or  material symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' At the same time, the same example shows the absence of strong  local Lipschitz minimizers which are not global minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Our proof that every  strong local minimizer in the corresponding variational problem is also a global one  hinges on the new general sufficiency theorem, proved by means of the novel nonlinear  generalization of the Clapeyron theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  1  Introduction  The phenomenon of metastability in elastostatics, manifesting itself through the existence of  strong local minimizers which are not global, is usually associated with a Neumann problem  (soft device) and is linked to the incompatibility of the energy wells [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' In this paper we  present an analytically transparent example of an energy with incompatible wells for which  one can prove the absence of strong local minimizers which are not global on any domain  and for any Dirichlet boundary conditions (hard device).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Moreover, for the same energy one can prove dramatic nonuniqueness of global mini-  mizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Previously, it has been understood that to ensure uniqueness, the use of Dirichlet  boundary conditions and topological, or even geometric simplicity of the domain are essential  [41, 48, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Uniqueness has been established for star-shaped domains, affine displacement  boundary conditions, and strictly quasiconvex stored energy functions [29, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Whether  uniqueness holds may also depend on the regularity class in which one looks for a minimizer  [30], or even on its integrability class [4, 44, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' For mixed boundary-value problems of  nonlinear elasticity nonuniqueness is common with the most familiar examples being those  associated with buckling, due to the emergence of multiple symmetry-related energy minima  [14, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The possibility of nonuniqueness with Dirichlet boundary conditions was shown for  non-homogeneous problems [12, 34, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Our simple example shows that the multiplicity of  1  f( )  Figure 1: Double-well nonlinearity in a geometrically linear bi-quadratic Hadamard material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  global minimizers in a hard device problem can be obtained even in the absence of geomet-  rical complexity of the domain, non-homogeneity, and can be unrelated to either objectivity  or material symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' It is very similar to examples related to the nonuniqueness of optimal  microstructures in composites [3, 38, 16, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  More specifically, we consider a “geometrically linearized” Hadamard material which is  a simplification of the fully nonlinear Hadamard material [21, 25] whose energy density  function is of the form  W(F ) = µ|F |2 + h(det F ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1)  where µ is the measure of rigidity and the non-negative function h(d) is defined on (0, +∞)  and has the property that h(d) → ∞, as d → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' If we use the “geometric” approximation  det F ≈ 1 + Tr (F − I), which is valid in the limit F → I, such formal asymptotic expan-  sion with respect to a small parameter would also induce physical linearization and would  trivialize the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' To retain physical non-linearity we consider the energy  W(H) = g(Tr H) + µ|H|2,  H = F − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2)  We may now perform geometric linearization in the second term as well [27, 9, 26, 1, 2],  replacing H with its symmetric part ε = (H +Ht)/2 and, to emphasise its isotropic nature,  writing the energy density as  W(ε) = f(Tr ε) + µ|dev(ε)|2,  dev(ε) = ε − 1  3Tr (ε)I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3)  Mathematically, the analyses of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3) are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' We will therefore, focus on  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3), because it preserves at least the linearized version of the frame indifference property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  We assume that µ > 0 and the function f has the “double-well” shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' For analytical  transparency we use in our explicit constructions the bi-quadratic potential  f(θ) = min{κ0θ2, κ0(θ − θp)2 + f0},  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4)  illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The ensuing model describes a material capable of undergoing a purely  dilatational phase transformation between two phases which are both linearly elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' They  have the same moduli and differ only by the transformation strain and the reference (chemi-  cal) energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The relaxation of such energies has been always considered as a benchmark case  for all theories of elastic phase transitions [27, 31, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  2  We recall that if the wells are rank-one connected, a quasiconvex energy would have to  be convex, due to the rank-one conexity of quasiconvex functions [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' On the other hand,  the incompatibility of the wells can be responsible for quasiconvexity of a non-convex energy  density, as we will see in our example, where the energy wells are not rank one connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  In our example, despite the fact that the non-convexity enters (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3) only through a scalar  potential f(θ), the energy density (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3) is non-quasiconvex for all µ > 0, if f(θ) is given  by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4), [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Therefore, even if subjected to homogeneous loading in a hard device, the  materials described by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4) must undergo non-homogeneous deformations by developing mi-  crostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' In the presence of local minima, such systems can be expected to experience  hysteresis [5, 24, 42, 43, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' However, as we have already mentioned, the local minima in  this model are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Then, the macroscopic quasistatic mechanical response in the hard  device corresponding to global minimization of the total energy at each value of the loading  parameter is defined uniquely, and as a result, the deformation path is reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The asso-  ciated stress-strain relation can be obtained from the knowledge of the quasiconvex envelope  of the energy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3) that can be characterized explicitly in the whole range of parameters and  for all double-well shaped potentials f(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' In this paper we perform the implied relaxation  of the energy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3) using simple laminates as the optimal microstructure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' a different proof  based on matching of upper and lower bounds can be found in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  One of the general results of this paper is the formulation of the necessary and sufficient  conditions for global minimizers in a star-shaped domain with affine boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  This creates a tool to characterize energy minimizing configurations in examples where the  energy relaxation, or at least the elastic binodal, is explicitly known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' An important technical  tool in this analysis is a fundamental nonlinear generalization of the Clapeyron theorem  which is presented here for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' These results can be viewed as a generalization of  optimality conditions for extremal microstructures in composites (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  We show that using our necessary and sufficient conditions one can reduce the problem of  finding a global minimizer to the solution of a nonlinear free boundary problem involving a  system of linear PDEs in a finite domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' It is remarkable that for the energy density (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3) the  solution of the ensuing problem can be found explicitly, for any double-well scalar potential  f(θ), providing us with an explicit example of multiplicity of global minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' In particular,  we were able to compute explicitly a one parameter family of low surface energy non-affine  configurations in the square domain, whose boundary is piecewise smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' In this way we  showed the possibility to always generate in this class of problems an optimal microstructure  with low surface area, which is physically advantageous comparing, for instance, to multiscale  lamination with infinitely large surface area [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  The analysis of optimal microstructures suggests that, despite the absence of the total  energy hysteresis in this system, the direct and the reverse transformation may follow dif-  ferent transformation paths in the configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Therefore in this problem, in the  absence of metastability and “constitutive hysteresis”, we encounter a phenomenon of “mor-  phological hysteresis”, whereby the forward and reverse transformations occur along different  morphological paths, while traversing energetically equivalent configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' In Section 2 we discuss the restrictions on the structure  of strong local minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The necessary and sufficient conditions for global minimizers are  formulated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The issue of attainment and the multiplicity of global minimizers  3  in our model are discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  An explicit construction of a nontrivial global  minimizer in a square domain which has expressly low surface area is presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  The last Section 6 contains our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' An Appendix A contains a technical discussion  of the generic loss of ellipticity for the energy represented by a rank one convex envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  2  Local minimizers  Given that the case of interest involves hard device loading, our goal is to study strong local  and global minima of the functional  E0(u) =  �  Ω  �  f(∇ · u) + µ  ����e(u) − 1  3(∇ · u)I  ����  2�  dx  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1)  among all Lipschitz displacement vector fields u ∈ W 1,∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' R3) subject to the constraint  u(x) = u0(x),  x ∈ ∂Ω,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2)  where u0(x) is a given Lipschitz function on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Restricting attention to Lipschitz minimiz-  ers allows us to focus on instabilities caused by failure of quasiconvexity [39, 10] and exclude  some other instabilities related, for instance, to the mismatch between the integrability of  the minimizing sequences and the growth of the energy density at infinity [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  First we recall that one of the necessary conditions of metastability, understood here as  the lack of strong local Lipschitz minimizers which are not also global minimizers, is stability  with respect to nucleation of a coherent precipitate of the new phase in the interior of the  old phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' That means that we must have ∇u(x) ∈ A, for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' x ∈ Ω, [48, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1],  where  A = {H ∈ M : W(H) = QW(H)},  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3)  where M denotes the set of all 3 × 3 matrices and QW(H) is the quasiconvexification of W  [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  In this paper we show that if u(x) is a weak solution of the Euler-Lagrange equation  for the energy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1) with the boundary conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2) and satisfies ∇u(x) ∈ A for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  x ∈ Ω then, it is necessarily a global minimizer for E0(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' However, before we move to the  actual analysis, it is appropriate to mention that we are not aware of any examples of proper  metastable states in a hard device in domains with trivial topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Moreover it has been  proved for general energies that every strong local minimizer is global in the hard device with  affine Dirichlet boundary conditions in star-shaped domains [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' In our special example,  the same result turns out to be true for all Dirichlet boundary conditions in domains with  piecewise smooth boundaries and arbitrary topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  The first step of the analysis is to characterize the set A of admissible displacement  gradients for the chosen material model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' It can result from the computation of QW(H)  which in our case can be done explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Suppose that W(H) is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Then,  QW(H) = Φ∗∗(Tr H) + µ  4|H − Ht|2 − 2µJ2(H),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4)  4  where  Φ(θ) = f(θ) + 2  3µθ2,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='5)  2J2(H) = (Tr H)2 − Tr (H2) = 2  �  i<j  ����  hii  hij  hji  hjj  ����  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='6)  is a quadratic null-Lagrangian, and Φ∗∗ denotes the convexification of Φ(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' It will be convenient to use the definition of quasiconvexification based on periodic  functions [10]:  QW(H) =  inf  φ−Q-periodic −  �  Q  W(H + ∇φ(x))dx,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='7)  where Q = [0, 1]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The computation of QW is based on the following formula  ����e(u) − 1  3(∇ · u)I  ����  2  = 2  3(∇ · u)2 + 1  2|∇ × u|2 − 2J2(∇u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='8)  Let u(x) = Hx + φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Then  −  �  Q  W(∇u)dx = −2µJ2(H) + −  �  Q  �  Φ(∇ · u) + µ  2|∇ × u|2�  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='9)  Since Φ∗∗(θ) ≤ Φ(θ) for all θ ∈ R, we obtain the inequality  −  �  Q  W(∇u)dx ≥ −2µJ2(H) + −  �  Q  �  Φ∗∗(∇ · u) + µ  2|∇ × u|2�  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Now applying the Jensen’s inequality we get:  −  �  Q  W(∇u)dx ≥ −2µJ2(H) + Φ∗∗(Tr H) + µ  4 |H − Ht|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='10)  Therefore,  QW(H) ≥ −2µJ2(H) + Φ∗∗(Tr H) + µ  4 |H − Ht|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='11)  In order to prove equality, we need to exhibit a periodic function φ such that ∇ × φ = 0  and  −  �  Q  Φ(∇ · φ(z) + Tr H)dz = Φ∗∗(Tr H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='12)  If Φ(θ), given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='5) is convex1, then the original energy W(H) is quasiconvex (polyconvex,  actually), and formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The case of interest is therefore when Φ(θ) retains the  double-well shape, in which case Φ∗∗(θ) = Φ(θ) if and only if θ ̸∈ (θ1, θ2), as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' For  example, for the potential (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4) we find  θ1 =  f0  2κ0θp  +  µθp  3κ0 + 2µ,  θ2 = θp +  f0  2κ0θp  −  µθp  3κ0 + 2µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='13)  1This is never the case for the potential (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  5  1  2  ( )  Figure 2: Definition of θ1 < θ2 via the common tangent construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Thus, if Tr H ̸∈ (θ1, θ2), then φ = 0 attains equality in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' If Tr H ∈ (θ1, θ2), then there  exists ω ∈ (0, 1) such that  Tr H = ωθ1 + (1 − ω)θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='14)  In order to attain equality in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='10) we need to split the period cell Q = [0, 1]3 into two  subsets A1, of volume ω and A2 of volume 1 − ω such that  ∇ · u(x) = θ1χA1(x) + θ2χA2(x)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='15)  for all x ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' This is easily achieved with a laminate construction with A1 = [0, ω] × [0, 1]2  and A2 = [1 − ω, 1] × [0, 1]2 and function ∇φ being symmetric and piecewise constant:  ∇φ(x) = M1χA1(x) + M2χA2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  The symmetric matrices M1 and M2 are easily found from the periodicity of φ(x), that is  equivalent to −�  Q ∇φ(x)dx = 0, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='15):  M1 = (1 − ω)[[θ]]e1 ⊗ e1,  M2 = −ω[[θ]]e1 ⊗ e1,  where [[θ]] = θ1 − θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' These choices guarantee equality in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='12) and the symmetry of ∇φ  guarantees ∇ × φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  A = {H ∈ M : Tr H ̸∈ (θ1, θ2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='16)  The constructed energy density QW(H) is obviously polyconvex, and coincides with  RW(H), the rank-1 convexification of W(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Indeed, as we see from the proof of The-  orem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1, the infimum in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='7) can be achieved by a simple laminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  In view of the  simplicity of the optimal microstructures, one can post factum conclude that computing  QW(H) explicitly was not really necessary for the characterization of the set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' We could  also delineate it by using inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='10) and relying on the knowledge and properties of  6  the jump set introduced in [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Indeed, inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='10) implies that all H satisfying  Φ∗∗(Tr H) = Φ(Tr H) are admissible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' {H ∈ M : Tr H ̸∈ (θ1, θ2)} ⊂ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' We can then  show that all H with θ1 < Tr H < θ2 fail rank-one convexity by looking for pairs H± of the  displacement gradient values that can occur on the two sides of a smooth jump discontinuity  representing a phase boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' We recall, [18], that these matrices satisfy the equations  [[H]] = a ⊗ n,  [[WH]]n = 0,  [[WH]]ta = 0,  [[W]] = ⟨{{WH }}, [[H]]⟩,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='17)  where  [[H]] = H+ − H−,  {{WH }} = 1  2(WH(H+) + WH(H−)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  We have shown in [20] that the jump set for the energy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3) consists of the union of two  hyperplanes  J− = {H ∈ M : Tr H = θ1},  J+ = {H ∈ M : Tr H = θ2},  so that if Tr H− = θ1, then the set of corresponding H+ is a projective plane worth of points  H+ = H− + (θ2 − θ1)n ⊗ n,  |n| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  As we have already remarked, inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='10) implies that J± ⊂ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  In this case the  following lemma shows that our results from [19] imply that all matrices Ht = tH+ + (1 −  t)H− would also fail to be rank-one convex for all t ∈ (0, 1), proving equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='16), as the  line segments Ht cover all points H, satisfying θ1 < Tr H < θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Suppose H± is a corresponding pair of points on the jump set J of the energy  density function W(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Suppose that  (i) W(H) ≥ W(H−) + ⟨P−, H − H−⟩ for any matrix H, such that rank(H − H−) = 1,  where P− = WH(H−);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (ii) None of the matrices Ht = tH+ +(1−t)H−, t ∈ (0, 1) satisfies the jump set equations  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='17) with H− as the corresponding pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Then all of the matrices Ht, t ∈ (0, 1) must fail rank-one convexity in the sense that they  must fail to satisfy the inequality  W(H) ≥ W(Ht) + ⟨Pt, H − Ht⟩  ∀H : rank(H − Ht) = 1,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='18)  where Pt = WH(Ht).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Suppose, with the goal of getting a contradiction, that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='18) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Then, since  matrices H−, Ht and H+ all lie on the same rank-one line, we obtain  W(H−) + t⟨P−, [[H]]⟩ ≤ W(Ht) ≤ W(H+) − (1 − t)⟨Pt, [[H]]⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Now, using the jump set equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='17), we obtain  (1 − t)(⟨P−, [[H]]⟩ − ⟨Pt, [[H]]⟩) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  7  Using the fact that Ht − H− = t[[H]] we obtain  ⟨Pt − P−, Ht − H−⟩ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='19)  The variational significance of inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='19), called the phase interchange stability in-  equality, was understood in [19], where, according to [19, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1], the pair of rank-one  related matrices H− and Ht would have to satisfy the jump set equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='17), in con-  tradiction with the assumption (ii) of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Thus, the assumption that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='18) holds  cannot be true, and all matrices Ht, t ∈ (0, 1), must fail to be rank-one convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The energy density QW(H) = RW(H) is non-convex as it has a double-well  shape along the multiples of I, specifically, QW(εI) = Φ∗∗(3ε)−6µε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' This is clear from the  fact that its bulk modulus is negative in the interval θ ∈ (θ1, θ2), where θ1 and θ2 are given  by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='13) for the bi-quadratic material model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Moreover, the value of this modulus is  constant and equal to −(4/3)µ, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' [9], which corresponds exactly to the threshold for the  loss of strong ellipticity of the equilibrium equations (saturation of the Legendre-Hadamard  conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' This is associated with the generic degeneration of the acoustic tensor along  rank one envelopes as we explain in our Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The vanishing of the velocity of the  longitudinal waves in our example turns the phase-transforming material into an elastic  aether [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  We are now ready to show that for a class of energies to which our example belongs all  strong local minimizers in a hard device (Dirichlet boundary conditions) must be global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  The first observation, leading to this result is that any two energies W and W ′ that have one  and the same quasiconvex envelope are equivalent as far as existence of metastable states  are concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' This is because for any metastable state u  �  Ω  W(∇u)dx =  �  Ω  QW(∇u)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Thus, any metastable configuration for W is also metastable for QW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1 shows  that our energy has the property  QW(H) = N(H) + C(H),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='20)  where N(H) is a null-Lagrangian and C(H) is convex and C1 smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Let W(H) be the energy density satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Then any Lipschitz  strong local minimizer of  �  Ω W(∇u)dx with prescribed Dirichlet boundary conditions is a  global minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Let v be a Lipschitz competitor that agrees with u on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Then  �  Ω  N(∇v)dx =  �  Ω  N(∇u)dx,  C(∇v) ≥ C(∇u) + ⟨CF (∇u), ∇(v − u)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='21)  Since u is an equilibrium we have  0 = ∇ · WF (∇u) = ∇ · QWF (∇u) = ∇ · NF (∇u) + ∇ · CF (∇u) = ∇ · CF (∇u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  8  The equality of WF (H) = QWF (H) for any H ∈ A follows from the fact that the function  W(H) − QW(H) is nonnegative, C1 smooth and attains its minimum value of 0 at all  F ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Now, the integration by parts in the second term on the right-hand side in the  inequality in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='21) implies  �  Ω C(∇v)dx ≥  �  Ω C(∇u)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' It follows that  �  Ω  QW(∇v)dx ≥  �  Ω  QW(∇u)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' An important observation was made by Sivaloganathan and Spector in [45]  that C(H) does not need to be convex for inequality in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='21) to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' In fact, as the authors  show, it is only required that C∗∗(∇u(x)) = C(∇u(x)) for all x ∈ Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' C(F ) agrees with  its convex hull at all values of ∇u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  3  Global minimizers  Consider next the problem of attainment in the definition of the quasiconvex envelope  QW(F ) =  inf  y∈F x+W 1,∞  0  (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='Rm)  −  �  Ω  W(∇y)dx,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1)  where Ω ⊂ Rd is a star-shaped Lipschitz domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Below we show that Lipschitz equilib-  rium configurations for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1) must be global minimizers, provided they satisfy the necessary  conditions for metastability and are sufficiently regular near ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Recall first the well-known necessary conditions for Lipschitz strong local minima of  variational functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  The first classical necessary condition is the Euler-Lagrange equation  ∇ · P (∇y) = 0,  P (F ) = WF (F ),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2)  which even weak local minimizers have to satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The second, is the Noether equation  ∇ · P ∗(∇y) = 0,  P ∗(F ) = W(F )Id − F tP (F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3)  We recall that if y(x) is of class C2 then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3) is a consequence of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2), while there are  Lipschitz configurations satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2), but not (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' For instance, the Maxwell relation,  which is the last equation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='17), is a consequence of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3), but not of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' A configuration y ∈ W 1,∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Rm) is called a stationary (or an equilibrium)  state if it satisfies both (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  The second classical necessary condition is quasiconvexity ∇y ∈ A for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' x ∈ Ω, where  A is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' It was shown in [19] that for configurations, whose only singularities are  smooth phase boundaries the combination of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2) and quasiconvexity implies stationarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  In general, however, an example in [33] shows that there could be weak local minimizers  9  for strictly quasiconvex energies that are not strong local minimizers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' most probably, the  configuration in [33] fails stationarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  It turns out that in the case affine boundary conditions in a star-shaped domain, the  two necessary conditions formulated above are sufficient for y(x) to be a global minimizer,  provided we assume its regularity near the boundary of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Let Ω ⊂ Rd be a star-shaped domain with a Lipschitz boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Assume  that  (i) y : Ω → Rm is Lipschitz continuous,  (ii) y solves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3) in the sense of distributions,  (iii) ∇y(x) is locally stable for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' x ∈ Ω,  (iv) y(x) = F x for all x ∈ ∂Ω,  (v) There exists ǫ > 0, such that y ∈ C1(Ωǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Rm), where Ωǫ = {x ∈ Ω : dist(x, ∂Ω) < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Then y(x) is the global minimizer in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  �  Ω  W(∇y)dx = |Ω|QW(F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' While each ingredient of the proof presented below was in one way or another already  present in the uniqueness proof of Knops and Stuart [29], to express the energy of an equilib-  rium configuration as a boundary integral, we use a far reaching generalization of a related  relation in [29] which, by itself, can be viewed as a nontrivial nonlinear generalization of the  Clapeyron theorem [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Suppose y(x) is a Lipschitz stationary equilibrium in a Lipschitz domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Then  E[y] =  �  Ω  W(∇y) = 1  d  �  ∂Ω  {P n · y + P ∗n · x}dx,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='5)  where P n and P ∗n can be regarded as trace functionals, since they act on Lipschitz functions  y(x) and x, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  �  ∂Ω  P ∗n · xdS =  �  Ω  ⟨P ∗, ∇x⟩dx =  �  Ω  Tr (P ∗)dx = d  �  Ω  W(∇y)dx −  �  Ω  ⟨P , ∇y⟩dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='5) follows from  �  Ω  ⟨P , ∇y⟩dx =  �  ∂Ω  P n · ydS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  10  We now apply formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='5) for the energy of y(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Using the boundary condition (iv)  and the assumption (v) we have at all x ∈ ∂Ω  ∇y = F + a ⊗ n,  a = ∂y  ∂n − F n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  We therefore compute  dE[y] =  �  ∂Ω  (W(F + a ⊗ n) − P (F + a ⊗ n)n · a)(n · x)dS  next, we use the local stability condition (iii) and appeal to [19, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' We quote the  part of the lemma we need for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Let V (F ) be a rank-one convex function such that V (F ) ≤ W(F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Let  AV = {F ∈ O : W(F ) = V (F )},  where O is an open subset of M on which W(F ) is of class C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Then for every F ∈ AV ,  u ∈ Rm, and v ∈ Rd  V (F + u ⊗ v) ≥ W(F ) + P (F )v · u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='6)  We now apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4 by choosing V (F ) to be QW(F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' By assumption (iii) for each  x ∈ ∂Ω the field ∇y(x) = F + a ⊗ n is in AV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Choosing u ⊗ v = −a ⊗ n, inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='6)  becomes  QW(F ) ≥ W(F + a ⊗ n) − P (F + a ⊗ n)n · a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='7)  Finally, we use the assumption of Ω being star-shaped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' If we choose the origin at the star  point, then the function n(x) · x is always non-negative at all points on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Therefore,  inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='7) implies  dE[y] ≤ QW(F )  �  ∂Ω  (n · x)dS = d|Ω|QW(F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Since |Ω|QW(F ) is the minimal value of E[y] we conclude that E[y] = QW(F ), and y(x)  is the global minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  4  Multiplicity of global minimizers  We remark that the construction of the periodic laminate in Section 2 provides us with a  minimizing sequence in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1) in arbitrary domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' In this section we raise the question of  attainability of the minimum in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' In this regard, laminates can no longer be used as  the boundary conditions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2) are inconsistent with ∇u taking exactly two specific values  everywhere in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' We will therefore attempt to find energy minimizers using the sufficiency  conditions from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Accordingly, we only need to find solutions u(x) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2) that  satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2) while respecting the stability condition  ∇ · u(x) ̸∈ (θ1, θ2) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1)  11  In what follows, in the interest of analytical transparency, we restrict our analysis to  specific categories of bounded domains Ω and assume affine boundary conditions  u(x) = H0x,  x ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2)  If Tr H0 ̸∈ [θ1, θ2] then, at least for star-shaped domain the theorem of Knops and Stuart  [29] (see also [47]) ensures that the solution u(x) = H0x is the unique minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Therefore  we only need to consider the case Tr H0 ∈ (θ1, θ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Let φ(x) = u(x) − H0x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2) can be written in terms of φ as follows  µ∆φ + ∇Φ′(∇ · u) = µ∇(∇ · φ),  x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3)  Integrating the dot product of φ(x) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3) by parts we obtain  �  Ω  �  µ(|∇ · φ|2 − |∇φ|2) − Φ′(∇ · u)∇ · φ  �  dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Now we use the algebraic identity  |∇ · φ|2 − |∇φ|2 = 2J2(∇φ) − |∇ × φ|2  and the fact that J2(∇φ) given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='6) is a null-Lagrangian to obtain  �  Ω  �  µ|∇ × φ|2 + Φ′(∇ · u)∇ · φ  �  dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4)  Now let us show that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1) implies that  �  Ω  Φ′(∇ · u)∇ · φ(x)dx ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='5)  Indeed, from the geometric interpretation of θ1 and θ2 (as the points of common tangency )  we conclude that in view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1) we can write  Φ(∇ · u(x)) = Φ∗∗(∇ · u(x))  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='6)  for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' For a convex function Φ∗∗(θ) the tangent line to the graph of that function  lies always below the graph which means that  Φ∗∗(η) ≥ Φ′(θ)(η − θ) + Φ(θ)  for any η ∈ R and any θ ̸∈ (θ1, θ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Substituting η = Tr H0 and θ = ∇ · u(x) we obtain  Φ′(∇ · u(x))∇ · φ(x) ≥ Φ(∇ · u(x)) − Φ∗∗(Tr H0) = Φ∗∗(∇ · u(x)) − Φ∗∗(Tr H0)  for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Now (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='5) follows from Jensen’s inequality for a convex function Φ∗∗(θ) and  the fact that  1  |Ω|  �  Ω  ∇ · u(x)dx = Tr H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='7)  12  Now, inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='5) together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4) imply that ∇ × φ = 0, and in simply connected  regions we may use the representation φ(x) = ∇h(x), where h ∈ W 2,2  0 (Ω) is some scalar  potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Substituting u = H0x + ∇h into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3), we obtain ∇Φ′(∇ · u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Thus, there  exists a constant P0, such that  Φ′(∇ · u) = P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='8)  The constraint (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1) then implies that either ∇ · u = Tr H0 + ∆h is constant in Ω, which  means that u(x) = H0x + a0, or that P0 = Φ′(θ1) = Φ′(θ2), in which case there exists a  subset A of Ω such that  ∇ · u(x) = θ1χA(x) + θ2χΩ\\A(x),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='9)  where θ1 < θ2 are the endpoints of the interval {θ ∈ R : Φ∗∗(θ) < Φ(θ)} (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  In terms of the potential h we obtain a free boundary problem  �  ∆h = (θ1 − θ2)(χA(x) − ω),  x ∈ Ω  h ∈ W 2,2  0 (Ω),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='10)  where  ω = |A|  |Ω| = Tr H0 − θ2  θ1 − θ2  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='11)  is the volume fraction of the phase in which ∇ · u(x) = θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' As formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='11) indicates, the  volume fraction ω of the precipitate is uniquely determined by the hard device loading H0,  whose shear component has no effect on the precipitate morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  We remark that only the divergence of the displacement gradient is constrained by the  requirement of optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' This explains a virtual cornucopia of optimal microstructures, the  simplest of which we are going to exhibit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' A solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='10) can be easily found in explicit  form for some particularly simple domain geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' For instance, it is easy to construct  a radially symmetric solution when Ω is the unit ball, taking as the set A to be either the  concentric ball of radius r0 = ω1/3, or its complement, in which case r0 = (1 − ω)1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Then  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='10) is solved by h(x) = hω(|x|), given by  hω(r) =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f3  1  6(θ1 − θ2)(1 − ω)  �  r2 − 3ω(1 − ω1/3)  ω1/3(1 − ω)  �  ,  0 ≤ r ≤ ω1/3  −1  6(θ1 − θ2)ω  �  r2 + 2  r − 3  �  ,  ω1/3 ≤ r ≤ 1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='12)  in the former case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' In the latter case, h(x) is given by the same formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='12) where θ1 and  θ2 are interchanged and ω is replaced by 1 − ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' h(x) = −h1−ω(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  It is not clear from the free boundary problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='10) if solutions exist in less symmetric  domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  In fact, Hashin’s “concentric sphere” construction, [22], shows that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='10) has  solutions in any Lipschitz domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' In this construction the set A is a countable union  of variously scaled copies of the radially symmetric solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='12) filling Ω up to a set of  Lebesgue measure zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Let B(xi, ai) ⊂ B(xi, Ri), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' are the concentric balls used  in Hashin’s construction, where  a3  i  R3  i  = ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  13  The inner balls B(xi, ai) in the construction belong to the set A, while the spherical shells  B(xi, Ri) \\ B(xi, ai) belong to Ω \\ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The function h(x) restricted to the ball B(xi, Ri) is  given by  h(x) = R2  i hω  �|x − xi|  Ri  �  ,  x ∈ B(xi, Ri),  where hω is given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The function h(x) defined like this on each of the concentric  balls does indeed solve (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='10), since on the boundary of each ball B(xi, Ri) both h(x) and  ∇h(x) are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  The problem with the above “concentric sphere” construction is the same one as with  the laminate-based optimal microstructures: they all have an infinite surface area (for the  “concentric spheres” see the formal proof in [36, 51]), which is unacceptable in physical  problems of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' However, we also expect (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='10) to have solutions with finite surface  area in all piecewise smooth domains2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' For example, if Ω is an ellipsoid, then A would be a  confocal ellipsoid (or its complement) [8, 37, 49, 53, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' In the next section we present a more  complex solution with finite surface area in a two dimensional domain with piecewise-smooth  boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  5  A finite perimeter global minimizer  For a square, the 2D version of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='10) can be solved using complex variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The actual  geometry of the phase boundary can be then found numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  To obtain the shape of phase boundary we used the conventional complex potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Indeed, in view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='10) the functions  �h+(x, y) = h(x, y) + 1  4(θ1 − θ2)ω(x2 + y2),  �h−(x, y) = h(x, y) − 1  4(θ1 − θ2)(1 − ω)(x2 + y2)  are harmonic in Ω\\A and A respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' We can then conclude that the functions ∂�h±/∂x−  i∂�h±/∂y are analytic in the complex variable z = x + iy on their respective domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Thus,  the functions  H+(z) =  2  θ1 − θ2  �∂h  ∂x − i∂h  ∂y  �  + ωz,  z ∈ Ω \\ A,  H−(z) =  2  θ1 − θ2  �∂h  ∂x − i∂h  ∂y  �  − (1 − ω)z,  z ∈ A  are also analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The boundary conditions in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='10) and the continuity of ∇h across ∂A,  representing the kinematic compatibility of the displacement, imply that  �  H+(z) = ωz,  z ∈ ∂Ω,  H+(z) − H−(z) = z,  z ∈ ∂A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1)  The knowledge of Ω allows one to determine H+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' If Ω is a square centered at the origin  with diagonal of length 2, then along the bottom side of the square we have H+(x−i/  √  2) =  2We can say that this is a conjecture, as we have no formal proof of this statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='6  Figure 3: A coherent precipitate in the square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  ω(x + i/  √  2) = ω(z + i  √  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Therefore H+(z) = ω(z + i  √  2), while along the right side of the  square we have H+(1/  √  2+iy) = ω(1/  √  2−iy) = ω(  √  2−z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Thus, H+(z) = ω(  √  2−z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' These  contradictory expressions for H+(z) can only be reconciled by a structure with topology  indicated in Figure 3, where the set A, in which ∇ · u = θ1, is shaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  The square symmetry of a problem also suggests that we should look for a structure with  square symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Thus we just need to find the curve Γ that joins the top and bottom ends  of the right side of the square, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Knowing H+(z) in all four regions adjacent to  the sides of the square gives the boundary values for H−(z) on ∂A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The square symmetry  of A together with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1) implies that  H−(iz) = −iH−(z),  z ∈ ∂A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2)  There is a holomorphic function H− in A with given boundary values if and only if for every  n ≥ 0  �  ∂A  H−(z)zndz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3)  The integral in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3) can easily be written as an integral along Γ because of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' If n is  not a multiple of 4 then the integral in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3) will evaluate to zero by virtue of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2) alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' If  n = 4k, then we require that  �  Γ  H−(z)z4kdz = 0,  k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4)  On Γ we have H−(z) = ω(  √  2 − z) − z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Therefore, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4) is equivalent to  �  Γ  zz4kdz =  (−1)kiω  (4k + 1)(2k + 1),  k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='5)  If we integrate by parts in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='5):  �  Γ  zz4kdz =  �  Γ  z d  � z4k+1  4k + 1  �  = −  �  Γ  z4k+1  4k + 1dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  and parametrize the curve Γ by z(t) = a(t) + it, t ∈ [−1/  √  2, 1/  √  2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Then z′(t) = z′(t) − 2i,  we obtain  �  Γ  zz4kdz =  (−1)k+1i  (4k + 1)(2k + 1) +  2i  4k + 1  � 1/  √  2  −1/  √  2  z(t)4k+1dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  15  Thus, the equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='5) becomes  � 1/  √  2  −1/  √  2  z(t)4k+1dt = (−1)k(ω + 1)  2(2k + 1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Due to the symmetry we have z(−t) = z(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Therefore, we finally obtain  ℜe  �� 1/  √  2  0  (a(t) + it)4k+1dt  �  = (−1)k(ω + 1)  4(2k + 1)  ,  k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='6)  Observe that |a(t) + it| < 1 for all t ∈ [0, 1/  √  2) and |a(t) + it| = 1 for t = 1/  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Thus, for  large k the principal contribution to the integral (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='6) comes only from the neighborhood of  t = 1/  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Therefore, we approximate  a(t) = 1  √  2  + m(t − 1  √  2  ) + O((t − 1  √  2  )2), as t → 1  √  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='7)  Substituting the leading term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='6) and computing the integral explicitly we obtain  (−1)k  (4k + 2)(m2 + 1) −  (1 − m)4k+2m  22k+1(4k + 2)(m2 + 1) ≈ (−1)k(ω + 1)  2(4k + 2)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='8)  If we choose  m =  �  1 − ω  1 + ω  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='9)  then for large k we have equality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='6) up to an exponentially small error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Thus, we have  determined the slope m with which the curve Γ enters the corner of the square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  In order to find the function a(t) numerically, we approximate it by even polynomials  at t = 0 and enforce (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='7) at every approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Notice that the error term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='8)  decays very fast with k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Hence, satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='6) only for small values of k should give a very  good approximate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' With the knowledge of this, we approximate a(t) by an even  polynomial  an(t) =  n  �  j=0  ajt2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  In order to satisfy (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='8) we require that  n  �  j=0  aj  2j = 1  √  2,  n  �  j=1  jaj  2j−1 = m  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='10)  Observe that equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='6) for k = 0 is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Hence,  n  �  j=0  aj  2j(2j + 1) = ω + 1  2  √  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='11)  16  To solve the system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='6) for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' , K we set n = K +2 which leaves us with an easily  solvable system of K polynomial equations in K unknowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  The configurations of the sets A and Ω \\ A for ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3 (left panel) and ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='8 (right  panel) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' When ω is almost zero, we effectively deal with phase nucleation  associated with the “direct” (1 → 2) transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Phase “1” appears first in the form of  two infinitely thin intersecting segments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' the subsequent thickening of these segments starts  first around their intersection point at the center of the square domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' As ω increases, the  segments turn into a 4-cornered star, which eventually takes over the whole body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Instead,  at ω almost one, we deal with the nucleation of phase “2” which appears first in the form of  a thin “film” along the boundary of the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' In other words, during the “reverse” (1 → 2)  transformation a new phase grows away from the boundary and eventually takes over the  whole body leaving phase “1” only as a star shaped subdomain centered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  However, it is important to keep in mind that phases “1” and “2” in these scenaria can be  always interchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The complete constitutive symmetry between the two phases (the fact  that θ1 < θ2 is irrelevant) implies that the new phase may first appear either as an infinitely  thin coating of the sides of the square (unshaded regions in the right panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' 3) or as  a cross located at the center of the domain and extending its arms to its corners (shaded  regions in the left panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  6  Conclusions  In this paper we considered a classical problem of nonlinear elastostatics for a material  undergoing phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Mathematically it reduces to a non-convex vectorial problem  of the calculus of variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' In this context we presented a simple, yet non-trivial example  of an energy density (of a material) for which a large class of local minimizers could be made  explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' In particular, we exhibited a material model, where the absence of metastability  coexists with a wild nonuniqueness of global minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Metastability in elastostatics, understood as the existence of strong local but not global  minimizers, is an important problem in nonlinear elasticity because of the ubiquity of hys-  teresis in martensitic phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' While in Neumann problem the existence of such  local minimizers has been linked to the incompatibility of the energy wells, here we showed  that in Dirichlet problem the incompatibility of the wells is not sufficient for metastability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  More specifically, we presented an example of the energy density of a hyperelastic material  with non-rank-one convex, double well energy, for which we could prove the lack of strong  local minimizers which are not global on any domain and for any Dirichlet boundary con-  ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The analytical transparency of our arguments was due to the choice of the energy  density which is geometrically linear and isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  For the same material model we could fully characterize the necessary and sufficient  conditions for global energy minima in hard device loading, which allowed us to discover  the multiplicity of global minimizers with nonuniqueness unrelated to either objectivity or  material symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' As an important element of this analysis we used a novel way of ex-  pressing the energy of equilibrium configurations as boundary integrals which can be viewed  as a nontrivial nonlinear generalization of the classical Clapeyron theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' First of all, we  17  showed that the quasiconvex envelope in our model can be achieved by simple lamination  which can be characterized explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The problem with this construction is that in real  physical situations surface energy plays the role of the selection mechanism of the otherwise  energetically equivalent configurations [28, 32, 11], ruling out such well-known optimal mi-  crogeometrs as laminates and coated sphere constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Therefore, of particular interest  in physical applications are global minimizers with finite surface area, and we exhibited their  existence for our material model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  An interesting aspect of our highly degenerate problem with elastically indistinguish-  able phases is a sensitivity of the morphology of the interface between the phases to the  global shape of the transforming body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' To illustrate this effect, we explicitly computed a  one-parameter family of non-affine energy minimizing configurations for the case of a finite  domain with piecewise-smooth boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Our analysis of this example suggests that de-  spite the absence of metastability and the associated constitutive hysteresis in this model,  the direct and the reverse transformation may follow different morphological paths while  traversing energetically equivalent configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  YG was supported by the National Science Foundation under  Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' DMS-2005538.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The work of LT was supported by the French grant ANR-10-  IDEX-0001-02 PSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' YG is grateful to Pavel Etingof who pointed out the references for the  proof of the infinity of the surface area of Hashin’s concentric sphere construction in 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  A  The degeneracy of acoustic tensors of rank-1 en-  velopes  Here we prove that the acoustic tensor of a rank-one convex envelope RW of the non rank-one  convex energy must have a degenerate direction at all points F0 where RW(F0) < W(F0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  We also show that in particularly simple situations, such as the one discussed in this paper,  this property may be even sufficient to compute the whole rank-one convex envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  We recall that the acoustic tensor of the energy W(F ) at F = F0 in the direction n is a  quadratic form A(n) defined by  A(n)a · a = ⟨WF F (F0)(a ⊗ n), a ⊗ n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Let F0 be fixed and suppose RW(F0) < W(F0), where RW denotes the  rank-1 convexification of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Assume further that W and RW are C2 near F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Let A0(n)  be the acoustic tensor of RW at F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Then there is a direction n such that det A0(n) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Assume that there is no such direction n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Since RW is necessarily rank-1 convex we  can conclude that for any direction n the matrix A0(n) is positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Since the  function n �→ A0(n) is continuous there exists a positive number α such that for every unit  vector n  A0(n) ≥ αI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Since RW ∈ C2 near F0 there exists a number δ > 0 such that for every F satisfying  |F − F0| < δ the following inequalities hold:  18  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' A(n) ≥ 1  2αI, where A(n) is the acoustic tensor of RW at F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' W(F ) − RW(F ) ≥ 1  2(W(F0) − RW(F0)) = β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Now consider a function  Tǫ(F ) = RW(F ) + ǫφ(F − F0  4√ǫ  ),  where φ is a smooth nonnegative function supported on the unit ball in M and such that  φ(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' We can choose ǫ so small that the following inequalities hold:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' ∥ǫφ∥L∞ < 1  2β,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' ∥√ǫ∇F ∇F φ∥L∞ ≤ 1  4α,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' supp φ( F −F0  4√ǫ ) ⊂ B(F0, δ), where B(F0, δ) is the ball of radius δ around F0 in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Then Tǫ(F ) ≤ W(F ) and the acoustic tensor Aǫ(n) of Tǫ(F ) satisfies  Aǫ(n) ≥ 1  4αI  in the sense of quadratic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Thus Tǫ(F ) is rank-1 convex and Tǫ(F ) ≤ W(F ) but  Tǫ(F0) > RW(F0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Thus our assumption is false and at every point F0  where RW(F0) < W(F0) there is a direction n such that the acoustic tensor A0(n) is  degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' It follows that materials that transform by forming microstructures with sharp  phase boundaries to accommodate deformations produced by the propagation of sound waves  will have a direction with a zero sound speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  We can apply Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1 to the energy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Since our material is isotropic, the  degeneration of the acoustic tensor may be either through µ = 0 or through λ + 2µ = 0 with  the latter also meaning that the bulk modulus κ = −(4/3)µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The possibility that µ = 0 is  excluded because the tangential shear modulus is the same at every deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Suppose that we have somehow guessed that if  W(H) = f(Tr ε) + µ|dev(ε)|2,  then  RW(H) = F(Tr ε) + µ|dev(ε)|2  for some function F, yet to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' In that case Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1 will let us determine  the function F(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' It is easy to compute that for any unit vector n and any vector a ∈ R3  ⟨RWHH(H)(n ⊗ a), n ⊗ a⟩ = F ′′(Tr ε)(a, n)2 + 2µ  ����  1  2(n ⊗ a + a ⊗ n) − 1  3(a, n)I  ����  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  19  Therefore, we get a formula for the acoustic tensor A(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  A(n) = (F ′′(Tr H) + 1  3µ)n ⊗ n + µI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='1)  We see that det A(n) = µ2(F ′′(Tr ε) + 4µ/3) for all directions n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Thus, for all H for which  RW(H) < W(H) we get F ′′(Tr H) = −4µ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' The continuity of RWH implies that at  the boundary points θ1 and θ2 of the binodal region we have Φ′(θ1) = Φ′(θ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Further, the  continuity of W implies that the affine function3 y(θ) = F(θ)+2µθ2/3 would be the equation  of the common tangent to the graph of Φ(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Thus, we obtain that F(θ) + 2µθ2/3 = Φ∗∗(θ),  and we recover the rank-one convex envelope of W(H), which in this case is seen to coincide  with its quasiconvexification (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  References  [1] Rohan Abeyaratne and Jiang Guo-Hua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Dilatationally nonlinear elastic materials—I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Some theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' International Journal of Solids and Structures, 25(10):1201 – 1219, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  [2] Rohan Abeyaratne and Jiang Guo-Hua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' Dilatationally nonlinear elastic materials—II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  an example illustrating stress concentration reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' International Journal of Solids  and Structures, 25(10):1221 – 1233, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
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+page_content=' English  translation by Oldfather, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content=' and Ellis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
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+page_content='  Estimates for the homogenized matrix and the homogenized tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  Russian Math Surveys, 46(3):65–136, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
+page_content='  23' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQf5_7j/content/2301.01870v1.pdf'}
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+Prepared for submission to JHEP
+UUITP-62/22
+NORDITA 2022-155
+Lagrangians Manifesting Color-Kinematics Duality
+in the NMHV Sector of Yang-Mills
+Maor Ben-Shahar,a Lucia Garozzoa and Henrik Johanssona,b
+aDepartment of Physics and Astronomy, Uppsala University,
+Box 516, 75120 Uppsala, Sweden
+bNordita, Stockholm University and KTH Royal Institute of Technology,
+Hannes Alfvéns väg 12, 10691 Stockholm, Sweden
+E-mail: benshahar.maor@physics.uu.se, garozzo.lucia@physics.uu.se,
+henrik.johansson@physics.uu.se
+Abstract: Scattering amplitudes in Yang-Mills theory are known to exhibit kinematic
+structures which hint to an underlying kinematic algebra that is dual to the gauge group
+color algebra. This color-kinematics duality is still poorly understood in terms of conven-
+tional Feynman rules, or from a Lagrangian formalism. In this work, we present explicit
+Lagrangians whose Feynman rules generate duality-satisfying tree-level BCJ numerators,
+to any multiplicity in the next-to-MHV sector of pure Yang Mills theory. Our Lagrangians
+make use of at most three pairs of auxiliary fields (2,1,0-forms) – surprisingly few compared
+to previous attempts of Lagrangians at low multiplicities. To restrict the Lagrangian free-
+dom it is necessary to make several non-trivial assumptions regarding field content, kinetic
+terms, and interactions, which we discuss in some detail. Future progress likely hinges on
+relaxing these assumptions.
+arXiv:2301.00233v1  [hep-th]  31 Dec 2022
+
+Contents
+1
+Introduction
+1
+2
+Graphs, Numerators and Tree Amplitudes
+4
+2.1
+Color-kinematics duality and double copy
+4
+2.2
+Decompositions for amplitudes and numerators
+7
+3
+MHV Lagrangian
+11
+4
+NMHV Lagrangian
+13
+4.1
+The four-point NMHV Lagrangian
+13
+4.2
+The five-point NMHV Lagrangian
+14
+4.3
+Closed form representation of the bi-scalar sector numerators
+15
+4.4
+NMHV Lagrangian beyond the bi-scalar sector
+17
+4.5
+How unique is the NMHV Lagrangian?
+20
+4.6
+Comments on the N2MHV sector
+22
+5
+Towards MHV numerators at one loop
+22
+5.1
+Polarization power zero at one loop
+23
+5.2
+Comment on one-loop MHV numerators
+23
+6
+Conclusions
+24
+1
+Introduction
+Scattering amplitudes offer valuable insight to the mathematical structures of quantum field
+theory and gravity by uncovering patterns that are not readily apparent through standard
+Lagrangian techniques. The Bern-Carrasco-Johansson (BCJ) duality between color and
+kinematics [1–3] is a clear example where on-shell formulations preceded any Lagrangian
+understanding. According to the duality, scattering amplitudes in many gauge theories can
+be represented using cubic diagrams, each consisting of a kinematic numerator and a color
+factor that obey isomorphic relations. The color factors draw their characteristics from the
+Lie algebra of the gauge group, whereas the mathematical structure of the numerators is
+believed to originate from an unknown kinematic Lie algebra. See recent reviews [3–9].
+In massless purely adjoint gauge theories, such as pure Yang-Mills (YM) theory, color-
+kinematics duality can be rephrased at tree level as the presence of BCJ amplitude rela-
+tions [1, 10–13]. The duality and amplitude relations were initially identified in pure YM [1]
+and its supersymmetric generalizations [2, 10, 11, 14], but have since been discovered in a
+– 1 –
+
+range of other gauge theories, including matter representations [15–23], higher-derivative in-
+teractions [24–35] or Chern-Simons fields [36–40], and massive gauge theories [18, 19, 21, 41–
+55]. Surprisingly, also certain scalar effective field theories [56–65] obey the duality. The
+duality has been extended to loop-level amplitudes [2, 66–91], form factors [92–101], and
+even to curved-space correlators [102–120].
+There have been various approaches to un-
+derstanding the existence of color-kinematics duality and BCJ relations in massless gauge
+theories, including string theory, scattering equations, positive geometry [10, 11, 121–126].
+Color-kinematics duality is intimately connected to the double-copy structure of grav-
+ity [1, 2]. For non-abelian gauge theories that obey color-kinematics duality, and contain
+physical spin-1 gluons, the double copy gives gravitational interactions after taking each
+cubic diagram and replacing the color factor by a second numerator copy. The double copy
+provides a versatile generalization of the Kawai-Lewellen-Tye (KLT) relations [127], which
+were originally derived for string theory at genus zero (for higher-genus generalizations
+see e.g. [82, 84, 128–134]). Through color-kinematics duality the double copy permits the
+construction of gravity loop amplitudes [2, 75, 135–142], more general gravity theories [15–
+27, 143, 144], classical solutions [145–168] and other non-perturbative solutions [169, 170].
+Recently black-hole scattering and gravitational wave physics have been studied using the
+double copy [86, 171–191].
+Despite extensive progress, the mathematical details of color-kinematics duality remain
+enigmatic even for the case of pure YM theory. While duality-satisfying YM numerators can
+be straightforwardly computed to any multiplicity at tree level [13, 192–203], the building
+blocks and underlying symmetry principles of these objects are poorly understood. Realiz-
+ing the hidden YM structure as a kinematic Lie algebra or a cubic Lagrangian may shine
+light on the duality. Explicit kinematic-algebra constructions have been realized for the self-
+dual YM sector [68, 204, 205], the maximally-helicity-violating (MHV) sector [57, 204, 206]
+and the next-to-MHV (NMHV) sector [206, 207]. Self-dual YM amplitudes vanish except
+for at one loop [208] and the kinematic algebra is identified with area-preserving diffeomor-
+phisms [204]. The MHV sector of YM gives the simplest non-vanishing 4D tree amplitudes,
+and while ref. [204] considered it through a non-local gauge choice, local formulations of
+the MHV kinematic algebra was discussed in refs. [57, 206].
+A cubic duality-satisfying Lagrangian for the non-linear-sigma model was considered
+in ref. [57], but via dimensional-reduction operations [209, 210] it is directly related to
+the MVH sector of YM. (It is also related to a non-abelian generalization of the Navier-
+Stokes equation that obeys color-kinematics duality [211–213].) Both the MHV and NMHV
+kinematic algebra was described in some detail in refs. [206, 207]. A natural extension of
+the area-preserving diffeomorphism algebra remains to be found in those sectors. However,
+in 3D Chern-Simons theory the complete kinematic algebra is now identified with volume-
+preserving diffeomorphisms [40]. In section 3, we briefly comment on the appearance of 4D
+volume-preserving diffeomorphism generators in the MHV sector. Recently, kinematic Hopf
+algebras have been introduced for YM and related theories, which gives gauge-invariant BCJ
+numerators for all tree-level sectors at the expense of additional heavy-mass poles [100, 200–
+203, 214, 215].
+Duality-satisfying YM Lagrangians have been sporadically studied for more than a
+– 2 –
+
+decade [14, 216–222] using non-local interaction terms or auxiliary fields. The early at-
+tempts produced Feynman rules capable of computing local BCJ numerators up to five [14]
+and six points [216]. However, the non-uniqueness of the BCJ numerators led to a prolif-
+eration of ambiguities in the resulting Lagrangians. In hindsight, it is clear that in a local
+formalism the ambiguities are absent in the MHV sector and first appear in the NMHV
+sector [206]. They can be traced back to the so-called generalized gauge freedom of the
+BCJ numerators [1, 2], which include the standard gauge freedom and non-linear field re-
+definitions. Understanding how to constrain this freedom and write down broadly valid and
+practical YM Lagrangians that manifest the duality is the topic of this paper.
+Recently there has been an upswing in formulations that attempts to approach the
+problem of realizing YM color-kinematics duality and double copy off shell [40, 169, 200,
+218, 219, 222–234]. This involves understanding the role of BRST symmetries, homotopy
+algebras, double field theory, equations of motions and other off-shell Langrangian perspec-
+tives. We note that, in principle, the problem has been solved using the 10D pure-spinor
+formalism [222]; however, the practical usefulness of this duality-satisfying supersymmetric
+YM Lagrangian for advanced calculations remains to be understood.
+In this paper, we approach the problem of finding duality-satisfying YM Lagrangians
+head on, by making a suitable small ansatz for the NMHV sector Lagrangian where the first
+ambiguities show up. We work in general dimension using covariant building blocks, and
+thus we generalize the 4D notion of helicity sectors by grading the tree-level numerators by
+the “polarization power” [206, 207], i.e. the number of inner products between polarization
+vectors εi·εj. Thus the NMHV sector corresponds to numerators with at most two polar-
+ization powers (εi·εj)2; or, equivalently, one power of momentum inner products pi·pj. The
+construction of the ansatz is aided by the observation that there exist a bi-scalar subsector1
+of the NMHV sector that has particularly simple BCJ numerators. This bi-scalar subsector
+was also considered in ref. [206] since it generates master numerators from which all other
+NMHV numerators can be determined. We find that the bi-scalar numerators can be com-
+puted using an exact rewriting of the standard YM tree-level Lagrangian, by “integrating
+in” a pair of auxiliary 2-form fields {Bµν, ˜Bµν} with additional simple cubic interactions.
+From the bi-scalar numerators one can obtain the remaining contributions to the
+NMHV numerators using a simple formula; however, we also demand that those contri-
+butions come directly from a Lagrangian. This required us to either use clever inspection
+and identification of needed new interactions terms aided by pictorial diagrams that ex-
+pose the tensors structures of intermediate fields, or alternatively, a brute-force Lagrangian
+ansatz where the assumptions are more clearly spelled out. We find several interesting
+solutions which involve a pair of auxiliary vector fields {Zµ, ˜Zµ}, and a pair of auxiliary
+scalar fields {X, ˜X}. We show that by considering an extended ansatz the need for scalar
+auxiliary fields is not obvious, as there exists interesting Lagrangian solutions where they
+are absent. Further relaxing the assumptions that went into the construction may provide
+simpler NMHV Lagrangians, but we leave this for future work. We briefely comment on
+1Terms in the half-ladder BCJ numerator proportional to a fixed polarization product ε1·εn, which
+effectively becomes two scalars interacting with the YM field.
+– 3 –
+
+the N2MHV sector, and also spell out some details about the possible generalization of the
+Lagrangian Feynman rules to one-loop calculations.
+The paper is organized as follows: In section 2, we introduce the necessary notation used
+for describing amplitudes and numerators that satisfy color-kinematics duality, including
+decomposition into partial amplitudes, bi-scalar half-ladder numerators and useful pictorial
+diagrams. In section 3, we consider a cubic Lagrangian for computing MHV numerators,
+obtained by truncating the standard YM action. In section 4, we transform the standard
+YM Lagrangian by integrating in a pair of two-form auxiliary fields and then we add
+interactions and further auxiliary fields using both diagrammatic and ansatz approaches.
+In section 5, we briefely discuss consequences for one-loop numerators. Conclusions and
+outlook are given in section 6.
+2
+Graphs, Numerators and Tree Amplitudes
+Here we will set the notation used for the amplitude and numerator building blocks, and
+discuss the decompositions and diagrammatic notation that are convenient for later sections.
+2.1
+Color-kinematics duality and double copy
+Scattering amplitudes in YM theory can be represented diagrammatically as a sum over
+cubic Feynman-like graphs [1, 2], for n-point tree level amplitudes this takes the form
+An = gn−2 �
+Γ∈Gn
+CΓNΓ
+DΓ
+.
+(2.1)
+Each cubic graph Γ is associated with a color factor CΓ, kinematic numerator NΓ and
+propagator denominator DΓ. Here the set of cubic n-point graphs is denoted by Gn, and it
+can be constructed recursively starting from the three-point case, G3, which contains only
+one such graph Γ = [1, 2]. At n points the set of cubic graphs is given by
+Gn =
+�
+Γ|ℓ→[ℓ, n−1]
+��� ℓ ∈ Γ ∈ Gn−1
+�
+,
+(2.2)
+where ℓ denotes a Lie-valued element of the graph Γ. For example, the graph Γ = [1, 2] has
+three such elements, the external leg labels ℓ = 1, ℓ = 2 and the commutator of the labels
+ℓ = [1, 2]. Applying the rule ℓ → [ℓ, 3] then gives three four-point graphs represented by
+nested commutators [207, 235, 236]
+G4 =
+�
+[[1, 3], 2], [1, [2, 3]], [[1, 2], 3]
+�
+.
+(2.3)
+Each one of the four-point graphs have five Lie-valued elements (three labels, and two
+commutators), thus the total number of five-point graphs is |G5| = 3 × 5. In general, at n
+points, the number of cubic graphs is |Gn| = 3 × 5 × 7 × · · · × (2n − 5) = (2n − 5)!!.
+With the above graph notation the propagator denominator can be written as
+DΓ =
+�
+ℓ∈Γ
+ℓ̸=N,ℓ̸=Γ
+p2
+ℓ ,
+(2.4)
+– 4 –
+
+where pℓ is the sum of the momentum of all the external legs in the nested commutator ℓ.
+For example, p[1,2] = p1 + p2 and p[[1,3],2] = p1 + p2 + p3, etc. Since the external legs are on
+shell p2
+1, . . . , p2
+n = 0, we do not include cases where ℓ is an integer, nor where it is the graph
+Γ, since by momentum conservation pΓ = −pn.
+The color factors for purely-adjoint YM theory correspond to rank-n tensors in the
+gauge-group Lie algebra, which can be constructed through contraction of structure con-
+stants, fabc, or as traces of products of generators T a. The latter case is most transparent
+in our graph notation, and an explicit formula can be given
+CΓ = Ca1a2···an
+Γ
+= Tr
+�
+(Γ|i→T ai) T an�
+,
+(2.5)
+where the external legs i = 1, . . . n−1 of the graph Γ are replaced by Lie-algebra generators,
+thus justifying the graph notation as nested commutators of Lie-valued elements.
+For
+example, the three-point graph Γ = [1, 2] has the color factor
+C[1,2] = Tr
+�
+[T a1, T a2]T a3�
+= fa1a2a3 .
+(2.6)
+Here, and throughout the paper, we use the normalization conventions [T a, T b] = fabcT c
+and Tr
+�
+T aT b�
+= δab.
+Finally the numerators NΓ capture all the remaining dependence on the kinematic data
+of the YM tree amplitude, and in this paper we assume that they are local polynomials of
+the polarization vectors εi and momenta pi. To describe gluon scattering in pure YM theory,
+the polynomials must be multi-linear in the individual polarizations and of degree-(n − 2)
+in the momenta. It implies that the contributing monomial terms have the schematic form
+NΓ ∼
+⌊n/2⌋
+�
+k=1
+(ε · ε)k (p · p)k−1 (ε · p)n−2k .
+(2.7)
+For example, for n = 3 points the numerator is equivalent to the color-stripped cubic
+Feynman rule of YM,
+N[1,2] = ε1 · ε2 ε3 · (p1 − p2) + cyclic(1, 2, 3) .
+(2.8)
+It satisfies the same dihedral permutation symmetries N[2,1] = −N[1,2], N[2,3] = N[3,1] =
+N[1,2] as the color factor C[1,2] = fa1a2a3.
+Due to the Jacobi identity of the gauge-group Lie algebra, there are certain cubic
+relations between color factors. The statement of the color-kinematics duality [1, 2] is that
+for a large class of gauge theories there exists a choice of so-called BCJ numerators that
+obey the same relations as the color factors,
+C···[[X,Y ],Z]··· + C···[[Y,Z],X]··· + C···[[Z,X],Y ]··· = 0
+⇔
+N···[[X,Y ],Z]··· + N···[[Y,Z],X]··· + N···[[Z,X],Y ]··· = 0 .
+(2.9)
+A four-point YM numerator that satisfies this property is [3]
+N[[1,2],3] =
+�
+ε1·ε2pµ
+1 + 2ε1·p2εµ
+2 − (1 ↔ 2)
+��
+ε3·ε4p3µ + 2ε3·p4ε4µ − (3 ↔ 4)
+�
++ s12(ε1·ε3ε2·ε4 − ε1·ε4ε2·ε3) .
+(2.10)
+– 5 –
+
+It is straightforward to check that, subject to on-shell conditions and momentum conserva-
+tion, the numerator satisfies the Jacobi relation
+N[[1,2],3] + N[[2,3],1] + N[[3,1],2] = 0 ,
+(2.11)
+as well as the permutation symmetries
+N[[2,1],3] = −N[[1,2],3] ,
+N[[4,3],2] = N[[1,2],3] .
+(2.12)
+In general, it is a difficult task to find YM numerators that satisfy color-kinematics dual-
+ity. However, efforts during the last decade have led to several different constructions of
+all-multiplicity numerators with various properties [13, 126, 192–196, 198–202, 237]. Nev-
+ertheless, until now it has not been known how to construct such numerators directly from
+a duality-satisfying Lagrangian or Feynman rules. We will attempt address this problem in
+section 4.
+The color factors in eq. (2.1) can be substituted by a second copy of color-kinematics
+satisfying numerators CΓ → ˜NΓ, which gives the double-copy formula [1, 2]
+Mn =
+�κ
+2
+�n−2 �
+Γ∈Gn
+NΓ ˜NΓ
+DΓ
+,
+(2.13)
+where, for dimensional consistency, we also substituted the couplings g → κ/2. The two sets
+of numerators can belong to either the same or different theories. If both sets of numerators
+belong to theories that contain physical spin-1 gauge fields, then the double-copy formula
+gives an amplitude for spin-2 fields in some theory of gravity. This follows from the fact
+that the double copy automatically gives diffeomorphism invariant amplitudes. To see this,
+consider a linearized gauge transformation acting on one of the polarizations of the gauge
+theory amplitude, δεi = pi. By definition, we expect that δAn = 0, from which it follows
+that it must be true that
+�
+Γ∈Gn
+CΓ δNΓ
+DΓ
+= 0 ,
+(2.14)
+where δNΓ are the gauge transformed numerators. The fact that this vanishes does not
+depend on the details of the color factors, only on the fact that they come from a Lie algebra
+and thus satisfy Jacobi identities. If we have kinematic numerators that satisfy the same
+Jacobi identities, then it also follows that [1, 2, 238]
+�
+Γ∈Gn
+˜NΓ δNΓ
+DΓ
+= 0 ,
+(2.15)
+and likewise for NΓ and ˜NΓ swapped. Hence, it follows that δMn = 0, and thus the double-
+copy formula is invariant under linearized diffeomorphisms [238]. Given that all amplitudes
+have this invariance, it follows that the theory enjoys non-linear diffeomorphism symmetry,
+so it can be interpreted as a theory of gravity. When both numerators come from four-
+dimensional pure YM, the double copy gives amplitudes in Einstein-dilaton-axion gravity,
+with Lagrangian [3]
+L = −
+√−g
+κ2
+�
+2R − ∂µτ∂µ¯τ
+(Im τ)2
+�
+,
+(2.16)
+– 6 –
+
+where the complex field τ = ie−φ + χ contains the dilaton φ and axion χ scalars. In D
+dimensions the axion is promoted to a two-form Bµν and the Lagragian is also well known,
+see e.g. [3].
+2.2
+Decompositions for amplitudes and numerators
+Color-ordered partial amplitudes are defined as the gauge-invariant coefficients of the trace
+decomposition of the full amplitude,
+An = gn−2
+�
+σ∈Sn−1
+Tr(T a1T aσ(2) . . . T aσ(n))An(1, σ2, . . . , σn) ,
+(2.17)
+where the sum over permutations σ runs over the symmetric group Sn−1 with (n − 1)!
+elements. The partial amplitudes can be computed directly using color-ordered Feynman
+rules, or from eq. (2.1) after expanding out the color factors CΓ into the basis of ordered
+traces of generators.
+Alternatively, repeatedly using the Jacobi identity (2.9) to obtain a smaller basis of
+color factors, gives the Del-Duca-Dixon-Maltoni (DDM) decomposition [239],
+An = gn−2
+�
+σ∈Sn−2
+C(1, σ2, . . . , σn−1, n)An(1, σ2, . . . , σn−1, n) ,
+(2.18)
+which uses a subset of the same partial amplitudes, and the independent color factors are
+C(1, σ2, . . . , σn−1, n) ≡ C[[···[[1,ρ2],ρ3],...],ρn−1] = Tr([[· · · [T a1, T aσ(2)], . . .], T aσ(n−1)]T an)
+(2.19)
+=
+�
+faσ(2) . . . faσ(n−1))a1an .
+(2.20)
+The (n − 2)!-fold DDM basis of color factors CΓ makes use of the so-called half-ladder
+graphs
+Γhalf-ladder(ρ) ≡ [[· · · [[1, ρ2], ρ3], . . .], ρn−1] ,
+(2.21)
+which will serve as our canonical basis choice of graphs. Similarly, repeatedly using the
+kinematic Jacobi identity (2.9) on the BCJ numerators gives the decomposition [1, 240]
+An =
+�
+Γ∈Gn
+CΓ NΓ
+DΓ
+=
+�
+σ,ρ∈Sn−2
+C(1, σ2, · · · , σn−1, n)m(σ|ρ)N(1, ρ2 · · · , ρn−1, n) ,
+(2.22)
+where we have set g = 1 and the independent half-ladder numerators are written as
+N(1, ρ2 · · · , ρn−1, n) ≡ N[[···[[1,ρ2],ρ3],...],ρn−1] .
+(2.23)
+Assuming that the half-ladder numerators come from a manifestly crossing-symmetric con-
+struction, they obey a reflection symmetry N(1, 2, 3 . . . , n) = (−1)nN(n, . . . , 3, 2, 1), as well
+as anti-symmetry in the first and last pairs of arguments: N(2, 1 . . .) = −N(1, 2 . . .) and
+N(. . . , n, n − 1) = −N(. . . , n − 1, n).
+The matrix m(σ|ρ) in eq. (2.22) is of size (n−2)! × (n−2)! and is built out of linear
+combinations of the scalar-type propagators, 1/DΓ, as can be worked out from the above
+– 7 –
+
+two DDM decompositions of the BCJ numerators and color factors. The matrix m(σ|ρ)
+has appeared in many places in the literature, and it goes by a variety of different names,
+such as “propagator matrix” [240], the “bi-adjoint scalar amplitude” [241], or “inverse of
+the KLT kernel” [127, 241]. Is is also equivalent to the color-stripped partial amplitudes of
+“dual-scalar theory” [14, 242], “color-scalar theory” [243] or “scalar φ3 theory” [17, 244].
+Using the propagator matrix the color-ordered partial amplitudes can be written as
+A(1, σ, n) =
+�
+ρ∈Sn−2
+m(σ|ρ)N(1, ρ, n) ,
+(2.24)
+which gives a (non-invertible) map between BCJ numerators and partial amplitudes [1].
+The propagator matrix is not invertible for on-shell conserved momenta and hence it has a
+kernel, or null space. This implies that BCJ numerators contain unphysical contributions
+that live in this kernel. This explains why BCJ numerators are in general not unique. The
+ambiguity is called generalized gauge freedom [1, 2] and it corresponds to the freedom of
+shifting the BCJ numerators by pure gauge contributions
+N(1, ρ, n) ∼ N(1, ρ, n) + Ngauge(1, ρ, n) ,
+(2.25)
+where pure gauge numerators are annihilated by the propagator matrix,
+�
+ρ∈Sn−2
+m(σ|ρ)Ngauge(1, ρ, n) = 0 .
+(2.26)
+The generalized gauge freedom includes both standard gauge freedom and field redefinitions,
+and more generally any operation that changes the cubic diagram numerators while leaving
+the partial amplitudes invariant.
+As illustrated in eq. (2.7), tree-level YM numerators are polynomial functions of the
+independent Lorentz contractions of polarizations and momenta {εi ·εj, εi ·pj, pi ·pj}. Most
+numerator operations that we are interested in do not mix terms that contains different
+powers ∼ (εi · εj)k, so it is convenient to decompose the numerators into sectors defined by
+their polarization power k [206]. Thus from the general structure (2.7), we decompose the
+numerators as
+N(1, . . . , n) =
+⌊n/2⌋
+�
+k=1
+N(k)(1, . . . , n) ,
+(2.27)
+where the polarization power k keeps track of how many εi·εj-contractions are present in
+each monomial. This decomposition also makes physical sense since by a suitable choice of
+the reference momenta qµ
+i in the polarizations εi = εi(pi, qi) one can show [206] that only
+the numerators N(1), N(2), . . . , N(k) contribute to the Nk−1MHV sector of YM. Therefore
+when we refer to the Nk−1MHV sector in this paper, we implicitly mean that we consider
+the N(≤k) numerators that contribute to this sector.
+Alternatively, if one considers a dimensional reduction of YM, SO(1, D−1) → SO(1, 3)×
+SO(D−4), then the N(k) numerators give rise to tree amplitudes with at most 2k external
+scalars. This can be formalized by considering derivative operators
+∂
+∂εi·εj that can act on
+the numerators and convert a pair of gluons to a pair of scalars [209, 210].
+– 8 –
+
+We often work with particles 1 and n being scalars, so it is convenient to introduce a
+bar notation on the half-ladder numerators to indicate that we are considering a bi-scalar
+YM sector [206],
+N
+(k)(1, . . . , n) ≡
+∂
+∂ε1·εn
+N(k)(1, . . . , n) .
+(2.28)
+The complete bi-scalar sector numerator is then N(1, . . . , n) = �⌊n/2⌋
+k=1 N
+(k)(1, . . . , n).
+It is possible to invert the operation in eq. (2.28) and construct the YM numerators
+from the bi-scalar numerators [206],
+N(k)(1, . . . , n) = 1
+k
+�
+1≤i<j≤n
+εi·εjN
+(k)(i, αi, i + 1, . . . , j − 1, βj, j) ,
+(2.29)
+where αi = [· · · [1, 2], 3], . . . , i − 1] and βj = [j + 1, [. . . , n − 2, [n − 1, n]] · · · ] are nested
+commutators.
+Numerators of commutators distribute over their arguments to be con-
+sistent with the Lie-algebraic interpretation, N
+(k)(. . . , [X, Y ], . . .) ≡ N
+(k)(. . . , X, Y, . . .) −
+N
+(k)(. . . , Y, X, . . .). The boundary cases of the sum are handled through the identifications
+α2 = [1] = 1, βn−1 = [n] = n, and when either bracket is empty α1 = βn = [ ] → (−1) the
+numerator is multiplied by a minus sign.
+We can demonstrate eq. (2.29) using the four-point numerator as an example. The
+bi-scalar numerator is unique (given that N(1, 2, 3, 4) = N(4, 3, 2, 1)),
+N(1, 2, 3, 4) = 4ε2·p1ε3·p1 + 4ε2·p1ε3·p2 − ε2·ε3s12 ,
+(2.30)
+and can be further decomposed into polarization powers
+N
+(1)(1, 2, 3, 4) = 4ε2·p1ε3·p1 + 4ε2·p1ε3·p2 ,
+N
+(2)(1, 2, 3, 4) = −ε2·ε3s12 .
+(2.31)
+Note that the polarization-power labels also keep track of the overall ε1·ε4 that was removed
+by the derivative
+∂
+∂ε1·εn . Using eq. (2.29) one can verify that
+N(k)(1, 2, 3, 4) =1
+k
+�
+(ε1·ε4)N
+(k)(1, 2, 3, 4) − (ε1·ε3)N
+(k)(1, 2, 4, 3) − (ε1·ε2)N
+(k)(1, [3, 4], 2)
+− (ε2·ε4)N
+(k)(2, 1, 3, 4) + (ε2·ε3)N
+(k)(2, 1, 4, 3) − (ε3 · ε4)N
+(k)(3, [1, 2], 4)
+�
+(2.32)
+matches the numerator in eq. (2.10), given that N[[1,2],3] = N(1)(1, 2, 3, 4) + N(2)(1, 2, 3, 4).
+For later purposes, it is convenient to introduce shorthand notation for the kinematic
+variables that frequently appear in the numerators. We define the variables2
+xµ
+i ≡
+i
+�
+j=1
+pµ
+j ,
+ui ≡ 2εi · xi ,
+(2.33)
+2The xi are sometimes called “region momenta”, and the ui are equivalent to cubic scalar-gluon vertices.
+– 9 –
+
+which make the bi-scalar half-ladder numerators simpler to work with. For example, the
+four-point bi-scalar numerator (2.30) now takes the simple form
+N(1, 2, 3, 4) = u2u3 − ε2 · ε3x2
+2 ,
+(2.34)
+and the polarization-power components are then N
+(1) = u2u3 and N
+(2) = −ε2 · ε3x2
+2. Note
+that this notation obscures the relabeling symmetries of the numerators so it should be
+used with some care, and we will mainly use it for bi-scalar half-ladder numerators.
+It is illuminating to attempt to represent the contributions to the numerators using
+diagrammatic notation. As a general guiding principle, we track inner products between
+polarization vectors using solid lines and wavy lines represents other contributions (typically
+inner products between polarizations and momenta). For example, the four-point bi-scalar
+numerators are represented by the two diagrams
+N
+(1) =
+1
+2
+3
+4
+= u2u3 ,
+N
+(2) =
+1
+2
+3
+4
+= −ε2 · ε3 x2
+2 .
+(2.35)
+This notation makes it clear that the numerator N
+(2) represents a contribution coming from
+an auxiliary rank-2 field in the intermediate channel. Indeed, we will see in section 4 that
+this interpretation is correct.
+At five points, the half-ladder BCJ numerator can again be decomposed into two bi-
+scalar numerators, N
+(1) and N
+(2), of different polarization powers. Focusing on N
+(2), we
+realize even before having an explicit formula that there are three different dot products
+ε3·ε4 , ε2·ε3 and ε2·ε4, which can be represented by the schematic diagrams
+N
+(2)(1, 2, 3, 4, 5) =
+1
+2
+3
+4
+5
++
+1
+2
+3
+4
+5
++
+1
+2
+3
+4
+5
+.
+(2.36)
+The explicit expressions for these diagrams will be given in section 4.
+Likewise at n points, the half-ladder bi-scalar numerator N
+(2) can be schematically
+written down as a sum over (n − 2)(n − 3)/2 different diagrams,
+N
+(2)(1, . . . , n) =
+�
+1<i<j<n
+. . .
+. . .
+. . .
+1
+n
+j
+i
+,
+(2.37)
+and in section 4 we will find an explicit formula for the diagrams using a Lagrangian.
+Before considering these NMHV amplitude contributions, we will have a short section on
+the Lagrangian that generates the N(1) numerators which gives MHV amplitudes.
+– 10 –
+
+3
+MHV Lagrangian
+Tree-level MHV amplitudes of YM can be computed from the BCJ numerators with the
+lowest polarization power N(1) [206]. For example, if legs 1 and 2 carry negative helicity
+and the remaining legs have positive helicity, then choosing reference null momenta qµ
+i as
+ε−
+1 (p1, q1 = p2) ,
+ε−
+2 (p2, q2 = p1) ,
+ε+
+i>2(pi, qi = p1)
+(3.1)
+ensures that the only non-vanishing polarization products are ε2·εi. Because they all contain
+ε2 these factors can at most appear linearly in the numerator, hence N(1) is sufficient.
+By dimensional analysis the numerators N(1) cannot contain products of momenta
+pi · pj hence they cannot give rise to inverse propagators p2 ∼ □, and thus there are no
+hidden contact terms inside the numerators. This implies that N(1) must originate from
+a YM Lagrangian that neither has contact terms nor auxiliary fields. Thus we can simply
+truncate the YM Lagrangian3 to cubic order, which defines the MHV Lagrangian
+LMHV = LYM
+���
+cubic = Tr
+�1
+2Aµ□Aµ − ∂µAν[Aµ, Aν]
+�
+.
+(3.2)
+For simplicity, we have set the gauge coupling to unity, g = 1, which we will do henceforth.
+The Lagrangian (3.2) gives standard cubic YM Feynman rules that produce numerators
+of all polarization power sectors k ≥ 1, but for the purpose of this section we can assume
+that the resulting numerators are truncated to N(1) because of the conditions (3.1) imposed
+on the polarization vectors. That this Lagrangian is invalid beyond this sector is obvious
+from the fact that the truncation of the quartic term is not a valid (gauge-fixing) operation.
+Let us now show that the numerators N(1) computed from the Lagrangian (3.2) obey
+color-kinematics duality. We use the diagrammatic notation from the previous section to
+make the argument simpler. The N(1) numerators are computed from sums of diagrams
+each having just one solid line (representing the contracted polarization vectors). We can
+think of the solid line as a scalar line, and we have to sum over all possible pairs of external
+states that are joined by this line. The scalar line can be identified from the equations of
+motion of the MHV Lagrangian,
+□Aµ = −2[Aν, ∂νAµ] + [Aν, ∂µAν] ,
+(3.3)
+where the second term produces contractions that ultimately connect external polarization
+vectors. The first term corresponds to cubic interactions were no scalar line is present, thus
+we think of it as consisting of only wavy lines. It is straightforward to then study the triplet
+sum of off-shell numerators obtained using these two interactions. One can show that the
+equation
+1
+2
+3
+4
+−
+1
+2
+3
+4
+=
+1
+2
+3
+4
+,
+(3.4)
+3We assume Lorenz gauge ∂ · A = 0 (or, more accurately, Feynman gauge) since the BCJ numerators
+are defined to sit on top of propagators that are in Feynman gauge, see eq. (2.4).
+– 11 –
+
+holds up to terms proportional to ε2·p2 and ε3·p3. Such terms vanish on-shell because of
+the transversality of εi, but we need to show that the above three-term identity holds for
+a generic off-shell numerator embedded into a larger tree diagram. Thus we need to show
+that transversality holds for all off-shell wavy lines in the this MHV sector.
+We can show this using induction, essentially feeding an arbitrary number of wavy-line
+interactions into eq. (3.4). Suppose that to some wavy line i we attach a cubic diagram
+generated from the first interaction term in eq. (3.3), producing two new wavy lines j, k, so
+i → [jk]. Then its effective polarization vector4 takes the form
+εµ
+i = −2(εj·pkεµ
+k − εk·pjεµ
+j ) ,
+(3.5)
+in terms of the effective polarizations of the lines j, k. By the induction hypothesis, we
+assume that the latter polarizations are transverse, εj·pj = εk·pk = 0. Using this, and the
+antisymmetry in the labels j and k, eq. (3.5) implies that εi·pi = 0. Thus the wavy-line
+interaction preserves transversality, and since the wavy lines will eventually terminate in
+proper external states a tree level the induction hypothesis is correct. This completes the
+argument that eq. (3.4) holds off shell.
+We also need to consider kinematic Jacobi identities of diagrams containing only the
+first term in eq. (3.3), which is the wavy-line interaction (3.5). However, note that eq. (3.5)
+is essentially a momentum-space Lie bracket of plane-wave vector fields Aµ
+i = εµ
+i eipi·x,
+Ai · ∂ = −2
+�
+Aj · ∂, Ak · ∂
+�
+,
+(3.6)
+and so it automatically obeys the Jacobi identity. Since the vector fields are transverse
+∂ · Ai = 0, we have thus exposed a kinematic sub-algebra that corresponds to volume-
+preserving diffeomorphisms (in analogy with the previously found 2D area-preserving [204]
+and 3D volume-preserving diffeomorphisms [40]). However, we cannot reproduce the full
+N(1) numerators from commutators of only these diffeomorphism generators, since the full
+numerator is a superposition of all diagrams with a single solid line going between pairs of
+external states. Nevertheless, by linearity of the kinematic Jacobi identity, the superposition
+of all such diagrams will give N(1) numerators that obey color-kinematics duality.
+To summarize, for tree level diagrams obtained from the standard cubic YM La-
+grangian, color-kinematics duality is satisfied at polarization power one, or equivalently,
+for MHV amplitudes. Note that the off-shell argument does not automatically extend to
+loop level since it was important that the recursive transversality argument terminates with
+external on-shell legs, which is always true at tree level, but not at loop level.
+Finally, let us give the half-ladder bi-scalar numerators that are generated from the
+Feynman rules of the MHV Lagrangian (3.2). They are simply
+N
+(1)(1, 2 . . . , n − 2, n) =
+. . .
+1
+n
+2
+n − 1
+=
+n−1
+�
+i=2
+ui ,
+(3.7)
+4Note that this is not to be confused with an external properly normalized polarization vector.
+– 12 –
+
+and the remaining contributions to the half-ladder numerator N(1) can be obtained using
+eq. (2.29), or from the Feynman rules of the Lagrangian (3.2). The non-half-ladder diagrams
+are similarly computed either from commutators of the half-ladder numerator N(1), or from
+the same Feynman rules.
+4
+NMHV Lagrangian
+In this section we construct several cubic Lagrangians that produce BCJ numerators at
+polarization power two, which allow us to compute helicity amplitudes up to the NMHV
+sector of YM. We begin the construction at four points, and work our way upwards in
+multiplicity by adding terms to the Lagrangian, until the corrections terminate at seven
+points.
+4.1
+The four-point NMHV Lagrangian
+Let us start from the standard YM Lagrangian (subject to Lorenz gauge ∂ · A = 0),
+LYM = −1
+4Tr (F µν)2 = Tr
+�1
+2Aµ□Aµ − ∂µAν[Aµ, Aν] − 1
+4[Aµ, Aν][Aµ, Aν]
+�
+,
+(4.1)
+where the field strength is F µν = ∂µAν − ∂νAµ + [Aµ, Aν]. For later purposes, let us also
+quote the corresponding equations of motion
+□Aν = −2[Aµ, ∂µAν] + [Aµ, ∂νAµ] − [Aµ, [Aµ, Aν]] .
+(4.2)
+We seek to find a cubic rewriting of this Lagrangian that generates BCJ numerators at
+polarization power two. As a first step, we can resolve the quartic interaction by introducing
+propagating auxiliary fields. There are several ways to do this. We find that an elegant
+solution is to reinterpret the nested commutator, appearing in the last term of the above
+equations of motion, to contain a two-form tensor Bµν = −1/2[Aµ, Aν]. In order to allow
+for a suitable kinetic term of correct mass dimension, we introduce a companion field ˜Bµν
+of mass dimension zero, into which Bµν propagates. The Lagrangian we obtain is
+L4 = Tr
+�1
+2Aµ□Aµ − ∂µAν[Aµ, Aν] + Bµν□ ˜Bµν + 1
+2[Aµ, Aν](Bµν + □ ˜Bµν)
+�
+,
+(4.3)
+and the equations of motion for Bµν and ˜Bµν are given by
+Bµν = □ ˜Bµν = −1
+2[Aµ, Aν] ,
+(4.4)
+which we can plug into the Lagrangian above to recover LYM as in eq. (4.1). To compute
+amplitudes from this Lagrangian one has to restrict the external legs to be vectors Aµ, since
+the B and ˜B fields are composite (the linearized excitations of B vanish). As discussed
+in the previous section, the cubic-in-Aµ interactions of this Lagrangian correctly reproduce
+the MHV sector of YM, while the double line in the polarization-power two sector (see for
+example the second graph in eq. (2.35)) corresponds to propagation of the Bµν and ˜Bµν
+fields. Since these fields can be integrated out yielding the standard YM Lagrangian, the
+– 13 –
+
+Feynman rules of eq. (4.3) reproduce standard YM tree amplitudes. At four points, the
+BCJ numerator is unique [207] (up to an overall normalization) upon imposing relabeling
+symmetry and reflection symmetry, as given in eq. (2.10), and it is also correctly reproduced
+by the Lagrangian (4.3). In fact, any cubic action for YM theory would reproduce the correct
+four-point numerator, but we find that the above choice of L4 with both the B and ˜B fields
+allows for the necessary freedom to proceed to generating BCJ numerators at higher points.
+4.2
+The five-point NMHV Lagrangian
+Going to five points, we can immediately see that L4 is unable to generate half-ladder
+numerators N(2) proportional to ε1·ε5 ε2·ε4, which is the last diagram pictured in eq. (2.36),
+and corresponds to the double-line field emitting a gluon line. To generate this kind of graph
+we need new interaction terms of the schematic form ∂AB ˜B. In principle, introducing new
+interactions could change the equations of motion, change the four-point amplitude, and
+break gauge-invariance of YM theory. Thus one can expect it to be a delicate procedure.
+Let consider adding new linear-in-Bµν or linear-in- ˜Bµν terms on the right-hand side of
+their corresponding equations of motion. By repeated insertion of the equations of motion
+into themselves, we would generate a set of non-local higher order Aµ interactions (similar
+to refs. [14, 216]). Can we make sure that such new interactions always cancel out in any
+tree-level amplitude computation? Let us start, for the sake of simplicity, by requiring
+something more specific, namely that the deformation introduced by the Lagrangian terms
+∂AB ˜B vanish within the subsystem of equations of motion for the auxiliary fields. This
+can be easily achieved if they take the following form
+Bµν = −1
+2[Aµ, Aν] + α∂ρ
+□ ([Aµ, Bρν] + cyclic(µρν)) ,
+(4.5)
+where α is a free parameter for now.
+When these equations of motion are repeatedly
+substituted back into themselves n times, the sum over cyclic permutations of Lorentz
+indices in the commutator manifestly vanishes by Jacobi identity at any order in A, yielding
+Bµν,(n) = −1
+2[Aµ, Aν] + O(αn) .
+(4.6)
+where the last term is O(αn) ∼ αn(Aµ)nBνρ,(0), and it vanishes in perturbative tree-level
+computations since we take Bνρ,(0) = 0 for external states. The equations of motion in
+eq. (4.5) must arise from introducing the following interactions in the Lagrangian:
+∆L5 = −α Tr ∂ρ ˜Bµν
+�
+[Aµ, Bνρ] + cyclic(µνρ)
+�
+.
+(4.7)
+where L5 = L4 +∆L5 is the total Lagrangian. As argued, the term in parentheses does not
+contribute to on-shell tree-level scattering amplitudes by the Jacobi identity, but it does
+alter the individual N(2) numerators.
+With the new Lagrangian L5 the equations of motion for ˜Bµν now read
+˜Bµν = −1
+2
+1
+□[Aµ, Aν] + α 1
+□
+�
+Aρ , ∂µ ˜Bρν + cyclic(µρν)
+�
+.
+(4.8)
+– 14 –
+
+Note that repeated insertion of the ˜B-field equations of motion into themselves generates an
+infinite set of contact terms of increasing A order. However, from the B-field equations of
+motion is clear that after integrating out the combined pair of auxiliary fields, the standard
+YM tree-amplitudes amplitudes are recovered.
+We now need to probe whether the deformed numerators enjoy color-kinematics duality
+for some choice of α. We can do so by explicitly checking Jacobi relations between the N(2)
+numerators, or checking that the DDM-decomposed partial amplitude (2.24) returns a gauge
+invariant quantity. At five points, we confirm that BCJ numerators are obtained after fixing
+the free parameter in ∆L5 to the value α = 2, giving the duality-satisfying Lagrangian
+L5 = Tr
+�1
+2Aµ□Aµ − ∂µAν[Aµ, Aν] + Bµν□ ˜Bµν + 1
+2[Aµ, Aν](Bµν + □ ˜Bµν)
++ 4∂ν ˜Bµρ[Aµ, Bνρ] − 2∂ρ ˜Bµν[Aρ, Bµν]
+�
+.
+(4.9)
+We thus find the following NMHV contributions to the bi-scalar sector diagrams that
+we introduced in section 2,
+1
+2
+3
+4
+5
+=
+ε3 · ε4(u2 x2
+3 − ε2·x3 x2
+2) ,
+(4.10)
+1
+2
+3
+4
+5
+=
+1
+2ε2 · ε4u3(x2
+2 + x2
+3) .
+(4.11)
+As before, the remaining N(2) contributions to the half-ladder diagrams can be obtained
+using eq. (2.29), or from the Feynman rules of the Lagrangian (4.9). Since we are at five
+points, all cubic diagrams are of the half-ladder type, and thus this completes the five-point
+construction.
+Before proceeding to discuss the six-point case, let us point out a remarkable fact about
+the Lagrangian (4.9). We have checked through multiplicity n = 11 that its Feynman rules
+correctly compute half-ladder BCJ numerators in the bi-scalar sector N
+(2). Inserting those
+into eq. (2.29), gives all half-ladder numerators N(2), and further use of Jacobi relations
+give all non-half-ladder numerators in the NMHV sector, up to the multiplicity we checked.
+Based on this rigorous pattern, we conjecture that the Lagrangian (4.9) correctly computes
+the bi-scalar NMHV numerators N(2) to any multiplicity at tree level, and furthermore
+implicitly provides a good BCJ representation for the remaining diagrams in this sector.
+4.3
+Closed form representation of the bi-scalar sector numerators
+Let us summarize what we have achieved thus far by giving a closed form expression for the
+bi-scalar numerators. As shown in eq. (3.7), the MHV bi-scalar numerators take a simple
+form
+N
+(1)(1, 2, . . . , n − 1, n) = u2u3 . . . un−1 .
+(4.12)
+– 15 –
+
+Next, we gave a schematic diagram representation for the bi-scalar sector NMHV numera-
+tors in eq. (2.37), and with the Lagrangian L5 in eq. (4.9) we can make this more precise.
+Since we are considering numerators at polarization power two, each contribution must
+have exactly one vertex of type AAB and one of type AA□ ˜B in the diagram. Consider a
+half-ladder diagram where the B field is first sourced at position i to the left of the final ˜B
+field at position k. Between these two vertices a chain of AB∂ ˜B-type vertices is inserted,
+giving the below diagram
+Dijkn =
+. . .
+. . .
+. . .
+. . .
+1
+n
+j
+i
+k
+□
+,
+(4.13)
+where j labels a scalar line (which need not be at position k), and the □ indicates the
+location of the inverse propagator x2
+k−1.
+The diagram where ˜B is sourced first can be
+obtained from reflection of this diagram. Since a solid line stretches all the way between i
+and j, the intermediate vertices must be of the form AµBνρ∂µBνρ, which gives a product
+of ul variables. Vertices between legs j and k can either be of the same type, or of the
+type AµBνρ∂ν ˜Bµρ. The vertices before i or after k are of the type AA∂A and can only give
+rise to ul variables in order to not increase the polarization power. Thus, all together, each
+diagram is given by the expression
+Dijkn = (−1)nεi·εjx2
+k−1U2,i−1Ui+1,j−1
+�
+xj−1 · Vj+1 · · · Vk−1 · εk
+�
+Uk+1,n−1 ,
+(4.14)
+where the matrices V µν
+i
+are given by
+V µν
+i
+= uiηµν − 2εµ
+i xν
+i−1 ,
+(4.15)
+and the U’s are products of consecutive ui, defined by
+Ui,j = uiui+1 · · · uj .
+(4.16)
+We set the boundary case j = k of the expression in parenthesis (xj−1 ·Vj · · · Vk−1 ·ϵk) to be
+equal to the number (−1/2) since otherwise the polarization vector εk is double counted.
+Finally, using the diagrams Dijkn the bi-scalar numerator in this sector is
+N
+(2)(1, . . . , n) =
+n−1
+�
+1<i<j≤k
+Dijkn + reflection .
+(4.17)
+The reflected diagram is obtained by reversing the labels {1, 2, . . . , n} on the momenta pi
+and polarizations εi, not the region momenta xi. It is interesting to note that the matrices
+Vi play a similar role to the Gij matrices used for NMHV numerators in ref. [207], except
+that the former depend on region momenta xi and the latter on particle momenta pi.
+– 16 –
+
+4.4
+NMHV Lagrangian beyond the bi-scalar sector
+As detailed in the previous two subsections, we conjectured that the bi-scalar sector numer-
+ators (4.17) generated by the five-point Lagrangian L5 (4.9) give valid BCJ numerators to
+all multiplicity at tree level. From these we can then uniquely compute all NMHV contri-
+butions beyond the bi-scalar sector. We now wish to find a Lagrangian description of such
+contributions, which starts at six points.
+First, let us write down an explicit expression for the six-point bi-scalar numerator
+(summing over polarization power one and two),
+N(1, 2, 3, 4, 5, 6) = U2,5 + (D2336 + D2346 + D2356 + D2446 + D2456 + D2556 + D3446
++ D3456 + D3556 + D4556 + reflections)
+= u2u3u4u5 − 2ε2·ε3
+�
+2ε4·x2ε5·x3x2
+4 − ε5·x2u4x2
+4 − ε4·x2u5x2
+3 + u4u5x2
+2
+�
++ ε2·ε4
+�
+2ε5·x3u3x2
+4 − (x2
+2 + x2
+3)u3u5
+�
+− ε2·ε5
+�
+x2
+2 + x2
+4
+�
+u3u4
++ 2ε3·ε4
+�
+ε5·x3u2x2
+4 + ε2·x3u5x2
+2 − u2u5x2
+3
+�
++ ε3·ε5
+�
+2ε2·x3u4x2
+2 − (x2
+4 + x2
+3)u2u4
+�
+− 2ε4·ε5
+�
+2ε2·x3ε3·x4x2
+2 − ε3·x4u2x2
+3 − ε2·x4u3x2
+2 + u2u3x2
+4
+�
+.
+(4.18)
+Plugging this numerator into the formula (2.29) gives the remaining NMHV numerator
+contributions N(2) at six points, which do not match (diagram-by-diagram) the same con-
+tributions computed from the Lagrangian L5.
+It is clear that the numerators obtained
+from eq. (2.29) are the ones we need since they satisfy color-kinematics duality, thus we
+must modify the Lagrangian L5 with new six-point contributions.
+We proceed by looking at individual kinematic monomials in N(2) that pinpoint the mis-
+match, and then infer what are the simplest interactions that can restore color-kinematics
+duality. For example, considering the term u3ε2·ε4ε5·ε6ε3·(p5 − p6)x2
+2, we find that it is
+necessary to introduce a pair of new vector fields Zµ and ˜Zµ of mass dimension one, which
+interact with Aµ, Bµν and ˜Bµν. To ensure that the new auxiliary vectors do not pollute the
+four and five-point construction, we must constrain certain interactions. Specifically we will
+only allow an interaction of the form AAZ but no conjugate AA ˜Z interaction, thus ensuring
+that at most the Z field can be sourced at four points, and hence it cannot propagate to a
+˜Z. Furthermore, we require that the Z, ˜Z fields can only source the linear combination of
+the tensor field (Bµν − □ ˜Bµν), which is specifically not sourced by Aµ fields at five points,
+hence the Z, ˜Z field cannot propagate at five points.
+Let us see how this example plays out in detail. Adding two interactions of the form
+Aµ ˜ZνBµν and ∂νAµAµZν generates a new diagram contribution of the schematic form
+u3 ε2·ε4 ε5·ε6 ε1·(p5 − p6)x2
+2
+−→
+1
+2
+3
+4
+5
+6
+∂
+□ ˜B B ˜B
+B ˜Z
+Z
+,
+(4.19)
+– 17 –
+
+where we have have placed the previously mentioned offending monomial to the left and
+the corresponding schematic diagram with new vertices A ˜ZB and AAZ to the right. The
+solid line that ends on a derivative indicates the ε1·(p5 − p6) contraction.
+Similarly, flipping the tilde and non-tilde fields yields a correspondence between the
+following offending monomial and new diagram
+u3 ε1·ε4 ε5·ε6 ε2·(p5 − p6)x2
+3
+−→
+1
+2
+3
+4
+5
+6
+∂
+B
+˜BB □ ˜B ˜Z
+Z
+.
+(4.20)
+It is clear that the needed modification to the Lagrangian coming from these terms takes
+the following form:
+∆L6 ∼ Tr
+�
+Zµ□ ˜Zµ + [∂νAµ, Aµ]Zν + [Aµ, ˜Zν]
+�
+Bµν − □ ˜Bµν
+��
+.
+(4.21)
+where we fixed the relative couplings by using the normalization freedom of the kinetic
+term, as well as the rescaling freedom Z → αZ and ˜Z → ˜Z/α that leaves the kinetic term
+invariant. There is only one free parameter, which we can take to the overall normalization,
+and we find that it is equal to unity in order to match the correct N(2) contribution.
+There are further mismatching monomials that we need to deal with. The N(2) numer-
+ators contain problematic terms such as ε2·ε3 ε5·ε6 ε1·x2 ε4·(p6 − p5) x2
+2, where no Lorentz
+indices are crossing the central half-ladder propagator, and also the mass dimensions are
+unbalanced. This suggests that we need to introduce a pair of scalar fields X and ˜X of
+mass dimension two and zero respectively. Again we need to ensure that the new fields
+do not modify the five-point numerators. Consider the new interactions AA ˜X and AX ˜Z
+which contribute to the mentioned monomial through the diagram
+ε2·ε3ε5·ε6ε1·x2ε4·(p5 − p6) x2
+2
+−→
+1
+2
+3
+4
+5
+6
+∂A □A ˜X X ˜Z
+Z
+∂
+∂
+.
+(4.22)
+As mentioned, there are more derivatives on the left half of the diagram than the right half,
+thus the imbalance of the dimensions of the X and ˜X fields.
+Additionally, we add A ˜X ˜Z interaction to address the offending term
+ε1·ε2ε5·ε6ε3·(p1 − p2)ε4·(p6 − p5)x2
+2
+−→
+1
+2
+3
+4
+5
+6
+Z
+□ ˜ZX
+˜X ˜Z
+Z
+←
+∂
+∂
+.
+(4.23)
+We also need ˜X to interact with the B field, to address offending terms of the form
+ε2·ε3 ε5·ε6 ε1·x3 ε4·(p5 − p6) x2
+2
+−→
+1
+2
+3
+4
+5
+6
+□ ˜B B∂X
+˜X ˜Z
+Z
+←
+∂
+∂
+.
+(4.24)
+– 18 –
+
+The pair of scalar fields enjoys the same rescaling freedom as the auxiliary vectors, which
+we use to fix the coefficient of the AA ˜X interaction. The above three monmial structures
+are then enough to constrain all remaining coefficients of the needed interactions, which we
+find to be
+∆L6 ∼ Tr
+�
+X□ ˜X + [Aµ, □Aµ] ˜X − [Aµ, X] ˜Zµ + 1
+2[Aµ, □ ˜X] ˜Zµ − 2[Aµ, Bµν]∂ν ˜X
+�
+. (4.25)
+This completes the construction of interactions that contribute to the six-point half-ladder
+numerator.
+As a side remark that we will come back to later, note that one can rearrange the
+flow of Lorentz indices in the half-ladder diagrams using conservation of momentum, and it
+turns out that it is not strictly necessary to introduce a pair of scalar fields. Indeed, in the
+next subsection we show that it is possible to formulate a completion of the L5 Lagrangian
+in the NMHV sector using only the pair of vectors Z and ˜Z.
+Finally, we need to address those diagrams that contribute to the non-half-ladder topol-
+ogy5 appearing at six points. Our Lagrangian does not yet get those contributions to match
+with what is predicted from the N(2) numerator. Assuming that no new fields are needed,
+the possible interactions are highly constrained by dimensional analysis and by the require-
+ment that they must involve three auxiliary fields in order to not spoil the half-ladder
+numerators. Consider the following example of a missing term and corresponding diagram
+with a new B ˜Z ˜Z interaction:
+s12ε3·ε4ε5·ε6ε1·(p5 − p6)ε2·(p3 − p4)
+−→
+1
+2
+3
+4
+5
+6
+□ ˜B
+B ˜Z ˜Z
+Z
+∂
+∂
+.
+(4.26)
+We find that a suitable interaction has the form
+Tr
+�
+[ ˜Zµ, ˜Zν]
+�
+βBµν + (1 − β)□ ˜Bµν��
+,
+(4.27)
+but at this multiplicity we can not yet fix the free parameter β. There is one more needed
+interaction which is rather simple, and fully constrained at six points, it is
+4Tr
+�
+[Bµν, ∂µ ˜Bνρ] ˜Zρ�
+.
+(4.28)
+This finishes the construction of the duality-satisfying six-point NMHV Lagrangian
+L6 = L5 + ∆L6, where all the new terms are assembled as
+∆L6 = Tr
+�
+Zµ□ ˜Zµ + X□ ˜X + [∂νAµ, Aµ]Zν + [Aµ, □Aµ] ˜X + [Aµ, ˜Zν]
+�
+Bµν − □ ˜Bµν�
+− [Aµ, X] ˜Zµ + 1
+2[Aµ, □ ˜X] ˜Zµ − 2[Aµ, Bµν]∂ν ˜X + [ ˜Zµ, ˜Zν]
+�
+βBµν + (1 − β)□ ˜Bµν�
++ 4[Bµν, ∂µ ˜Bνρ] ˜Zρ�
+,
+(4.29)
+5Sometimes called star or Mercedes topology.
+– 19 –
+
+and, as already mentioned, the β parameter is not yet fixed.
+Moving on to seven points, interactions of the form ∂AZ ˜Z and ∂AX ˜X can now partake
+in the half-ladder diagrams. These rather simple contributions are needed for generating
+correct half-ladder factors ui, which in hindsight is not surprising. Their coefficients in
+the Lagrangian can be fixed by computing the terms ε1·ε2ε6·ε7ε3·(p1 − p2)ε4·(p6 − p7)u5x2
+2
+and ε1·ε2ε6·ε7ε3·(p1 − p2)ε5·(p6 − p7)u4x2
+2, and comparing the result to the predicted N(2)
+numerator. Furthermore, a new interaction B ˜X ˜Z is required for the non-half-ladder graphs,
+and finally the unknown parameter β from ∆L6 is now constrained to β = 1/2.
+Thus we conclude that the seven-point corrections to the duality-satisfying NMHV
+Lagrangian consists of the three terms
+∆L7 = Tr
+�
+− 2[Aµ, Zν]∂µ ˜Zν − 2[Aµ, X]∂µ ˜X − 2[Bµν, ∂ν ˜X] ˜Zµ
+�
+.
+(4.30)
+No further corrections are needed at higher multiplicity.
+We explicitly computed and
+checked the properties of color-kinematics duality and gauge invariance for all NMHV nu-
+merators and amplitudes through ten points, which worked flawlessly.
+Based on the robust patterns observed up to multiplicity ten, we conjecture that the fol-
+lowing assembled NMHV Lagrangian computes all BCJ numerators and NMHV amplitudes
+to any multiplicity at tree level:
+L = L5 + Tr
+�
+Zµ□ ˜Zµ + X□ ˜X + [∂νAµ, Aµ]Zν + [Aµ, □Aµ] ˜X − 2[Aµ, Zν]∂µ ˜Zν
+− 2[Aµ, X]∂µ ˜X + [Aµ, ˜Zν]
+�
+Bµν − □ ˜Bµν�
+− [Aµ, X] ˜Zµ + 1
+2[Aµ, □ ˜X] ˜Zµ
+− 2[Aµ, Bµν]∂ν ˜X + 1
+2[ ˜Zµ, ˜Zν]
+�
+Bµν + □ ˜Bµν�
++ 4[Bµν, ∂µ ˜Bνρ] ˜Zρ
+− 2[Bµν, ∂ν ˜X] ˜Zµ
+�
+.
+(4.31)
+This is the simplest Lagrangian that we have found in this paper. A natural question to
+ask next is: how unique is it?
+4.5
+How unique is the NMHV Lagrangian?
+The way in which we obtained the NMHV Lagrangian (4.31) does not give strong evidence
+for its uniqueness. To check how unique it really is, we constructed a more general ansatz
+that is vastly larger in complexity compared to the above construction. Again we con-
+strained the results by checking color-kinematics duality and gauge invariance, but we were
+more meticulous in keeping track of our assumptions. This is useful for future work were
+these assumptions can be further relaxed.
+Our enlarged ansatz for the NMHV Lagrangian is subject to the following assumptions:
+1. The simple Lagrangian L5 (4.9) is still assumed to give the bi-scalar numerators.
+2. No additional fields beyond those in the previous sections are used. That is, only the
+tensors B, ˜B, vectors Z, ˜Z and scalars X, ˜X appear as auxiliary fields.
+3. Kinetic terms do not mix fields: L2 = Tr
+�
+1
+2Aµ□Aµ + Bµν□ ˜Bµν + Zµ□ ˜Zµ + X□ ˜X
+�
+.
+– 20 –
+
+4. To preserve the four-point numerator, a pair of external A’s can either source Z or
+˜Z. As before, we choose to exclude AA ˜Z interactions.
+5. Two-derivative interactions always appear as a d’Alembertian □. This makes it man-
+ifest that the interaction contributes at most to polarization-power two.
+6. We make an ad hoc simplifying choice to exclude interactions AZZ and A ˜Z ˜Z, in order
+the make the ansatz space more manageable.
+After taking care of the rescaling freedom of the auxiliary fields, the Lagrangian ansatz we
+obtain with the above constraints has 174 free parameters. We constrain the numerators
+generated by the ansatz up to eight points by comparison with the predicted numerators
+N(2), as obtained from linear combinations of the bi-scalar numerators coming from L5.
+Note these constrains are non-linear equations in the free parameters of the Lagrangian
+ansatz, hence the equation system is non-trivial to deal with.
+With the constraints imposed up to eight points, this only fixes a subset of all param-
+eters and we are left with 129 free coefficients. At nine points, we find that all numerator
+topologies generated by the Lagrangian ansatz are independent of the leftover coefficients.
+It is difficult to go to higher points due to the non-linearities, and we stop trying to find
+further constraints. Instead, we now seek solutions that minimizes the number of terms
+in the Lagrangian. We find four solutions which all give 13-term Lagrangians. Provided
+that ∂ · A = 0, all solutions coincide (up to total derivatives) to give the same Lagrangian
+(4.31) we found in the previous section. Thus it seems reasonable to think that this is the
+simplest NMHV Lagrangian, given the above list of assumptions.
+For example, another simple solution gives a 14-term Lagrangian. It is reachable by
+deforming the 13 term Lagrangian L (4.31) by the following interactions:
+L − L14-term = Tr
+�
+[Aµ, □Aµ] ˜X + 4[Aµ, ∂νBµν] ˜X + [Aµ, ˜X]□ ˜Zµ
+�
+.
+(4.32)
+A more interesting scenario would arise if we managed to find a solution that makes use
+of fewer auxiliary fields. We find no solutions that discards of the vectors Z and ˜Z, but
+interestingly, the scalars X and ˜X do seem to not be strictly necessary for all solutions.
+Consider replacing the above ad hoc assumption 6, by the following new assumption:
+6’. Exclude the scalar fields X, ˜X, but now include all interactions AZZ and A ˜Z ˜Z.
+We start with a 95 parameter ansatz and constrain it by comparing to the predicted N(2)
+numerators up to eight points.
+At nine points, there is a single non-half-ladder graph
+depending on one free parameter. To fix this parameter, we use the unique half-ladder
+graphs generated by our ansatz and through Jacobi identities construct the graph containing
+the free parameter. At this stage, we have 50 free coefficients. Assuming they do not enter
+the numerators at higher multiplicity, we look for the smallest possible Lagrangian and
+find six solutions with 20 terms each. To select one of the solutions, we will assume that
+– 21 –
+
+∂ · A = 0. The solution is not particularly simple but we report it below for completeness:
+L′ = L5 + Tr
+�
+Zµ□ ˜Zµ − [Aµ, ∂νAµ]Zν + [Aµ, Zν]∂νZµ − 3
+4[Aµ, Zµ]∂ · Z − [Aµ, ˜Zµ]∂ · ˜Z
+− 2[Aµ, Zν]∂µ ˜Zν + 1
+2[Aµ, ∂ · Z] ˜Zµ − 2[Aµ, ∂νZµ] ˜Zν − 3
+2[Aµ, Zµ]∂ · ˜Z
+− 3
+2[Aµ, Bµν]Zν − [Aµ, Bµν] ˜Zν − 1
+2[Aµ, □ ˜Bµν]Zν + [Aµ, □ ˜Bµν] ˜Zν
++ 9
+8[Bµν, Zµ]Zν + 3
+2[Bµν, Zµ] ˜Zν + 1
+2[Bµν, ˜Zµ] ˜Zν + 1
+8[□ ˜Bµν, Zµ]Zν
+− 1
+2[□ ˜Bµν, Zµ] ˜Zν + 1
+2[□ ˜Bµν, ˜Zµ] ˜Zν − 2[Bµν, ∂µ ˜Bνρ]Zρ + 4[Bµν, ∂µ ˜Bνρ] ˜Zρ�
+.
+(4.33)
+It would be interesting to more broadly explore the ansatz space of NMHV Lagrangians,
+with the above six assumptions further relaxed, but we leave it for future work.
+4.6
+Comments on the N2MHV sector
+The complete six-point BCJ numerator must contain also terms of polarization power three,
+which are needed for the N2MHV amplitudes. Unfortunately, the Lagrangian ansatz space
+we have considered does not permit solutions which correctly reproduces this sector. This
+is not very surprising. Since, as a minimal extension of our ansätze, we would need to
+introduce a three-form auxiliary field (or even rank-three tensors of mixed symmetries).
+This is clear from considering a six point BCJ numerator that necessarily [207] contains
+terms of the form
+1
+2
+3
+4
+5
+6
+∼
+ε1·ε6 ε2·ε5 ε3·ε4 x2
+i pj·pk ,
+(4.34)
+which in D-dimensions cannot be removed by generalized gauge transformations that main-
+tain locality. The generalized gauge freedom of local crossing-symmetric D-dimensional
+six-point numerators were analysed in ref. [207]. From the triple-solid line in the middle
+of the above diagram, it is clear that a rank-three auxiliary field is propagating. There
+could in principle also exist higher-rank fields since momentum dot products pj·pk may be
+contracted across this central propagator. Also, the type of tensors needed is unclear, it
+could be forms or tensors of mixed symmetries. We leave the problem of reproducing the
+N2MHV and higher sectors to future work.
+5
+Towards MHV numerators at one loop
+Having obtained Lagrangians that generate BCJ numerators it is interesting to check if
+they can also generate BCJ numerators at loop level. Loop color-kinematics duality does
+not necessarily follow from tree level, since loops also contain off-shell internal states that
+are absent at tree level by the equations of motion.
+– 22 –
+
+5.1
+Polarization power zero at one loop
+To obtain the one-loop polarization-power zero numerators (valid for all-plus and one-minus
+helicity YM sectors), we glue legs 1 and n of the tree-level half-ladder numerators, with
+appropriate contributions from Faddeev–Popov ghosts c, ¯c to remove unphysical degrees of
+freedom,
+N(0)
+1-loop(1, 2, . . . , n) = N(1)(ℓ, 1, 2, . . . , n, −ℓ)+N(c, 1, 2, . . . , n, ¯c)+N(¯c, 1, 2, . . . , n, c) . (5.1)
+It should be understood that the states labeled by the loop momentum ℓ and −ℓ are
+contracted, and the two ghost numerators come with a minus sign after the sewing of the
+states since the c and ¯c fields are fermionic. As we have shown in section 3, the Feynman
+rules for the bi-scalar sector automatically obey the color-kinematics duality so long as the
+external vector fields are transverse. This assumption holds for external states as well as
+subdiagrams that are of polarization power zero, just as argued below eq. (3.5).
+All we have to do is add the ghosts in such a way that they respect the color-kinematics
+duality, and this is achieved by the standard Faddeev-Popov ghost Lagrangian
+L(0)
+ghost = Tr
+�
+c□˜c + Aµc∂µ¯c
+�
+.
+(5.2)
+The interaction term here is identical to the effective interactions of the bi-scalar sector so
+color-kinematics duality works out automatically when external states are transverse. Since
+the diagrams that contribute here come from standard YM Feynman rules, the auxiliary
+fields are not yet present.
+Let us give the duality-satisfying n-gon master numerators that are valid for the all-plus
+and one-minus helicity one-loop amplitudes (see also refs. [68, 91]),
+N(0)
+1-loop(1, . . . , n) = Tr(W1 · · · Wn) + Tr(�
+W1 · · · �
+Wn) − (2 + D)U1,n ,
+(5.3)
+where D = Tr(1) is the dimension, U1,n is defined in eq. (4.16) and the matrices are
+(Wi)µν = uiηµν + 2εµ
+i pν
+i ,
+(�
+Wi)µν = uiηµν − 2pµ
+i εν
+i .
+(5.4)
+The region momenta present in ui = 2εi·xi is here defined to include the loop momentum
+xi = ℓ + �i
+j=1 pj. The above numerators can be checked to give the correct maximal [91]
+and non-maximal cuts, since by construction they make use of the standard YM three-point
+vertex and there are no contact terms present in the amplitudes.
+5.2
+Comment on one-loop MHV numerators
+As already emphasized, tree-level all-multiplicity color-kinematics duality is in general in-
+sufficient to infer that the loop-level duality holds. Indeed, we find obstructions in realizing
+one-loop BCJ numerators in the MHV sector from the Feynman rules derived in previous
+sections. At polarization-power one, in particular, the numerators receive contributions
+from the new fields we added to the Lagrangian. As before, we have contributions from the
+tree-level polarization-power one sector after gluing half-ladder numerators,
+N(1)
+1-loop(1, 2, . . . , n) ⊃ N(1)(ℓ, 1, 2, . . . , n, −ℓ) ,
+(5.5)
+– 23 –
+
+but now we must add the glued half-ladder numerator at polarization power two as well,
+N(1)
+1-loop(1, 2, . . . , n) ⊃ N(2)(ℓ, 1, 2, . . . , n, −ℓ) .
+(5.6)
+In addition, to not break crossing symmetry we should allow for the B, Z and X fields to
+cross the sewn loop line. This makes it possible for the auxiliary fields to propagate all the
+way around the loop, which is not expected to give reasonable contributions. Thus, we may
+attempt to project out certain contributions by adding new ghosts for the auxiliary fields.
+The contributions we need to remove are very similar to the bi-scalar sector contributions
+(as already noted in the previous section), hence we add new ghost Lagrangian terms
+corresponding to the B, ˜B fields,
+L(1)
+ghost = Tr
+�
+c□˜c + bµν□˜bµν + Aµc∂µ¯c + 4∂ν˜bµρ[Aµ, bνρ] − 2∂ρ˜bµν[Aρ, bµν]
+�
+.
+(5.7)
+As usual, the ghost fields cannot be sourced and so they only contribute through propagating
+in a complete closed loop.
+We performed a few crude tests to the one-loop numerators as obtained from our
+Lagrangians, and found that for the 13-term Lagrangian in eq. (4.31) already the three-
+point one-loop numerators are not well behaved, meaning they do not give gauge-invariant
+unitarity cuts. For the 14-term Lagrangian obtained by the deformation (4.32), the maximal
+cuts work as tested up to four points, but the next-to maximal cuts do not. For example,
+the following contribution to the box diagram appears to not be correct:
+1
+2
+3
+4
+↑
+□ ˜B
+B
+X
+˜X
+˜Z
+Z
+A
+A
+∼
+(ε3 · x3)x2
+1ε[µ
+4 εν]
+1 (2p2µ + x2µ)ε2ν .
+(5.8)
+Specifically, the term that spoils gauge invariance is proportional to ε1 · ε2. It is clear that
+the ghost Lagrangian (5.7) does not remove this diagram, nor is it removed by introducing
+further obvious vector-ghost interactions, inspired by the fields already present. While it
+should be possible to introduce fine-tuned ghosts and interactions that precisely cancels this
+diagram, it is non-trivial task to ensure the new terms are consistent with color-kinematics
+duality and gauge invariance for all higher-multiplicity MHV numerators. We leave the
+problem of formulating duality-satisfying one-loop-compatible Lagrangians to future work.
+6
+Conclusions
+In this paper, we considered the problem of constructing Lagrangians that manifest color-
+kinematics duality for YM theory.
+Such explicit Lagrangians can be used to compute
+BCJ numerators, as well as give non-trivial clues to the mathematical structure underlying
+color-kinematics duality. While duality-satisfying tree-level numerators are known to any
+multiplicity, finding corresponding Lagrangian descriptions appears to be more challenging.
+– 24 –
+
+The problem simplifies by restricting to helicity sectors of YM, and in this paper we fully
+address the NMHV sector.
+As a first step, we found a simple Lagrangian (4.9) that is fully equivalent to the
+standard YM Lagrangian at tree level, and it computes BCJ numerators in the bi-scalar
+subsector of the NMHV sector. This Lagrangian was constructed by first resolving the four-
+gluon contact term using a pair of auxiliary two-forms fields, and subsequently deforming
+with new cubic interactions involving these auxiliary fields. The bi-scalar numerators can
+be computed to any multiplicity, and they provide complete information via eq. (2.29) to
+obtain all tree-level NMHV numerators. These provide a clear target for what a complete
+NMHV Lagrangian should reproduce.
+Next we searched for a complete NMHV Lagrangian, and we found that by introducing
+at most two additional pairs of auxiliary fields, of vector and scalar type, there are several
+solutions for such Lagrangians. Using a larger ansatz, we found that solutions also exists
+if the scalar is removed at the cost of additional interactions between the remaining fields.
+Because of the large freedoms of the Lagrangians, we chose to present the simplest solutions,
+given in eqs. (4.31) and (4.33), which were explicitly tested through ten points and conjec-
+tured to work to all multiplicities at tree level. With the our current limited Lagrangian
+ansatz space, we cannot obtain the N2MHV and higher sector contributions to the BCJ
+numerators. Nor are the presented duality-satisfying NMHV Lagrangians equivalent to YM
+in the N2MHV sector and beyond, unlike the simple Lagrangian (4.9) first found. It would
+be desirable to revisit this problem in the future, and repeat the ansatz construction of
+duality-satisfying NMHV Lagrangian while maintaining gauge covariance at intermediate
+steps such that the N2MHV and higher sectors are not spoiled, even if they might not fully
+enjoy color-kinematics duality.
+We briefly discussed the need of higher-rank tensor auxiliary fields to reproduce N2MHV
+sector BCJ numerators. It is clear that in a covariant (D-dimensional) and local formalism
+it is unavoidable to encounter, at the very minimum, three-form fields in a duality satisfying
+N2MHV Lagrangian. However, the minimal set of needed auxiliary fields in the N2MHV
+sector is something that needs further studies. Currently the main challenge in brute-force
+constructions of duality-satisfying Lagrangians is to predict the number and types of needed
+auxiliary fields. Clearly it would be desirable to better understand the general structure and
+need of such fields such that more refined Lagrangian ansätze can be constructed. Small
+finely tuned ansätze would speed up progress because the equation systems encountered are
+non-linear in the ansatz parameters, and the solutions contains large redundancy making
+them difficult to analyse.
+Even in the NMHV sector there are more questions to be answered. Can one remove
+some of the assumptions that went into our constructions and perhaps obtain much simpler
+Lagrangians? For example, hints from Chern-Simons-type Lagrangians [40, 222] suggest
+that it should be beneficial to look for more intricate kinetic terms than the diagonal ones
+used in this paper. Furthermore, gauge covariance plays no role in the current construction
+and this is likely an oversight that should be addressed in more refined attempts. While
+our NMHV Lagrangians fit on a few lines, it is fair to say that that their complexity are
+likely artificially high compared to more optimal duality-satisfying rewritings of the YM
+– 25 –
+
+Lagrangian that might be found in the future. Nevertheless, we have taken critical steps
+in this research program by finding the first examples of NMHV Lagrangians that: 1) uses
+very few auxiliary fields 2), have very simple structure in the bi-scalar subsector, and 3)
+gives local all-multiplicity BCJ numerators.
+Acknowledgments
+We thank Zvi Bern, Lucile Cangemi, Gang Chen, Paolo Pichini, Oliver Schlotterer, Fei Teng,
+Tianheng Wang and Maxim Zabzine for enlightening discussions related to this work. This
+research was supported in part by the Knut and Alice Wallenberg Foundation under grants
+KAW 2018.0116 (From Scattering Amplitudes to Gravitational Waves) and KAW 2018.0162
+(Exploring a Web of Gravitational Theories through Gauge-Theory Methods), as well as the
+Ragnar Söderberg Foundation (Swedish Foundations’ Starting Grant).
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diff --git a/s9AyT4oBgHgl3EQfZ_c4/content/tmp_files/load_file.txt b/s9AyT4oBgHgl3EQfZ_c4/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..667da4fcb0c888b1be296dc1433f9ea701d7dcbd
--- /dev/null
+++ b/s9AyT4oBgHgl3EQfZ_c4/content/tmp_files/load_file.txt
@@ -0,0 +1,2457 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf,len=2456
+page_content='Prepared for submission to JHEP  UUITP-62/22  NORDITA 2022-155  Lagrangians Manifesting Color-Kinematics Duality  in the NMHV Sector of Yang-Mills  Maor Ben-Shahar,a Lucia Garozzoa and Henrik Johanssona,b  aDepartment of Physics and Astronomy, Uppsala University,  Box 516, 75120 Uppsala, Sweden  bNordita, Stockholm University and KTH Royal Institute of Technology,  Hannes Alfvéns väg 12, 10691 Stockholm, Sweden  E-mail: benshahar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='maor@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='se, garozzo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='lucia@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='se,  henrik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='johansson@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='se  Abstract: Scattering amplitudes in Yang-Mills theory are known to exhibit kinematic  structures which hint to an underlying kinematic algebra that is dual to the gauge group  color algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' This color-kinematics duality is still poorly understood in terms of conven-  tional Feynman rules, or from a Lagrangian formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' In this work, we present explicit  Lagrangians whose Feynman rules generate duality-satisfying tree-level BCJ numerators,  to any multiplicity in the next-to-MHV sector of pure Yang Mills theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Our Lagrangians  make use of at most three pairs of auxiliary fields (2,1,0-forms) – surprisingly few compared  to previous attempts of Lagrangians at low multiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' To restrict the Lagrangian free-  dom it is necessary to make several non-trivial assumptions regarding field content, kinetic  terms, and interactions, which we discuss in some detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Future progress likely hinges on  relaxing these assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='00233v1  [hep-th]  31 Dec 2022  Contents  1  Introduction  1  2  Graphs, Numerators and Tree Amplitudes  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='1  Color-kinematics duality and double copy  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='2  Decompositions for amplitudes and numerators  7  3  MHV Lagrangian  11  4  NMHV Lagrangian  13  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='1  The four-point NMHV Lagrangian  13  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='2  The five-point NMHV Lagrangian  14  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='3  Closed form representation of the bi-scalar sector numerators  15  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='4  NMHV Lagrangian beyond the bi-scalar sector  17  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='5  How unique is the NMHV Lagrangian?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  20  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='6  Comments on the N2MHV sector  22  5  Towards MHV numerators at one loop  22  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='1  Polarization power zero at one loop  23  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='2  Comment on one-loop MHV numerators  23  6  Conclusions  24  1  Introduction  Scattering amplitudes offer valuable insight to the mathematical structures of quantum field  theory and gravity by uncovering patterns that are not readily apparent through standard  Lagrangian techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The Bern-Carrasco-Johansson (BCJ) duality between color and  kinematics [1–3] is a clear example where on-shell formulations preceded any Lagrangian  understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' According to the duality, scattering amplitudes in many gauge theories can  be represented using cubic diagrams, each consisting of a kinematic numerator and a color  factor that obey isomorphic relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The color factors draw their characteristics from the  Lie algebra of the gauge group, whereas the mathematical structure of the numerators is  believed to originate from an unknown kinematic Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' See recent reviews [3–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  In massless purely adjoint gauge theories, such as pure Yang-Mills (YM) theory, color-  kinematics duality can be rephrased at tree level as the presence of BCJ amplitude rela-  tions [1, 10–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The duality and amplitude relations were initially identified in pure YM [1]  and its supersymmetric generalizations [2, 10, 11, 14], but have since been discovered in a  – 1 –  range of other gauge theories, including matter representations [15–23], higher-derivative in-  teractions [24–35] or Chern-Simons fields [36–40], and massive gauge theories [18, 19, 21, 41–  55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Surprisingly, also certain scalar effective field theories [56–65] obey the duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The  duality has been extended to loop-level amplitudes [2, 66–91], form factors [92–101], and  even to curved-space correlators [102–120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  There have been various approaches to un-  derstanding the existence of color-kinematics duality and BCJ relations in massless gauge  theories, including string theory, scattering equations, positive geometry [10, 11, 121–126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Color-kinematics duality is intimately connected to the double-copy structure of grav-  ity [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' For non-abelian gauge theories that obey color-kinematics duality, and contain  physical spin-1 gluons, the double copy gives gravitational interactions after taking each  cubic diagram and replacing the color factor by a second numerator copy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The double copy  provides a versatile generalization of the Kawai-Lewellen-Tye (KLT) relations [127], which  were originally derived for string theory at genus zero (for higher-genus generalizations  see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' [82, 84, 128–134]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Through color-kinematics duality the double copy permits the  construction of gravity loop amplitudes [2, 75, 135–142], more general gravity theories [15–  27, 143, 144], classical solutions [145–168] and other non-perturbative solutions [169, 170].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Recently black-hole scattering and gravitational wave physics have been studied using the  double copy [86, 171–191].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Despite extensive progress, the mathematical details of color-kinematics duality remain  enigmatic even for the case of pure YM theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' While duality-satisfying YM numerators can  be straightforwardly computed to any multiplicity at tree level [13, 192–203], the building  blocks and underlying symmetry principles of these objects are poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Realiz-  ing the hidden YM structure as a kinematic Lie algebra or a cubic Lagrangian may shine  light on the duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Explicit kinematic-algebra constructions have been realized for the self-  dual YM sector [68, 204, 205], the maximally-helicity-violating (MHV) sector [57, 204, 206]  and the next-to-MHV (NMHV) sector [206, 207].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Self-dual YM amplitudes vanish except  for at one loop [208] and the kinematic algebra is identified with area-preserving diffeomor-  phisms [204].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The MHV sector of YM gives the simplest non-vanishing 4D tree amplitudes,  and while ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' [204] considered it through a non-local gauge choice, local formulations of  the MHV kinematic algebra was discussed in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' [57, 206].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  A cubic duality-satisfying Lagrangian for the non-linear-sigma model was considered  in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' [57], but via dimensional-reduction operations [209, 210] it is directly related to  the MVH sector of YM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (It is also related to a non-abelian generalization of the Navier-  Stokes equation that obeys color-kinematics duality [211–213].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=') Both the MHV and NMHV  kinematic algebra was described in some detail in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' [206, 207].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' A natural extension of  the area-preserving diffeomorphism algebra remains to be found in those sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' However,  in 3D Chern-Simons theory the complete kinematic algebra is now identified with volume-  preserving diffeomorphisms [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' In section 3, we briefly comment on the appearance of 4D  volume-preserving diffeomorphism generators in the MHV sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Recently, kinematic Hopf  algebras have been introduced for YM and related theories, which gives gauge-invariant BCJ  numerators for all tree-level sectors at the expense of additional heavy-mass poles [100, 200–  203, 214, 215].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Duality-satisfying YM Lagrangians have been sporadically studied for more than a  – 2 –  decade [14, 216–222] using non-local interaction terms or auxiliary fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The early at-  tempts produced Feynman rules capable of computing local BCJ numerators up to five [14]  and six points [216].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' However, the non-uniqueness of the BCJ numerators led to a prolif-  eration of ambiguities in the resulting Lagrangians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' In hindsight, it is clear that in a local  formalism the ambiguities are absent in the MHV sector and first appear in the NMHV  sector [206].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' They can be traced back to the so-called generalized gauge freedom of the  BCJ numerators [1, 2], which include the standard gauge freedom and non-linear field re-  definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Understanding how to constrain this freedom and write down broadly valid and  practical YM Lagrangians that manifest the duality is the topic of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Recently there has been an upswing in formulations that attempts to approach the  problem of realizing YM color-kinematics duality and double copy off shell [40, 169, 200,  218, 219, 222–234].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' This involves understanding the role of BRST symmetries, homotopy  algebras, double field theory, equations of motions and other off-shell Langrangian perspec-  tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' We note that, in principle, the problem has been solved using the 10D pure-spinor  formalism [222];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' however, the practical usefulness of this duality-satisfying supersymmetric  YM Lagrangian for advanced calculations remains to be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  In this paper, we approach the problem of finding duality-satisfying YM Lagrangians  head on, by making a suitable small ansatz for the NMHV sector Lagrangian where the first  ambiguities show up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' We work in general dimension using covariant building blocks, and  thus we generalize the 4D notion of helicity sectors by grading the tree-level numerators by  the “polarization power” [206, 207], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' the number of inner products between polarization  vectors εi·εj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Thus the NMHV sector corresponds to numerators with at most two polar-  ization powers (εi·εj)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' or, equivalently, one power of momentum inner products pi·pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The  construction of the ansatz is aided by the observation that there exist a bi-scalar subsector1  of the NMHV sector that has particularly simple BCJ numerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' This bi-scalar subsector  was also considered in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' [206] since it generates master numerators from which all other  NMHV numerators can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' We find that the bi-scalar numerators can be com-  puted using an exact rewriting of the standard YM tree-level Lagrangian, by “integrating  in” a pair of auxiliary 2-form fields {Bµν, ˜Bµν} with additional simple cubic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  From the bi-scalar numerators one can obtain the remaining contributions to the  NMHV numerators using a simple formula;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' however, we also demand that those contri-  butions come directly from a Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' This required us to either use clever inspection  and identification of needed new interactions terms aided by pictorial diagrams that ex-  pose the tensors structures of intermediate fields, or alternatively, a brute-force Lagrangian  ansatz where the assumptions are more clearly spelled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' We find several interesting  solutions which involve a pair of auxiliary vector fields {Zµ, ˜Zµ}, and a pair of auxiliary  scalar fields {X, ˜X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' We show that by considering an extended ansatz the need for scalar  auxiliary fields is not obvious, as there exists interesting Lagrangian solutions where they  are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Further relaxing the assumptions that went into the construction may provide  simpler NMHV Lagrangians, but we leave this for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' We briefely comment on  1Terms in the half-ladder BCJ numerator proportional to a fixed polarization product ε1·εn, which  effectively becomes two scalars interacting with the YM field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  – 3 –  the N2MHV sector, and also spell out some details about the possible generalization of the  Lagrangian Feynman rules to one-loop calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  The paper is organized as follows: In section 2, we introduce the necessary notation used  for describing amplitudes and numerators that satisfy color-kinematics duality, including  decomposition into partial amplitudes, bi-scalar half-ladder numerators and useful pictorial  diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' In section 3, we consider a cubic Lagrangian for computing MHV numerators,  obtained by truncating the standard YM action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' In section 4, we transform the standard  YM Lagrangian by integrating in a pair of two-form auxiliary fields and then we add  interactions and further auxiliary fields using both diagrammatic and ansatz approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  In section 5, we briefely discuss consequences for one-loop numerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Conclusions and  outlook are given in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  2  Graphs, Numerators and Tree Amplitudes  Here we will set the notation used for the amplitude and numerator building blocks, and  discuss the decompositions and diagrammatic notation that are convenient for later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='1  Color-kinematics duality and double copy  Scattering amplitudes in YM theory can be represented diagrammatically as a sum over  cubic Feynman-like graphs [1, 2], for n-point tree level amplitudes this takes the form  An = gn−2 �  Γ∈Gn  CΓNΓ  DΓ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='1)  Each cubic graph Γ is associated with a color factor CΓ, kinematic numerator NΓ and  propagator denominator DΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Here the set of cubic n-point graphs is denoted by Gn, and it  can be constructed recursively starting from the three-point case, G3, which contains only  one such graph Γ = [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' At n points the set of cubic graphs is given by  Gn =  �  Γ|ℓ→[ℓ, n−1]  ��� ℓ ∈ Γ ∈ Gn−1  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='2)  where ℓ denotes a Lie-valued element of the graph Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' For example, the graph Γ = [1, 2] has  three such elements, the external leg labels ℓ = 1, ℓ = 2 and the commutator of the labels  ℓ = [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Applying the rule ℓ → [ℓ, 3] then gives three four-point graphs represented by  nested commutators [207, 235, 236]  G4 =  �  [[1, 3], 2], [1, [2, 3]], [[1, 2], 3]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='3)  Each one of the four-point graphs have five Lie-valued elements (three labels, and two  commutators), thus the total number of five-point graphs is |G5| = 3 × 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' In general, at n  points, the number of cubic graphs is |Gn| = 3 × 5 × 7 × · · · × (2n − 5) = (2n − 5)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='.  With the above graph notation the propagator denominator can be written as  DΓ =  �  ℓ∈Γ  ℓ̸=N,ℓ̸=Γ  p2  ℓ ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='4)  – 4 –  where pℓ is the sum of the momentum of all the external legs in the nested commutator ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  For example, p[1,2] = p1 + p2 and p[[1,3],2] = p1 + p2 + p3, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Since the external legs are on  shell p2  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , p2  n = 0, we do not include cases where ℓ is an integer, nor where it is the graph  Γ, since by momentum conservation pΓ = −pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  The color factors for purely-adjoint YM theory correspond to rank-n tensors in the  gauge-group Lie algebra, which can be constructed through contraction of structure con-  stants, fabc, or as traces of products of generators T a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The latter case is most transparent  in our graph notation, and an explicit formula can be given  CΓ = Ca1a2···an  Γ  = Tr  �  (Γ|i→T ai) T an�  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='5)  where the external legs i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' n−1 of the graph Γ are replaced by Lie-algebra generators,  thus justifying the graph notation as nested commutators of Lie-valued elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  For  example, the three-point graph Γ = [1, 2] has the color factor  C[1,2] = Tr  �  [T a1, T a2]T a3�  = fa1a2a3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='6)  Here, and throughout the paper, we use the normalization conventions [T a, T b] = fabcT c  and Tr  �  T aT b�  = δab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Finally the numerators NΓ capture all the remaining dependence on the kinematic data  of the YM tree amplitude, and in this paper we assume that they are local polynomials of  the polarization vectors εi and momenta pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' To describe gluon scattering in pure YM theory,  the polynomials must be multi-linear in the individual polarizations and of degree-(n − 2)  in the momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' It implies that the contributing monomial terms have the schematic form  NΓ ∼  ⌊n/2⌋  �  k=1  (ε · ε)k (p · p)k−1 (ε · p)n−2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='7)  For example, for n = 3 points the numerator is equivalent to the color-stripped cubic  Feynman rule of YM,  N[1,2] = ε1 · ε2 ε3 · (p1 − p2) + cyclic(1, 2, 3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='8)  It satisfies the same dihedral permutation symmetries N[2,1] = −N[1,2], N[2,3] = N[3,1] =  N[1,2] as the color factor C[1,2] = fa1a2a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Due to the Jacobi identity of the gauge-group Lie algebra, there are certain cubic  relations between color factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The statement of the color-kinematics duality [1, 2] is that  for a large class of gauge theories there exists a choice of so-called BCJ numerators that  obey the same relations as the color factors,  C···[[X,Y ],Z]··· + C···[[Y,Z],X]··· + C···[[Z,X],Y ]··· = 0  ⇔  N···[[X,Y ],Z]··· + N···[[Y,Z],X]··· + N···[[Z,X],Y ]··· = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='9)  A four-point YM numerator that satisfies this property is [3]  N[[1,2],3] =  �  ε1·ε2pµ  1 + 2ε1·p2εµ  2 − (1 ↔ 2)  ��  ε3·ε4p3µ + 2ε3·p4ε4µ − (3 ↔ 4)  �  + s12(ε1·ε3ε2·ε4 − ε1·ε4ε2·ε3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='10)  – 5 –  It is straightforward to check that, subject to on-shell conditions and momentum conserva-  tion, the numerator satisfies the Jacobi relation  N[[1,2],3] + N[[2,3],1] + N[[3,1],2] = 0 ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='11)  as well as the permutation symmetries  N[[2,1],3] = −N[[1,2],3] ,  N[[4,3],2] = N[[1,2],3] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='12)  In general, it is a difficult task to find YM numerators that satisfy color-kinematics dual-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' However, efforts during the last decade have led to several different constructions of  all-multiplicity numerators with various properties [13, 126, 192–196, 198–202, 237].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Nev-  ertheless, until now it has not been known how to construct such numerators directly from  a duality-satisfying Lagrangian or Feynman rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' We will attempt address this problem in  section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  The color factors in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='1) can be substituted by a second copy of color-kinematics  satisfying numerators CΓ → ˜NΓ, which gives the double-copy formula [1, 2]  Mn =  �κ  2  �n−2 �  Γ∈Gn  NΓ ˜NΓ  DΓ  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='13)  where, for dimensional consistency, we also substituted the couplings g → κ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The two sets  of numerators can belong to either the same or different theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' If both sets of numerators  belong to theories that contain physical spin-1 gauge fields, then the double-copy formula  gives an amplitude for spin-2 fields in some theory of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' This follows from the fact  that the double copy automatically gives diffeomorphism invariant amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' To see this,  consider a linearized gauge transformation acting on one of the polarizations of the gauge  theory amplitude, δεi = pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' By definition, we expect that δAn = 0, from which it follows  that it must be true that  �  Γ∈Gn  CΓ δNΓ  DΓ  = 0 ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='14)  where δNΓ are the gauge transformed numerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The fact that this vanishes does not  depend on the details of the color factors, only on the fact that they come from a Lie algebra  and thus satisfy Jacobi identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' If we have kinematic numerators that satisfy the same  Jacobi identities, then it also follows that [1, 2, 238]  �  Γ∈Gn  ˜NΓ δNΓ  DΓ  = 0 ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='15)  and likewise for NΓ and ˜NΓ swapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Hence, it follows that δMn = 0, and thus the double-  copy formula is invariant under linearized diffeomorphisms [238].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Given that all amplitudes  have this invariance, it follows that the theory enjoys non-linear diffeomorphism symmetry,  so it can be interpreted as a theory of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' When both numerators come from four-  dimensional pure YM, the double copy gives amplitudes in Einstein-dilaton-axion gravity,  with Lagrangian [3]  L = −  √−g  κ2  �  2R − ∂µτ∂µ¯τ  (Im τ)2  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='16)  – 6 –  where the complex field τ = ie−φ + χ contains the dilaton φ and axion χ scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' In D  dimensions the axion is promoted to a two-form Bµν and the Lagragian is also well known,  see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='2  Decompositions for amplitudes and numerators  Color-ordered partial amplitudes are defined as the gauge-invariant coefficients of the trace  decomposition of the full amplitude,  An = gn−2  �  σ∈Sn−1  Tr(T a1T aσ(2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' T aσ(n))An(1, σ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , σn) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='17)  where the sum over permutations σ runs over the symmetric group Sn−1 with (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The partial amplitudes can be computed directly using color-ordered Feynman  rules, or from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='1) after expanding out the color factors CΓ into the basis of ordered  traces of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Alternatively, repeatedly using the Jacobi identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='9) to obtain a smaller basis of  color factors, gives the Del-Duca-Dixon-Maltoni (DDM) decomposition [239],  An = gn−2  �  σ∈Sn−2  C(1, σ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , σn−1, n)An(1, σ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , σn−1, n) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='18)  which uses a subset of the same partial amplitudes, and the independent color factors are  C(1, σ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , σn−1, n) ≡ C[[···[[1,ρ2],ρ3],.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='],ρn−1] = Tr([[· · · [T a1, T aσ(2)], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' ], T aσ(n−1)]T an)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='19)  =  �  faσ(2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' faσ(n−1))a1an .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='20)  The (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='-fold DDM basis of color factors CΓ makes use of the so-called half-ladder  graphs  Γhalf-ladder(ρ) ≡ [[· · · [[1, ρ2], ρ3], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' ], ρn−1] ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='21)  which will serve as our canonical basis choice of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Similarly, repeatedly using the  kinematic Jacobi identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='9) on the BCJ numerators gives the decomposition [1, 240]  An =  �  Γ∈Gn  CΓ NΓ  DΓ  =  �  σ,ρ∈Sn−2  C(1, σ2, · · · , σn−1, n)m(σ|ρ)N(1, ρ2 · · · , ρn−1, n) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='22)  where we have set g = 1 and the independent half-ladder numerators are written as  N(1, ρ2 · · · , ρn−1, n) ≡ N[[···[[1,ρ2],ρ3],.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='],ρn−1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='23)  Assuming that the half-ladder numerators come from a manifestly crossing-symmetric con-  struction, they obey a reflection symmetry N(1, 2, 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n) = (−1)nN(n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , 3, 2, 1), as well  as anti-symmetry in the first and last pairs of arguments: N(2, 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=') = −N(1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=') and  N(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n, n − 1) = −N(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n − 1, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  The matrix m(σ|ρ) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='22) is of size (n−2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' × (n−2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' and is built out of linear  combinations of the scalar-type propagators, 1/DΓ, as can be worked out from the above  – 7 –  two DDM decompositions of the BCJ numerators and color factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The matrix m(σ|ρ)  has appeared in many places in the literature, and it goes by a variety of different names,  such as “propagator matrix” [240], the “bi-adjoint scalar amplitude” [241], or “inverse of  the KLT kernel” [127, 241].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Is is also equivalent to the color-stripped partial amplitudes of  “dual-scalar theory” [14, 242], “color-scalar theory” [243] or “scalar φ3 theory” [17, 244].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Using the propagator matrix the color-ordered partial amplitudes can be written as  A(1, σ, n) =  �  ρ∈Sn−2  m(σ|ρ)N(1, ρ, n) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='24)  which gives a (non-invertible) map between BCJ numerators and partial amplitudes [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  The propagator matrix is not invertible for on-shell conserved momenta and hence it has a  kernel, or null space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' This implies that BCJ numerators contain unphysical contributions  that live in this kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' This explains why BCJ numerators are in general not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The  ambiguity is called generalized gauge freedom [1, 2] and it corresponds to the freedom of  shifting the BCJ numerators by pure gauge contributions  N(1, ρ, n) ∼ N(1, ρ, n) + Ngauge(1, ρ, n) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='25)  where pure gauge numerators are annihilated by the propagator matrix,  �  ρ∈Sn−2  m(σ|ρ)Ngauge(1, ρ, n) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='26)  The generalized gauge freedom includes both standard gauge freedom and field redefinitions,  and more generally any operation that changes the cubic diagram numerators while leaving  the partial amplitudes invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  As illustrated in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='7), tree-level YM numerators are polynomial functions of the  independent Lorentz contractions of polarizations and momenta {εi ·εj, εi ·pj, pi ·pj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Most  numerator operations that we are interested in do not mix terms that contains different  powers ∼ (εi · εj)k, so it is convenient to decompose the numerators into sectors defined by  their polarization power k [206].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Thus from the general structure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='7), we decompose the  numerators as  N(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n) =  ⌊n/2⌋  �  k=1  N(k)(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='27)  where the polarization power k keeps track of how many εi·εj-contractions are present in  each monomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' This decomposition also makes physical sense since by a suitable choice of  the reference momenta qµ  i in the polarizations εi = εi(pi, qi) one can show [206] that only  the numerators N(1), N(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , N(k) contribute to the Nk−1MHV sector of YM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Therefore  when we refer to the Nk−1MHV sector in this paper, we implicitly mean that we consider  the N(≤k) numerators that contribute to this sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Alternatively, if one considers a dimensional reduction of YM, SO(1, D−1) → SO(1, 3)×  SO(D−4), then the N(k) numerators give rise to tree amplitudes with at most 2k external  scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' This can be formalized by considering derivative operators  ∂  ∂εi·εj that can act on  the numerators and convert a pair of gluons to a pair of scalars [209, 210].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  – 8 –  We often work with particles 1 and n being scalars, so it is convenient to introduce a  bar notation on the half-ladder numerators to indicate that we are considering a bi-scalar  YM sector [206],  N  (k)(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n) ≡  ∂  ∂ε1·εn  N(k)(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='28)  The complete bi-scalar sector numerator is then N(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n) = �⌊n/2⌋  k=1 N  (k)(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  It is possible to invert the operation in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='28) and construct the YM numerators  from the bi-scalar numerators [206],  N(k)(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n) = 1  k  �  1≤i<j≤n  εi·εjN  (k)(i, αi, i + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , j − 1, βj, j) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='29)  where αi = [· · · [1, 2], 3], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , i − 1] and βj = [j + 1, [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n − 2, [n − 1, n]] · · · ] are nested  commutators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Numerators of commutators distribute over their arguments to be con-  sistent with the Lie-algebraic interpretation, N  (k)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , [X, Y ], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=') ≡ N  (k)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , X, Y, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=') −  N  (k)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , Y, X, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The boundary cases of the sum are handled through the identifications  α2 = [1] = 1, βn−1 = [n] = n, and when either bracket is empty α1 = βn = [ ] → (−1) the  numerator is multiplied by a minus sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  We can demonstrate eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='29) using the four-point numerator as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The  bi-scalar numerator is unique (given that N(1, 2, 3, 4) = N(4, 3, 2, 1)),  N(1, 2, 3, 4) = 4ε2·p1ε3·p1 + 4ε2·p1ε3·p2 − ε2·ε3s12 ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='30)  and can be further decomposed into polarization powers  N  (1)(1, 2, 3, 4) = 4ε2·p1ε3·p1 + 4ε2·p1ε3·p2 ,  N  (2)(1, 2, 3, 4) = −ε2·ε3s12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='31)  Note that the polarization-power labels also keep track of the overall ε1·ε4 that was removed  by the derivative  ∂  ∂ε1·εn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='29) one can verify that  N(k)(1, 2, 3, 4) =1  k  �  (ε1·ε4)N  (k)(1, 2, 3, 4) − (ε1·ε3)N  (k)(1, 2, 4, 3) − (ε1·ε2)N  (k)(1, [3, 4], 2)  − (ε2·ε4)N  (k)(2, 1, 3, 4) + (ε2·ε3)N  (k)(2, 1, 4, 3) − (ε3 · ε4)N  (k)(3, [1, 2], 4)  �  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='32)  matches the numerator in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='10), given that N[[1,2],3] = N(1)(1, 2, 3, 4) + N(2)(1, 2, 3, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  For later purposes, it is convenient to introduce shorthand notation for the kinematic  variables that frequently appear in the numerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' We define the variables2  xµ  i ≡  i  �  j=1  pµ  j ,  ui ≡ 2εi · xi ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='33)  2The xi are sometimes called “region momenta”, and the ui are equivalent to cubic scalar-gluon vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  – 9 –  which make the bi-scalar half-ladder numerators simpler to work with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' For example, the  four-point bi-scalar numerator (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='30) now takes the simple form  N(1, 2, 3, 4) = u2u3 − ε2 · ε3x2  2 ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='34)  and the polarization-power components are then N  (1) = u2u3 and N  (2) = −ε2 · ε3x2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Note  that this notation obscures the relabeling symmetries of the numerators so it should be  used with some care, and we will mainly use it for bi-scalar half-ladder numerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  It is illuminating to attempt to represent the contributions to the numerators using  diagrammatic notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' As a general guiding principle, we track inner products between  polarization vectors using solid lines and wavy lines represents other contributions (typically  inner products between polarizations and momenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' For example, the four-point bi-scalar  numerators are represented by the two diagrams  N  (1) =  1  2  3  4  = u2u3 ,  N  (2) =  1  2  3  4  = −ε2 · ε3 x2  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='35)  This notation makes it clear that the numerator N  (2) represents a contribution coming from  an auxiliary rank-2 field in the intermediate channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Indeed, we will see in section 4 that  this interpretation is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  At five points, the half-ladder BCJ numerator can again be decomposed into two bi-  scalar numerators, N  (1) and N  (2), of different polarization powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Focusing on N  (2), we  realize even before having an explicit formula that there are three different dot products  ε3·ε4 , ε2·ε3 and ε2·ε4, which can be represented by the schematic diagrams  N  (2)(1, 2, 3, 4, 5) =  1  2  3  4  5  +  1  2  3  4  5  +  1  2  3  4  5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='36)  The explicit expressions for these diagrams will be given in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Likewise at n points, the half-ladder bi-scalar numerator N  (2) can be schematically  written down as a sum over (n − 2)(n − 3)/2 different diagrams,  N  (2)(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n) =  �  1<i<j<n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  1  n  j  i  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='37)  and in section 4 we will find an explicit formula for the diagrams using a Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Before considering these NMHV amplitude contributions, we will have a short section on  the Lagrangian that generates the N(1) numerators which gives MHV amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  – 10 –  3  MHV Lagrangian  Tree-level MHV amplitudes of YM can be computed from the BCJ numerators with the  lowest polarization power N(1) [206].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' For example, if legs 1 and 2 carry negative helicity  and the remaining legs have positive helicity, then choosing reference null momenta qµ  i as  ε−  1 (p1, q1 = p2) ,  ε−  2 (p2, q2 = p1) ,  ε+  i>2(pi, qi = p1)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='1)  ensures that the only non-vanishing polarization products are ε2·εi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Because they all contain  ε2 these factors can at most appear linearly in the numerator, hence N(1) is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  By dimensional analysis the numerators N(1) cannot contain products of momenta  pi · pj hence they cannot give rise to inverse propagators p2 ∼ □, and thus there are no  hidden contact terms inside the numerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' This implies that N(1) must originate from  a YM Lagrangian that neither has contact terms nor auxiliary fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Thus we can simply  truncate the YM Lagrangian3 to cubic order, which defines the MHV Lagrangian  LMHV = LYM  ���  cubic = Tr  �1  2Aµ□Aµ − ∂µAν[Aµ, Aν]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='2)  For simplicity, we have set the gauge coupling to unity, g = 1, which we will do henceforth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  The Lagrangian (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='2) gives standard cubic YM Feynman rules that produce numerators  of all polarization power sectors k ≥ 1, but for the purpose of this section we can assume  that the resulting numerators are truncated to N(1) because of the conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='1) imposed  on the polarization vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' That this Lagrangian is invalid beyond this sector is obvious  from the fact that the truncation of the quartic term is not a valid (gauge-fixing) operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Let us now show that the numerators N(1) computed from the Lagrangian (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='2) obey  color-kinematics duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' We use the diagrammatic notation from the previous section to  make the argument simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The N(1) numerators are computed from sums of diagrams  each having just one solid line (representing the contracted polarization vectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' We can  think of the solid line as a scalar line, and we have to sum over all possible pairs of external  states that are joined by this line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The scalar line can be identified from the equations of  motion of the MHV Lagrangian,  □Aµ = −2[Aν, ∂νAµ] + [Aν, ∂µAν] ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='3)  where the second term produces contractions that ultimately connect external polarization  vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The first term corresponds to cubic interactions were no scalar line is present, thus  we think of it as consisting of only wavy lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' It is straightforward to then study the triplet  sum of off-shell numerators obtained using these two interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' One can show that the  equation  1  2  3  4  −  1  2  3  4  =  1  2  3  4  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='4)  3We assume Lorenz gauge ∂ · A = 0 (or, more accurately, Feynman gauge) since the BCJ numerators  are defined to sit on top of propagators that are in Feynman gauge, see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  – 11 –  holds up to terms proportional to ε2·p2 and ε3·p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Such terms vanish on-shell because of  the transversality of εi, but we need to show that the above three-term identity holds for  a generic off-shell numerator embedded into a larger tree diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Thus we need to show  that transversality holds for all off-shell wavy lines in the this MHV sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  We can show this using induction, essentially feeding an arbitrary number of wavy-line  interactions into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Suppose that to some wavy line i we attach a cubic diagram  generated from the first interaction term in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='3), producing two new wavy lines j, k, so  i → [jk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Then its effective polarization vector4 takes the form  εµ  i = −2(εj·pkεµ  k − εk·pjεµ  j ) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='5)  in terms of the effective polarizations of the lines j, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' By the induction hypothesis, we  assume that the latter polarizations are transverse, εj·pj = εk·pk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Using this, and the  antisymmetry in the labels j and k, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='5) implies that εi·pi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Thus the wavy-line  interaction preserves transversality, and since the wavy lines will eventually terminate in  proper external states a tree level the induction hypothesis is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' This completes the  argument that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='4) holds off shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  We also need to consider kinematic Jacobi identities of diagrams containing only the  first term in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='3), which is the wavy-line interaction (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' However, note that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='5)  is essentially a momentum-space Lie bracket of plane-wave vector fields Aµ  i = εµ  i eipi·x,  Ai · ∂ = −2  �  Aj · ∂, Ak · ∂  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='6)  and so it automatically obeys the Jacobi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Since the vector fields are transverse  ∂ · Ai = 0, we have thus exposed a kinematic sub-algebra that corresponds to volume-  preserving diffeomorphisms (in analogy with the previously found 2D area-preserving [204]  and 3D volume-preserving diffeomorphisms [40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' However, we cannot reproduce the full  N(1) numerators from commutators of only these diffeomorphism generators, since the full  numerator is a superposition of all diagrams with a single solid line going between pairs of  external states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Nevertheless, by linearity of the kinematic Jacobi identity, the superposition  of all such diagrams will give N(1) numerators that obey color-kinematics duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  To summarize, for tree level diagrams obtained from the standard cubic YM La-  grangian, color-kinematics duality is satisfied at polarization power one, or equivalently,  for MHV amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Note that the off-shell argument does not automatically extend to  loop level since it was important that the recursive transversality argument terminates with  external on-shell legs, which is always true at tree level, but not at loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Finally, let us give the half-ladder bi-scalar numerators that are generated from the  Feynman rules of the MHV Lagrangian (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' They are simply  N  (1)(1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n − 2, n) =  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  1  n  2  n − 1  =  n−1  �  i=2  ui ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='7)  4Note that this is not to be confused with an external properly normalized polarization vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  – 12 –  and the remaining contributions to the half-ladder numerator N(1) can be obtained using  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='29), or from the Feynman rules of the Lagrangian (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The non-half-ladder diagrams  are similarly computed either from commutators of the half-ladder numerator N(1), or from  the same Feynman rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  4  NMHV Lagrangian  In this section we construct several cubic Lagrangians that produce BCJ numerators at  polarization power two, which allow us to compute helicity amplitudes up to the NMHV  sector of YM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' We begin the construction at four points, and work our way upwards in  multiplicity by adding terms to the Lagrangian, until the corrections terminate at seven  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='1  The four-point NMHV Lagrangian  Let us start from the standard YM Lagrangian (subject to Lorenz gauge ∂ · A = 0),  LYM = −1  4Tr (F µν)2 = Tr  �1  2Aµ□Aµ − ∂µAν[Aµ, Aν] − 1  4[Aµ, Aν][Aµ, Aν]  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='1)  where the field strength is F µν = ∂µAν − ∂νAµ + [Aµ, Aν].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' For later purposes, let us also  quote the corresponding equations of motion  □Aν = −2[Aµ, ∂µAν] + [Aµ, ∂νAµ] − [Aµ, [Aµ, Aν]] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='2)  We seek to find a cubic rewriting of this Lagrangian that generates BCJ numerators at  polarization power two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' As a first step, we can resolve the quartic interaction by introducing  propagating auxiliary fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' There are several ways to do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' We find that an elegant  solution is to reinterpret the nested commutator, appearing in the last term of the above  equations of motion, to contain a two-form tensor Bµν = −1/2[Aµ, Aν].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' In order to allow  for a suitable kinetic term of correct mass dimension, we introduce a companion field ˜Bµν  of mass dimension zero, into which Bµν propagates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The Lagrangian we obtain is  L4 = Tr  �1  2Aµ□Aµ − ∂µAν[Aµ, Aν] + Bµν□ ˜Bµν + 1  2[Aµ, Aν](Bµν + □ ˜Bµν)  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='3)  and the equations of motion for Bµν and ˜Bµν are given by  Bµν = □ ˜Bµν = −1  2[Aµ, Aν] ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='4)  which we can plug into the Lagrangian above to recover LYM as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' To compute  amplitudes from this Lagrangian one has to restrict the external legs to be vectors Aµ, since  the B and ˜B fields are composite (the linearized excitations of B vanish).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' As discussed  in the previous section, the cubic-in-Aµ interactions of this Lagrangian correctly reproduce  the MHV sector of YM, while the double line in the polarization-power two sector (see for  example the second graph in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='35)) corresponds to propagation of the Bµν and ˜Bµν  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Since these fields can be integrated out yielding the standard YM Lagrangian, the  – 13 –  Feynman rules of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='3) reproduce standard YM tree amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' At four points, the  BCJ numerator is unique [207] (up to an overall normalization) upon imposing relabeling  symmetry and reflection symmetry, as given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='10), and it is also correctly reproduced  by the Lagrangian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' In fact, any cubic action for YM theory would reproduce the correct  four-point numerator, but we find that the above choice of L4 with both the B and ˜B fields  allows for the necessary freedom to proceed to generating BCJ numerators at higher points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='2  The five-point NMHV Lagrangian  Going to five points, we can immediately see that L4 is unable to generate half-ladder  numerators N(2) proportional to ε1·ε5 ε2·ε4, which is the last diagram pictured in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='36),  and corresponds to the double-line field emitting a gluon line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' To generate this kind of graph  we need new interaction terms of the schematic form ∂AB ˜B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' In principle, introducing new  interactions could change the equations of motion, change the four-point amplitude, and  break gauge-invariance of YM theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Thus one can expect it to be a delicate procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Let consider adding new linear-in-Bµν or linear-in- ˜Bµν terms on the right-hand side of  their corresponding equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' By repeated insertion of the equations of motion  into themselves, we would generate a set of non-local higher order Aµ interactions (similar  to refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' [14, 216]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Can we make sure that such new interactions always cancel out in any  tree-level amplitude computation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Let us start, for the sake of simplicity, by requiring  something more specific, namely that the deformation introduced by the Lagrangian terms  ∂AB ˜B vanish within the subsystem of equations of motion for the auxiliary fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' This  can be easily achieved if they take the following form  Bµν = −1  2[Aµ, Aν] + α∂ρ  □ ([Aµ, Bρν] + cyclic(µρν)) ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='5)  where α is a free parameter for now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  When these equations of motion are repeatedly  substituted back into themselves n times, the sum over cyclic permutations of Lorentz  indices in the commutator manifestly vanishes by Jacobi identity at any order in A, yielding  Bµν,(n) = −1  2[Aµ, Aν] + O(αn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='6)  where the last term is O(αn) ∼ αn(Aµ)nBνρ,(0), and it vanishes in perturbative tree-level  computations since we take Bνρ,(0) = 0 for external states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The equations of motion in  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='5) must arise from introducing the following interactions in the Lagrangian:  ∆L5 = −α Tr ∂ρ ˜Bµν  �  [Aµ, Bνρ] + cyclic(µνρ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='7)  where L5 = L4 +∆L5 is the total Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' As argued, the term in parentheses does not  contribute to on-shell tree-level scattering amplitudes by the Jacobi identity, but it does  alter the individual N(2) numerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  With the new Lagrangian L5 the equations of motion for ˜Bµν now read  ˜Bµν = −1  2  1  □[Aµ, Aν] + α 1  □  �  Aρ , ∂µ ˜Bρν + cyclic(µρν)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='8)  – 14 –  Note that repeated insertion of the ˜B-field equations of motion into themselves generates an  infinite set of contact terms of increasing A order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' However, from the B-field equations of  motion is clear that after integrating out the combined pair of auxiliary fields, the standard  YM tree-amplitudes amplitudes are recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  We now need to probe whether the deformed numerators enjoy color-kinematics duality  for some choice of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' We can do so by explicitly checking Jacobi relations between the N(2)  numerators, or checking that the DDM-decomposed partial amplitude (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='24) returns a gauge  invariant quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' At five points, we confirm that BCJ numerators are obtained after fixing  the free parameter in ∆L5 to the value α = 2, giving the duality-satisfying Lagrangian  L5 = Tr  �1  2Aµ□Aµ − ∂µAν[Aµ, Aν] + Bµν□ ˜Bµν + 1  2[Aµ, Aν](Bµν + □ ˜Bµν)  + 4∂ν ˜Bµρ[Aµ, Bνρ] − 2∂ρ ˜Bµν[Aρ, Bµν]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='9)  We thus find the following NMHV contributions to the bi-scalar sector diagrams that  we introduced in section 2,  1  2  3  4  5  =  ε3 · ε4(u2 x2  3 − ε2·x3 x2  2) ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='10)  1  2  3  4  5  =  1  2ε2 · ε4u3(x2  2 + x2  3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='11)  As before, the remaining N(2) contributions to the half-ladder diagrams can be obtained  using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='29), or from the Feynman rules of the Lagrangian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Since we are at five  points, all cubic diagrams are of the half-ladder type, and thus this completes the five-point  construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Before proceeding to discuss the six-point case, let us point out a remarkable fact about  the Lagrangian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' We have checked through multiplicity n = 11 that its Feynman rules  correctly compute half-ladder BCJ numerators in the bi-scalar sector N  (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Inserting those  into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='29), gives all half-ladder numerators N(2), and further use of Jacobi relations  give all non-half-ladder numerators in the NMHV sector, up to the multiplicity we checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Based on this rigorous pattern, we conjecture that the Lagrangian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='9) correctly computes  the bi-scalar NMHV numerators N(2) to any multiplicity at tree level, and furthermore  implicitly provides a good BCJ representation for the remaining diagrams in this sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='3  Closed form representation of the bi-scalar sector numerators  Let us summarize what we have achieved thus far by giving a closed form expression for the  bi-scalar numerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' As shown in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='7), the MHV bi-scalar numerators take a simple  form  N  (1)(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n − 1, n) = u2u3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' un−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='12)  – 15 –  Next, we gave a schematic diagram representation for the bi-scalar sector NMHV numera-  tors in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='37), and with the Lagrangian L5 in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='9) we can make this more precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Since we are considering numerators at polarization power two, each contribution must  have exactly one vertex of type AAB and one of type AA□ ˜B in the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Consider a  half-ladder diagram where the B field is first sourced at position i to the left of the final ˜B  field at position k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Between these two vertices a chain of AB∂ ˜B-type vertices is inserted,  giving the below diagram  Dijkn =  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  1  n  j  i  k  □  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='13)  where j labels a scalar line (which need not be at position k), and the □ indicates the  location of the inverse propagator x2  k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  The diagram where ˜B is sourced first can be  obtained from reflection of this diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Since a solid line stretches all the way between i  and j, the intermediate vertices must be of the form AµBνρ∂µBνρ, which gives a product  of ul variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Vertices between legs j and k can either be of the same type, or of the  type AµBνρ∂ν ˜Bµρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The vertices before i or after k are of the type AA∂A and can only give  rise to ul variables in order to not increase the polarization power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Thus, all together, each  diagram is given by the expression  Dijkn = (−1)nεi·εjx2  k−1U2,i−1Ui+1,j−1  �  xj−1 · Vj+1 · · · Vk−1 · εk  �  Uk+1,n−1 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='14)  where the matrices V µν  i  are given by  V µν  i  = uiηµν − 2εµ  i xν  i−1 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='15)  and the U’s are products of consecutive ui, defined by  Ui,j = uiui+1 · · · uj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='16)  We set the boundary case j = k of the expression in parenthesis (xj−1 ·Vj · · · Vk−1 ·ϵk) to be  equal to the number (−1/2) since otherwise the polarization vector εk is double counted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Finally, using the diagrams Dijkn the bi-scalar numerator in this sector is  N  (2)(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n) =  n−1  �  1<i<j≤k  Dijkn + reflection .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='17)  The reflected diagram is obtained by reversing the labels {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n} on the momenta pi  and polarizations εi, not the region momenta xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' It is interesting to note that the matrices  Vi play a similar role to the Gij matrices used for NMHV numerators in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' [207], except  that the former depend on region momenta xi and the latter on particle momenta pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  – 16 –  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='4  NMHV Lagrangian beyond the bi-scalar sector  As detailed in the previous two subsections, we conjectured that the bi-scalar sector numer-  ators (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='17) generated by the five-point Lagrangian L5 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='9) give valid BCJ numerators to  all multiplicity at tree level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' From these we can then uniquely compute all NMHV contri-  butions beyond the bi-scalar sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' We now wish to find a Lagrangian description of such  contributions, which starts at six points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  First,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' let us write down an explicit expression for the six-point bi-scalar numerator  (summing over polarization power one and two),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  N(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' 6) = U2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='5 + (D2336 + D2346 + D2356 + D2446 + D2456 + D2556 + D3446  + D3456 + D3556 + D4556 + reflections)  = u2u3u4u5 − 2ε2·ε3  �  2ε4·x2ε5·x3x2  4 − ε5·x2u4x2  4 − ε4·x2u5x2  3 + u4u5x2  2  �  + ε2·ε4  �  2ε5·x3u3x2  4 − (x2  2 + x2  3)u3u5  �  − ε2·ε5  �  x2  2 + x2  4  �  u3u4  + 2ε3·ε4  �  ε5·x3u2x2  4 + ε2·x3u5x2  2 − u2u5x2  3  �  + ε3·ε5  �  2ε2·x3u4x2  2 − (x2  4 + x2  3)u2u4  �  − 2ε4·ε5  �  2ε2·x3ε3·x4x2  2 − ε3·x4u2x2  3 − ε2·x4u3x2  2 + u2u3x2  4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='18)  Plugging this numerator into the formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='29) gives the remaining NMHV numerator  contributions N(2) at six points, which do not match (diagram-by-diagram) the same con-  tributions computed from the Lagrangian L5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  It is clear that the numerators obtained  from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='29) are the ones we need since they satisfy color-kinematics duality, thus we  must modify the Lagrangian L5 with new six-point contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  We proceed by looking at individual kinematic monomials in N(2) that pinpoint the mis-  match, and then infer what are the simplest interactions that can restore color-kinematics  duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' For example, considering the term u3ε2·ε4ε5·ε6ε3·(p5 − p6)x2  2, we find that it is  necessary to introduce a pair of new vector fields Zµ and ˜Zµ of mass dimension one, which  interact with Aµ, Bµν and ˜Bµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' To ensure that the new auxiliary vectors do not pollute the  four and five-point construction, we must constrain certain interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Specifically we will  only allow an interaction of the form AAZ but no conjugate AA ˜Z interaction, thus ensuring  that at most the Z field can be sourced at four points, and hence it cannot propagate to a  ˜Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Furthermore, we require that the Z, ˜Z fields can only source the linear combination of  the tensor field (Bµν − □ ˜Bµν), which is specifically not sourced by Aµ fields at five points,  hence the Z, ˜Z field cannot propagate at five points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Let us see how this example plays out in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Adding two interactions of the form  Aµ ˜ZνBµν and ∂νAµAµZν generates a new diagram contribution of the schematic form  u3 ε2·ε4 ε5·ε6 ε1·(p5 − p6)x2  2  −→  1  2  3  4  5  6  ∂  □ ˜B B ˜B  B ˜Z  Z  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='19)  – 17 –  where we have have placed the previously mentioned offending monomial to the left and  the corresponding schematic diagram with new vertices A ˜ZB and AAZ to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The  solid line that ends on a derivative indicates the ε1·(p5 − p6) contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Similarly, flipping the tilde and non-tilde fields yields a correspondence between the  following offending monomial and new diagram  u3 ε1·ε4 ε5·ε6 ε2·(p5 − p6)x2  3  −→  1  2  3  4  5  6  ∂  B  ˜BB □ ˜B ˜Z  Z  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='20)  It is clear that the needed modification to the Lagrangian coming from these terms takes  the following form:  ∆L6 ∼ Tr  �  Zµ□ ˜Zµ + [∂νAµ, Aµ]Zν + [Aµ, ˜Zν]  �  Bµν − □ ˜Bµν  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='21)  where we fixed the relative couplings by using the normalization freedom of the kinetic  term, as well as the rescaling freedom Z → αZ and ˜Z → ˜Z/α that leaves the kinetic term  invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' There is only one free parameter, which we can take to the overall normalization,  and we find that it is equal to unity in order to match the correct N(2) contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  There are further mismatching monomials that we need to deal with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The N(2) numer-  ators contain problematic terms such as ε2·ε3 ε5·ε6 ε1·x2 ε4·(p6 − p5) x2  2, where no Lorentz  indices are crossing the central half-ladder propagator, and also the mass dimensions are  unbalanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' This suggests that we need to introduce a pair of scalar fields X and ˜X of  mass dimension two and zero respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Again we need to ensure that the new fields  do not modify the five-point numerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Consider the new interactions AA ˜X and AX ˜Z  which contribute to the mentioned monomial through the diagram  ε2·ε3ε5·ε6ε1·x2ε4·(p5 − p6) x2  2  −→  1  2  3  4  5  6  ∂A □A ˜X X ˜Z  Z  ∂  ∂  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='22)  As mentioned, there are more derivatives on the left half of the diagram than the right half,  thus the imbalance of the dimensions of the X and ˜X fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Additionally, we add A ˜X ˜Z interaction to address the offending term  ε1·ε2ε5·ε6ε3·(p1 − p2)ε4·(p6 − p5)x2  2  −→  1  2  3  4  5  6  Z  □ ˜ZX  ˜X ˜Z  Z  ←  ∂  ∂  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='23)  We also need ˜X to interact with the B field, to address offending terms of the form  ε2·ε3 ε5·ε6 ε1·x3 ε4·(p5 − p6) x2  2  −→  1  2  3  4  5  6  □ ˜B B∂X  ˜X ˜Z  Z  ←  ∂  ∂  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='24)  – 18 –  The pair of scalar fields enjoys the same rescaling freedom as the auxiliary vectors, which  we use to fix the coefficient of the AA ˜X interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The above three monmial structures  are then enough to constrain all remaining coefficients of the needed interactions, which we  find to be  ∆L6 ∼ Tr  �  X□ ˜X + [Aµ, □Aµ] ˜X − [Aµ, X] ˜Zµ + 1  2[Aµ, □ ˜X] ˜Zµ − 2[Aµ, Bµν]∂ν ˜X  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='25)  This completes the construction of interactions that contribute to the six-point half-ladder  numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  As a side remark that we will come back to later, note that one can rearrange the  flow of Lorentz indices in the half-ladder diagrams using conservation of momentum, and it  turns out that it is not strictly necessary to introduce a pair of scalar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Indeed, in the  next subsection we show that it is possible to formulate a completion of the L5 Lagrangian  in the NMHV sector using only the pair of vectors Z and ˜Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Finally, we need to address those diagrams that contribute to the non-half-ladder topol-  ogy5 appearing at six points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Our Lagrangian does not yet get those contributions to match  with what is predicted from the N(2) numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Assuming that no new fields are needed,  the possible interactions are highly constrained by dimensional analysis and by the require-  ment that they must involve three auxiliary fields in order to not spoil the half-ladder  numerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Consider the following example of a missing term and corresponding diagram  with a new B ˜Z ˜Z interaction:  s12ε3·ε4ε5·ε6ε1·(p5 − p6)ε2·(p3 − p4)  −→  1  2  3  4  5  6  □ ˜B  B ˜Z ˜Z  Z  ∂  ∂  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='26)  We find that a suitable interaction has the form  Tr  �  [ ˜Zµ, ˜Zν]  �  βBµν + (1 − β)□ ˜Bµν��  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='27)  but at this multiplicity we can not yet fix the free parameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' There is one more needed  interaction which is rather simple, and fully constrained at six points, it is  4Tr  �  [Bµν, ∂µ ˜Bνρ] ˜Zρ�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='28)  This finishes the construction of the duality-satisfying six-point NMHV Lagrangian  L6 = L5 + ∆L6, where all the new terms are assembled as  ∆L6 = Tr  �  Zµ□ ˜Zµ + X□ ˜X + [∂νAµ, Aµ]Zν + [Aµ, □Aµ] ˜X + [Aµ, ˜Zν]  �  Bµν − □ ˜Bµν�  − [Aµ, X] ˜Zµ + 1  2[Aµ, □ ˜X] ˜Zµ − 2[Aµ, Bµν]∂ν ˜X + [ ˜Zµ, ˜Zν]  �  βBµν + (1 − β)□ ˜Bµν�  + 4[Bµν, ∂µ ˜Bνρ] ˜Zρ�  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='29)  5Sometimes called star or Mercedes topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  – 19 –  and, as already mentioned, the β parameter is not yet fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Moving on to seven points, interactions of the form ∂AZ ˜Z and ∂AX ˜X can now partake  in the half-ladder diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' These rather simple contributions are needed for generating  correct half-ladder factors ui, which in hindsight is not surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Their coefficients in  the Lagrangian can be fixed by computing the terms ε1·ε2ε6·ε7ε3·(p1 − p2)ε4·(p6 − p7)u5x2  2  and ε1·ε2ε6·ε7ε3·(p1 − p2)ε5·(p6 − p7)u4x2  2, and comparing the result to the predicted N(2)  numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Furthermore, a new interaction B ˜X ˜Z is required for the non-half-ladder graphs,  and finally the unknown parameter β from ∆L6 is now constrained to β = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Thus we conclude that the seven-point corrections to the duality-satisfying NMHV  Lagrangian consists of the three terms  ∆L7 = Tr  �  − 2[Aµ, Zν]∂µ ˜Zν − 2[Aµ, X]∂µ ˜X − 2[Bµν, ∂ν ˜X] ˜Zµ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='30)  No further corrections are needed at higher multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  We explicitly computed and  checked the properties of color-kinematics duality and gauge invariance for all NMHV nu-  merators and amplitudes through ten points, which worked flawlessly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Based on the robust patterns observed up to multiplicity ten, we conjecture that the fol-  lowing assembled NMHV Lagrangian computes all BCJ numerators and NMHV amplitudes  to any multiplicity at tree level:  L = L5 + Tr  �  Zµ□ ˜Zµ + X□ ˜X + [∂νAµ, Aµ]Zν + [Aµ, □Aµ] ˜X − 2[Aµ, Zν]∂µ ˜Zν  − 2[Aµ, X]∂µ ˜X + [Aµ, ˜Zν]  �  Bµν − □ ˜Bµν�  − [Aµ, X] ˜Zµ + 1  2[Aµ, □ ˜X] ˜Zµ  − 2[Aµ, Bµν]∂ν ˜X + 1  2[ ˜Zµ, ˜Zν]  �  Bµν + □ ˜Bµν�  + 4[Bµν, ∂µ ˜Bνρ] ˜Zρ  − 2[Bµν, ∂ν ˜X] ˜Zµ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='31)  This is the simplest Lagrangian that we have found in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' A natural question to  ask next is: how unique is it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='5  How unique is the NMHV Lagrangian?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  The way in which we obtained the NMHV Lagrangian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='31) does not give strong evidence  for its uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' To check how unique it really is, we constructed a more general ansatz  that is vastly larger in complexity compared to the above construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Again we con-  strained the results by checking color-kinematics duality and gauge invariance, but we were  more meticulous in keeping track of our assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' This is useful for future work were  these assumptions can be further relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Our enlarged ansatz for the NMHV Lagrangian is subject to the following assumptions:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The simple Lagrangian L5 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='9) is still assumed to give the bi-scalar numerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' No additional fields beyond those in the previous sections are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' That is, only the  tensors B, ˜B, vectors Z, ˜Z and scalars X, ˜X appear as auxiliary fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Kinetic terms do not mix fields: L2 = Tr  �  1  2Aµ□Aµ + Bµν□ ˜Bµν + Zµ□ ˜Zµ + X□ ˜X  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  – 20 –  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' To preserve the four-point numerator, a pair of external A’s can either source Z or  ˜Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' As before, we choose to exclude AA ˜Z interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Two-derivative interactions always appear as a d’Alembertian □.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' This makes it man-  ifest that the interaction contributes at most to polarization-power two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' We make an ad hoc simplifying choice to exclude interactions AZZ and A ˜Z ˜Z, in order  the make the ansatz space more manageable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  After taking care of the rescaling freedom of the auxiliary fields, the Lagrangian ansatz we  obtain with the above constraints has 174 free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' We constrain the numerators  generated by the ansatz up to eight points by comparison with the predicted numerators  N(2), as obtained from linear combinations of the bi-scalar numerators coming from L5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Note these constrains are non-linear equations in the free parameters of the Lagrangian  ansatz, hence the equation system is non-trivial to deal with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  With the constraints imposed up to eight points, this only fixes a subset of all param-  eters and we are left with 129 free coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' At nine points, we find that all numerator  topologies generated by the Lagrangian ansatz are independent of the leftover coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  It is difficult to go to higher points due to the non-linearities, and we stop trying to find  further constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Instead, we now seek solutions that minimizes the number of terms  in the Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' We find four solutions which all give 13-term Lagrangians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Provided  that ∂ · A = 0, all solutions coincide (up to total derivatives) to give the same Lagrangian  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='31) we found in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Thus it seems reasonable to think that this is the  simplest NMHV Lagrangian, given the above list of assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  For example, another simple solution gives a 14-term Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' It is reachable by  deforming the 13 term Lagrangian L (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='31) by the following interactions:  L − L14-term = Tr  �  [Aµ, □Aµ] ˜X + 4[Aµ, ∂νBµν] ˜X + [Aµ, ˜X]□ ˜Zµ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='32)  A more interesting scenario would arise if we managed to find a solution that makes use  of fewer auxiliary fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' We find no solutions that discards of the vectors Z and ˜Z, but  interestingly, the scalars X and ˜X do seem to not be strictly necessary for all solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Consider replacing the above ad hoc assumption 6, by the following new assumption:  6’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Exclude the scalar fields X, ˜X, but now include all interactions AZZ and A ˜Z ˜Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  We start with a 95 parameter ansatz and constrain it by comparing to the predicted N(2)  numerators up to eight points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  At nine points, there is a single non-half-ladder graph  depending on one free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' To fix this parameter, we use the unique half-ladder  graphs generated by our ansatz and through Jacobi identities construct the graph containing  the free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' At this stage, we have 50 free coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Assuming they do not enter  the numerators at higher multiplicity, we look for the smallest possible Lagrangian and  find six solutions with 20 terms each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' To select one of the solutions, we will assume that  – 21 –  ∂ · A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The solution is not particularly simple but we report it below for completeness:  L′ = L5 + Tr  �  Zµ□ ˜Zµ − [Aµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' ∂νAµ]Zν + [Aµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Zν]∂νZµ − 3  4[Aµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Zµ]∂ · Z − [Aµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' ˜Zµ]∂ · ˜Z  − 2[Aµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Zν]∂µ ˜Zν + 1  2[Aµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' ∂ · Z] ˜Zµ − 2[Aµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' ∂νZµ] ˜Zν − 3  2[Aµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Zµ]∂ · ˜Z  − 3  2[Aµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Bµν]Zν − [Aµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Bµν] ˜Zν − 1  2[Aµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' □ ˜Bµν]Zν + [Aµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' □ ˜Bµν] ˜Zν  + 9  8[Bµν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Zµ]Zν + 3  2[Bµν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Zµ] ˜Zν + 1  2[Bµν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' ˜Zµ] ˜Zν + 1  8[□ ˜Bµν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Zµ]Zν  − 1  2[□ ˜Bµν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Zµ] ˜Zν + 1  2[□ ˜Bµν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' ˜Zµ] ˜Zν − 2[Bµν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' ∂µ ˜Bνρ]Zρ + 4[Bµν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' ∂µ ˜Bνρ] ˜Zρ�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='33)  It would be interesting to more broadly explore the ansatz space of NMHV Lagrangians,  with the above six assumptions further relaxed, but we leave it for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='6  Comments on the N2MHV sector  The complete six-point BCJ numerator must contain also terms of polarization power three,  which are needed for the N2MHV amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Unfortunately, the Lagrangian ansatz space  we have considered does not permit solutions which correctly reproduces this sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' This  is not very surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Since, as a minimal extension of our ansätze, we would need to  introduce a three-form auxiliary field (or even rank-three tensors of mixed symmetries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  This is clear from considering a six point BCJ numerator that necessarily [207] contains  terms of the form  1  2  3  4  5  6  ∼  ε1·ε6 ε2·ε5 ε3·ε4 x2  i pj·pk ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='34)  which in D-dimensions cannot be removed by generalized gauge transformations that main-  tain locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The generalized gauge freedom of local crossing-symmetric D-dimensional  six-point numerators were analysed in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' [207].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' From the triple-solid line in the middle  of the above diagram, it is clear that a rank-three auxiliary field is propagating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' There  could in principle also exist higher-rank fields since momentum dot products pj·pk may be  contracted across this central propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Also, the type of tensors needed is unclear, it  could be forms or tensors of mixed symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' We leave the problem of reproducing the  N2MHV and higher sectors to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  5  Towards MHV numerators at one loop  Having obtained Lagrangians that generate BCJ numerators it is interesting to check if  they can also generate BCJ numerators at loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Loop color-kinematics duality does  not necessarily follow from tree level, since loops also contain off-shell internal states that  are absent at tree level by the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  – 22 –  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='1  Polarization power zero at one loop  To obtain the one-loop polarization-power zero numerators (valid for all-plus and one-minus  helicity YM sectors), we glue legs 1 and n of the tree-level half-ladder numerators, with  appropriate contributions from Faddeev–Popov ghosts c, ¯c to remove unphysical degrees of  freedom,  N(0)  1-loop(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n) = N(1)(ℓ, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n, −ℓ)+N(c, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n, ¯c)+N(¯c, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n, c) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='1)  It should be understood that the states labeled by the loop momentum ℓ and −ℓ are  contracted, and the two ghost numerators come with a minus sign after the sewing of the  states since the c and ¯c fields are fermionic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' As we have shown in section 3, the Feynman  rules for the bi-scalar sector automatically obey the color-kinematics duality so long as the  external vector fields are transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' This assumption holds for external states as well as  subdiagrams that are of polarization power zero, just as argued below eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  All we have to do is add the ghosts in such a way that they respect the color-kinematics  duality, and this is achieved by the standard Faddeev-Popov ghost Lagrangian  L(0)  ghost = Tr  �  c□˜c + Aµc∂µ¯c  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='2)  The interaction term here is identical to the effective interactions of the bi-scalar sector so  color-kinematics duality works out automatically when external states are transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Since  the diagrams that contribute here come from standard YM Feynman rules, the auxiliary  fields are not yet present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Let us give the duality-satisfying n-gon master numerators that are valid for the all-plus  and one-minus helicity one-loop amplitudes (see also refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' [68, 91]),  N(0)  1-loop(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n) = Tr(W1 · · · Wn) + Tr(�  W1 · · · �  Wn) − (2 + D)U1,n ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='3)  where D = Tr(1) is the dimension, U1,n is defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='16) and the matrices are  (Wi)µν = uiηµν + 2εµ  i pν  i ,  (�  Wi)µν = uiηµν − 2pµ  i εν  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='4)  The region momenta present in ui = 2εi·xi is here defined to include the loop momentum  xi = ℓ + �i  j=1 pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The above numerators can be checked to give the correct maximal [91]  and non-maximal cuts, since by construction they make use of the standard YM three-point  vertex and there are no contact terms present in the amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='2  Comment on one-loop MHV numerators  As already emphasized, tree-level all-multiplicity color-kinematics duality is in general in-  sufficient to infer that the loop-level duality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Indeed, we find obstructions in realizing  one-loop BCJ numerators in the MHV sector from the Feynman rules derived in previous  sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' At polarization-power one, in particular, the numerators receive contributions  from the new fields we added to the Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' As before, we have contributions from the  tree-level polarization-power one sector after gluing half-ladder numerators,  N(1)  1-loop(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n) ⊃ N(1)(ℓ, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n, −ℓ) ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='5)  – 23 –  but now we must add the glued half-ladder numerator at polarization power two as well,  N(1)  1-loop(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n) ⊃ N(2)(ℓ, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' , n, −ℓ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='6)  In addition, to not break crossing symmetry we should allow for the B, Z and X fields to  cross the sewn loop line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' This makes it possible for the auxiliary fields to propagate all the  way around the loop, which is not expected to give reasonable contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Thus, we may  attempt to project out certain contributions by adding new ghosts for the auxiliary fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  The contributions we need to remove are very similar to the bi-scalar sector contributions  (as already noted in the previous section), hence we add new ghost Lagrangian terms  corresponding to the B, ˜B fields,  L(1)  ghost = Tr  �  c□˜c + bµν□˜bµν + Aµc∂µ¯c + 4∂ν˜bµρ[Aµ, bνρ] − 2∂ρ˜bµν[Aρ, bµν]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='7)  As usual, the ghost fields cannot be sourced and so they only contribute through propagating  in a complete closed loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  We performed a few crude tests to the one-loop numerators as obtained from our  Lagrangians, and found that for the 13-term Lagrangian in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='31) already the three-  point one-loop numerators are not well behaved, meaning they do not give gauge-invariant  unitarity cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' For the 14-term Lagrangian obtained by the deformation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='32), the maximal  cuts work as tested up to four points, but the next-to maximal cuts do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' For example,  the following contribution to the box diagram appears to not be correct:  1  2  3  4  ↑  □ ˜B  B  X  ˜X  ˜Z  Z  A  A  ∼  (ε3 · x3)x2  1ε[µ  4 εν]  1 (2p2µ + x2µ)ε2ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='8)  Specifically, the term that spoils gauge invariance is proportional to ε1 · ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' It is clear that  the ghost Lagrangian (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='7) does not remove this diagram, nor is it removed by introducing  further obvious vector-ghost interactions, inspired by the fields already present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' While it  should be possible to introduce fine-tuned ghosts and interactions that precisely cancels this  diagram, it is non-trivial task to ensure the new terms are consistent with color-kinematics  duality and gauge invariance for all higher-multiplicity MHV numerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' We leave the  problem of formulating duality-satisfying one-loop-compatible Lagrangians to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  6  Conclusions  In this paper, we considered the problem of constructing Lagrangians that manifest color-  kinematics duality for YM theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Such explicit Lagrangians can be used to compute  BCJ numerators, as well as give non-trivial clues to the mathematical structure underlying  color-kinematics duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' While duality-satisfying tree-level numerators are known to any  multiplicity, finding corresponding Lagrangian descriptions appears to be more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  – 24 –  The problem simplifies by restricting to helicity sectors of YM, and in this paper we fully  address the NMHV sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  As a first step, we found a simple Lagrangian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='9) that is fully equivalent to the  standard YM Lagrangian at tree level, and it computes BCJ numerators in the bi-scalar  subsector of the NMHV sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' This Lagrangian was constructed by first resolving the four-  gluon contact term using a pair of auxiliary two-forms fields, and subsequently deforming  with new cubic interactions involving these auxiliary fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' The bi-scalar numerators can  be computed to any multiplicity, and they provide complete information via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='29) to  obtain all tree-level NMHV numerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' These provide a clear target for what a complete  NMHV Lagrangian should reproduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Next we searched for a complete NMHV Lagrangian, and we found that by introducing  at most two additional pairs of auxiliary fields, of vector and scalar type, there are several  solutions for such Lagrangians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Using a larger ansatz, we found that solutions also exists  if the scalar is removed at the cost of additional interactions between the remaining fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Because of the large freedoms of the Lagrangians, we chose to present the simplest solutions,  given in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='31) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='33), which were explicitly tested through ten points and conjec-  tured to work to all multiplicities at tree level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' With the our current limited Lagrangian  ansatz space, we cannot obtain the N2MHV and higher sector contributions to the BCJ  numerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Nor are the presented duality-satisfying NMHV Lagrangians equivalent to YM  in the N2MHV sector and beyond, unlike the simple Lagrangian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='9) first found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' It would  be desirable to revisit this problem in the future, and repeat the ansatz construction of  duality-satisfying NMHV Lagrangian while maintaining gauge covariance at intermediate  steps such that the N2MHV and higher sectors are not spoiled, even if they might not fully  enjoy color-kinematics duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  We briefly discussed the need of higher-rank tensor auxiliary fields to reproduce N2MHV  sector BCJ numerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' It is clear that in a covariant (D-dimensional) and local formalism  it is unavoidable to encounter, at the very minimum, three-form fields in a duality satisfying  N2MHV Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' However, the minimal set of needed auxiliary fields in the N2MHV  sector is something that needs further studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Currently the main challenge in brute-force  constructions of duality-satisfying Lagrangians is to predict the number and types of needed  auxiliary fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Clearly it would be desirable to better understand the general structure and  need of such fields such that more refined Lagrangian ansätze can be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Small  finely tuned ansätze would speed up progress because the equation systems encountered are  non-linear in the ansatz parameters, and the solutions contains large redundancy making  them difficult to analyse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Even in the NMHV sector there are more questions to be answered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Can one remove  some of the assumptions that went into our constructions and perhaps obtain much simpler  Lagrangians?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' For example, hints from Chern-Simons-type Lagrangians [40, 222] suggest  that it should be beneficial to look for more intricate kinetic terms than the diagonal ones  used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Furthermore, gauge covariance plays no role in the current construction  and this is likely an oversight that should be addressed in more refined attempts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' While  our NMHV Lagrangians fit on a few lines, it is fair to say that that their complexity are  likely artificially high compared to more optimal duality-satisfying rewritings of the YM  – 25 –  Lagrangian that might be found in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' Nevertheless, we have taken critical steps  in this research program by finding the first examples of NMHV Lagrangians that: 1) uses  very few auxiliary fields 2), have very simple structure in the bi-scalar subsector, and 3)  gives local all-multiplicity BCJ numerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='  Acknowledgments  We thank Zvi Bern, Lucile Cangemi, Gang Chen, Paolo Pichini, Oliver Schlotterer, Fei Teng,  Tianheng Wang and Maxim Zabzine for enlightening discussions related to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content=' This  research was supported in part by the Knut and Alice Wallenberg Foundation under grants  KAW 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='0116 (From Scattering Amplitudes to Gravitational Waves) and KAW 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
+page_content='0162  (Exploring a Web of Gravitational Theories through Gauge-Theory Methods), as well as the  Ragnar Söderberg Foundation (Swedish Foundations’ Starting Grant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AyT4oBgHgl3EQfZ_c4/content/2301.00233v1.pdf'}
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+The Democratic Illusion through the Technological Illusion: a Case Study 
+of the Implementation of a Blockchain to Support an E-voting Platform in 
+Moscow (Active Citizen)
+Hugo Estecahandy,
+September 2022
+From 1 to 4 July 2022, Muscovites were called upon to vote electronically for the location of
+the "Luhansk People's Republic" park in the capital. The vote was carried out via the online
+platform Active Citizen, a project of the Moscow City Hall that allows citizens to vote for or
+against urban development projects concerning the municipality. In this case, the vote did not
+ask Muscovites to choose between several names for a particular park, but rather to choose
+which of three proposed areas of the capital would be given a name that would resonate with
+current events in the country. Indeed, one of the reasons invoked by Russia to justify the
+invasion of Ukraine by its army, from 24 February 2022, is the protection and liberation of
+Donbass, a region in the east of Ukraine, notably composed of the People's Republic of
+Luhansk  and  the  People's  Republic  of  Donetsk.  These  Russian-speaking  territories,
+separatists from Ukraine since 2014, were officially recognised by Russia the day after the
+surprise attack, and are at the center of the Kremlin's war communication to justify its attack,
+both to the international community and to its own population. Thus, it is a name laden with
+heavy political symbolism that Muscovites can assign to a piece of their city.
+The location of the three proposed territories was indicated by geographical markers,
+mostly  the  mention  of  adjacent  streets,  and  were  mostly  enriched  by  specific  details,
+indicated in brackets. Thus, among the nearly 110,000 voters, 56.85% chose the plot "close to
+the British embassy", while 13.78% opted for a location "between the Lithuanian and Belgian
+embassies", 10.64% for a territory "close to the German consulate", while 18.73% relied on
+"the decision of specialists". While voters represented only 5% of Active Citizen registrants
+(about 2.2 million in total) and less than 1% of the city's total population (about 12.5 million),
+it is interesting to note that this highly political decision on urban planning was done using an
+electronic voting platform.
+Launched in 2014, Active citizen is one of three e-democracy platforms, a set of digital tools
+and technologies meant to consolidate the place of citizens in the governance of their
+territory, set up by the Moscow City Hall. The capital's inhabitants can also express certain
+problems encountered in their experience of the city on "Moscow - Our city", or propose
+solutions on a crowdsourcing platform. This use of digital participatory democracy solutions
+is in line with many projects in other major urban centers around the world that embrace the
+concept  of  "e-governance"  or  "smart  city".  Furthermore,  in  2017,  Moscow  City  Hall
+announced that this local digital democracy solution would be supported by a "blockchain", a
+technology underlying cryptocurrencies, which, according to the city hall, makes voting data
+immutable and freely searchable, as well as the source code, and that every Internet user can
+verify the integrity of the voting data. The communication states that voting in Active Citizen
+"is even more open" and that "it will be difficult to say that the administration misinterprets
+the answers, changes  the results  of the vote, when citizens  themselves  can verify this
+
+Different kinds of authority in Distributed Ledger Technologies
+Blockchain
+Consortium DLT
+Private DLT
+(Distributed authority)
+(Shared authority)
+(Centralized authority)
+Node with authority
+Authority connection
+Hugo Estecahandy
+Node without authority
+No authority
+September 2022Active citizen's e-voting platform's Distributed Ledger Technology
+Voting
+Active Citizen
+Node 5
+user
+Active Citizen website or
+smartphone application
+Node 1
+<DLT)
+Right of reading
+Master node
+Node 4
+Node 2
+Node 3
+Hugo Estecahandy,September 2022Class 5
+Class 4
+Class 3
+Class 2
+Class 1
+Service /
+City /
+Functioning /
+Benefits /
+Technology
+Actors
+Urbanism
+Uses
+Improvement
+13.84
+22.5%
+25.9
+network
+active citize belozerov
+service
+street
+host
+datum
+transport
+user
+trust
+offer
+store
+million
+great
+count
+system
+node
+park
+scale
+concern
+electronic
+peer-to-peer
+believe
+limit
+cast
+capital
+database
+peak
+result
+route
+farmer
+authority
+increase
+initiative
+minute
+department
+process
+Improve
+zone
+mln
+contractor
+vote
+cio
+play
+launch
+build
+website
+work
+state
+place
+pedestrian
+platform
+blockchain
+record
+public
+sign
+moscows
+technology
+test
+owner
+project
+influence
+e-voting
+speed
+market
+help
+government
+adoption
+list
+report
+forum
+election
+introduce
+project
+russian
+create
+challenge
+innovation
+work
+develop
+world
+attract
+aim
+urban
+develobment
+transparency
+enrnurane
+IASAnode
+opinion
+peer
+change
+state
+include
+database
+offerdevelopment
+innovation
+metro
+network
+resident
+serVice host
+place
+train
+list
+hope
+matter
+allow
+tech] street
+910
+influeneafficial
+park
+m/n
+server
+city
+limit
+wehsitsuncemepbte
+election
+initiative
+issue
+launch
+ governmerarge
+speed
+cast
+count
+capital
+policy
+improve
+moscow, smart
+ great
+lote
+help
+work
+pilot
+time
+datum
+scale
+challenge
+increase
+public
+transparency
+urban
+route
+blockchain
+create
+transport
+market
+interest
+million
+technologyl
+lauthority
+farmer
+world
+moscows
+transparent
+belozeroy
+concern
+result
+develop
+record
+democracy
+decide
+administrative
+test
+implemertctive citizen
+process
+architecture
+implementation
+ethereum
+bike
+platform
+system
+local
+poll
+introduce
+attract
+expansion
+contractor
+build
+choose
+muscovite
+user
+sign
+report
+announce
+account
+base
+encourage
+information
+conduct
+cost
+electronic
+open
+department
+pwa
+Hugo Estecahandy, September 2022
+avditinformation". Here, the Moscow authorities are playing on a widespread representation, in
+Russia and around the world, that direct-democratic voting in the Federation is biased and
+altered by corrupt authorities. As researcher Vladimir Pawlotsky notes, the city is "regularly
+plagued by corruption cases" (Pawlotsky, 2020a), particularly in the area of urban planning, a
+sector in which Active Citizen is supposed to strengthen the democratic power of Muscovites.
+This specialist in urban planning in the Russian capital also explains that when Sergei
+Sobyanin took over as mayor in 2010, "this unconditional supporter of Russian President
+Vladimir Putin wanted to improve Moscow's image so that the city could compete with New
+York, London, Tokyo, Paris and other major capitals. Thus, by the mere implementation, or
+even mention of the implementation, of a technology, the Moscow City Hall seems to be
+offering arguments in favor of its desire for transparency and participatory democracy, on the
+one hand, and its technological advance, on the other.
+The question then arises as to whether this implementation of a blockchain to support an
+online voting solution does not reinforce the political power centralized by the Moscow City
+Hall more than the distribution of this power among the population. This is also shown by the
+research of Jacob Zionts (2018) and Caroline Schlaufer (2020), to which this article will
+return. Similarly, the very use of blockchain to support electronic voting is questioned by
+some researchers, while explaining that other solutions are more appropriate (Çabuk et al.,
+2020, Blanchard & al., 2022).
+The aim of this article is therefore, firstly, to present the technical contours of Active
+Citizen and the place of blockchain technology in its operation. Then, we will expose the
+limits, both technical and ethical, of using a blockchain to support this voting system. Finally,
+thanks to a lexicometric analysis, we will try to analyze the potential gains for the Moscow
+City Hall that the use of a blockchain, or rather its announcement effect, entails, especially at
+the international level.
+I.
+A very limited technological and democratic contribution
+While  in 2017 the Moscow City Hall team  indicated  that the Active  Citizen  e-voting
+ecosystem would be supported by a blockchain, the reality is a bit more complex than that. A
+blockchain is a decentralized digital ledger, which can be replicated in as many copies as
+there  are  users,  and  which  binds  the  recorded  data  with  cryptographic  processes  that
+guarantee their inalterability. Created to support the cryptocurrency bitcoin, the blockchain is
+also characterized by a distribution of authority among all the nodes of the network: all
+participants theoretically have equivalent rights (to read, modify, decide, etc.). Moreover, the
+blockchain  used  within  Active  Citizen  is  an  adaptation  of  the  Ethereum  blockchain,
+supporting the ether cryptocurrency, promoting the same concepts of decentralization. Thus,
+without going into the technical details, the use of a voting register that is readable by all and
+secured, not by an institution to which the population must give its trust, but by mathematical
+algorithms, could theoretically bring a consolidation of the democratic tool. Following this
+logic, the assertion of researcher Leonid Smorgunov is acceptable. He noted in 2018 that "in
+the theory of public policy, there is an eternal gap of fair procedures that could be used to
+
+Different kinds of authority in Distributed Ledger Technologies
+Blockchain
+Consortium DLT
+Private DLT
+(Distributed authority)
+(Shared authority)
+(Centralized authority)
+Node with authority
+Authority connection
+Hugo Estecahandy
+Node without authority
+No authority
+September 2022Active citizen's e-voting platform's Distributed Ledger Technology
+Voting
+Active Citizen
+Node 5
+user
+Active Citizen website or
+smartphone application
+Node 1
+<DLT)
+Right of reading
+Master node
+Node 4
+Node 2
+Node 3
+Hugo Estecahandy,September 2022Class 5
+Class 4
+Class 3
+Class 2
+Class 1
+Service /
+City /
+Functioning /
+Benefits /
+Technology
+Actors
+Urbanism
+Uses
+Improvement
+13.84
+22.5%
+25.9
+network
+active citize belozerov
+service
+street
+host
+datum
+transport
+user
+trust
+offer
+store
+million
+great
+count
+system
+node
+park
+scale
+concern
+electronic
+peer-to-peer
+believe
+limit
+cast
+capital
+database
+peak
+result
+route
+farmer
+authority
+increase
+initiative
+minute
+department
+process
+Improve
+zone
+mln
+contractor
+vote
+cio
+play
+launch
+build
+website
+work
+state
+place
+pedestrian
+platform
+blockchain
+record
+public
+sign
+moscows
+technology
+test
+owner
+project
+influence
+e-voting
+speed
+market
+help
+government
+adoption
+list
+report
+forum
+election
+introduce
+project
+russian
+create
+challenge
+innovation
+work
+develop
+world
+attract
+aim
+urban
+develobment
+transparency
+enrnurane
+IASAnode
+opinion
+peer
+change
+state
+include
+database
+offerdevelopment
+innovation
+metro
+network
+resident
+serVice host
+place
+train
+list
+hope
+matter
+allow
+tech] street
+910
+influeneafficial
+park
+m/n
+server
+city
+limit
+wehsitsuncemepbte
+election
+initiative
+issue
+launch
+ governmerarge
+speed
+cast
+count
+capital
+policy
+improve
+moscow, smart
+ great
+lote
+help
+work
+pilot
+time
+datum
+scale
+challenge
+increase
+public
+transparency
+urban
+route
+blockchain
+create
+transport
+market
+interest
+million
+technologyl
+lauthority
+farmer
+world
+moscows
+transparent
+belozeroy
+concern
+result
+develop
+record
+democracy
+decide
+administrative
+test
+implemertctive citizen
+process
+architecture
+implementation
+ethereum
+bike
+platform
+system
+local
+poll
+introduce
+attract
+expansion
+contractor
+build
+choose
+muscovite
+user
+sign
+report
+announce
+account
+base
+encourage
+information
+conduct
+cost
+electronic
+open
+department
+pwa
+Hugo Estecahandy, September 2022
+avditachieve a fair outcome in the decision. Blockchain, as a network of distributed registers, often
+positions itself as an institution ensuring fairness in decisions by voting based on a consensus
+procedure" (Smorgunov, 2018.)
+ Since the first blockchain in 2008, and especially since the technology started to be
+industrially adapted in 2015/2016, derivatives of the original concept have been created.
+Thus, it was the appearance of distributed ledger technologies (DLT), proposing blockchains
+with a distribution of authority no longer distributed but shared, where only a few nodes of
+the network formed by the copies of the ledger had the maximum level of authority, or
+centralized in a single node that has authority over the others.
+ 
+In fact, the blockchain that supports online voting on Active Citizen seems to belong to the
+last of these categories. Indeed, if all the nodes of the network can read the data recorded on
+the register, there is an entity, in this case the Moscow City Hall - or rather the Internet or
+mobile application of Active Citizen via which the Internet users vote - which can write
+information, in this case the vote, its timestamp, the secret identifier of the voter. There is
+thus an important link in the Moscow City Hall's implementation of a blockchain, since the
+voting information is written on a node (Master node) that it controls, from which the other
+nodes are updated and can check that no information is subsequently altered.
+Figure 1
+
+Different kinds of authority in Distributed Ledger Technologies
+Blockchain
+Consortium DLT
+Private DLT
+(Distributed authority)
+(Shared authority)
+(Centralized authority)
+Node with authority
+Authority connection
+Hugo Estecahandy
+Node without authority
+No authority
+September 2022Active citizen's e-voting platform's Distributed Ledger Technology
+Voting
+Active Citizen
+Node 5
+user
+Active Citizen website or
+smartphone application
+Node 1
+<DLT)
+Right of reading
+Master node
+Node 4
+Node 2
+Node 3
+Hugo Estecahandy,September 2022Class 5
+Class 4
+Class 3
+Class 2
+Class 1
+Service /
+City /
+Functioning /
+Benefits /
+Technology
+Actors
+Urbanism
+Uses
+Improvement
+13.84
+22.5%
+25.9
+network
+active citize belozerov
+service
+street
+host
+datum
+transport
+user
+trust
+offer
+store
+million
+great
+count
+system
+node
+park
+scale
+concern
+electronic
+peer-to-peer
+believe
+limit
+cast
+capital
+database
+peak
+result
+route
+farmer
+authority
+increase
+initiative
+minute
+department
+process
+Improve
+zone
+mln
+contractor
+vote
+cio
+play
+launch
+build
+website
+work
+state
+place
+pedestrian
+platform
+blockchain
+record
+public
+sign
+moscows
+technology
+test
+owner
+project
+influence
+e-voting
+speed
+market
+help
+government
+adoption
+list
+report
+forum
+election
+introduce
+project
+russian
+create
+challenge
+innovation
+work
+develop
+world
+attract
+aim
+urban
+develobment
+transparency
+enrnurane
+IASAnode
+opinion
+peer
+change
+state
+include
+database
+offerdevelopment
+innovation
+metro
+network
+resident
+serVice host
+place
+train
+list
+hope
+matter
+allow
+tech] street
+910
+influeneafficial
+park
+m/n
+server
+city
+limit
+wehsitsuncemepbte
+election
+initiative
+issue
+launch
+ governmerarge
+speed
+cast
+count
+capital
+policy
+improve
+moscow, smart
+ great
+lote
+help
+work
+pilot
+time
+datum
+scale
+challenge
+increase
+public
+transparency
+urban
+route
+blockchain
+create
+transport
+market
+interest
+million
+technologyl
+lauthority
+farmer
+world
+moscows
+transparent
+belozeroy
+concern
+result
+develop
+record
+democracy
+decide
+administrative
+test
+implemertctive citizen
+process
+architecture
+implementation
+ethereum
+bike
+platform
+system
+local
+poll
+introduce
+attract
+expansion
+contractor
+build
+choose
+muscovite
+user
+sign
+report
+announce
+account
+base
+encourage
+information
+conduct
+cost
+electronic
+open
+department
+pwa
+Hugo Estecahandy, September 2022
+avditActive Citizen's DTL ultimately allows some observers to ensure that no information in the
+registry is altered, but the validity of this information is the responsibility of one actor:
+Moscow City Hall. As Blanchard et al. (2022) write, "the link between the real world and the
+blockchain depends on the fact that all parties commit to rules. The only truth that is
+guaranteed by writing information on a blockchain is that the information in question is
+written on the blockchain in question. There is no evidence here that the data originally
+written is reliable.
+II.
+Active citizen or the democratic illusion
+Thus, as we have just seen, the implementation of a blockchain to support the Active Citizen
+voting system is  only very limited,  both technologically  and democratically.  However,
+Moscow City Hall has an interest in offering this service to its citizens, whether it is secured
+by a blockchain or not. Researcher Caroline Schlaufer has looked at the case of Active
+Citizen, asking "Why do non-democratic regimes promote e-participation" (Schlaufer, 2020).
+She  argues  that  simply  offering  an  online  voting  system,  "can  accommodate  different
+legitimation strategies into one single online tool. A well-designed e-participation tool serves,
+first, as an input channel  for citizen preferences, thereby enhancing input legitimation.
+Second, it can increase output-based legitimation by increasing citizen satisfaction with
+Figure 2
+
+Different kinds of authority in Distributed Ledger Technologies
+Blockchain
+Consortium DLT
+Private DLT
+(Distributed authority)
+(Shared authority)
+(Centralized authority)
+Node with authority
+Authority connection
+Hugo Estecahandy
+Node without authority
+No authority
+September 2022Active citizen's e-voting platform's Distributed Ledger Technology
+Voting
+Active Citizen
+Node 5
+user
+Active Citizen website or
+smartphone application
+Node 1
+<DLT)
+Right of reading
+Master node
+Node 4
+Node 2
+Node 3
+Hugo Estecahandy,September 2022Class 5
+Class 4
+Class 3
+Class 2
+Class 1
+Service /
+City /
+Functioning /
+Benefits /
+Technology
+Actors
+Urbanism
+Uses
+Improvement
+13.84
+22.5%
+25.9
+network
+active citize belozerov
+service
+street
+host
+datum
+transport
+user
+trust
+offer
+store
+million
+great
+count
+system
+node
+park
+scale
+concern
+electronic
+peer-to-peer
+believe
+limit
+cast
+capital
+database
+peak
+result
+route
+farmer
+authority
+increase
+initiative
+minute
+department
+process
+Improve
+zone
+mln
+contractor
+vote
+cio
+play
+launch
+build
+website
+work
+state
+place
+pedestrian
+platform
+blockchain
+record
+public
+sign
+moscows
+technology
+test
+owner
+project
+influence
+e-voting
+speed
+market
+help
+government
+adoption
+list
+report
+forum
+election
+introduce
+project
+russian
+create
+challenge
+innovation
+work
+develop
+world
+attract
+aim
+urban
+develobment
+transparency
+enrnurane
+IASAnode
+opinion
+peer
+change
+state
+include
+database
+offerdevelopment
+innovation
+metro
+network
+resident
+serVice host
+place
+train
+list
+hope
+matter
+allow
+tech] street
+910
+influeneafficial
+park
+m/n
+server
+city
+limit
+wehsitsuncemepbte
+election
+initiative
+issue
+launch
+ governmerarge
+speed
+cast
+count
+capital
+policy
+improve
+moscow, smart
+ great
+lote
+help
+work
+pilot
+time
+datum
+scale
+challenge
+increase
+public
+transparency
+urban
+route
+blockchain
+create
+transport
+market
+interest
+million
+technologyl
+lauthority
+farmer
+world
+moscows
+transparent
+belozeroy
+concern
+result
+develop
+record
+democracy
+decide
+administrative
+test
+implemertctive citizen
+process
+architecture
+implementation
+ethereum
+bike
+platform
+system
+local
+poll
+introduce
+attract
+expansion
+contractor
+build
+choose
+muscovite
+user
+sign
+report
+announce
+account
+base
+encourage
+information
+conduct
+cost
+electronic
+open
+department
+pwa
+Hugo Estecahandy, September 2022
+avditpolicy performance. Third, an e-participation platform also disseminates the narrative of a
+modern, inclusive and responsive government. Thus, Active Citizen would above all give
+Muscovites the illusion that they are integrated into a more democratic process thought out by
+an administration at the cutting edge of new technologies.
+This democratic illusion is even more perceptible when we look at the types of votes
+that are proposed. For example, between January 2019 and mid-April of the same year, the
+five projects with the most votes were the opening of a pediatric operating theater in a city
+hospital, a 24-hour helicopter medical evacuation service, the creation of a "protected nature
+zone" in the Bratayevskaya Poima Park to preserve local wildlife, free admission to the city's
+museums once every three weeks and, finally, the introduction of modern trains to connect
+the Moscow Belarusskaya station with the city of Odintsovo (Moscow Oblast). Moreover, as
+Jacob Zionts writes, "Active Citizen does little to change the underlying distribution of power
+in Moscow politics: voting on the name of a new metro line does not disrupt endemic
+corruption; selecting new seat colors for the Luzhniki Big Sports Arena will not empower
+investigative journalists. Active Citizen grants Muscovites limited self-expression but only in
+a safe and sanitized form's (Zionts, 2018).
+The researcher draws a parallel between the use of the platform and the legitimization
+of urban projects that could have led to public rejection. In 2017, with the blessing of Russian
+President Vladimir Putin, Moscow Mayor Sergei Sobyanin planned the destruction of several
+thousand homes: Soviet buildings that he wanted to replace with modern real estate projects
+by wealthy developers. As anger began to grow among the population of the capital, a
+"referendum" was proposed to the inhabitants via Active citizens. They could vote for or
+against the 'renovation' of their buildings, to which 90% of them replied in the affirmative.
+Then, "officials quickly pointed to the overwhelming support among residents as support for
+the demolition."  As one of the interviewed people said to Schlaufer during her research, 'of
+course, if a majority of citizens agreed with Active Citizen about something, they will,
+afterwards,  not  complain  about  it.  And  it  leads  to  a  higher  satisfaction  of  the  city
+government's activities' (Schlaufer, 2020). In addition to giving a democratic illusion to some
+Muscovites, the voting platform would allow the mayor's office to make them accept, in spite
+of themselves, policies that they would logically have opposed.
+III.
+An international digital showcase for Moscow?
+But this use of an electronic voting platform does not only seem to allow the Moscow
+mayor's office to consolidate its power and its image in Moscow or in Russia, but also
+internationally. As Vladimir Pawlotsky explains, Mayor Sobyanin's aim is to make Moscow
+"a global city" and "a world center of financial stability and innovation" that "an economy
+dependent on the - very volatile - price of hydrocarbons" requires. Moscow sought - since the
+war in Ukraine has surely put this strategy into question - 'as a privileged platform for foreign
+investment, to capture international economic and financial flows by hosting the headquarters
+and  subsidiaries  of  transnational  corporations,  developing  cultural  activities  as  well  as
+tourism' (Pawlotsky, 2020a). Thus, Moscow, and by extension Russia, are in a logic of city
+branding and nation branding internationally. The 2018 Football World Cup helped to present
+the city and the rest of the country to the world.  
+
+Different kinds of authority in Distributed Ledger Technologies
+Blockchain
+Consortium DLT
+Private DLT
+(Distributed authority)
+(Shared authority)
+(Centralized authority)
+Node with authority
+Authority connection
+Hugo Estecahandy
+Node without authority
+No authority
+September 2022Active citizen's e-voting platform's Distributed Ledger Technology
+Voting
+Active Citizen
+Node 5
+user
+Active Citizen website or
+smartphone application
+Node 1
+<DLT)
+Right of reading
+Master node
+Node 4
+Node 2
+Node 3
+Hugo Estecahandy,September 2022Class 5
+Class 4
+Class 3
+Class 2
+Class 1
+Service /
+City /
+Functioning /
+Benefits /
+Technology
+Actors
+Urbanism
+Uses
+Improvement
+13.84
+22.5%
+25.9
+network
+active citize belozerov
+service
+street
+host
+datum
+transport
+user
+trust
+offer
+store
+million
+great
+count
+system
+node
+park
+scale
+concern
+electronic
+peer-to-peer
+believe
+limit
+cast
+capital
+database
+peak
+result
+route
+farmer
+authority
+increase
+initiative
+minute
+department
+process
+Improve
+zone
+mln
+contractor
+vote
+cio
+play
+launch
+build
+website
+work
+state
+place
+pedestrian
+platform
+blockchain
+record
+public
+sign
+moscows
+technology
+test
+owner
+project
+influence
+e-voting
+speed
+market
+help
+government
+adoption
+list
+report
+forum
+election
+introduce
+project
+russian
+create
+challenge
+innovation
+work
+develop
+world
+attract
+aim
+urban
+develobment
+transparency
+enrnurane
+IASAnode
+opinion
+peer
+change
+state
+include
+database
+offerdevelopment
+innovation
+metro
+network
+resident
+serVice host
+place
+train
+list
+hope
+matter
+allow
+tech] street
+910
+influeneafficial
+park
+m/n
+server
+city
+limit
+wehsitsuncemepbte
+election
+initiative
+issue
+launch
+ governmerarge
+speed
+cast
+count
+capital
+policy
+improve
+moscow, smart
+ great
+lote
+help
+work
+pilot
+time
+datum
+scale
+challenge
+increase
+public
+transparency
+urban
+route
+blockchain
+create
+transport
+market
+interest
+million
+technologyl
+lauthority
+farmer
+world
+moscows
+transparent
+belozeroy
+concern
+result
+develop
+record
+democracy
+decide
+administrative
+test
+implemertctive citizen
+process
+architecture
+implementation
+ethereum
+bike
+platform
+system
+local
+poll
+introduce
+attract
+expansion
+contractor
+build
+choose
+muscovite
+user
+sign
+report
+announce
+account
+base
+encourage
+information
+conduct
+cost
+electronic
+open
+department
+pwa
+Hugo Estecahandy, September 2022
+avditThe Active Citizen project and the announcement of the implementation of a blockchain also
+meet the objectives of this strategy, communicating a democratic and innovative image of
+Moscow to the rest of the world. Thus, the announcement of the addition of a blockchain to
+Active citizen was shared internationally, by more or less specialized English-speaking
+media. It is interesting to note that most of these articles repeat the December 2017 press
+release from the city hall and its language. Let's remember that this communiqué made much
+of the codes dear to cryptoanarchists and other defenders of a free Internet, some of which are
+founding principles of blockchain: distribution of the verification of the voting process, free
+access to the underlying protocol or incorruptibility of the decisions taken by Muscovites
+concerning the future of their city.
+In order to test our hypothesis that the foreign media treatment of the implementation of a
+blockchain within Active Citizen has helped to project a democratic and modern image of
+Moscow, we conducted a lexicometric analysis on 6 of these English-language press articles
+(see Appendix 1 for the article references). This analysis has been conducted through the
+software IRaMuTeQ, developed at Toulouse's University of Mirail (France). Above all, the
+software allows us to highlight classes of words that it will automatically detect using a
+classification method called the Reinert method. As it can be seen on the Figure 3, the Reinert
+classification brings out 5 classes of words, out of the text corpus formed by our 6 articles.
+Figure 3
+
+Different kinds of authority in Distributed Ledger Technologies
+Blockchain
+Consortium DLT
+Private DLT
+(Distributed authority)
+(Shared authority)
+(Centralized authority)
+Node with authority
+Authority connection
+Hugo Estecahandy
+Node without authority
+No authority
+September 2022Active citizen's e-voting platform's Distributed Ledger Technology
+Voting
+Active Citizen
+Node 5
+user
+Active Citizen website or
+smartphone application
+Node 1
+<DLT)
+Right of reading
+Master node
+Node 4
+Node 2
+Node 3
+Hugo Estecahandy,September 2022Class 5
+Class 4
+Class 3
+Class 2
+Class 1
+Service /
+City /
+Functioning /
+Benefits /
+Technology
+Actors
+Urbanism
+Uses
+Improvement
+13.84
+22.5%
+25.9
+network
+active citize belozerov
+service
+street
+host
+datum
+transport
+user
+trust
+offer
+store
+million
+great
+count
+system
+node
+park
+scale
+concern
+electronic
+peer-to-peer
+believe
+limit
+cast
+capital
+database
+peak
+result
+route
+farmer
+authority
+increase
+initiative
+minute
+department
+process
+Improve
+zone
+mln
+contractor
+vote
+cio
+play
+launch
+build
+website
+work
+state
+place
+pedestrian
+platform
+blockchain
+record
+public
+sign
+moscows
+technology
+test
+owner
+project
+influence
+e-voting
+speed
+market
+help
+government
+adoption
+list
+report
+forum
+election
+introduce
+project
+russian
+create
+challenge
+innovation
+work
+develop
+world
+attract
+aim
+urban
+develobment
+transparency
+enrnurane
+IASAnode
+opinion
+peer
+change
+state
+include
+database
+offerdevelopment
+innovation
+metro
+network
+resident
+serVice host
+place
+train
+list
+hope
+matter
+allow
+tech] street
+910
+influeneafficial
+park
+m/n
+server
+city
+limit
+wehsitsuncemepbte
+election
+initiative
+issue
+launch
+ governmerarge
+speed
+cast
+count
+capital
+policy
+improve
+moscow, smart
+ great
+lote
+help
+work
+pilot
+time
+datum
+scale
+challenge
+increase
+public
+transparency
+urban
+route
+blockchain
+create
+transport
+market
+interest
+million
+technologyl
+lauthority
+farmer
+world
+moscows
+transparent
+belozeroy
+concern
+result
+develop
+record
+democracy
+decide
+administrative
+test
+implemertctive citizen
+process
+architecture
+implementation
+ethereum
+bike
+platform
+system
+local
+poll
+introduce
+attract
+expansion
+contractor
+build
+choose
+muscovite
+user
+sign
+report
+announce
+account
+base
+encourage
+information
+conduct
+cost
+electronic
+open
+department
+pwa
+Hugo Estecahandy, September 2022
+avditClass 1 is the most interesting for us, as it refers to all the words that refer to the positive
+impact of the implementation of blockchain by the Moscow City Hall. And among these
+words we find several terms that are part of the lexical field of blockchain technology:
+"trust," "transparency," "innovation," "improve," "increase."
+Then we also performed a similarity analysis on the same body of text. This graph shows the
+words and their frequency. The larger the word, the more often it appears in the text. The
+words are spatialized according to their proximity to the text.
+Figure 4
+Blockchain is the most present word as we have seen on the cloud and it is in the center of
+this graph. What we see here is interesting for two reasons: many words in the lexical field of
+democracy (in red) are directly related to the term blockchain, a term to which the words
+"Moscow" and "Activie_Citzen" are also strongly related. The lexical field of innovation is
+also very present, with terms directly linked to "Moscow." This graph allows us to see how
+the media’s treatment of the implementation of a blockchain by the Moscow City Council
+served its international communication objectives.
+
+Different kinds of authority in Distributed Ledger Technologies
+Blockchain
+Consortium DLT
+Private DLT
+(Distributed authority)
+(Shared authority)
+(Centralized authority)
+Node with authority
+Authority connection
+Hugo Estecahandy
+Node without authority
+No authority
+September 2022Active citizen's e-voting platform's Distributed Ledger Technology
+Voting
+Active Citizen
+Node 5
+user
+Active Citizen website or
+smartphone application
+Node 1
+<DLT)
+Right of reading
+Master node
+Node 4
+Node 2
+Node 3
+Hugo Estecahandy,September 2022Class 5
+Class 4
+Class 3
+Class 2
+Class 1
+Service /
+City /
+Functioning /
+Benefits /
+Technology
+Actors
+Urbanism
+Uses
+Improvement
+13.84
+22.5%
+25.9
+network
+active citize belozerov
+service
+street
+host
+datum
+transport
+user
+trust
+offer
+store
+million
+great
+count
+system
+node
+park
+scale
+concern
+electronic
+peer-to-peer
+believe
+limit
+cast
+capital
+database
+peak
+result
+route
+farmer
+authority
+increase
+initiative
+minute
+department
+process
+Improve
+zone
+mln
+contractor
+vote
+cio
+play
+launch
+build
+website
+work
+state
+place
+pedestrian
+platform
+blockchain
+record
+public
+sign
+moscows
+technology
+test
+owner
+project
+influence
+e-voting
+speed
+market
+help
+government
+adoption
+list
+report
+forum
+election
+introduce
+project
+russian
+create
+challenge
+innovation
+work
+develop
+world
+attract
+aim
+urban
+develobment
+transparency
+enrnurane
+IASAnode
+opinion
+peer
+change
+state
+include
+database
+offerdevelopment
+innovation
+metro
+network
+resident
+serVice host
+place
+train
+list
+hope
+matter
+allow
+tech] street
+910
+influeneafficial
+park
+m/n
+server
+city
+limit
+wehsitsuncemepbte
+election
+initiative
+issue
+launch
+ governmerarge
+speed
+cast
+count
+capital
+policy
+improve
+moscow, smart
+ great
+lote
+help
+work
+pilot
+time
+datum
+scale
+challenge
+increase
+public
+transparency
+urban
+route
+blockchain
+create
+transport
+market
+interest
+million
+technologyl
+lauthority
+farmer
+world
+moscows
+transparent
+belozeroy
+concern
+result
+develop
+record
+democracy
+decide
+administrative
+test
+implemertctive citizen
+process
+architecture
+implementation
+ethereum
+bike
+platform
+system
+local
+poll
+introduce
+attract
+expansion
+contractor
+build
+choose
+muscovite
+user
+sign
+report
+announce
+account
+base
+encourage
+information
+conduct
+cost
+electronic
+open
+department
+pwa
+Hugo Estecahandy, September 2022
+avditConclusion
+The implementation of a blockchain in the voting system of Active Citizen has above all
+allowed to consolidate the discourse that the Moscow City Hall wishes to attach to this
+application. Thus, with the Active cities project "the idea is not to make Muscovites vote on
+major issues - far from it - but to ask them questions designed to elicit answers that
+correspond to the objectives of the city government, thus helping to legitimize its policies
+(Pawlotsky, 2020b). From what we have seen, these goals were also a consolidation of the
+positive feelings of its population towards a democratic and modern city. It also participated
+in the creation of an international brand image of Moscow as we could understand through
+the analysis of press materials of foreign media. Recently, we could observe what could be a
+new type of recourse to Active Citizen for the authorities, which allowed renaming a part of
+the urban area in front of the British embassy and thus artificially correlate what would
+represent the ideas of Muscovites with the geopolitical ambitions of the Kremlin and its
+discourse of justification of the war in Ukraine. In addition to the above-mentioned benefits,
+Active Citizen seems to allow the Moscow City Hall to attract even more favors from the
+Kremlin  under  the  cover  of  a  completely  illusory  participatory  democracy.  And  the
+announcement of the implementation of blockchain has contributed to this.
+Through the common thought that "blockchain  technology provides the kind of
+transparency and immutability that could sustain democracy, Moscow's implementation of
+the technology through Active Citizen has provided only the illusion of empowerment; the
+political system in Moscow remains autocratic" (Zionts, 2018). And this allows us to go
+further and quote Schlaufer as our results also showed "that the introduction of e-participation
+in a non-democratic regime is not a contradiction. Therefore, controlled e-participation is not
+a challenge to the underlying distribution of power between governments and citizens"
+(Schlaufer, 2020).
+References
+●
+Alexandrov D. A., Boukhanovsky A. V., Chugunov A. V., Kabanov Y., Koltsova O. 
+(éd.). Digital Transformation and Global Society: Third International Conference, 
+DTGS 2018, St. Petersburg, Russia, May 30 – June 2, 2018, Revised Selected Papers, 
+Part I [En ligne]. Cham : Springer International Publishing, 2018. (Communications 
+in Computer and Information Science.) Disponible sur : < 
+https://doi.org/10.1007/978-3-030-02843-5 > (consulté le 13 septembre 2022)ISBN : 
+978-3-030-02842-8.
+●
+Blanchard E., Vigni F. L., Rauzy P. « Critique de l’article de Serge Surin “Le 
+renouveau de la promesse du vote électronique. Étude du vote par chaîne de blocs au 
+regard du système électoral français” ». p. 20.
+●
+Çabuk U. C., Adıgüzel E., Karaarslan E. « A Survey on Feasibility and Suitability of 
+Blockchain Techniques for the E-Voting Systems ». International Journal of 
+Advanced Research in Computer and Communication Engineering [En ligne]. Thirty 
+
+Different kinds of authority in Distributed Ledger Technologies
+Blockchain
+Consortium DLT
+Private DLT
+(Distributed authority)
+(Shared authority)
+(Centralized authority)
+Node with authority
+Authority connection
+Hugo Estecahandy
+Node without authority
+No authority
+September 2022Active citizen's e-voting platform's Distributed Ledger Technology
+Voting
+Active Citizen
+Node 5
+user
+Active Citizen website or
+smartphone application
+Node 1
+<DLT)
+Right of reading
+Master node
+Node 4
+Node 2
+Node 3
+Hugo Estecahandy,September 2022Class 5
+Class 4
+Class 3
+Class 2
+Class 1
+Service /
+City /
+Functioning /
+Benefits /
+Technology
+Actors
+Urbanism
+Uses
+Improvement
+13.84
+22.5%
+25.9
+network
+active citize belozerov
+service
+street
+host
+datum
+transport
+user
+trust
+offer
+store
+million
+great
+count
+system
+node
+park
+scale
+concern
+electronic
+peer-to-peer
+believe
+limit
+cast
+capital
+database
+peak
+result
+route
+farmer
+authority
+increase
+initiative
+minute
+department
+process
+Improve
+zone
+mln
+contractor
+vote
+cio
+play
+launch
+build
+website
+work
+state
+place
+pedestrian
+platform
+blockchain
+record
+public
+sign
+moscows
+technology
+test
+owner
+project
+influence
+e-voting
+speed
+market
+help
+government
+adoption
+list
+report
+forum
+election
+introduce
+project
+russian
+create
+challenge
+innovation
+work
+develop
+world
+attract
+aim
+urban
+develobment
+transparency
+enrnurane
+IASAnode
+opinion
+peer
+change
+state
+include
+database
+offerdevelopment
+innovation
+metro
+network
+resident
+serVice host
+place
+train
+list
+hope
+matter
+allow
+tech] street
+910
+influeneafficial
+park
+m/n
+server
+city
+limit
+wehsitsuncemepbte
+election
+initiative
+issue
+launch
+ governmerarge
+speed
+cast
+count
+capital
+policy
+improve
+moscow, smart
+ great
+lote
+help
+work
+pilot
+time
+datum
+scale
+challenge
+increase
+public
+transparency
+urban
+route
+blockchain
+create
+transport
+market
+interest
+million
+technologyl
+lauthority
+farmer
+world
+moscows
+transparent
+belozeroy
+concern
+result
+develop
+record
+democracy
+decide
+administrative
+test
+implemertctive citizen
+process
+architecture
+implementation
+ethereum
+bike
+platform
+system
+local
+poll
+introduce
+attract
+expansion
+contractor
+build
+choose
+muscovite
+user
+sign
+report
+announce
+account
+base
+encourage
+information
+conduct
+cost
+electronic
+open
+department
+pwa
+Hugo Estecahandy, September 2022
+avditmars 2018. Vol. 7, n°3, p. 124-134. Disponible sur : < 
+https://doi.org/10.17148/IJARCCE.2018.7324 >
+●
+N. S. Vinogradovа, O. A. Moiseeva. « Open Government and “e-Government” in 
+Russia ». SS [En ligne]. 28 janvier 2015. Vol. 5, n°1,. Disponible sur : < 
+https://doi.org/10.17265/2159-5526/2015.01.004 > (consulté le 5 septembre 2022)
+●
+Pawlotsky V. « Moscou se rêve en « ville globale » ». Le Monde diplomatique. 
+novembre 2020a.
+●
+Pawlotsky V. « Moscou se rêve en « ville globale » ». Le Monde diplomatique. 
+novembre 2020b.
+●
+Schlaufer C. « Why do nondemocratic regimes promote E‐PARTICIPATION ? The case 
+of Moscow’s active citizen online voting platform ». Governance [En ligne]. 23 août 
+2020. Disponible sur : < https://doi.org/10.1111/gove.12531 > (consulté le 8 juin 
+2021)
+●
+Smorgunov L. « Blockchain and a Problem of Procedural Justice of Public Choice ». 
+In : Alexandrov DA, Boukhanovsky AV, Chugunov AV, Kabanov Y, Koltsova O 
+(éd.). Digital Transformation and Global Society: Third International Conference, 
+DTGS 2018, St. Petersburg, Russia, May 30 – June 2, 2018, Revised Selected Papers, 
+Part I [En ligne]. Cham : Springer International Publishing, 2018. Disponible sur : < 
+https://doi.org/10.1007/978-3-030-02843-5 > (consulté le 13 septembre 2022)ISBN : 
+978-3-030-02842-8.
+●
+Zionts J. « “Blockchain Democracy” and the Misleading Promise of Moscow’s Active
+Citizen Program ». In : Medium [En ligne]. [s.l.] : [s.n.], 2018. Disponible sur : < 
+https://media.consensys.net/blockchain-democracy-and-the-misleading-promise-of-
+moscows-active-citizen-program-2eda7141c5b9 > (consulté le 7 juin 2021)
+Annexe 1 (Analyzed articles references) :
+1. « Moscow introduces blockchain e-voting », SmartCitiesWorld, 12/05/2017.
+2. « Moscow to Develop a Blockchain System for Transparent City Services », CoinDesk, 
+08/15/2019.
+3. « Moscow to pilot blockchain platform for e-voting », FinExtra, 12/04/2017.
+4. « Russia Is Leading the Push for Blockchain Democracy », CoinDesk, 02/18/2018.
+5. « How Moscow is pioneering the rise of smart cities », CIO, 11/06/2017.
+6. « This is How Russia Utilizing Blockchain to Enhance Democracy », Sputnik News, 
+03/02/2018.
+
+Different kinds of authority in Distributed Ledger Technologies
+Blockchain
+Consortium DLT
+Private DLT
+(Distributed authority)
+(Shared authority)
+(Centralized authority)
+Node with authority
+Authority connection
+Hugo Estecahandy
+Node without authority
+No authority
+September 2022Active citizen's e-voting platform's Distributed Ledger Technology
+Voting
+Active Citizen
+Node 5
+user
+Active Citizen website or
+smartphone application
+Node 1
+<DLT)
+Right of reading
+Master node
+Node 4
+Node 2
+Node 3
+Hugo Estecahandy,September 2022Class 5
+Class 4
+Class 3
+Class 2
+Class 1
+Service /
+City /
+Functioning /
+Benefits /
+Technology
+Actors
+Urbanism
+Uses
+Improvement
+13.84
+22.5%
+25.9
+network
+active citize belozerov
+service
+street
+host
+datum
+transport
+user
+trust
+offer
+store
+million
+great
+count
+system
+node
+park
+scale
+concern
+electronic
+peer-to-peer
+believe
+limit
+cast
+capital
+database
+peak
+result
+route
+farmer
+authority
+increase
+initiative
+minute
+department
+process
+Improve
+zone
+mln
+contractor
+vote
+cio
+play
+launch
+build
+website
+work
+state
+place
+pedestrian
+platform
+blockchain
+record
+public
+sign
+moscows
+technology
+test
+owner
+project
+influence
+e-voting
+speed
+market
+help
+government
+adoption
+list
+report
+forum
+election
+introduce
+project
+russian
+create
+challenge
+innovation
+work
+develop
+world
+attract
+aim
+urban
+develobment
+transparency
+enrnurane
+IASAnode
+opinion
+peer
+change
+state
+include
+database
+offerdevelopment
+innovation
+metro
+network
+resident
+serVice host
+place
+train
+list
+hope
+matter
+allow
+tech] street
+910
+influeneafficial
+park
+m/n
+server
+city
+limit
+wehsitsuncemepbte
+election
+initiative
+issue
+launch
+ governmerarge
+speed
+cast
+count
+capital
+policy
+improve
+moscow, smart
+ great
+lote
+help
+work
+pilot
+time
+datum
+scale
+challenge
+increase
+public
+transparency
+urban
+route
+blockchain
+create
+transport
+market
+interest
+million
+technologyl
+lauthority
+farmer
+world
+moscows
+transparent
+belozeroy
+concern
+result
+develop
+record
+democracy
+decide
+administrative
+test
+implemertctive citizen
+process
+architecture
+implementation
+ethereum
+bike
+platform
+system
+local
+poll
+introduce
+attract
+expansion
+contractor
+build
+choose
+muscovite
+user
+sign
+report
+announce
+account
+base
+encourage
+information
+conduct
+cost
+electronic
+open
+department
+pwa
+Hugo Estecahandy, September 2022
+avdit
\ No newline at end of file
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new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf,len=2265
+page_content='The Democratic Illusion through the Technological Illusion: a Case Study  of the Implementation of a Blockchain to Support an E-voting Platform in  Moscow (Active Citizen)  Hugo Estecahandy,  September 2022  From 1 to 4 July 2022, Muscovites were called upon to vote electronically for the location of  the "Luhansk People\'s Republic" park in the capital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' The vote was carried out via the online  platform Active Citizen, a project of the Moscow City Hall that allows citizens to vote for or  against urban development projects concerning the municipality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' In this case, the vote did not  ask Muscovites to choose between several names for a particular park, but rather to choose  which of three proposed areas of the capital would be given a name that would resonate with  current events in the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=" Indeed, one of the reasons invoked by Russia to justify the  invasion of Ukraine by its army, from 24 February 2022, is the protection and liberation of  Donbass, a region in the east of Ukraine, notably composed of the People's Republic of  Luhansk  and  the  People's  Republic  of  Donetsk." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=" These  Russian-speaking  territories,  separatists from Ukraine since 2014, were officially recognised by Russia the day after the  surprise attack, and are at the center of the Kremlin's war communication to justify its attack,  both to the international community and to its own population." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Thus, it is a name laden with  heavy political symbolism that Muscovites can assign to a piece of their city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  The location of the three proposed territories was indicated by geographical markers,  mostly  the  mention  of  adjacent  streets,  and  were  mostly  enriched  by  specific  details,  indicated in brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Thus, among the nearly 110,000 voters, 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='85% chose the plot "close to  the British embassy", while 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='78% opted for a location "between the Lithuanian and Belgian  embassies", 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='64% for a territory "close to the German consulate", while 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='73% relied on  "the decision of specialists".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' While voters represented only 5% of Active Citizen registrants  (about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content="2 million in total) and less than 1% of the city's total population (about 12." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='5 million),  it is interesting to note that this highly political decision on urban planning was done using an  electronic voting platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  Launched in 2014, Active citizen is one of three e-democracy platforms, a set of digital tools  and technologies meant to consolidate the place of citizens in the governance of their  territory, set up by the Moscow City Hall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' The capital\'s inhabitants can also express certain  problems encountered in their experience of the city on "Moscow - Our city", or propose  solutions on a crowdsourcing platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' This use of digital participatory democracy solutions  is in line with many projects in other major urban centers around the world that embrace the  concept  of  "e-governance"  or  "smart  city".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Furthermore,  in  2017,  Moscow  City  Hall  announced that this local digital democracy solution would be supported by a "blockchain", a  technology underlying cryptocurrencies, which, according to the city hall, makes voting data  immutable and freely searchable, as well as the source code, and that every Internet user can  verify the integrity of the voting data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' The communication states that voting in Active Citizen  "is even more open" and that "it will be difficult to say that the administration misinterprets  the answers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' changes  the results  of the vote,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' when citizens  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='themselves  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='can verify this  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Different kinds of authority in Distributed Ledger Technologies  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Blockchain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Consortium DLT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Private DLT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='(Distributed authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='(Shared authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='(Centralized authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node with authority  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Authority connection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Hugo Estecahandy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node without authority  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='No authority  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content="September 2022Active citizen's e-voting platform's Distributed Ledger Technology  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='Active Citizen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='Node 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='<DLT)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Right of reading  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Master node  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='Hugo Estecahandy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='September 2022Class 5  Class 4  Class 3  Class 2  Class 1  Service /  City /  Functioning /  Benefits /  Technology  Actors  Urbanism  Uses  Improvement  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='84  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='5%  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content=' As researcher Vladimir Pawlotsky notes, the city is "regularly  plagued by corruption cases" (Pawlotsky, 2020a), particularly in the area of urban planning, a  sector in which Active Citizen is supposed to strengthen the democratic power of Muscovites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  This specialist in urban planning in the Russian capital also explains that when Sergei  Sobyanin took over as mayor in 2010, "this unconditional supporter of Russian President  Vladimir Putin wanted to improve Moscow\'s image so that the city could compete with New  York, London, Tokyo, Paris and other major capitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Thus, by the mere implementation, or  even mention of the implementation, of a technology, the Moscow City Hall seems to be  offering arguments in favor of its desire for transparency and participatory democracy, on the  one hand, and its technological advance, on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  The question then arises as to whether this implementation of a blockchain to support an  online voting solution does not reinforce the political power centralized by the Moscow City  Hall more than the distribution of this power among the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' This is also shown by the  research of Jacob Zionts (2018) and Caroline Schlaufer (2020), to which this article will  return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Similarly, the very use of blockchain to support electronic voting is questioned by  some researchers, while explaining that other solutions are more appropriate (Çabuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=',  2020, Blanchard & al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  The aim of this article is therefore, firstly, to present the technical contours of Active  Citizen and the place of blockchain technology in its operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Then, we will expose the  limits, both technical and ethical, of using a blockchain to support this voting system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Finally,  thanks to a lexicometric analysis, we will try to analyze the potential gains for the Moscow  City Hall that the use of a blockchain, or rather its announcement effect, entails, especially at  the international level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  A very limited technological and democratic contribution  While  in 2017 the Moscow City Hall team  indicated  that the Active  Citizen  e-voting  ecosystem would be supported by a blockchain, the reality is a bit more complex than that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' A  blockchain is a decentralized digital ledger, which can be replicated in as many copies as  there  are  users,  and  which  binds  the  recorded  data  with  cryptographic  processes  that  guarantee their inalterability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Created to support the cryptocurrency bitcoin, the blockchain is  also characterized by a distribution of authority among all the nodes of the network: all  participants theoretically have equivalent rights (to read, modify, decide, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Moreover, the  blockchain  used  within  Active  Citizen  is  an  adaptation  of  the  Ethereum  blockchain,  supporting the ether cryptocurrency, promoting the same concepts of decentralization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Thus,  without going into the technical details, the use of a voting register that is readable by all and  secured, not by an institution to which the population must give its trust, but by mathematical  algorithms, could theoretically bring a consolidation of the democratic tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Following this  logic, the assertion of researcher Leonid Smorgunov is acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' He noted in 2018 that "in  the theory of public policy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' there is an eternal gap of fair procedures that could be used to  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Different kinds of authority in Distributed Ledger Technologies  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Blockchain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Consortium DLT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Private DLT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='(Distributed authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='(Shared authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='(Centralized authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node with authority  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Authority connection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Hugo Estecahandy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node without authority  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='No authority  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content="September 2022Active citizen's e-voting platform's Distributed Ledger Technology  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Voting  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Active Citizen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='user  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Active Citizen website or  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='smartphone application  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='<DLT)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Right of reading  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Master node  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Hugo Estecahandy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='September 2022Class 5  Class 4  Class 3  Class 2  Class 1  Service /  City /  Functioning /  Benefits /  Technology  Actors  Urbanism  Uses  Improvement  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='84  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='5%  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='Hugo Estecahandy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' September 2022  avditachieve a fair outcome in the decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Blockchain, as a network of distributed registers, often  positions itself as an institution ensuring fairness in decisions by voting based on a consensus  procedure" (Smorgunov, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=')  Since the first blockchain in 2008, and especially since the technology started to be  industrially adapted in 2015/2016, derivatives of the original concept have been created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  Thus, it was the appearance of distributed ledger technologies (DLT), proposing blockchains  with a distribution of authority no longer distributed but shared, where only a few nodes of  the network formed by the copies of the ledger had the maximum level of authority, or  centralized in a single node that has authority over the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  In fact, the blockchain that supports online voting on Active Citizen seems to belong to the  last of these categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Indeed, if all the nodes of the network can read the data recorded on  the register, there is an entity, in this case the Moscow City Hall - or rather the Internet or  mobile application of Active Citizen via which the Internet users vote - which can write  information, in this case the vote, its timestamp, the secret identifier of the voter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=" There is  thus an important link in the Moscow City Hall's implementation of a blockchain, since the  voting information is written on a node (Master node) that it controls, from which the other  nodes are updated and can check that no information is subsequently altered." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Figure 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Different kinds of authority in Distributed Ledger Technologies  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Blockchain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Consortium DLT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Private DLT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='(Distributed authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='(Shared authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='(Centralized authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node with authority  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Authority connection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Hugo Estecahandy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node without authority  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='No authority  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content="September 2022Active citizen's e-voting platform's Distributed Ledger Technology  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Voting  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Active Citizen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='Active Citizen website or  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='smartphone application  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='<DLT)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Right of reading  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Master node  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='Hugo Estecahandy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='September 2022Class 5  Class 4  Class 3  Class 2  Class 1  Service /  City /  Functioning /  Benefits /  Technology  Actors  Urbanism  Uses  Improvement  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='84  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='5%  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='  Active citizen or the democratic illusion  Thus, as we have just seen, the implementation of a blockchain to support the Active Citizen  voting system is  only very limited,  both technologically  and democratically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' However,  Moscow City Hall has an interest in offering this service to its citizens, whether it is secured  by a blockchain or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Researcher Caroline Schlaufer has looked at the case of Active  Citizen, asking "Why do non-democratic regimes promote e-participation" (Schlaufer, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  She  argues  that  simply  offering  an  online  voting  system,  "can  accommodate  different  legitimation strategies into one single online tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' A well-designed e-participation tool serves,  first, as an input channel  for citizen preferences, thereby enhancing input legitimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  Second,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' it can increase output-based legitimation by increasing citizen satisfaction with  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Figure 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Different kinds of authority in Distributed Ledger Technologies  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Blockchain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Consortium DLT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='(Shared authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='(Centralized authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node with authority  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='Hugo Estecahandy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content="September 2022Active citizen's e-voting platform's Distributed Ledger Technology  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='<DLT)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='Master node  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='September 2022Class 5  Class 4  Class 3  Class 2  Class 1  Service /  City /  Functioning /  Benefits /  Technology  Actors  Urbanism  Uses  Improvement  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='84  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content=' the introduction of modern trains to connect  the Moscow Belarusskaya station with the city of Odintsovo (Moscow Oblast).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Moreover, as  Jacob Zionts writes, "Active Citizen does little to change the underlying distribution of power  in Moscow politics: voting on the name of a new metro line does not disrupt endemic  corruption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' selecting new seat colors for the Luzhniki Big Sports Arena will not empower  investigative journalists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=" Active Citizen grants Muscovites limited self-expression but only in  a safe and sanitized form's (Zionts, 2018)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  The researcher draws a parallel between the use of the platform and the legitimization  of urban projects that could have led to public rejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' In 2017, with the blessing of Russian  President Vladimir Putin, Moscow Mayor Sergei Sobyanin planned the destruction of several  thousand homes: Soviet buildings that he wanted to replace with modern real estate projects  by wealthy developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' As anger began to grow among the population of the capital, a  "referendum" was proposed to the inhabitants via Active citizens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=" They could vote for or  against the 'renovation' of their buildings, to which 90% of them replied in the affirmative." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  Then, "officials quickly pointed to the overwhelming support among residents as support for  the demolition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='" As one of the interviewed people said to Schlaufer during her research, \'of  course, if a majority of citizens agreed with Active Citizen about something, they will,  afterwards,  not  complain  about  it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=" And  it  leads  to  a  higher  satisfaction  of  the  city  government's activities' (Schlaufer, 2020)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=" In addition to giving a democratic illusion to some  Muscovites, the voting platform would allow the mayor's office to make them accept, in spite  of themselves, policies that they would logically have opposed." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  An international digital showcase for Moscow?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content="  But this use of an electronic voting platform does not only seem to allow the Moscow  mayor's office to consolidate its power and its image in Moscow or in Russia, but also  internationally." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' As Vladimir Pawlotsky explains, Mayor Sobyanin\'s aim is to make Moscow  "a global city" and "a world center of financial stability and innovation" that "an economy  dependent on the - very volatile - price of hydrocarbons" requires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=" Moscow sought - since the  war in Ukraine has surely put this strategy into question - 'as a privileged platform for foreign  investment, to capture international economic and financial flows by hosting the headquarters  and  subsidiaries  of  transnational  corporations,  developing  cultural  activities  as  well  as  tourism' (Pawlotsky, 2020a)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Thus, Moscow, and by extension Russia, are in a logic of city  branding and nation branding internationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' The 2018 Football World Cup helped to present  the city and the rest of the country to the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Different kinds of authority in Distributed Ledger Technologies  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Blockchain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Consortium DLT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Private DLT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='(Distributed authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='(Shared authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='(Centralized authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node with authority  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Authority connection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Hugo Estecahandy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node without authority  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='No authority  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content="September 2022Active citizen's e-voting platform's Distributed Ledger Technology  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Voting  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Active Citizen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='user  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Active Citizen website or  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='smartphone application  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='<DLT)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Right of reading  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Master node  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Hugo Estecahandy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='September 2022Class 5  Class 4  Class 3  Class 2  Class 1  Service /  City /  Functioning /  Benefits /  Technology  Actors  Urbanism  Uses  Improvement  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='84  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='5%  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='network  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='active citize belozerov  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='service  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='pwa  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Hugo Estecahandy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' September 2022  avditThe Active Citizen project and the announcement of the implementation of a blockchain also  meet the objectives of this strategy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' communicating a democratic and innovative image of  Moscow to the rest of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Thus, the announcement of the addition of a blockchain to  Active citizen was shared internationally, by more or less specialized English-speaking  media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' It is interesting to note that most of these articles repeat the December 2017 press  release from the city hall and its language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=" Let's remember that this communiqué made much  of the codes dear to cryptoanarchists and other defenders of a free Internet, some of which are  founding principles of blockchain: distribution of the verification of the voting process, free  access to the underlying protocol or incorruptibility of the decisions taken by Muscovites  concerning the future of their city." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  In order to test our hypothesis that the foreign media treatment of the implementation of a  blockchain within Active Citizen has helped to project a democratic and modern image of  Moscow, we conducted a lexicometric analysis on 6 of these English-language press articles  (see Appendix 1 for the article references).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=" This analysis has been conducted through the  software IRaMuTeQ, developed at Toulouse's University of Mirail (France)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Above all, the  software allows us to highlight classes of words that it will automatically detect using a  classification method called the Reinert method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' As it can be seen on the Figure 3, the Reinert  classification brings out 5 classes of words, out of the text corpus formed by our 6 articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Figure 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Different kinds of authority in Distributed Ledger Technologies  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Blockchain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Consortium DLT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Private DLT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='(Distributed authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='(Shared authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='(Centralized authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node with authority  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Authority connection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Hugo Estecahandy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node without authority  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='No authority  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content="September 2022Active citizen's e-voting platform's Distributed Ledger Technology  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Voting  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Active Citizen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='user  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Active Citizen website or  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='smartphone application  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='<DLT)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Right of reading  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Master node  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Hugo Estecahandy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='September 2022Class 5  Class 4  Class 3  Class 2  Class 1  Service /  City /  Functioning /  Benefits /  Technology  Actors  Urbanism  Uses  Improvement  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='84  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='5%  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='network  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='active citize belozerov  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='service  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='street  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='host  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='datum  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='transport  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='user  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='trust  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='offer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='store  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='million  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='scale  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content=' This graph shows the  words and their frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' The larger the word, the more often it appears in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' The  words are spatialized according to their proximity to the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  Figure 4  Blockchain is the most present word as we have seen on the cloud and it is in the center of  this graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' What we see here is interesting for two reasons: many words in the lexical field of  democracy (in red) are directly related to the term blockchain, a term to which the words  "Moscow" and "Activie_Citzen" are also strongly related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' The lexical field of innovation is  also very present, with terms directly linked to "Moscow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='" This graph allows us to see how  the media’s treatment of the implementation of a blockchain by the Moscow City Council  served its international communication objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Different kinds of authority in Distributed Ledger Technologies  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Blockchain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='(Distributed authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='(Shared authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='(Centralized authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node with authority  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='Hugo Estecahandy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='No authority  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content="September 2022Active citizen's e-voting platform's Distributed Ledger Technology  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='Active Citizen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='Active Citizen website or  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='smartphone application  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='<DLT)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='Master node  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Node 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='September 2022Class 5  Class 4  Class 3  Class 2  Class 1  Service /  City /  Functioning /  Benefits /  Technology  Actors  Urbanism  Uses  Improvement  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='84  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='5%  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='Hugo Estecahandy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' September 2022  avditConclusion  The implementation of a blockchain in the voting system of Active Citizen has above all  allowed to consolidate the discourse that the Moscow City Hall wishes to attach to this  application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Thus, with the Active cities project "the idea is not to make Muscovites vote on  major issues - far from it - but to ask them questions designed to elicit answers that  correspond to the objectives of the city government, thus helping to legitimize its policies  (Pawlotsky, 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' From what we have seen, these goals were also a consolidation of the  positive feelings of its population towards a democratic and modern city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' It also participated  in the creation of an international brand image of Moscow as we could understand through  the analysis of press materials of foreign media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Recently, we could observe what could be a  new type of recourse to Active Citizen for the authorities, which allowed renaming a part of  the urban area in front of the British embassy and thus artificially correlate what would  represent the ideas of Muscovites with the geopolitical ambitions of the Kremlin and its  discourse of justification of the war in Ukraine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' In addition to the above-mentioned benefits,  Active Citizen seems to allow the Moscow City Hall to attract even more favors from the  Kremlin  under  the  cover  of  a  completely  illusory  participatory  democracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' And  the  announcement of the implementation of blockchain has contributed to this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  Through the common thought that "blockchain  technology provides the kind of  transparency and immutability that could sustain democracy, Moscow\'s implementation of  the technology through Active Citizen has provided only the illusion of empowerment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' the  political system in Moscow remains autocratic" (Zionts, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' And this allows us to go  further and quote Schlaufer as our results also showed "that the introduction of e-participation  in a non-democratic regime is not a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Therefore, controlled e-participation is not  a challenge to the underlying distribution of power between governments and citizens"  (Schlaufer, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  References Alexandrov D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=', Boukhanovsky A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=', Chugunov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=', Kabanov Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=', Koltsova O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  (éd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Digital Transformation and Global Society: Third International Conference,  DTGS 2018, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Petersburg, Russia, May 30 – June 2, 2018, Revised Selected Papers,  Part I [En ligne].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Cham : Springer International Publishing, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' (Communications  in Computer and Information Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=') Disponible sur : <  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='1007/978-3-030-02843-5 > (consulté le 13 septembre 2022)ISBN :  978-3-030-02842-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Blanchard E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=', Vigni F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=', Rauzy P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' « Critique de l’article de Serge Surin “Le  renouveau de la promesse du vote électronique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Çabuk U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content=' International Journal of  Advanced Research in Computer and Communication Engineering [En ligne].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content=' September 2022  avditmars 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' 7, n°3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' 124-134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Disponible sur : <  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='17148/IJARCCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='7324 > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Moiseeva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' « Open Government and “e-Government” in  Russia ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' SS [En ligne].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' 28 janvier 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' 5, n°1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='17265/2159-5526/2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='004 > (consulté le 5 septembre 2022) Pawlotsky V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' « Moscou se rêve en « ville globale » ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Le Monde diplomatique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  novembre 2020a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Pawlotsky V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' « Moscou se rêve en « ville globale » ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Le Monde diplomatique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  novembre 2020b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Schlaufer C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' « Why do nondemocratic regimes promote E‐PARTICIPATION ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' The case  of Moscow’s active citizen online voting platform ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Governance [En ligne].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' 23 août  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Disponible sur : < https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='12531 > (consulté le 8 juin  2021) Smorgunov L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' « Blockchain and a Problem of Procedural Justice of Public Choice ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  In : Alexandrov DA, Boukhanovsky AV, Chugunov AV, Kabanov Y, Koltsova O  (éd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Digital Transformation and Global Society: Third International Conference,  DTGS 2018, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Petersburg, Russia, May 30 – June 2, 2018, Revised Selected Papers,  Part I [En ligne].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Cham : Springer International Publishing, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Disponible sur : <  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='1007/978-3-030-02843-5 > (consulté le 13 septembre 2022)ISBN :  978-3-030-02842-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Zionts J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' « “Blockchain Democracy” and the Misleading Promise of Moscow’s Active  Citizen Program ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' In : Medium [En ligne].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' [s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='] : [s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' ], 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' Disponible sur : <  https://media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='consensys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='net/blockchain-democracy-and-the-misleading-promise-of-  moscows-active-citizen-program-2eda7141c5b9 > (consulté le 7 juin 2021)  Annexe 1 (Analyzed articles references) :  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' « Moscow introduces blockchain e-voting », SmartCitiesWorld, 12/05/2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' « Moscow to Develop a Blockchain System for Transparent City Services », CoinDesk,  08/15/2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' « Moscow to pilot blockchain platform for e-voting », FinExtra, 12/04/2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' « Russia Is Leading the Push for Blockchain Democracy », CoinDesk, 02/18/2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' « How Moscow is pioneering the rise of smart cities », CIO, 11/06/2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content=' « This is How Russia Utilizing Blockchain to Enhance Democracy », Sputnik News,  03/02/2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Different kinds of authority in Distributed Ledger Technologies  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Blockchain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Consortium DLT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='Private DLT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='(Distributed authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='(Shared authority)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='Authority connection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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+page_content='September 2022Class 5  Class 4  Class 3  Class 2  Class 1  Service /  City /  Functioning /  Benefits /  Technology  Actors  Urbanism  Uses  Improvement  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
+page_content='84  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E2T4oBgHgl3EQfiAeb/content/2301.03954v1.pdf'}
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diff --git a/zNFIT4oBgHgl3EQf1ysJ/content/tmp_files/2301.11374v1.pdf.txt b/zNFIT4oBgHgl3EQf1ysJ/content/tmp_files/2301.11374v1.pdf.txt
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@@ -0,0 +1,1821 @@
+Policy Optimization with Robustness Certificates
+Chenxi Yang 1 Greg Anderson 1 Swarat Chaudhuri 1
+Abstract
+We present a policy optimization framework in
+which the learned policy comes with a machine-
+checkable certificate of adversarial robustness.
+Our approach, called CAROL, learns a model of
+the environment. In each learning iteration, it uses
+the current version of this model and an external
+abstract interpreter to construct a differentiable
+signal for provable robustness. This signal is used
+to guide policy learning, and the abstract inter-
+pretation used to construct it directly leads to the
+robustness certificate returned at convergence. We
+give a theoretical analysis that bounds the worst-
+case accumulative reward of CAROL. We also
+experimentally evaluate CAROL on four MuJoCo
+environments. On these tasks, which involve con-
+tinuous state and action spaces, CAROL learns
+certified policies that have performance compa-
+rable to the (non-certified) policies learned using
+state-of-the-art robust RL methods.
+1. Introduction
+Reinforcement learning (RL) is an established approach to
+control tasks (Polydoros & Nalpantidis, 2017; Mnih et al.,
+2015), including ones in which safety is critical (Cheng et al.,
+2019a; Sallab et al., 2017). However, state-of-the-art RL
+methods use deep neural networks as policy representations.
+This makes them vulnerable to adversarial attacks in which
+carefully crafted perturbations to a policy’s inputs cause it to
+behave incorrectly. These problems are even more serious in
+RL than in supervised learning, as the effects of successive
+mistakes can cascade over a long time horizon.
+These challenges have motivated research on RL algorithms
+that are robust to adversarial perturbations. In general, meth-
+ods for adversarial learning can be divided into best-effort
+heuristic defenses and certified approaches that guarantee
+provable adversarial robustness. The latter are preferable
+1UT
+Austin.
+Correspondence
+to:
+Chenxi
+Yang
+<cxyang@cs.utexas.edu>,
+Greg
+Anderson
+<ganderso@cs.utexas.edu>,
+Swarat
+Chaudhuri
+<swarat@cs.utexas.edu>.
+Preprint. Under review.
+as heuristic defenses are often defeated by counterattacks
+(Russo & Proutiere, 2019). In the supervised learning set-
+ting, many certified learning techniques have been proposed
+over the years (Mirman et al., 2018; Cohen et al., 2019;
+Wong & Kolter, 2018). However, developing such meth-
+ods for the RL setting has been difficult because of the
+presence of a blackbox environment. To ensure the robust-
+ness of an RL policy, one needs to reason about repeated
+interactions between the policy, the environment, and the
+adversary. There is no general approach to doing so. Ex-
+isting approaches to deep certified RL typically sidestep
+the challenge through various simplifying assumptions, for
+example, that the perturbations are stochastic rather than
+adversarial (Kumar et al., 2021), that the certificate only
+applies to one-shot interactions between the policy and the
+environment (Oikarinen et al., 2021; Zhang et al., 2020), or
+that the action space is discrete (L¨utjens et al., 2020).
+In this paper, we develop a framework for certifiably robust
+policy optimization that fills this gap in the literature. We
+observe that we can reason about adversarial dynamics over
+entire episodes by learning a model of the environment and
+repeatedly composing it with the policy and the adversary.
+To this end, we consider a state-adversarial Markov Deci-
+sion Process (Zhang et al., 2020) in which the observed
+states are adversarially attacked states of the original envi-
+ronment. During exploration, our algorithm learns a model
+of the environment using an existing model-based policy
+optimization algorithm (Janner et al., 2019). We perform
+abstract interpretation (Cousot & Cousot, 1977; Mirman
+et al., 2018) over compositions of the current policy and the
+learned environment model to estimate worst-case bounds
+on the agent’s adversarial reward. The lower bound on the
+reward is then used to guide policy optimization.
+A key benefit of abstract interpretation is that it not only
+computes bounds on a policy’s worst-case reward but also
+offers a proof of this fact if it holds. A certificate of robust-
+ness in our framework consists of such a proof.
+Our results include a theoretical analysis of our learning
+algorithm, which shows that our learned certificates give
+probabilistically sound lower bounds on the accumulative re-
+ward of any allowed adversary. We also empirically evaluate
+CAROL over four high-dimensional MuJoCo environments
+(Hopper, Walker2d, Halfcheetah, and Ant). We demonstrate
+arXiv:2301.11374v1  [cs.LG]  26 Jan 2023
+
+Policy Optimization with Robustness Certificates
+that CAROL is able to successfully learn certified policies
+for these environments and that our strong certification re-
+quirements do not compromise empirical performance.
+To summarize, our main contributions are as follows:
+• We offer CAROL, the first RL framework to guarantee
+episode-level certifiable adversarial robustness in the pres-
+ence of continuous states and actions. The framework is
+based on a new combination of model-based learning and
+abstract interpretation that can be of independent interest.
+• We give a rigorous theoretical analysis that establishes
+the (probabilistic) soundness of CAROL.
+• We give an empirical evaluation that establishes CAROL
+as a new state of the art for certifiably robust RL.
+2. Background
+Markov Decision Processes (MDPs).
+We start with the
+standard definition of an Markov Decision Process (MDP)
+M = (S, A, r, P, S0), where S is a set of states, A is a set
+of actions, S0 is a distribution of initial states, P(s′ | s, a)
+for s, s′ ∈ S and a ∈ A is a probabilistic transition func-
+tion, and r(s, a) for s ∈ S, a ∈ A is a real-valued reward
+function. Our method assumes an additional property that
+is commonly satisfied in practice: that P(s′ | s, a) has
+the form µP (s, a) + fP (s′), where fP (s′) is a distribution
+independent of (s, a) and µP is deterministic.
+A policy in M is a distribution π(a | s) with s ∈ S and a ∈
+A. A (finite) trajectory τ is a sequence s0, a0, s1, a1, . . .
+such that s0 ∼ S0, each ai ∼ π(si), and each si+1 ∼ P(s′ |
+si, ai). We denote by R(τ) = �
+i r(si, ai) the aggregate
+(undiscounted) reward along a trajectory τ, and by R(π) the
+expected reward of trajectories unrolled under π.
+State-Adversarial MDPs. We model adversarial dynam-
+ics using state-adversarial MDPs (Zhang et al., 2020).
+Such a structure is a pair Mν = (M, B), where M =
+(S, A, r, P, S0) is an MDP, and B : S → 2S is a perturba-
+tion map.
+Suppose we have a policy π in the underlying MDP M. In
+an attack scenario, an adversary ν perturbs the observations
+of the agent at a state s. As a result, rather than choosing
+an action from π(a | s), the agent now chooses an action
+from π(a | ν(s)). However, the environment transition is
+still sampled from P(s′ | s, a) and not P(s′ | ν(s), a), as
+the ground-truth state does not change under the attack. We
+denote by π ◦ ν the state-action mapping that results when
+π is used under this attack scenario.
+Naturally, if ν can arbitrarily perturb states, then adversar-
+ially robust learning is intractable. Consequently, we con-
+strain ν using B, requiring ν(s) ∈ B(s) for all s ∈ S.
+We denote the set of allowable adversaries in Mν as
+AB = {ν : S → S | ∀s ∈ S. ν(s) ∈ B(s)}.
+Abstract Interpretation.
+We certify adversarial robust-
+ness using abstract interpretation (Cousot & Cousot, 1977),
+a classic framework for worst-case safety analysis of sys-
+tems. Here, one represents sets of system states using sym-
+bolic representations (abstract states) in a predefined lan-
+guage. For example, we can set our abstract states to be
+hyperintervals that maintain upper and lower bounds in each
+dimension of the state space. We denote abstract values with
+the superscript #, and we use brackets �·�# to indicate that
+we are using abstract semantics. For a set of concrete states
+S, α(S) denotes the smallest abstract state which contains
+S. For an abstract state s#, we denote by β(s#) the set of
+concrete states represented by s#.
+The core of abstract interpretation is the propagation of
+abstract states s# through a function f(s) that captures
+single-step system dynamics. This function can be repre-
+sented in a variety of ways, including neural networks (Gehr
+et al., 2018). For propagation, we assume that we have ac-
+cess to a map f #(s#) that “lifts” f to abstract states. This
+function must satisfy the property β(f #(s#)) ⊇ {f(s) :
+s ∈ β(s#)}. Intuitively, f # overapproximates the behavior
+of f: while the abstract state f #(s#) may include some
+states that are not actually reachable through the application
+of f to states encoded by s#, it will at least include every
+state that is reachable this way.
+By starting with an abstraction s#
+0 of the initial states and
+using abstract interpretation to propagate this abstract state
+through the transition function f, we can obtain an abstract
+state s#
+i which includes all states of the system that are
+reachable in i steps, for increasing i. The sequence of ab-
+stract states τ # = s#
+0 s#
+1 s#
+2 . . . is called an abstract trace.
+3. Problem Formulation
+To formulate our problem, we first need to define robustness.
+Assume a fixed adversarial MDP Mν, a policy π, and a
+robustness threshold ∆ > 0. A robustness property is a
+constraint φ(π, ∆) of the form
+∀ν ∈ AB. R(π) − R(π ◦ ν) < ∆.
+(1)
+Intuitively, φ requires that no allowable adversary can re-
+duce the expected reward of the policy by more than ∆.
+Our goal in this paper is to learn policies that are not just
+robust but provably so. Accordingly, we expect our learning
+algorithm to produce, in addition to a policy π, a certificate,
+or proof, c of robustness.
+Formally, let Π be the universe of all policies in a given
+state-adversarial MDP Mν = (M, B). For a policy π and
+a robustness property φ, we write π ⊢c φ if π provably
+satisfies φ, and c is a proof of this fact.
+
+Policy Optimization with Robustness Certificates
+s0 = 1
+ν(s) = ⟨s0 + ϵ0, s1 + ϵ1⟩
+s0
+sobs0
+a0
+s1
+sobs1
+a1
+R(π ◦ ν)
+Eτ∼π◦ν[R]
+No-Adv
+ϵ0 = ϵ1 = 0.0
+1
+1
+1
+2 + e ∼ N(0, 1)
+2 + e
+2 + e
+6 + 2e
+6
+Adv-1
+ϵ0 = 0.1, ϵ1 = −0.4
+1
+1.1
+1.1
+2.1 + e ∼ N(0, 1)
+1.7 + e
+1.7 + e
+5.9 + 2e
+5.9
+Adv-2
+ϵ0 = −0.2, ϵ1 = −0.3
+1
+0.8
+0.8
+1.8 + e ∼ N(0, 1)
+1.5 + e
+1.5 + e
+5.1 + 2e
+5.1
+Reward Bound (R#)
+ϵt ∈ [−0.5, 0.5], ϵ#
+t = [−0.5, 0.5]
+1
+1 + [−0.5, 0.5]
+[0.5, 1.5]
+[1.5, 2.5] + e ∼ N(0, 1)
+[1, 3] + e
+[1 + e, 3 + e]
+[4 + 2e, 8 + 2e]
+[4, 8]
+Table 1: Example of reward bound calculation. The MDP in this example has initial state set S0 = {1}, white-box transition
+function P(s′|s, a) = s + a + N(0, 1), reward function r(s, a) = s + a, and adversary ν(s) ∈ [s − 0.5, s + 0.5]. ϵt denotes
+the disturbance added on step t. We aim to certify over the worst-case accumulative reward of a deterministic policy π
+defined as π(s) = s. We define the worst-case here by considering all potential adversaries while still considering the
+expected behavior over the stochastic environment, P. As shown in the above table, we first demonstrate three traces from
+fixed adversaries. In the last row, we demonstrate the way how we consider all the adversary behaviors through an abstract
+trace via abstract interpretation with intervals. The worst-case accumulative reward in this example is 4 as EN [e] = 0. The
+abstract trace over all the adversaries in the last row is our certificate which serves as a proof that the policy satisfies our
+property. We want to ensure the lower bound of the R# should not be lower than a threshold. In training, we use the abstract
+trace to compute a loss to guide the policy optimization process.
+The problem of policy optimization with robustness certifi-
+cates is now defined as:
+(π∗, c)
+=
+arg max
+π∈Π
+E
+τ∼(M,π) [R(τ)]
+s.t. π∗ ⊢c φ.
+(2)
+That is, we want to find a policy that maximizes the standard
+expected reward of the policy, but also ensures that the
+expected worst-case reward under adversarial dynamics is
+provably above a threshold.
+Our certificates can be constructed using a variety of sym-
+bolic or statistical techniques. In CAROL, certificates are
+constructed using an abstract interpreter. Suppose we have
+a policy π and an abstract trace τ # = s#
+0 s#
+1 . . . s#
+n such
+that for all length-n trajectories τ = s0 . . . sn and all i,
+si ∈ β(s#
+i ). The abstract trace allows us to compute a
+lower bound on the expected reward for π and also serves
+as a proof of this bound. We give an example of such certi-
+fication in a simple state-adversarial MDP, assumed to be
+available in white-box form, in Table 1.
+A challenge here is that abstract interpretation requires a
+white-box transition function, which is not available in RL.
+We overcome this challenge by learning a model of the
+environment during exploration. Model learning is a source
+of error, so our certificates are probabilistically sound, i.e.,
+they guarantee robustness with high probability. However,
+this error only depends on the underlying model-based RL
+algorithm and does not restrict the adversary.
+4. Learning Algorithm
+Now we present the CAROL framework. The framework
+(Figure 1) has two key components: a model-based learner
+and an abstract interpreter. During each training round, the
+learner maintains a model of the environment dynamics
+and a policy. These are sent to the abstract interpreter,
+which calculates a lower bound on the abstract reward. The
+Environment
+Model
+Policy
+Model#
+Policy#
+Loss
+Robustness Property
+Learner
+Abstract 
+Interpreter
+Certificate
+Figure 1: Schematic of CAROL
+lower bound is used to compute a differentiable loss that
+the learner uses in the next iteration of policy optimization.
+At convergence, the abstract trace computed during abstract
+interpretation is returned as a certificate of robustness.
+Abstract Interpretation in CAROL. Now we describe the
+abstract interpreter in CAROL in more detail. Recall that our
+definition of robustness compares the expected reward of the
+original policy to the expected reward of the policy under an
+adversarial perturbation. As a result, our verifier is designed
+to reason about the worst-case reward under adversarial
+perturbations, while considering average-case behavior for
+stochastic policies and environments. Algorithm 1 finds
+a lower bound on this worst-case expected reward using
+abstract interpretation to overapproximate the adversary’s
+possible behaviors along with sampling to approximate the
+average-case behavior of the policy and environment. We
+denote this lower bound from Algorithm 1 as worst-case ac-
+cumulative reward (WCAR), which is also used to measure
+the certified performance in our evaluation.
+In more detail, Algorithm 1 proceeds by sampling a starting
+state s0 ∼ S0. Then in Algorithm 2 for each time step, we
+find an overapproximation s#
+obsi which includes all of the
+possible ways the adversary may perturb si. Based on this
+approximation, we sample a new approximation from the
+policy π. Intuitively, this may be done by using a policy π
+whose randomness does not depend on the current state of
+
+Policy Optimization with Robustness Certificates
+Algorithm 1 Worst-Case Accumulative Reward (WCAR)
+1: Input: policy π, model E
+2: Output: worst case reward of π under any adversary
+3: for t from 1 to N do
+4:
+Sample an initial state s0 ∼ S0
+5:
+Get the worst case reward Rmint using Algorithm 2 over horizon T starting from s0
+6: end for
+7: return 1
+N
+�N
+t=1 Rmint
+Algorithm 2 Worst-case rollout under adversarial perturbation
+1: Input: Initial state s0, rollout horizon T
+2: Output: Worst-case reward of π starting from s0 over one random trajectory
+3: Abstract the initial state and reward: soriginal
+#
+0 ← α({s0}),
+R#
+minti ← α ({0})
+4: for i from 1 to T do
+5:
+Abstract over possible perturbations: s#
+obsi ←
+�
+B(s#
+originali)
+�#
+6:
+Calculate symbolic predicted actions: a#
+i ←
+�
+π(s#
+obsi)
+�#
+7:
+Calculate symbolic next-step states and rewards: s#
+originali+1, r#
+i ←
+�
+Eθ(s#
+originali, a#
+i ) + α({x | ∥x∥ ≤ εE})
+�#
+8:
+Update worst-case reward: R#
+mint ←
+�
+R#
+mint + r#
+i
+�#
+9: end for
+10: return inf β(R#
+mint)
+the system. More formally, π(a | s) = µπ(s)+fπ(a) where
+fπ(a) is a distribution with zero mean which is independent
+of s. Then a#
+i may be computed as �µπ(s#
+obsi) + α({e})�#
+where e ∼ fπ(a). Once the abstract action is computed,
+we may find the new (abstract) state and reward using the
+environment model E. The model is assumed to satisfy a
+PAC-style bound, i.e., there exist δE and εE such that with
+probability at least 1 − δE, ∥E(s, a) − P(s, a)∥ ≤ εE. The
+values of δE and εE can be measured at model construction.
+One way to understand Algorithm 1 is to consider pairs
+of abstract and concrete trajectories in which the random-
+ness is resolved in the same way. Specifically, if π(a |
+s) = µπ(s) + fπ(a) and E(s′ | s, a) = µ(s, a) + fE(s′),
+the initial state s0 combined with the sequence of values
+ei ∼ fπ(a) and e′
+i ∼ fE(s′) for 0 ≤ i ≤ T uniquely de-
+termine a trajectory. For a given set of values, the reward
+bound inf β(R#
+mint) represents the worst-case reward under
+any adversary for a particular resolution of the randomness
+in the environment and the policy. The outer loop of Algo-
+rithm 1 approximates the expectation over these different
+random values by sampling. Theorem 1 in Section 5 shows
+formally that with high probability, Algorithm 1 gives a
+lower bound on the true adversarial reward.
+Learning in CAROL Now we discuss how to learn a pol-
+icy and environment model which may be proven robust by
+Algorithm 1. At a high level, Algorithm 3 works by intro-
+ducing a symbolic loss term Lsymbolic
+ψ
+which measures the
+robustness of the policy. Because robustness is a constrained
+optimization problem, we use this symbolic loss together
+with a Lagrange multiplier in an alternating gradient descent
+scheme to find the optimal robust policy. Formally, for a
+given environment model E, the inner loop in Algorithm 3
+solves the optimization problem
+arg min
+ψ
+Lnormal(πψ, Dmodel)
+s.t.
+Lsymbolic(π, E) ≤ ∆
+via the Lagrangian
+arg min
+ψ
+max
+λ≥0 Lnormal(πψ, Dmodel)+λ(Lsymbolic(π, E)−∆).
+We ensure that solving this problem solves the certifi-
+able robustness problem by enforcing the following con-
+ditions: (i) E accurately models the environment and (ii)
+Lsymbolic(π, E) measures the “provable robustness” of π.
+Condition (i) is handled by alternating model updates with
+policy updates, in the style of Dyna (Sutton, 1990), so we
+will focus on condition (ii).
+The computation of Lsymbolic uses the same underlying ab-
+stract rollouts (Algorithm 2) as the verifier described in
+Algorithm 1. Once again, this algorithm estimates the re-
+ward achieved by a policy under worst-case adversarial
+perturbations but average-case policy actions and environ-
+ment transitions. We then define the robustness loss as the
+difference between the nominal loss Ro and the provable
+lower bound on the worst-case loss Rmin. Now as long as
+
+Policy Optimization with Robustness Certificates
+Algorithm 3 Policy Optimization for Certifiably Robust Reinforcement Learning
+1: Initialize a random policy πψ, random environment model Eθ, and empty model dataset Dmodel.
+2: Initialize an environment dataset Denv by unrolling trajectories under a random policy.
+3: for N epochs do
+4:
+Train model Eθ on Denv via maximum likelihood
+5:
+Unroll M trajectories int he model under πψ; add to Dmodel
+6:
+Take action in environment according to πψ; add to Denv
+7:
+for G gradient updates do
+8:
+Calculate normal policy loss Lnormal(πψ, Dmodel) as in MBPO (Janner et al., 2019)
+9:
+Sample ⟨st, at, st+1, rt⟩ uniformly from Dmodel
+10:
+Rollout π starting from st under Eθ for Ttrain steps and compute the total reward Ro
+11:
+Compute the worst-case reward Rmin using Algorithm 2 over horizon Ttrain.
+12:
+Compute the robustness loss Lsymbolic(πψ, Eθ) ← Ro − Rmin
+13:
+Update policy parameters: ψ ← ψ − α∇ψ(Lnormal(πψ, Dmodel) + λ(Lsymbolic(πψ, Eθ) − ∆))
+14:
+Update Lagrange multiplier: λ ← max(0, λ + α′(Lsymbolic(πψ, Eθ) − ∆))
+15:
+end for
+16:
+Unroll n trajectories in the true environment under πψ; add to Denv
+17: end for
+Lsymbolic < ∆, we satisfy the definition of robustness given
+in Section 3 for that specific trace. Repeating these gra-
+dient updates gives an approximation of the average-case
+behavior which is considered in Algorithm 1.
+5. Theoretical Analysis
+In this section, we explore some key theoretical properties
+of CAROL. Proofs are deferred to Appendix B.
+Theorem 1. Assume the environment transition distribution
+is P(s′ | s, a) = N(µP (s, a), ΣP ) and the environment
+model is E(s′ | s, a) = N(µE(s, a), ΣE) with ΣP , ΣE
+diagonal. Further, we assume that the model satisfies a
+PAC-style guarantee: for any state s, action a, and ϵ ∈ S,
+|(µP (s, a) + Σ1/2
+P ϵ) − (µE(s, a) + Σ1/2
+E ϵ)| ≤ εE with
+probability at least 1−δE. For any policy π, let the result of
+Algorithm 1 be ˆR# and let the reward of π under the optimal
+adversary ν∗ be R. Then for any δ > 0 with probability at
+least 1 − δ, we have
+R ≥ ˆR# − 1
+√
+δ
+�
+Var [R#]
+N
+−
+�
+1 − (1 − δE)T �
+C.
+where C is a constant (see Appendix B for details of C).
+Theorem 1 shows that our checker is a valid (probabilistic)
+proof strategy for determining if a policy is robust. That
+is, if we use Algorithm 1 to measure the reward of a policy
+under perturbation, the result is a lower bound of the true
+worst-case reward (minus a constant) with high probability,
+assuming an accurate environment model. The bound in
+Theorem 1 gives some interesting insights. First, the bound
+grows as δ shrinks, so as we consider higher confidence
+levels, we pay the price of a looser bound. Second, the
+bound depends on the variance of the abstract reward and
+the number of samples in an intuitive way — higher vari-
+ance makes it harder to measure the true reward, and more
+samples makes the bound tighter. Third, as δE increases,
+the last term of the bound grows, indicating that a less ac-
+curate environment model leads to a looser bound. Finally,
+the bound grows with T, indicating that over longer time
+horizons, our reward measurement gets less accurate. This
+is consistent with the intuition that the environment model
+may drift away from the true environment over long rollouts.
+Theorem 2. Algorithm 3 converges to a policy π which is
+robust and verifiable by Algorithm 1.
+Intuitively, the theorem shows that Algorithm 3 solves the
+certifiable robustness problem, i.e., it converges to a pol-
+icy which passes the check by Algorithm 1. The proof is
+straightforward because Algorithm 3 is a standard primal-
+dual approach to solve the constrained optimization problem
+outlined in Equation 2 (Nandwani et al., 2019).
+6. Evaluation
+We study the following experimental questions:
+RQ1: Can CAROL learn policies with nontrivial certified
+reward bounds?
+RQ2: How does CAROL compare with other (non-certified)
+robust RL methods in terms of certified bounds?
+RQ3: How does CAROL compare with other robust RL
+methods in terms of empirical adversarial performance?
+RQ4: How does CAROL’s model-based training approach
+affect performance?
+Environments and Setup. Our experiments consider l∞-
+
+Policy Optimization with Robustness Certificates
+Figure 2: Certified performance of policies π with the learned-together model, E. Each bar is an average of 20 starting
+states. nan denotes not a number, which means that (π, E) is not certifiable by a third-party verifier (Zhang et al., 2018). A
+higher value indicates a better certified worst-case performance.
+Figure 3: Certified performance of policies π under a set of separately learned models, {Ei}. Each bar averages the learned
+policies on each Ei of 20 starting states.
+norms perturbation of the state with radius ϵ: Bp(s, ϵ) :=
+{s′|∥s′ − s∥ ≤ ϵ}. We implement CAROL on top of the
+MBPO (Janner et al., 2019) model-based RL algorithm
+using the implementation from Pineda et al. (2021). For
+training, we use IBP (Gowal et al., 2018) as a scalable
+abstract interpretation mechanism. During the evaluation,
+we use CROWN (Zhang et al., 2018), which is a more
+expensive but accurate bound propagation method. We use
+a ϵ-schedule (Gowal et al., 2018; Zhang et al., 2020) during
+training to slowly increase the ϵt at each epoch within the
+perturbation budget until reaching ϵ. Note that the policies
+take action stochastically during training, but we set them
+to be deterministic during evaluation.
+We experiment on four MuJoCo environments in OpenAI
+Gym (Brockman et al., 2016). For CAROL, we use the
+same hyperparameters as in Pineda et al. (2021) without
+further tuning. Specifically, we do not use an ensemble of
+dynamics models. Instead, we use a single dynamic model,
+which is the case when the ensemble is of size 1. We use
+Gaussian distribution as the independent noise distribution,
+fπ(a), fE(s′) for both policy and model in the experiments.
+Concretely, the output of our policies are the parameters
+µπ, Σπ of a Gaussian, with Σπ being diagonal and inde-
+pendent of input state s. For the model, the output are the
+parameters µE, ΣE of a Gaussian, with ΣE being diagonal
+and independent of input s, a. In the implementation, we
+approximate the εE as zero when measuring the certified
+performance for all experiments.
+We run Hopper 5 × 105 steps, Walker2d 7.5 × 105 steps,
+Halfcheetah 8.5 × 105 steps, and Ant 9 × 105 steps for
+convergence. CAROL has a regularization parameter, λ, for
+regularizing the robust loss Lsymbolic, which we set to 1.0,
+1.0, 0.5, 0.1 for Hopper, Walker2d, HalfCheetah, and Ant,
+respectively. We set Ttrain = 1 for all the training of CAROL.
+We compare CAROL with the following methods: (1) MBPO
+(Janner et al., 2019), our base algorithm for policy optimiza-
+tion. (2) SA-PPO (Zhang et al., 2020), a robust RL algo-
+rithm bounding per-step action distance. (3) RADIAL-PPO
+(Oikarinen et al., 2021), a robust RL algorithm using lower
+bound PPO loss to update the policy. (4) CAROL-Separate
+Sampler(CAROL-SS), an ablation of CAROL. In CAROL, we
+update the policy loss Lnormal with the data sampled from the
+rollout between the learned model and the policy. While in
+CAROL-SS, the data for Lnormal is sampled from the rollout
+between the environment and the policy. The ϵtrain is 0.075,
+0.05, 0.075, 0.05 for Hopper, Walker2d, HalfCheetah, and
+Ant for CAROL, CAROL-SS, SA-PPO, and RADIAL-PPO
+in this section. More details of the training and setup are
+available in Appendix C.
+Evaluation. We evaluate the performance of policies with
+two metrics: (i). WCAR following Algorithm 1 for certified
+performance. (ii). total reward under MAD attacks (Zhang
+et al., 2020) for empirical performance.
+
+hopper
+ant
+halfcheetah
+walker2d
+1.5
+1.5
+1.5 
+3
+1.0 
+5
+1.0
+1.0
+1.0 -
+0.5
+3
+0+36"T-
+WCAR / T
+WCAR / T
+EO
+-2.5E+09
+1.7E+12
+WCAR / T
+SO
+0.5 
+0.5 -
++
+0.5
+WCAR
+0.0 -
+0.0 -
+0.0
+0.0
+0.5
+0.5-
+0.5
+0.5
+1.0
+1.0
+1.0
+1.0
+5
+15
+20
+25
+5
+7
+10
+5
+10
+15
+20
+5
+10
+15
+20
+Time Horizons (T)
+Time Horizons (T)
+Time Horizons (T)
+Time Horizons (T)
+CAROL
+MBPOhopper
+ant
+halfcheetah
+walker2d
+1.5
+1.5 
+1.3
+1.5 -
+2
+1.0 
+6
+6
+1.0 -
+1.0
+6'0
+1.0 -
+1.0
+0.5
+405
+1/
+1/
+WCAR / T
+3
+ /
+0.5
+0.5
+5.5
+0.5
+WCAR
+WCAR
+WCAR
+851
+四
+85
+48
+95
+0.0 -
+0.0 -
+0.0
+0.0
+0.5
+0.5-
+0.5
+0.5
+1.0
+1.0
+1.0
+1.0
+5
+15
+20
+25
+5
+6
+8
+10
+5
+10
+15
+20
+5
+10
+15
+20
+Time Horizons (T)
+Time Horizons (T)
+Time Horizons (T)
+Time Horizons (T)
+CAROL
+SA-PPO
+RADIAL-PPOPolicy Optimization with Robustness Certificates
+Figure 4: Training Curves of CAROL and CAROL-SS. The solid lines in the graph show the average natural rewards of five
+training trials, and the shaded areas represent the standard deviation among those trials.
+RQ1:
+Certified Performance with Learned-together
+Certificate.
+After training, we get a policy, π, and an
+environment model, E, trained with the policy. Then, we
+evaluate the WCAR following Algorithm 1 with π and E.
+Note that we use an ϵtest =
+1
+255 for the evaluation of prov-
+ability as certifying over traces of neural network models
+accurately is a challenging task for abstract interpreters due
+to accumulated approximation error. The proof becomes
+more challenging as the horizon increases as the impact from
+each step’s worst-case adversary accumulates. We vary the
+certified horizon under the ϵtest to exhibit the certified perfor-
+mance. To have a fair comparison across different horizons,
+we quantify the certified performance by WCAR/T.
+Figure 2 exhibits the certified performance of CAROL. Both
+CAROL and MBPO are evaluated with the model trained
+together. We are able to train a policy with better certified
+accumulative reward under the worst attacks compared to
+the base algorithm, MBPO, which does not use the regular-
+ization Lsymbolic. As the time horizon increases, it becomes
+harder to certify the accumulative reward. For example, in
+Ant and HalfCheetah, CAROL is not able to give a good
+certified performance when the horizon reaches 10 and 20
+respectively because of the accumulative influence from
+the worst-case attack and the overapproximation from the
+abstract interpreter. We also highlight that Ant is a challeng-
+ing task for certification due to the high-dimensional state
+space.
+RQ2: Comparison of Certified Performance with Other
+Methods. We compare CAROL with two robust RL meth-
+ods, SA-PPO (Zhang et al., 2020) and RADIAL-PPO
+(Oikarinen et al., 2021), which both bound the per-step per-
+formance of the policy during training. SA-PPO bounds the
+per-step action deviation under perturbation, and RADIAL-
+PPO bounds the one-step loss under perturbation. To have
+a fair comparison of the certified performance of policies
+and avoid model error bias, we separately train 5 additional
+environment models, {Ei}, with the trajectory datasets un-
+rolled from 5 additional policies (with regular RL training)
+and the environment. We truncate CAROL by extracting the
+policies from training and certify them with these separately
+Nominal
+Attack (MAD)
+Environment
+Model
+ϵ = 0
+ϵ = ϵtrain
+Hopper
+(ϵtrain = 0.075)
+MBPO
+3246.0±76.1
+2874.2±203.4
+SA-PPO
+3423.9±164.2
+3213.8±284.8
+RADIAL-PPO
+3547.0±166.9
+3100.3±368.3
+CAROL
+3290.1±104.9
+3201.4±100.5
+Ant
+(ϵtrain = 0.05)
+MBPO
+4051.9±526.2
+406.2±83.5
+SA-PPO
+5368.8±96.4
+5327.4±112.7
+RADIAL-PPO
+4694.1±219.5
+4478.9±232.8
+CAROL
+5696.6±277.9
+5362.2±242.8
+HalfCheetah
+(ϵtrain = 0.075)
+MBPO
+7706.3±710.1
+2314.6±566.7
+SA-PPO
+3193.9±650.7
+3231.6±659.9
+RADIAL-PPO
+3686.5±439.2
+3409.6±683.9
+CAROL
+5821.5±2401.9
+3961.6±899.5
+Walker2d
+(ϵtrain = 0.05)
+MBPO
+3815.6±211.9
+3616.5±228.2
+SA-PPO
+4271.7±222.2
+4444.4±286.0
+RADIAL-PPO
+2935.1±272.1
+3022.6±381.7
+CAROL
+3784.4±329.1
+3774.3±260.3
+Table 2: Average episodic reward ± standard deviation over
+100 episodes on three baselines and CAROL. We show natu-
+ral rewards (under no attack) and rewards under adversarial
+attacks. The best results over all methods are in bold.
+trained environment models. This setting is not completely
+inline with CAROL’s learned certificate and verification (see
+RQ1) but is designed for a fair comparison across policies.
+As shown in Figure 3, the CAROL’s certified performance
+with separately trained models is slightly worse yet compa-
+rable to its performance when using learned-together cer-
+tificates. When compared with non-certified RL policies,
+CAROL consistently exhibits better certifiable performance.
+It is worth noting that CAROL is able to provide increas-
+ing worst-case rewards over time for the Hopper, Ant, and
+HalfCheetah, which aligns with the reward mechanisms
+used in these environments. Thus, we believe these results
+demonstrate that CAROL is able to give a reasonable certi-
+fied performance, while the other methods, which are not
+specifically designed for worst-case accumulative reward
+certification, struggle to achieve the same goal.
+RQ3:
+Comparison of Empirical Performance with
+
+Hopper
+Ant
+HalfCheetah
+Walker2d
+5000
+8000
+3000
+3000
+wards
+Rewar
+4000
+0-
+0-
+0
+0
+200K
+400K
+400K
+800K
+0
+400K
+800K
+0
+300K
+600K
+Steps
+Steps
+Steps
+Steps
+CAROL-SS
+CAROLPolicy Optimization with Robustness Certificates
+Other Methods.
+Usually, there is a trade-off between
+certified robustness and empirical robustness. One can get
+good provability but may sacrifice empirical rewards. We
+show that policy from our algorithm shows comparable
+natural rewards (without attack) and adversarial rewards
+compared with other methods. In Table 2, we show results
+on 4 environments and comparison with MBPO, SA-PPO,
+and RADIAL-PPO. The policies are the same ones evalu-
+ated for RQ1 and RQ2. For each environment, we compare
+the performance under MAD attacks (Zhang et al., 2020).
+CAROL outperforms other methods on Ant and HalfCheetah
+under attack when the base algorithm, MBPO, is extremely
+not robust. For Hopper, CAROL has comparable adversarial
+rewards with the best methods. CAROL’s reward is worse
+on Walker2d though still reasonable.
+RQ4: Impact of Model-Based Training. In this part, we
+investigate the impact of our design choices for Lnormal on
+performance. We compare our framework, CAROL, with an
+ablation of it, CAROL-SS, to understand how rollout with the
+learned model for Lnormal matters in CAROL. We present a
+comparison of performance during training, as shown in Fig-
+ure 4. In the implementation, we set a smoother ϵ-schedule
+for CAROL-SS by allowing CAROL-SS to take longer steps
+from ϵ = 0 to the target ϵ. These results show that CAROL
+converges much faster while achieving a comparable or
+better final performance due to the benefits of the sample
+efficiency of MBRL. Additionally, the consistency between
+the rollout datasets for Lnormal and the ones for Lsymbolic also
+leads to a better natural reward at convergence in training.
+7. Related Work
+Adversarial RL. Adversarial attacks on RL systems have
+been extensively studied. Specific attacks that have been
+investigated include adversarial perturbations on agents’ ob-
+servations or actions (Huang et al., 2017; Lin et al., 2017;
+Weng et al., 2019), adversarial disturbance forces to the
+system (Pinto et al., 2017), and other adversarial policies
+in a multiagent setting (Gleave et al., 2019). Most recently,
+Zhang et al. (2021) and Sun et al. (2021) propose methods
+to train agents together with a learned optimal adversary in
+an online way to achieve better adversarial reward. These
+efforts are orthogonal to existing regularization-based tech-
+niques (including ours).
+Robust RL and Provable Robustness in RL. Multiple
+robust training methods have been applied to deep RL. Fis-
+cher et al. (2019) leverage additional student networks to
+help the robust Q learning, and Everett et al. (2021) enhance
+an agent’s robustness during testing time by computing the
+lower bound of each action’s Q value at each step. Zhang
+et al. (2020) and Oikarinen et al. (2021) leverage a bound
+propagation technique in a loss regularizer to encourage the
+agent to either follow its original actions or optimize over a
+loss lower bound. While these efforts achieve robustness by
+deterministic certification techniques for neural networks
+(Gowal et al., 2018; Xu et al., 2020), they mainly focus on
+the step-wise certification and are not able to give robustness
+certification if the impact from attacks accumulates across
+multiple steps. CAROL differs from these papers by offering
+certified robustness for the aggregate reward in an episode.
+We know of only two recent efforts that study robustness
+certification for cumulative rewards (Wu et al., 2021; Kumar
+et al., 2021). Of these, (Wu et al., 2021) is a framework
+for certification rather than certified learning. (Kumar et al.,
+2021) proposes a certified learning algorithm under the as-
+sumption that the adversarial perturbation is smoothed using
+random noise. The attack model here is weaker than the
+adversarial model assumed by CAROL and most other work
+on adversarial learning.
+Certified RL. Safe control with learned certificates is an
+active field (Dawson et al., 2022). A few efforts in this space
+have considered controllers discovered through RL. Many
+works use a given certificate with strong control-theoretic
+priors to constrain the actions of an RL agent (Cheng et al.,
+2019a; Li & Belta, 2019; Cheng et al., 2019b) or assume the
+full knowledge of the environment to yield the certificate
+during the training of an agent (Yang & Chaudhuri, 2021).
+Chow et al. (2019) and Chow et al. (2018) attempt to derive
+certificates from the structure of the constrained Markov
+decision process (Altman, 1999) for the safe control prob-
+lems. Chang & Gao (2021) incorporate Lyapunov methods
+in deep RL to learn a neural Lyapunov critic function to im-
+prove the stability of an RL agent. We differ from this work
+by focusing on adversarial robustness rather than stability.
+8. Conclusion and Future Work
+We have presented CAROL, the first RL framework with cer-
+tifiable episode-level robustness guarantees. Our approach
+is based on a new combination of model-based RL and ab-
+stract interpretation. We have given a theoretical analysis to
+justify the approach and also validated it empirically in four
+challenging continuous control tasks.
+A key challenge in CAROL is that our abstract interpreter
+may not be sufficiently precise, and attempts to increase pre-
+cision may compromise scalability. Future research should
+work to address this issue with more accurate and scalable
+verification techniques. Also, as seen in Figure 4, there is
+variance in the results due to the use of a single dynamics
+model. To address this issue, future work should explore
+ways to incorporate certificates across ensembles of models.
+Finally, because abstract interpretation of probabilistic sys-
+tems is difficult, our approach assumes that the randomness
+in the environment transitions is state-independent. Future
+work should try to eliminate this assumption through ab-
+stract interpreters tailored to probabilistic systems.
+
+Policy Optimization with Robustness Certificates
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+
+Policy Optimization with Robustness Certificates
+A. Symbols
+We give a summary of the symbols used in this paper below.
+Definition
+Symbol/Notation
+Policy
+πψ, π
+Model of the environment
+Eθ, E
+Environment transition
+P
+Parameters (mean, covariance) for Gaussian distribution for environment, model, and policy
+µP , ΣP , µE, ΣE, µπ, Σπ
+Distribution representing noise for environment, model, and policy
+fP , fE, fπ
+Adversary
+ν
+Regular policy loss
+Lnormal
+Robustness loss
+Lsymbolic
+Lipschitz constants for the environment model and policy mean
+LE, LP
+Model error
+εE
+PAC bound probability
+δE
+Abstract values and semantics
+·#, �·�#
+Abstraction
+α
+Concretization
+β
+Horizon for training and testing
+T, Ttrain, Ttest
+Disturbance for training and testing
+ϵ, ϵtrain, ϵtest
+B. Proofs
+In this section we present proofs of the theorems from Section 5.
+Assumption 3. The horizon of the MDP is bounded by T.
+Assumption 4. The environment transition distribution has the form P (s′ | s, a) = N (µP (s, a) , ΣP ) with ΣP diagonal
+and the environment model E has the form E (s′ | s, a) = N (µE(s, a), ΣE) with ΣE diagonal.
+Assumption 5. There exist values εE and δE such that for all s, a and for any fixed e with probability at least
+1 − δE,
+���
+�
+µP (s, a) + Σ1/2
+P e
+�
+−
+�
+µE (s, a) + Σ1/2
+E e
+���� ≤ εE. Further, there exists some dE such that for all s, a,
+���
+�
+µP (s, a) + Σ1/2
+P e
+�
+−
+�
+µE (s, a) + Σ1/2
+E e
+���� ≤ dE.
+Assumption 6. The environment model mean function µE (s, a) is LE-Lipschitz continuous, the immediate reward function
+r (s, a) is Lr-Lipschitz continuous, and the policy mean µπ (s) is Lπ-Lipschitz continuous.
+Assumption 7. For all s ∈ S, we have s ∈ B (s). That is, the adversary may choose not to perturb any state.
+Theorem 1. For any policy π, let the result of Algorithm 1 be ˆR#, let ν∗ be the optimal adversary (i.e., for all ν ∈ AB,
+R (π ◦ ν∗) ≤ R (π ◦ ν)), and let the reward of π ◦ ν∗ be R. Then for any δ > 0 with probability at least 1 − δ, we have
+E [R (τ)] ≥ ˆR# (τ) − 1
+√
+δ
+�
+Var [R# (τ)]
+N
+−
+�
+1 − (1 − δE)T �
+Lr(1 + Lπ)dE
+(LELπ)T + (1 − LELπ) T − 1
+(1 − LELπ)2
+.
+Proof. Recall that the environment transition P and policy π are assumed to be separable, i.e., P (s′ | s, a) = µP (s, a) +
+fP (s′) and π (a | s) = µπ (s) + fπ (a) with µP and µπ deterministic. As a result, a trajectory under policy π ◦ ν∗
+may be written τ = s0, a0, s1, a1, . . . , sn, an where s0 ∼ S0, each ai = µπ (ν∗ (si)) + eπ
+i for eπ
+i ∼ fπ (a), and each
+si = µP (si−1, ai−1) + eP
+i for eP
+i ∼ fP (s′). By Assumption 4, we know that eP
+i ∼ N (0, ΣP ) so that eP
+i = Σ1/2
+P ei
+where ei ∼ N (0, I). In particular, because each trajectory τ is uniquely determined by s0, {eπ
+i }n
+i=0, {ei}n
+i=1, we can write
+the reward of π ◦ ν∗ as
+R (π ◦ ν∗) =
+E
+s0∼S0,{eπ
+i ∼fπ(a)}
+n
+i=0,{ei∼N (0,I)}n
+i=1
+R (τ)
+Because this expectation ranges over the values of s0, {eπ
+i }n
+i=0 , {ei}n
+i=1, we will proceed by considering pairs of abstract
+and concrete trajectories unrolled with the same starting state and noise terms.
+
+Policy Optimization with Robustness Certificates
+To do this, we analyze Algorithm 2 for some fixed s0, {eπ
+i }n
+i=0 , {ei}n
+i=1. Let eE
+i
+= Σ1/2
+E ei. That is, given the same
+underlying sample from N (0, I), eP
+i is the noise in the true environment while eE
+i is the noise in the modeled environment.
+We show by induction that for all i, si ∈ β
+�
+s#
+originali
+�
+with probability at least (1 − δE)i. Note that, because abstract
+interpretation is sound, s0 ∈ β
+�
+s#
+original0
+�
+. Additionally, for all i if si ∈ β
+�
+s#
+originali
+�
+then ν∗ (si) ∈ β
+�
+s#
+obsi
+�
+. Moreover,
+since eπ
+i is fixed, we have
+�
+π
+�
+s#
+obsi
+��#
+=
+�
+µπ
+�
+s#
+obsi
+�
++ eπ
+i
+�#
+so that π (ν∗ (s)) ∈ β
+�
+a#
+i
+�
+. Similarly, because eE
+i is fixed, let ∆E = α ({x | ∥x∥ ≤ εE}) and we have
+�
+E
+�
+s#
+originali, a#
+i
+�
++ ∆E
+�#
+=
+�
+µE
+�
+s#
+originali, a#
+i
+�
++ eE
+i + ∆E
+�#
+.
+By the induction hypothesis, we know that si−1 ∈ β
+�
+s#
+originali−1
+�
+with probability at least (1 − δE)i−1 and therefore ai−1 ∈
+β
+�
+a#
+i−1
+�
+. By Assumption 5, we have that
+���
+µP (s, a) + eP
+i
+�
+−
+�
+µE (s, a) + eE
+i
+��� < εE with probability at least 1 − δE.
+In particular,
+�
+µP (s, a) + εP
+i
+�
+−
+�
+µE (s, a) + εE
+i
+�
+∈ ∆E, so that µP (s, a) + eP
+i ∈ β
+��
+E
+�
+s#
+originali, a#
+i
+�
++ ∆E
+�#�
+.
+Then with probability at least 1 − δE, if si−1 ∈ β
+�
+s#
+originali−1
+�
+then si ∈ β
+�
+s#
+originali
+�
+. As a result, si ∈ β
+�
+s#
+originali
+�
+with
+probability at least (1 − δE)i. In particular, by Assumption 3, n ≤ T so that for a fixed τ defined by s0, {eπ
+i }n
+i=0 , {ei}n
+i=1,
+we have that with probability at least (1 − δE)T , Algorithm 2 returns a lower bound on R (τ).
+Now we consider the case where Algorithm 2 does not return a lower bound of R (τ). In this case, we show (again by
+induction) that for all 0 ≤ i ≤ T, there exists a point s′
+i ∈ β
+�
+s#
+originali
+�
+such that
+∥si − s′
+i∥ ≤
+i−1
+�
+j=0
+(LELπ)j dE = dE
+�
+1 − (LELπ)i−1
+1 − LELπ
+�
+(when �−1
+j=0 (LELπ)j dE is taken to be zero). First, note that s0 ∈ β
+�
+s#
+original0
+�
+, so the base case is trivially true. Now by
+the induction hypothesis we have that there exists some s′
+i−1 ∈ β
+�
+s#
+originali−1
+�
+with
+��si−1 − s′
+i−1
+�� ≤ �i−2
+j=0 (LELπ)j dE.
+Notice that by Assumption 7, we also have s′
+i−1 ∈ β
+�
+s#
+obsi−1
+�
+. Now because abstract interpretation is sound, we have that
+µπ
+�
+s′
+i−1
+�
++eπ
+i−1 ∈ β
+�
+a#
+i−1
+�
+and by Assumption 6, ∥µπ (si−1)−µπ
+�
+s′
+i−1
+�
+∥ ≤ Lπ
+�i−2
+j=0 (LELπ)j dE. Similarly, we have
+µE
+�
+s′
+i−1, µπ
+�
+s′
+i−1
+�
++ eπ
+i−1
+�
++eE
+i ∈ β
+�
+s#
+originali
+�
+, and
+��µE
+�
+si−1, µπ (si−1) + eπ
+i−1
+�
+− µE
+�
+s′
+i−1, µπ
+�
+s′
+i−1
+�
++ eπ
+i−1
+��� ≤
+LELπ
+�i−2
+j=0 (LELπ)j dE. Let ˆsi = µE
+�
+si−1, µπ (si−1) + επ
+i−1
+�
++ εE
+i . Then by Assumption 5, we have ∥ˆsi − si∥ ≤
+dE, so that in particular
+��si − µE
+�
+s′
+i−1, µπ
+�
+s′
+i−1
+�
++ eπ
+i−1
+�
++ εE
+i
+�� ≤ dE + LELπ
+�i−2
+j=0 (LELπ)j dE. Letting s′
+i =
+µE
+�
+s′
+i−1, µπ
+�
+s′
+i−1
+�
++ eπ
+i−1
+�
++ εP
+i , we have the desired result.
+We use this result to bound the difference in reward between the abstract and concrete rollouts when Algorithm 2 does not
+return a lower bound. For each i, because s′
+i ∈ β
+�
+s#
+originali
+�
+and µπ (s′
+i) + eπ
+i ∈ a#
+i , we define r′
+i = r (s′
+i, µπ (s′
+i) + eπ
+i )
+and we know that r′ ∈ r#
+i . Because ∥si − s′
+i∥ ≤ dE
+�
+1−(LELπ)i−1
+1−LELπ
+�
+we have ∥ai − a′
+i∥ ≤ LπdE
+�
+1−(LELπ)i−1
+1−LELπ
+�
+and
+|r (si, ai) − r′
+i| ≤ Lr(1 + Lπ)dE
+�
+1−(LELπ)i−1
+1−LELπ
+�
+. In particular, let R′ = �
+i r′
+i and then
+|R (τ) − R′| ≤
+T
+�
+i=1
+Lr(1 + Lπ)dE
+�
+1 − (LELπ)i−1
+1 − LELπ
+�
+= Lr(1 + Lπ)dE
+(LELπ)T + (1 − LELπ) T − 1
+(1 − LELπ)2
+.
+We now combine these two cases to bound the expected difference between the reward returned by Algorithm 2, denoted
+R# (τ), and the reward of τ. Let D = R# (τ) − R (τ) be a random variable representing this difference. Then with
+
+Policy Optimization with Robustness Certificates
+probability at least (1 − δE)T , D ≤ 0 and in all other cases (i.e., with probability no greater than 1 − (1 − δE)T ),
+D ≤ LR(1 + Lπ)dE
+(LELπ)T +(1−LELπ)T −1
+(1−LELπ)2
+. In particular then,
+E [D] ≤
+�
+1 − (1 − δE)T �
+Lr(1 + Lπ)dE
+(LELπ)T + (1 − LELπ) T − 1
+(1 − LELπ)2
+.
+By definition E
+�
+R# (τ)
+�
+= E [R (τ)] + E [D]. Therefore, we have
+E [R (τ)] = E
+�
+R# (τ)
+�
+−E [D] ≥ E
+�
+R# (τ)
+�
+−
+�
+1 − (1 − δE)T �
+Lr(1+Lπ)dE
+(LELπ)T + (1 − LELπ) T − 1
+(1 − LELπ)2
+. (3)
+Algorithm 3 approximates E
+�
+R# (τ)
+�
+by sampling N values. Let ˆR# (τ) be the measured mean and recall E
+�
+ˆR# (τ)
+�
+=
+E
+�
+R# (τ)
+�
+and Var
+�
+ˆR# (τ)
+�
+= Var
+�
+R# (τ)
+�
+/N.
+Then by Chebyshev’s inequality we have the for all k > 0,
+Pr
+���� ˆR# (τ) − E
+�
+R# (τ)
+���� ≥ k
+�
+Var
+�
+ˆR# (τ)
+��
+≤ 1/k2. Then in particular, with probability at least 1 − 1/k2,
+ˆR# (τ) − k
+�
+Var [R# (τ)]
+N
+≤ R# (τ) .
+Combining this with Equation 3 above and letting k = 1/
+√
+δ, we have with probability at least 1 − δ,
+E [R (τ)] ≥ ˆR# (τ) − 1
+√
+δ
+�
+Var [R# (τ)]
+N
+−
+�
+1 − (1 − δE)T �
+Lr(1 + Lπ)dE
+(LELπ)T + (1 − LELπ) T − 1
+(1 − LELπ)2
+.
+C. Implementation Details
+C.1. Training Details
+We run our experiments on Quadro RTX 8000 and Nvidia T4 GPUs. We define the perturbation set B(s) to be an l∞
+norm perturbation of the state with radius ϵ: Bp(s, ϵ) := {s′|∥s′ − s∥ ≤ ϵ} in the experiments. We use a smoothed linear
+ϵ-schedule during training as in (Zhang et al., 2020; Oikarinen et al., 2021). For the environments, we use the MuJoCo
+environments in OpenAI Gym (Brockman et al., 2016). We use Hopper-v2, HalfCheetah-v2, Ant-v2, Walker2d-v2 with
+1000 trial length.
+Network Structures and Hyperparameters for Training
+For both policy networks and the model networks, we use the
+same network as in (Pineda et al., 2021). For both MBPO and CAROL, we use the optimal hyperpapramters in (Pineda et al.,
+2021). We mainly set two additional parameters, regularization parameters and the ϵ-schedule (Zhang et al., 2020; Oikarinen
+et al., 2021; Gowal et al., 2018) parameters for CAROL. The additional regularization parameter λ for regularizing Lsymbolic
+is chosen in {0.1, 0.5, 1.0}. The ϵ-schedule starts as an exponential growth from ϵ = 10−12 and transitions smoothly into
+a linear schedule until reaching ϵtrain. Then the schedule keeps ϵt = ϵtrain for the rest iterations. We set the temperature
+parameter controlling the exponential growth with 4.0 for all experiments. We have two other parameters to control the
+ϵ-schedule: endStep, and finalStep, where endStep is the step where ϵt reaches ϵtrain and finalStep is the steps for the total
+training. The midStep = 0.25 ∗ endStep is the turning point from the exponential growth to the linear growth. Table 3 shows
+the details of each parameters.
+C.2. Certificate Usage
+In evaluation, we compare the certified performance of policies trained from different algorithms with a set of environment
+models. We train these models based on the rollout trajectory dataset for 4 × 105 environment steps between 5 policies and
+the environment. Among these 5 policies, three of them are trained from MBPO and the remaining are trained from SAC
+(Haarnoja et al., 2018), which is the base algorithm without model for MBPO.
+
+Policy Optimization with Robustness Certificates
+Environments
+Methods
+endStep
+finalStep
+Hopper
+CAROL
+4 × 105
+5 × 105
+CAROL-SS
+4 × 105
+5 × 105
+Ant
+CAROL
+8 × 105
+9 × 105
+CAROL-SS
+4 × 106
+5 × 106
+Walker2d
+CAROL
+7 × 105
+7.5 × 105
+CAROL-SS
+1.5 × 106
+2 × 106
+HalfCheetah
+CAROL
+7.5 × 105
+8.5 × 105
+CAROL-SS
+7.5 × 105
+8.5 × 105
+Table 3: Parameters for ϵ-schedule.
+
diff --git a/zNFIT4oBgHgl3EQf1ysJ/content/tmp_files/load_file.txt b/zNFIT4oBgHgl3EQf1ysJ/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..32d95612745a00bf4b0f49a1627d81c1b515d0fc
--- /dev/null
+++ b/zNFIT4oBgHgl3EQf1ysJ/content/tmp_files/load_file.txt
@@ -0,0 +1,1028 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf,len=1027
+page_content='Policy Optimization with Robustness Certificates  Chenxi Yang 1 Greg Anderson 1 Swarat Chaudhuri 1  Abstract  We present a policy optimization framework in  which the learned policy comes with a machine-  checkable certificate of adversarial robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Our approach, called CAROL, learns a model of  the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' In each learning iteration, it uses  the current version of this model and an external  abstract interpreter to construct a differentiable  signal for provable robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' This signal is used  to guide policy learning, and the abstract inter-  pretation used to construct it directly leads to the  robustness certificate returned at convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We  give a theoretical analysis that bounds the worst-  case accumulative reward of CAROL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We also  experimentally evaluate CAROL on four MuJoCo  environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' On these tasks, which involve con-  tinuous state and action spaces, CAROL learns  certified policies that have performance compa-  rable to the (non-certified) policies learned using  state-of-the-art robust RL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Introduction  Reinforcement learning (RL) is an established approach to  control tasks (Polydoros & Nalpantidis, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=',  2015), including ones in which safety is critical (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=',  2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Sallab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' However, state-of-the-art RL  methods use deep neural networks as policy representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  This makes them vulnerable to adversarial attacks in which  carefully crafted perturbations to a policy’s inputs cause it to  behave incorrectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' These problems are even more serious in  RL than in supervised learning, as the effects of successive  mistakes can cascade over a long time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  These challenges have motivated research on RL algorithms  that are robust to adversarial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' In general, meth-  ods for adversarial learning can be divided into best-effort  heuristic defenses and certified approaches that guarantee  provable adversarial robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The latter are preferable  1UT  Austin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Correspondence  to:  Chenxi  Yang  <cxyang@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='edu>,  Greg  Anderson  <ganderso@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='edu>,  Swarat  Chaudhuri  <swarat@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='edu>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  as heuristic defenses are often defeated by counterattacks  (Russo & Proutiere, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' In the supervised learning set-  ting, many certified learning techniques have been proposed  over the years (Mirman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Wong & Kolter, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' However, developing such meth-  ods for the RL setting has been difficult because of the  presence of a blackbox environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' To ensure the robust-  ness of an RL policy, one needs to reason about repeated  interactions between the policy, the environment, and the  adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' There is no general approach to doing so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Ex-  isting approaches to deep certified RL typically sidestep  the challenge through various simplifying assumptions, for  example, that the perturbations are stochastic rather than  adversarial (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2021), that the certificate only  applies to one-shot interactions between the policy and the  environment (Oikarinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2020), or  that the action space is discrete (L¨utjens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  In this paper, we develop a framework for certifiably robust  policy optimization that fills this gap in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We  observe that we can reason about adversarial dynamics over  entire episodes by learning a model of the environment and  repeatedly composing it with the policy and the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  To this end, we consider a state-adversarial Markov Deci-  sion Process (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2020) in which the observed  states are adversarially attacked states of the original envi-  ronment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' During exploration, our algorithm learns a model  of the environment using an existing model-based policy  optimization algorithm (Janner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We perform  abstract interpretation (Cousot & Cousot, 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Mirman  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2018) over compositions of the current policy and the  learned environment model to estimate worst-case bounds  on the agent’s adversarial reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The lower bound on the  reward is then used to guide policy optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  A key benefit of abstract interpretation is that it not only  computes bounds on a policy’s worst-case reward but also  offers a proof of this fact if it holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' A certificate of robust-  ness in our framework consists of such a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Our results include a theoretical analysis of our learning  algorithm, which shows that our learned certificates give  probabilistically sound lower bounds on the accumulative re-  ward of any allowed adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We also empirically evaluate  CAROL over four high-dimensional MuJoCo environments  (Hopper, Walker2d, Halfcheetah, and Ant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We demonstrate  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='11374v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='LG]  26 Jan 2023  Policy Optimization with Robustness Certificates  that CAROL is able to successfully learn certified policies  for these environments and that our strong certification re-  quirements do not compromise empirical performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  To summarize, our main contributions are as follows:  We offer CAROL, the first RL framework to guarantee  episode-level certifiable adversarial robustness in the pres-  ence of continuous states and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The framework is  based on a new combination of model-based learning and  abstract interpretation that can be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  We give a rigorous theoretical analysis that establishes  the (probabilistic) soundness of CAROL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  We give an empirical evaluation that establishes CAROL  as a new state of the art for certifiably robust RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Background  Markov Decision Processes (MDPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  We start with the  standard definition of an Markov Decision Process (MDP)  M = (S, A, r, P, S0), where S is a set of states, A is a set  of actions, S0 is a distribution of initial states, P(s′ | s, a)  for s, s′ ∈ S and a ∈ A is a probabilistic transition func-  tion, and r(s, a) for s ∈ S, a ∈ A is a real-valued reward  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Our method assumes an additional property that  is commonly satisfied in practice: that P(s′ | s, a) has  the form µP (s, a) + fP (s′), where fP (s′) is a distribution  independent of (s, a) and µP is deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  A policy in M is a distribution π(a | s) with s ∈ S and a ∈  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' A (finite) trajectory τ is a sequence s0, a0, s1, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  such that s0 ∼ S0, each ai ∼ π(si), and each si+1 ∼ P(s′ |  si, ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We denote by R(τ) = �  i r(si, ai) the aggregate  (undiscounted) reward along a trajectory τ, and by R(π) the  expected reward of trajectories unrolled under π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  State-Adversarial MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We model adversarial dynam-  ics using state-adversarial MDPs (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Such a structure is a pair Mν = (M, B), where M =  (S, A, r, P, S0) is an MDP, and B : S → 2S is a perturba-  tion map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Suppose we have a policy π in the underlying MDP M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' In  an attack scenario, an adversary ν perturbs the observations  of the agent at a state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' As a result, rather than choosing  an action from π(a | s), the agent now chooses an action  from π(a | ν(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' However, the environment transition is  still sampled from P(s′ | s, a) and not P(s′ | ν(s), a), as  the ground-truth state does not change under the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We  denote by π ◦ ν the state-action mapping that results when  π is used under this attack scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Naturally, if ν can arbitrarily perturb states, then adversar-  ially robust learning is intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Consequently, we con-  strain ν using B, requiring ν(s) ∈ B(s) for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  We denote the set of allowable adversaries in Mν as  AB = {ν : S → S | ∀s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' ν(s) ∈ B(s)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Abstract Interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  We certify adversarial robust-  ness using abstract interpretation (Cousot & Cousot, 1977),  a classic framework for worst-case safety analysis of sys-  tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Here, one represents sets of system states using sym-  bolic representations (abstract states) in a predefined lan-  guage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' For example, we can set our abstract states to be  hyperintervals that maintain upper and lower bounds in each  dimension of the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We denote abstract values with  the superscript #, and we use brackets �·�# to indicate that  we are using abstract semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' For a set of concrete states  S, α(S) denotes the smallest abstract state which contains  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' For an abstract state s#, we denote by β(s#) the set of  concrete states represented by s#.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  The core of abstract interpretation is the propagation of  abstract states s# through a function f(s) that captures  single-step system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' This function can be repre-  sented in a variety of ways, including neural networks (Gehr  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' For propagation, we assume that we have ac-  cess to a map f #(s#) that “lifts” f to abstract states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' This  function must satisfy the property β(f #(s#)) ⊇ {f(s) :  s ∈ β(s#)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Intuitively, f # overapproximates the behavior  of f: while the abstract state f #(s#) may include some  states that are not actually reachable through the application  of f to states encoded by s#, it will at least include every  state that is reachable this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  By starting with an abstraction s#  0 of the initial states and  using abstract interpretation to propagate this abstract state  through the transition function f, we can obtain an abstract  state s#  i which includes all states of the system that are  reachable in i steps, for increasing i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The sequence of ab-  stract states τ # = s#  0 s#  1 s#  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' is called an abstract trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Problem Formulation  To formulate our problem, we first need to define robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Assume a fixed adversarial MDP Mν, a policy π, and a  robustness threshold ∆ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' A robustness property is a  constraint φ(π, ∆) of the form  ∀ν ∈ AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' R(π) − R(π ◦ ν) < ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  (1)  Intuitively, φ requires that no allowable adversary can re-  duce the expected reward of the policy by more than ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Our goal in this paper is to learn policies that are not just  robust but provably so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Accordingly, we expect our learning  algorithm to produce, in addition to a policy π, a certificate,  or proof, c of robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Formally, let Π be the universe of all policies in a given  state-adversarial MDP Mν = (M, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' For a policy π and  a robustness property φ, we write π ⊢c φ if π provably  satisfies φ, and c is a proof of this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Policy Optimization with Robustness Certificates  s0 = 1  ν(s) = ⟨s0 + ϵ0, s1 + ϵ1⟩  s0  sobs0  a0  s1  sobs1  a1  R(π ◦ ν)  Eτ∼π◦ν[R]  No-Adv  ϵ0 = ϵ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0  1  1  1  2 + e ∼ N(0, 1)  2 + e  2 + e  6 + 2e  6  Adv-1  ϵ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='1, ϵ1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='4  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='1 + e ∼ N(0, 1)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='7 + e  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='7 + e  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='9 + 2e  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='9  Adv-2  ϵ0 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='2, ϵ1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='3  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='8 + e ∼ N(0, 1)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5 + e  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5 + e  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='1 + 2e  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='1  Reward Bound (R#)  ϵt ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5], ϵ#  t = [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5]  1  1 + [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5]  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5]  [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5] + e ∼ N(0, 1)  [1, 3] + e  [1 + e, 3 + e]  [4 + 2e, 8 + 2e]  [4, 8]  Table 1: Example of reward bound calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The MDP in this example has initial state set S0 = {1}, white-box transition  function P(s′|s, a) = s + a + N(0, 1), reward function r(s, a) = s + a, and adversary ν(s) ∈ [s − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5, s + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' ϵt denotes  the disturbance added on step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We aim to certify over the worst-case accumulative reward of a deterministic policy π  defined as π(s) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We define the worst-case here by considering all potential adversaries while still considering the  expected behavior over the stochastic environment, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' As shown in the above table, we first demonstrate three traces from  fixed adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' In the last row, we demonstrate the way how we consider all the adversary behaviors through an abstract  trace via abstract interpretation with intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The worst-case accumulative reward in this example is 4 as EN [e] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The  abstract trace over all the adversaries in the last row is our certificate which serves as a proof that the policy satisfies our  property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We want to ensure the lower bound of the R# should not be lower than a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' In training, we use the abstract  trace to compute a loss to guide the policy optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  The problem of policy optimization with robustness certifi-  cates is now defined as:  (π∗, c)  =  arg max  π∈Π  E  τ∼(M,π) [R(τ)]  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' π∗ ⊢c φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  (2)  That is, we want to find a policy that maximizes the standard  expected reward of the policy, but also ensures that the  expected worst-case reward under adversarial dynamics is  provably above a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Our certificates can be constructed using a variety of sym-  bolic or statistical techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' In CAROL, certificates are  constructed using an abstract interpreter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Suppose we have  a policy π and an abstract trace τ # = s#  0 s#  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' s#  n such  that for all length-n trajectories τ = s0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' sn and all i,  si ∈ β(s#  i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The abstract trace allows us to compute a  lower bound on the expected reward for π and also serves  as a proof of this bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We give an example of such certi-  fication in a simple state-adversarial MDP, assumed to be  available in white-box form, in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  A challenge here is that abstract interpretation requires a  white-box transition function, which is not available in RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  We overcome this challenge by learning a model of the  environment during exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Model learning is a source  of error, so our certificates are probabilistically sound, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=',  they guarantee robustness with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' However,  this error only depends on the underlying model-based RL  algorithm and does not restrict the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Learning Algorithm  Now we present the CAROL framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The framework  (Figure 1) has two key components: a model-based learner  and an abstract interpreter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' During each training round, the  learner maintains a model of the environment dynamics  and a policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' These are sent to the abstract interpreter,  which calculates a lower bound on the abstract reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The  Environment  Model  Policy  Model#  Policy#  Loss  Robustness Property  Learner  Abstract  Interpreter  Certificate  Figure 1: Schematic of CAROL  lower bound is used to compute a differentiable loss that  the learner uses in the next iteration of policy optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  At convergence, the abstract trace computed during abstract  interpretation is returned as a certificate of robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Abstract Interpretation in CAROL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Now we describe the  abstract interpreter in CAROL in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Recall that our  definition of robustness compares the expected reward of the  original policy to the expected reward of the policy under an  adversarial perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' As a result, our verifier is designed  to reason about the worst-case reward under adversarial  perturbations, while considering average-case behavior for  stochastic policies and environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Algorithm 1 finds  a lower bound on this worst-case expected reward using  abstract interpretation to overapproximate the adversary’s  possible behaviors along with sampling to approximate the  average-case behavior of the policy and environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We  denote this lower bound from Algorithm 1 as worst-case ac-  cumulative reward (WCAR), which is also used to measure  the certified performance in our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  In more detail, Algorithm 1 proceeds by sampling a starting  state s0 ∼ S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Then in Algorithm 2 for each time step, we  find an overapproximation s#  obsi which includes all of the  possible ways the adversary may perturb si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Based on this  approximation, we sample a new approximation from the  policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Intuitively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' this may be done by using a policy π  whose randomness does not depend on the current state of  Policy Optimization with Robustness Certificates  Algorithm 1 Worst-Case Accumulative Reward (WCAR)  1: Input: policy π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' model E  2: Output: worst case reward of π under any adversary  3: for t from 1 to N do  4:  Sample an initial state s0 ∼ S0  5:  Get the worst case reward Rmint using Algorithm 2 over horizon T starting from s0  6: end for  7: return 1  N  �N  t=1 Rmint  Algorithm 2 Worst-case rollout under adversarial perturbation  1: Input: Initial state s0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' rollout horizon T  2: Output: Worst-case reward of π starting from s0 over one random trajectory  3: Abstract the initial state and reward: soriginal  #  0 ← α({s0}),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  R#  minti ← α ({0})  4: for i from 1 to T do  5:  Abstract over possible perturbations: s#  obsi ←  �  B(s#  originali)  �#  6:  Calculate symbolic predicted actions: a#  i ←  �  π(s#  obsi)  �#  7:  Calculate symbolic next-step states and rewards: s#  originali+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' r#  i ←  �  Eθ(s#  originali,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' a#  i ) + α({x | ∥x∥ ≤ εE})  �#  8:  Update worst-case reward: R#  mint ←  �  R#  mint + r#  i  �#  9: end for  10: return inf β(R#  mint)  the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' More formally, π(a | s) = µπ(s)+fπ(a) where  fπ(a) is a distribution with zero mean which is independent  of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Then a#  i may be computed as �µπ(s#  obsi) + α({e})�#  where e ∼ fπ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Once the abstract action is computed,  we may find the new (abstract) state and reward using the  environment model E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The model is assumed to satisfy a  PAC-style bound, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', there exist δE and εE such that with  probability at least 1 − δE, ∥E(s, a) − P(s, a)∥ ≤ εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The  values of δE and εE can be measured at model construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  One way to understand Algorithm 1 is to consider pairs  of abstract and concrete trajectories in which the random-  ness is resolved in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Specifically, if π(a |  s) = µπ(s) + fπ(a) and E(s′ | s, a) = µ(s, a) + fE(s′),  the initial state s0 combined with the sequence of values  ei ∼ fπ(a) and e′  i ∼ fE(s′) for 0 ≤ i ≤ T uniquely de-  termine a trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' For a given set of values, the reward  bound inf β(R#  mint) represents the worst-case reward under  any adversary for a particular resolution of the randomness  in the environment and the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The outer loop of Algo-  rithm 1 approximates the expectation over these different  random values by sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Theorem 1 in Section 5 shows  formally that with high probability, Algorithm 1 gives a  lower bound on the true adversarial reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Learning in CAROL Now we discuss how to learn a pol-  icy and environment model which may be proven robust by  Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' At a high level, Algorithm 3 works by intro-  ducing a symbolic loss term Lsymbolic  ψ  which measures the  robustness of the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Because robustness is a constrained  optimization problem, we use this symbolic loss together  with a Lagrange multiplier in an alternating gradient descent  scheme to find the optimal robust policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Formally, for a  given environment model E, the inner loop in Algorithm 3  solves the optimization problem  arg min  ψ  Lnormal(πψ, Dmodel)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Lsymbolic(π, E) ≤ ∆  via the Lagrangian  arg min  ψ  max  λ≥0 Lnormal(πψ, Dmodel)+λ(Lsymbolic(π, E)−∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  We ensure that solving this problem solves the certifi-  able robustness problem by enforcing the following con-  ditions: (i) E accurately models the environment and (ii)  Lsymbolic(π, E) measures the “provable robustness” of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Condition (i) is handled by alternating model updates with  policy updates, in the style of Dyna (Sutton, 1990), so we  will focus on condition (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  The computation of Lsymbolic uses the same underlying ab-  stract rollouts (Algorithm 2) as the verifier described in  Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Once again, this algorithm estimates the re-  ward achieved by a policy under worst-case adversarial  perturbations but average-case policy actions and environ-  ment transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We then define the robustness loss as the  difference between the nominal loss Ro and the provable  lower bound on the worst-case loss Rmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Now as long as  Policy Optimization with Robustness Certificates  Algorithm 3 Policy Optimization for Certifiably Robust Reinforcement Learning  1: Initialize a random policy πψ, random environment model Eθ, and empty model dataset Dmodel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  2: Initialize an environment dataset Denv by unrolling trajectories under a random policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  3: for N epochs do  4:  Train model Eθ on Denv via maximum likelihood  5:  Unroll M trajectories int he model under πψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' add to Dmodel  6:  Take action in environment according to πψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' add to Denv  7:  for G gradient updates do  8:  Calculate normal policy loss Lnormal(πψ, Dmodel) as in MBPO (Janner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2019)  9:  Sample ⟨st, at, st+1, rt⟩ uniformly from Dmodel  10:  Rollout π starting from st under Eθ for Ttrain steps and compute the total reward Ro  11:  Compute the worst-case reward Rmin using Algorithm 2 over horizon Ttrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  12:  Compute the robustness loss Lsymbolic(πψ, Eθ) ← Ro − Rmin  13:  Update policy parameters: ψ ← ψ − α∇ψ(Lnormal(πψ, Dmodel) + λ(Lsymbolic(πψ, Eθ) − ∆))  14:  Update Lagrange multiplier: λ ← max(0, λ + α′(Lsymbolic(πψ, Eθ) − ∆))  15:  end for  16:  Unroll n trajectories in the true environment under πψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' add to Denv  17: end for  Lsymbolic < ∆, we satisfy the definition of robustness given  in Section 3 for that specific trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Repeating these gra-  dient updates gives an approximation of the average-case  behavior which is considered in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Theoretical Analysis  In this section, we explore some key theoretical properties  of CAROL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Proofs are deferred to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Assume the environment transition distribution  is P(s′ | s, a) = N(µP (s, a), ΣP ) and the environment  model is E(s′ | s, a) = N(µE(s, a), ΣE) with ΣP , ΣE  diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Further, we assume that the model satisfies a  PAC-style guarantee: for any state s, action a, and ϵ ∈ S,  |(µP (s, a) + Σ1/2  P ϵ) − (µE(s, a) + Σ1/2  E ϵ)| ≤ εE with  probability at least 1−δE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' For any policy π, let the result of  Algorithm 1 be ˆR# and let the reward of π under the optimal  adversary ν∗ be R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Then for any δ > 0 with probability at  least 1 − δ, we have  R ≥ ˆR# − 1  √  δ  �  Var [R#]  N  −  �  1 − (1 − δE)T �  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  where C is a constant (see Appendix B for details of C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Theorem 1 shows that our checker is a valid (probabilistic)  proof strategy for determining if a policy is robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' That  is, if we use Algorithm 1 to measure the reward of a policy  under perturbation, the result is a lower bound of the true  worst-case reward (minus a constant) with high probability,  assuming an accurate environment model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The bound in  Theorem 1 gives some interesting insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' First, the bound  grows as δ shrinks, so as we consider higher confidence  levels, we pay the price of a looser bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Second, the  bound depends on the variance of the abstract reward and  the number of samples in an intuitive way — higher vari-  ance makes it harder to measure the true reward, and more  samples makes the bound tighter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Third, as δE increases,  the last term of the bound grows, indicating that a less ac-  curate environment model leads to a looser bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Finally,  the bound grows with T, indicating that over longer time  horizons, our reward measurement gets less accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' This  is consistent with the intuition that the environment model  may drift away from the true environment over long rollouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Algorithm 3 converges to a policy π which is  robust and verifiable by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Intuitively, the theorem shows that Algorithm 3 solves the  certifiable robustness problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', it converges to a pol-  icy which passes the check by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The proof is  straightforward because Algorithm 3 is a standard primal-  dual approach to solve the constrained optimization problem  outlined in Equation 2 (Nandwani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Evaluation  We study the following experimental questions:  RQ1: Can CAROL learn policies with nontrivial certified  reward bounds?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  RQ2: How does CAROL compare with other (non-certified)  robust RL methods in terms of certified bounds?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  RQ3: How does CAROL compare with other robust RL  methods in terms of empirical adversarial performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  RQ4: How does CAROL’s model-based training approach  affect performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Environments and Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Our experiments consider l∞-  Policy Optimization with Robustness Certificates  Figure 2: Certified performance of policies π with the learned-together model, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Each bar is an average of 20 starting  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' nan denotes not a number, which means that (π, E) is not certifiable by a third-party verifier (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' A  higher value indicates a better certified worst-case performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Figure 3: Certified performance of policies π under a set of separately learned models, {Ei}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Each bar averages the learned  policies on each Ei of 20 starting states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  norms perturbation of the state with radius ϵ: Bp(s, ϵ) :=  {s′|∥s′ − s∥ ≤ ϵ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We implement CAROL on top of the  MBPO (Janner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2019) model-based RL algorithm  using the implementation from Pineda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' For  training, we use IBP (Gowal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2018) as a scalable  abstract interpretation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' During the evaluation,  we use CROWN (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2018), which is a more  expensive but accurate bound propagation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We use  a ϵ-schedule (Gowal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2020) during  training to slowly increase the ϵt at each epoch within the  perturbation budget until reaching ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Note that the policies  take action stochastically during training, but we set them  to be deterministic during evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  We experiment on four MuJoCo environments in OpenAI  Gym (Brockman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' For CAROL, we use the  same hyperparameters as in Pineda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' (2021) without  further tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Specifically, we do not use an ensemble of  dynamics models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Instead, we use a single dynamic model,  which is the case when the ensemble is of size 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We use  Gaussian distribution as the independent noise distribution,  fπ(a), fE(s′) for both policy and model in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Concretely, the output of our policies are the parameters  µπ, Σπ of a Gaussian, with Σπ being diagonal and inde-  pendent of input state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' For the model, the output are the  parameters µE, ΣE of a Gaussian, with ΣE being diagonal  and independent of input s, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' In the implementation, we  approximate the εE as zero when measuring the certified  performance for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  We run Hopper 5 × 105 steps, Walker2d 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5 × 105 steps,  Halfcheetah 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5 × 105 steps, and Ant 9 × 105 steps for  convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' CAROL has a regularization parameter, λ, for  regularizing the robust loss Lsymbolic, which we set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0,  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='1 for Hopper, Walker2d, HalfCheetah, and Ant,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We set Ttrain = 1 for all the training of CAROL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  We compare CAROL with the following methods: (1) MBPO  (Janner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2019), our base algorithm for policy optimiza-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' (2) SA-PPO (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2020), a robust RL algo-  rithm bounding per-step action distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' (3) RADIAL-PPO  (Oikarinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2021), a robust RL algorithm using lower  bound PPO loss to update the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' (4) CAROL-Separate  Sampler(CAROL-SS), an ablation of CAROL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' In CAROL, we  update the policy loss Lnormal with the data sampled from the  rollout between the learned model and the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' While in  CAROL-SS, the data for Lnormal is sampled from the rollout  between the environment and the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The ϵtrain is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='075,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='075, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='05 for Hopper, Walker2d, HalfCheetah, and  Ant for CAROL, CAROL-SS, SA-PPO, and RADIAL-PPO  in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' More details of the training and setup are  available in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We evaluate the performance of policies with  two metrics: (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' WCAR following Algorithm 1 for certified  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' total reward under MAD attacks (Zhang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2020) for empirical performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  hopper  ant  halfcheetah  walker2d  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5  3  0+36"T-  WCAR / T  WCAR / T  EO  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5E+09  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='7E+12  WCAR / T  SO  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5 -  +  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5  WCAR  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
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+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0  5  15  20  25  5  7  10  5  10  15  20  5  10  15  20  Time Horizons (T)  Time Horizons (T)  Time Horizons (T)  Time Horizons (T)  CAROL  MBPOhopper  ant  halfcheetah  walker2d  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
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+page_content='5 -  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0  6  6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
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+page_content="0  6'0  1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
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+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5  405  1/  1/  WCAR / T  3  /  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5  WCAR  WCAR  WCAR  851  四  85  48  95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
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+page_content='5-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0  5  15  20  25  5  6  8  10  5  10  15  20  5  10  15  20  Time Horizons (T)  Time Horizons (T)  Time Horizons (T)  Time Horizons (T)  CAROL  SA-PPO  RADIAL-PPOPolicy Optimization with Robustness Certificates  Figure 4: Training Curves of CAROL and CAROL-SS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The solid lines in the graph show the average natural rewards of five  training trials, and the shaded areas represent the standard deviation among those trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  RQ1:  Certified Performance with Learned-together  Certificate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  After training, we get a policy, π, and an  environment model, E, trained with the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Then, we  evaluate the WCAR following Algorithm 1 with π and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Note that we use an ϵtest =  1  255 for the evaluation of prov-  ability as certifying over traces of neural network models  accurately is a challenging task for abstract interpreters due  to accumulated approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The proof becomes  more challenging as the horizon increases as the impact from  each step’s worst-case adversary accumulates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We vary the  certified horizon under the ϵtest to exhibit the certified perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' To have a fair comparison across different horizons,  we quantify the certified performance by WCAR/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Figure 2 exhibits the certified performance of CAROL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Both  CAROL and MBPO are evaluated with the model trained  together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We are able to train a policy with better certified  accumulative reward under the worst attacks compared to  the base algorithm, MBPO, which does not use the regular-  ization Lsymbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' As the time horizon increases, it becomes  harder to certify the accumulative reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' For example, in  Ant and HalfCheetah, CAROL is not able to give a good  certified performance when the horizon reaches 10 and 20  respectively because of the accumulative influence from  the worst-case attack and the overapproximation from the  abstract interpreter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We also highlight that Ant is a challeng-  ing task for certification due to the high-dimensional state  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  RQ2: Comparison of Certified Performance with Other  Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We compare CAROL with two robust RL meth-  ods, SA-PPO (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2020) and RADIAL-PPO  (Oikarinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2021), which both bound the per-step per-  formance of the policy during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' SA-PPO bounds the  per-step action deviation under perturbation, and RADIAL-  PPO bounds the one-step loss under perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' To have  a fair comparison of the certified performance of policies  and avoid model error bias, we separately train 5 additional  environment models, {Ei}, with the trajectory datasets un-  rolled from 5 additional policies (with regular RL training)  and the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We truncate CAROL by extracting the  policies from training and certify them with these separately  Nominal  Attack (MAD)  Environment  Model  ϵ = 0  ϵ = ϵtrain  Hopper  (ϵtrain = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='075)  MBPO  3246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0±76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='1  2874.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='2±203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='4  SA-PPO  3423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='9±164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='2  3213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='8±284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='8  RADIAL-PPO  3547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0±166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='9  3100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='3±368.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='3  CAROL  3290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='1±104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='9  3201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='4±100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5  Ant  (ϵtrain = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='05)  MBPO  4051.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='9±526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='2  406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='2±83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5  SA-PPO  5368.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='8±96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='4  5327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='4±112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='7  RADIAL-PPO  4694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='1±219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5  4478.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='9±232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='8  CAROL  5696.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='6±277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='9  5362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='2±242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='8  HalfCheetah  (ϵtrain = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='075)  MBPO  7706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='3±710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='1  2314.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='6±566.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='7  SA-PPO  3193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='9±650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='7  3231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='6±659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='9  RADIAL-PPO  3686.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5±439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='2  3409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='6±683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='9  CAROL  5821.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5±2401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='9  3961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='6±899.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5  Walker2d  (ϵtrain = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='05)  MBPO  3815.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='6±211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='9  3616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5±228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='2  SA-PPO  4271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='7±222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='2  4444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='4±286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0  RADIAL-PPO  2935.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='1±272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='1  3022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='6±381.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='7  CAROL  3784.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='4±329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='1  3774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='3±260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='3  Table 2: Average episodic reward ± standard deviation over  100 episodes on three baselines and CAROL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We show natu-  ral rewards (under no attack) and rewards under adversarial  attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The best results over all methods are in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  trained environment models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' This setting is not completely  inline with CAROL’s learned certificate and verification (see  RQ1) but is designed for a fair comparison across policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  As shown in Figure 3, the CAROL’s certified performance  with separately trained models is slightly worse yet compa-  rable to its performance when using learned-together cer-  tificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' When compared with non-certified RL policies,  CAROL consistently exhibits better certifiable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  It is worth noting that CAROL is able to provide increas-  ing worst-case rewards over time for the Hopper, Ant, and  HalfCheetah, which aligns with the reward mechanisms  used in these environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Thus, we believe these results  demonstrate that CAROL is able to give a reasonable certi-  fied performance, while the other methods, which are not  specifically designed for worst-case accumulative reward  certification, struggle to achieve the same goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  RQ3:  Comparison of Empirical Performance with  Hopper  Ant  HalfCheetah  Walker2d  5000  8000  3000  3000  wards  Rewar  4000  0-  0-  0  0  200K  400K  400K  800K  0  400K  800K  0  300K  600K  Steps  Steps  Steps  Steps  CAROL-SS  CAROLPolicy Optimization with Robustness Certificates  Other Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Usually, there is a trade-off between  certified robustness and empirical robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' One can get  good provability but may sacrifice empirical rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We  show that policy from our algorithm shows comparable  natural rewards (without attack) and adversarial rewards  compared with other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' In Table 2, we show results  on 4 environments and comparison with MBPO, SA-PPO,  and RADIAL-PPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The policies are the same ones evalu-  ated for RQ1 and RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' For each environment, we compare  the performance under MAD attacks (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  CAROL outperforms other methods on Ant and HalfCheetah  under attack when the base algorithm, MBPO, is extremely  not robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' For Hopper, CAROL has comparable adversarial  rewards with the best methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' CAROL’s reward is worse  on Walker2d though still reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  RQ4: Impact of Model-Based Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' In this part, we  investigate the impact of our design choices for Lnormal on  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We compare our framework, CAROL, with an  ablation of it, CAROL-SS, to understand how rollout with the  learned model for Lnormal matters in CAROL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We present a  comparison of performance during training, as shown in Fig-  ure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' In the implementation, we set a smoother ϵ-schedule  for CAROL-SS by allowing CAROL-SS to take longer steps  from ϵ = 0 to the target ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' These results show that CAROL  converges much faster while achieving a comparable or  better final performance due to the benefits of the sample  efficiency of MBRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Additionally, the consistency between  the rollout datasets for Lnormal and the ones for Lsymbolic also  leads to a better natural reward at convergence in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Related Work  Adversarial RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Adversarial attacks on RL systems have  been extensively studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Specific attacks that have been  investigated include adversarial perturbations on agents’ ob-  servations or actions (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Weng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2019), adversarial disturbance forces to the  system (Pinto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2017), and other adversarial policies  in a multiagent setting (Gleave et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Most recently,  Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' (2021) and Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' (2021) propose methods  to train agents together with a learned optimal adversary in  an online way to achieve better adversarial reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' These  efforts are orthogonal to existing regularization-based tech-  niques (including ours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Robust RL and Provable Robustness in RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Multiple  robust training methods have been applied to deep RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Fis-  cher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' (2019) leverage additional student networks to  help the robust Q learning, and Everett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' (2021) enhance  an agent’s robustness during testing time by computing the  lower bound of each action’s Q value at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Zhang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' (2020) and Oikarinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' (2021) leverage a bound  propagation technique in a loss regularizer to encourage the  agent to either follow its original actions or optimize over a  loss lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' While these efforts achieve robustness by  deterministic certification techniques for neural networks  (Gowal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2020), they mainly focus on  the step-wise certification and are not able to give robustness  certification if the impact from attacks accumulates across  multiple steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' CAROL differs from these papers by offering  certified robustness for the aggregate reward in an episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  We know of only two recent efforts that study robustness  certification for cumulative rewards (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Kumar  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Of these, (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2021) is a framework  for certification rather than certified learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=',  2021) proposes a certified learning algorithm under the as-  sumption that the adversarial perturbation is smoothed using  random noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The attack model here is weaker than the  adversarial model assumed by CAROL and most other work  on adversarial learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Certified RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Safe control with learned certificates is an  active field (Dawson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' A few efforts in this space  have considered controllers discovered through RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Many  works use a given certificate with strong control-theoretic  priors to constrain the actions of an RL agent (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=',  2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Li & Belta, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2019b) or assume the  full knowledge of the environment to yield the certificate  during the training of an agent (Yang & Chaudhuri, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Chow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' (2019) and Chow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' (2018) attempt to derive  certificates from the structure of the constrained Markov  decision process (Altman, 1999) for the safe control prob-  lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Chang & Gao (2021) incorporate Lyapunov methods  in deep RL to learn a neural Lyapunov critic function to im-  prove the stability of an RL agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We differ from this work  by focusing on adversarial robustness rather than stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Conclusion and Future Work  We have presented CAROL, the first RL framework with cer-  tifiable episode-level robustness guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Our approach  is based on a new combination of model-based RL and ab-  stract interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We have given a theoretical analysis to  justify the approach and also validated it empirically in four  challenging continuous control tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  A key challenge in CAROL is that our abstract interpreter  may not be sufficiently precise, and attempts to increase pre-  cision may compromise scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Future research should  work to address this issue with more accurate and scalable  verification techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Also, as seen in Figure 4, there is  variance in the results due to the use of a single dynamics  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' To address this issue, future work should explore  ways to incorporate certificates across ensembles of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Finally, because abstract interpretation of probabilistic sys-  tems is difficult, our approach assumes that the randomness  in the environment transitions is state-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Future  work should try to eliminate this assumption through ab-  stract interpreters tailored to probabilistic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Policy Optimization with Robustness Certificates  References  Altman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Constrained Markov decision processes:  stochastic modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Routledge, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
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+page_content=' In  Proceedings of the AAAI Conference on Artificial Intelli-  gence, volume 33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
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+page_content=',  Duenez-Guzman,  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=',  and  Ghavamzadeh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' A lyapunov-based approach to safe  reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Advances in neural information  processing systems, 31, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Chow, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', Nachum, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', Faust, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', Duenez-Guzman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=',  and Ghavamzadeh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Lyapunov-based safe policy  optimization for continuous control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  arXiv preprint  arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='10031, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Cohen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
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+page_content=' Crop: Certifying robust policies for reinforcement  learning through functional smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' In International  Conference on Learning Representations, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Xu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', Shi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', Huang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', Chang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=',  Kailkhura, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', Lin, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', and Hsieh, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Automatic per-  turbation analysis on general computational graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Tech-  nical report, Lawrence Livermore National Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' (LLNL),  Livermore, CA (United States), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' and Chaudhuri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Safe neurosymbolic learning  with differentiable symbolic execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' In International  Conference on Learning Representations, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', Weng, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', Chen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', Hsieh, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', and  Daniel, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Efficient neural network robustness certifica-  tion with general activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Advances in neural  information processing systems, 31, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', Xiao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', Boning,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', and Hsieh, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Robust deep reinforcement learning  against adversarial perturbations on state observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Advances in Neural Information Processing Systems, 33:  21024–21037, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', Boning, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', and Hsieh, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Robust  reinforcement learning on state observations with learned  optimal adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  arXiv preprint arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='08452,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Policy Optimization with Robustness Certificates  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Symbols  We give a summary of the symbols used in this paper below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Definition  Symbol/Notation  Policy  πψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' π  Model of the environment  Eθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' E  Environment transition  P  Parameters (mean,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' covariance) for Gaussian distribution for environment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' and policy  µP ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' ΣP ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' µE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' ΣE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' µπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Σπ  Distribution representing noise for environment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' and policy  fP ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' fE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' fπ  Adversary  ν  Regular policy loss  Lnormal  Robustness loss  Lsymbolic  Lipschitz constants for the environment model and policy mean  LE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' LP  Model error  εE  PAC bound probability  δE  Abstract values and semantics  #,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' �·�#  Abstraction  α  Concretization  β  Horizon for training and testing  T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Ttrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Ttest  Disturbance for training and testing  ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' ϵtrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' ϵtest  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Proofs  In this section we present proofs of the theorems from Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The horizon of the MDP is bounded by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The environment transition distribution has the form P (s′ | s, a) = N (µP (s, a) , ΣP ) with ΣP diagonal  and the environment model E has the form E (s′ | s, a) = N (µE(s, a), ΣE) with ΣE diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' There exist values εE and δE such that for all s, a and for any fixed e with probability at least  1 − δE,  ���  �  µP (s, a) + Σ1/2  P e  �  −  �  µE (s, a) + Σ1/2  E e  ���� ≤ εE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Further, there exists some dE such that for all s, a,  ���  �  µP (s, a) + Σ1/2  P e  �  −  �  µE (s, a) + Σ1/2  E e  ���� ≤ dE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The environment model mean function µE (s, a) is LE-Lipschitz continuous, the immediate reward function  r (s, a) is Lr-Lipschitz continuous, and the policy mean µπ (s) is Lπ-Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' For all s ∈ S, we have s ∈ B (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' That is, the adversary may choose not to perturb any state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' For any policy π, let the result of Algorithm 1 be ˆR#, let ν∗ be the optimal adversary (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', for all ν ∈ AB,  R (π ◦ ν∗) ≤ R (π ◦ ν)), and let the reward of π ◦ ν∗ be R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Then for any δ > 0 with probability at least 1 − δ, we have  E [R (τ)] ≥ ˆR# (τ) − 1  √  δ  �  Var [R# (τ)]  N  −  �  1 − (1 − δE)T �  Lr(1 + Lπ)dE  (LELπ)T + (1 − LELπ) T − 1  (1 − LELπ)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Recall that the environment transition P and policy π are assumed to be separable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', P (s′ | s, a) = µP (s, a) +  fP (s′) and π (a | s) = µπ (s) + fπ (a) with µP and µπ deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' As a result, a trajectory under policy π ◦ ν∗  may be written τ = s0, a0, s1, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' , sn, an where s0 ∼ S0, each ai = µπ (ν∗ (si)) + eπ  i for eπ  i ∼ fπ (a), and each  si = µP (si−1, ai−1) + eP  i for eP  i ∼ fP (s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' By Assumption 4, we know that eP  i ∼ N (0, ΣP ) so that eP  i = Σ1/2  P ei  where ei ∼ N (0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' In particular, because each trajectory τ is uniquely determined by s0, {eπ  i }n  i=0, {ei}n  i=1, we can write  the reward of π ◦ ν∗ as  R (π ◦ ν∗) =  E  s0∼S0,{eπ  i ∼fπ(a)}  n  i=0,{ei∼N (0,I)}n  i=1  R (τ)  Because this expectation ranges over the values of s0, {eπ  i }n  i=0 , {ei}n  i=1, we will proceed by considering pairs of abstract  and concrete trajectories unrolled with the same starting state and noise terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Policy Optimization with Robustness Certificates  To do this, we analyze Algorithm 2 for some fixed s0, {eπ  i }n  i=0 , {ei}n  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Let eE  i  = Σ1/2  E ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' That is, given the same  underlying sample from N (0, I), eP  i is the noise in the true environment while eE  i is the noise in the modeled environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  We show by induction that for all i, si ∈ β  �  s#  originali  �  with probability at least (1 − δE)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Note that, because abstract  interpretation is sound, s0 ∈ β  �  s#  original0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Additionally, for all i if si ∈ β  �  s#  originali  �  then ν∗ (si) ∈ β  �  s#  obsi  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Moreover,  since eπ  i is fixed, we have  �  π  �  s#  obsi  ��#  =  �  µπ  �  s#  obsi  �  + eπ  i  �#  so that π (ν∗ (s)) ∈ β  �  a#  i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Similarly, because eE  i is fixed, let ∆E = α ({x | ∥x∥ ≤ εE}) and we have  �  E  �  s#  originali, a#  i  �  + ∆E  �#  =  �  µE  �  s#  originali, a#  i  �  + eE  i + ∆E  �#  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  By the induction hypothesis, we know that si−1 ∈ β  �  s#  originali−1  �  with probability at least (1 − δE)i−1 and therefore ai−1 ∈  β  �  a#  i−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' By Assumption 5, we have that  ���  µP (s, a) + eP  i  �  −  �  µE (s, a) + eE  i  ��� < εE with probability at least 1 − δE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  In particular,  �  µP (s, a) + εP  i  �  −  �  µE (s, a) + εE  i  �  ∈ ∆E, so that µP (s, a) + eP  i ∈ β  ��  E  �  s#  originali, a#  i  �  + ∆E  �#�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Then with probability at least 1 − δE, if si−1 ∈ β  �  s#  originali−1  �  then si ∈ β  �  s#  originali  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' As a result, si ∈ β  �  s#  originali  �  with  probability at least (1 − δE)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' In particular, by Assumption 3, n ≤ T so that for a fixed τ defined by s0, {eπ  i }n  i=0 , {ei}n  i=1,  we have that with probability at least (1 − δE)T , Algorithm 2 returns a lower bound on R (τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Now we consider the case where Algorithm 2 does not return a lower bound of R (τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' In this case, we show (again by  induction) that for all 0 ≤ i ≤ T, there exists a point s′  i ∈ β  �  s#  originali  �  such that  ∥si − s′  i∥ ≤  i−1  �  j=0  (LELπ)j dE = dE  �  1 − (LELπ)i−1  1 − LELπ  �  (when �−1  j=0 (LELπ)j dE is taken to be zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' First, note that s0 ∈ β  �  s#  original0  �  , so the base case is trivially true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Now by  the induction hypothesis we have that there exists some s′  i−1 ∈ β  �  s#  originali−1  �  with  ��si−1 − s′  i−1  �� ≤ �i−2  j=0 (LELπ)j dE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Notice that by Assumption 7, we also have s′  i−1 ∈ β  �  s#  obsi−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Now because abstract interpretation is sound, we have that  µπ  �  s′  i−1  �  +eπ  i−1 ∈ β  �  a#  i−1  �  and by Assumption 6, ∥µπ (si−1)−µπ  �  s′  i−1  �  ∥ ≤ Lπ  �i−2  j=0 (LELπ)j dE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Similarly, we have  µE  �  s′  i−1, µπ  �  s′  i−1  �  + eπ  i−1  �  +eE  i ∈ β  �  s#  originali  �  , and  ��µE  �  si−1, µπ (si−1) + eπ  i−1  �  − µE  �  s′  i−1, µπ  �  s′  i−1  �  + eπ  i−1  ��� ≤  LELπ  �i−2  j=0 (LELπ)j dE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Let ˆsi = µE  �  si−1, µπ (si−1) + επ  i−1  �  + εE  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Then by Assumption 5, we have ∥ˆsi − si∥ ≤  dE, so that in particular  ��si − µE  �  s′  i−1, µπ  �  s′  i−1  �  + eπ  i−1  �  + εE  i  �� ≤ dE + LELπ  �i−2  j=0 (LELπ)j dE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Letting s′  i =  µE  �  s′  i−1, µπ  �  s′  i−1  �  + eπ  i−1  �  + εP  i , we have the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  We use this result to bound the difference in reward between the abstract and concrete rollouts when Algorithm 2 does not  return a lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' For each i, because s′  i ∈ β  �  s#  originali  �  and µπ (s′  i) + eπ  i ∈ a#  i , we define r′  i = r (s′  i, µπ (s′  i) + eπ  i )  and we know that r′ ∈ r#  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Because ∥si − s′  i∥ ≤ dE  �  1−(LELπ)i−1  1−LELπ  �  we have ∥ai − a′  i∥ ≤ LπdE  �  1−(LELπ)i−1  1−LELπ  �  and  |r (si, ai) − r′  i| ≤ Lr(1 + Lπ)dE  �  1−(LELπ)i−1  1−LELπ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' In particular, let R′ = �  i r′  i and then  |R (τ) − R′| ≤  T  �  i=1  Lr(1 + Lπ)dE  �  1 − (LELπ)i−1  1 − LELπ  �  = Lr(1 + Lπ)dE  (LELπ)T + (1 − LELπ) T − 1  (1 − LELπ)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  We now combine these two cases to bound the expected difference between the reward returned by Algorithm 2, denoted  R# (τ), and the reward of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Let D = R# (τ) − R (τ) be a random variable representing this difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Then with  Policy Optimization with Robustness Certificates  probability at least (1 − δE)T , D ≤ 0 and in all other cases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', with probability no greater than 1 − (1 − δE)T ),  D ≤ LR(1 + Lπ)dE  (LELπ)T +(1−LELπ)T −1  (1−LELπ)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' In particular then,  E [D] ≤  �  1 − (1 − δE)T �  Lr(1 + Lπ)dE  (LELπ)T + (1 − LELπ) T − 1  (1 − LELπ)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  By definition E  �  R# (τ)  �  = E [R (τ)] + E [D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Therefore, we have  E [R (τ)] = E  �  R# (τ)  �  −E [D] ≥ E  �  R# (τ)  �  −  �  1 − (1 − δE)T �  Lr(1+Lπ)dE  (LELπ)T + (1 − LELπ) T − 1  (1 − LELπ)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' (3)  Algorithm 3 approximates E  �  R# (τ)  �  by sampling N values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Let ˆR# (τ) be the measured mean and recall E  �  ˆR# (τ)  �  =  E  �  R# (τ)  �  and Var  �  ˆR# (τ)  �  = Var  �  R# (τ)  �  /N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Then by Chebyshev’s inequality we have the for all k > 0,  Pr  ���� ˆR# (τ) − E  �  R# (τ)  ���� ≥ k  �  Var  �  ˆR# (τ)  ��  ≤ 1/k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Then in particular, with probability at least 1 − 1/k2,  ˆR# (τ) − k  �  Var [R# (τ)]  N  ≤ R# (τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Combining this with Equation 3 above and letting k = 1/  √  δ, we have with probability at least 1 − δ,  E [R (τ)] ≥ ˆR# (τ) − 1  √  δ  �  Var [R# (τ)]  N  −  �  1 − (1 − δE)T �  Lr(1 + Lπ)dE  (LELπ)T + (1 − LELπ) T − 1  (1 − LELπ)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Implementation Details  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Training Details  We run our experiments on Quadro RTX 8000 and Nvidia T4 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We define the perturbation set B(s) to be an l∞  norm perturbation of the state with radius ϵ: Bp(s, ϵ) := {s′|∥s′ − s∥ ≤ ϵ} in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We use a smoothed linear  ϵ-schedule during training as in (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Oikarinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' For the environments, we use the MuJoCo  environments in OpenAI Gym (Brockman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We use Hopper-v2, HalfCheetah-v2, Ant-v2, Walker2d-v2 with  1000 trial length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Network Structures and Hyperparameters for Training  For both policy networks and the model networks, we use the  same network as in (Pineda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' For both MBPO and CAROL, we use the optimal hyperpapramters in (Pineda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=',  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We mainly set two additional parameters, regularization parameters and the ϵ-schedule (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Oikarinen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Gowal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2018) parameters for CAROL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The additional regularization parameter λ for regularizing Lsymbolic  is chosen in {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The ϵ-schedule starts as an exponential growth from ϵ = 10−12 and transitions smoothly into  a linear schedule until reaching ϵtrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Then the schedule keeps ϵt = ϵtrain for the rest iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We set the temperature  parameter controlling the exponential growth with 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='0 for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We have two other parameters to control the  ϵ-schedule: endStep, and finalStep, where endStep is the step where ϵt reaches ϵtrain and finalStep is the steps for the total  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' The midStep = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='25 ∗ endStep is the turning point from the exponential growth to the linear growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Table 3 shows  the details of each parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Certificate Usage  In evaluation, we compare the certified performance of policies trained from different algorithms with a set of environment  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' We train these models based on the rollout trajectory dataset for 4 × 105 environment steps between 5 policies and  the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=' Among these 5 policies, three of them are trained from MBPO and the remaining are trained from SAC  (Haarnoja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content=', 2018), which is the base algorithm without model for MBPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='  Policy Optimization with Robustness Certificates  Environments  Methods  endStep  finalStep  Hopper  CAROL  4 × 105  5 × 105  CAROL-SS  4 × 105  5 × 105  Ant  CAROL  8 × 105  9 × 105  CAROL-SS  4 × 106  5 × 106  Walker2d  CAROL  7 × 105  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5 × 105  CAROL-SS  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5 × 106  2 × 106  HalfCheetah  CAROL  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5 × 105  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5 × 105  CAROL-SS  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5 × 105  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}
+page_content='5 × 105  Table 3: Parameters for ϵ-schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFIT4oBgHgl3EQf1ysJ/content/2301.11374v1.pdf'}